author | Christian Urban <urbanc@in.tum.de> |
Mon, 26 Jun 2017 18:40:58 +0100 | |
changeset 250 | 8927b737936f |
parent 212 | 9fd41f224e8d |
child 311 | 8b8db9558ecf |
permissions | -rw-r--r-- |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
1 |
|
7
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
2 |
theory Re |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3 |
imports "Main" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
4 |
begin |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
5 |
|
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
6 |
|
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
7 |
section {* Sequential Composition of Sets *} |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
8 |
m |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
9 |
definition |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
10 |
Sequ :: "string set \<Rightarrow> string set \<Rightarrow> string set" ("_ ;; _" [100,100] 100) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
11 |
where |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
12 |
"A ;; B = {s1 @ s2 | s1 s2. s1 \<in> A \<and> s2 \<in> B}" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
13 |
|
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
14 |
text {* Two Simple Properties about Sequential Composition *} |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
15 |
|
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
16 |
lemma seq_empty [simp]: |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
17 |
shows "A ;; {[]} = A" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
18 |
and "{[]} ;; A = A" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
19 |
by (simp_all add: Sequ_def) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
20 |
|
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
21 |
lemma seq_null [simp]: |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
22 |
shows "A ;; {} = {}" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
23 |
and "{} ;; A = {}" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
24 |
by (simp_all add: Sequ_def) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
25 |
|
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
26 |
section {* Regular Expressions *} |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
27 |
|
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
28 |
datatype rexp = |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
29 |
NULL |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
30 |
| EMPTY |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
31 |
| CHAR char |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
32 |
| SEQ rexp rexp |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
33 |
| ALT rexp rexp |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
34 |
|
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
35 |
section {* Semantics of Regular Expressions *} |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
36 |
|
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
37 |
fun |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
38 |
L :: "rexp \<Rightarrow> string set" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
39 |
where |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
40 |
"L (NULL) = {}" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
41 |
| "L (EMPTY) = {[]}" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
42 |
| "L (CHAR c) = {[c]}" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
43 |
| "L (SEQ r1 r2) = (L r1) ;; (L r2)" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
44 |
| "L (ALT r1 r2) = (L r1) \<union> (L r2)" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
45 |
|
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
46 |
fun |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
47 |
nullable :: "rexp \<Rightarrow> bool" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
48 |
where |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
49 |
"nullable (NULL) = False" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
50 |
| "nullable (EMPTY) = True" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
51 |
| "nullable (CHAR c) = False" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
52 |
| "nullable (ALT r1 r2) = (nullable r1 \<or> nullable r2)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
53 |
| "nullable (SEQ r1 r2) = (nullable r1 \<and> nullable r2)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
54 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
55 |
lemma nullable_correctness: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
56 |
shows "nullable r \<longleftrightarrow> [] \<in> (L r)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
57 |
apply (induct r) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
58 |
apply(auto simp add: Sequ_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
59 |
done |
7
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
60 |
|
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
61 |
section {* Values *} |
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
62 |
|
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
63 |
datatype val = |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
64 |
Void |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
65 |
| Char char |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
66 |
| Seq val val |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
67 |
| Right val |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
68 |
| Left val |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
69 |
|
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
70 |
section {* The string behind a value *} |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
71 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
72 |
fun |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
73 |
flat :: "val \<Rightarrow> string" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
74 |
where |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
75 |
"flat(Void) = []" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
76 |
| "flat(Char c) = [c]" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
77 |
| "flat(Left v) = flat(v)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
78 |
| "flat(Right v) = flat(v)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
79 |
| "flat(Seq v1 v2) = flat(v1) @ flat(v2)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
80 |
|
7
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
81 |
section {* Relation between values and regular expressions *} |
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
82 |
|
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
83 |
inductive |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
84 |
Prf :: "val \<Rightarrow> rexp \<Rightarrow> bool" ("\<turnstile> _ : _" [100, 100] 100) |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
85 |
where |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
86 |
"\<lbrakk>\<turnstile> v1 : r1; \<turnstile> v2 : r2\<rbrakk> \<Longrightarrow> \<turnstile> Seq v1 v2 : SEQ r1 r2" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
87 |
| "\<turnstile> v1 : r1 \<Longrightarrow> \<turnstile> Left v1 : ALT r1 r2" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
88 |
| "\<turnstile> v2 : r2 \<Longrightarrow> \<turnstile> Right v2 : ALT r1 r2" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
89 |
| "\<turnstile> Void : EMPTY" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
90 |
| "\<turnstile> Char c : CHAR c" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
91 |
|
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
92 |
lemma not_nullable_flat: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
93 |
assumes "\<turnstile> v : r" "\<not>nullable r" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
94 |
shows "flat v \<noteq> []" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
95 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
96 |
apply(induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
97 |
apply(auto) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
98 |
done |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
99 |
|
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
100 |
lemma Prf_flat_L: |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
101 |
assumes "\<turnstile> v : r" shows "flat v \<in> L r" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
102 |
using assms |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
103 |
apply(induct v r rule: Prf.induct) |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
104 |
apply(auto simp add: Sequ_def) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
105 |
done |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
106 |
|
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
107 |
lemma L_flat_Prf: |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
108 |
"L(r) = {flat v | v. \<turnstile> v : r}" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
109 |
apply(induct r) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
110 |
apply(auto dest: Prf_flat_L simp add: Sequ_def) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
111 |
apply (metis Prf.intros(4) flat.simps(1)) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
112 |
apply (metis Prf.intros(5) flat.simps(2)) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
113 |
apply (metis Prf.intros(1) flat.simps(5)) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
114 |
apply (metis Prf.intros(2) flat.simps(3)) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
115 |
apply (metis Prf.intros(3) flat.simps(4)) |
7
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
116 |
apply(erule Prf.cases) |
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
117 |
apply(auto) |
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
118 |
done |
6
87618dae0e04
getting back the original version by Sulzmann
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
5
diff
changeset
|
119 |
|
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
120 |
section {* Greedy Ordering according to Frisch/Cardelli *} |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
121 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
122 |
inductive |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
123 |
GrOrd :: "val \<Rightarrow> val \<Rightarrow> bool" ("_ gr\<succ> _") |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
124 |
where |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
125 |
"v1 gr\<succ> v1' \<Longrightarrow> (Seq v1 v2) gr\<succ> (Seq v1' v2')" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
126 |
| "v2 gr\<succ> v2' \<Longrightarrow> (Seq v1 v2) gr\<succ> (Seq v1 v2')" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
127 |
| "v1 gr\<succ> v2 \<Longrightarrow> (Left v1) gr\<succ> (Left v2)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
128 |
| "v1 gr\<succ> v2 \<Longrightarrow> (Right v1) gr\<succ> (Right v2)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
129 |
| "(Left v2) gr\<succ>(Right v1)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
130 |
| "(Char c) gr\<succ> (Char c)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
131 |
| "(Void) gr\<succ> (Void)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
132 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
133 |
lemma Gr_refl: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
134 |
assumes "\<turnstile> v : r" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
135 |
shows "v gr\<succ> v" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
136 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
137 |
apply(induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
138 |
apply(auto intro: GrOrd.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
139 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
140 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
141 |
lemma Gr_total: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
142 |
assumes "\<turnstile> v1 : r" "\<turnstile> v2 : r" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
143 |
shows "v1 gr\<succ> v2 \<or> v2 gr\<succ> v1" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
144 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
145 |
apply(induct v1 r arbitrary: v2 rule: Prf.induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
146 |
apply(rotate_tac 4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
147 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
148 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
149 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
150 |
apply (metis GrOrd.intros(1) GrOrd.intros(2)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
151 |
apply(rotate_tac 2) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
152 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
153 |
apply(simp_all) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
154 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
155 |
apply (metis GrOrd.intros(3)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
156 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
157 |
apply (metis GrOrd.intros(5)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
158 |
apply(rotate_tac 2) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
159 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
160 |
apply(simp_all) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
161 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
162 |
apply (metis GrOrd.intros(5)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
163 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
164 |
apply (metis GrOrd.intros(4)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
165 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
166 |
apply(simp_all) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
167 |
apply (metis GrOrd.intros(7)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
168 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
169 |
apply(simp_all) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
170 |
apply (metis GrOrd.intros(6)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
171 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
172 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
173 |
lemma Gr_trans: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
174 |
assumes "v1 gr\<succ> v2" "v2 gr\<succ> v3" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
175 |
and "\<turnstile> v1 : r" "\<turnstile> v2 : r" "\<turnstile> v3 : r" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
176 |
shows "v1 gr\<succ> v3" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
177 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
178 |
apply(induct r arbitrary: v1 v2 v3) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
179 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
180 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
181 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
182 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
183 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
184 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
185 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
186 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
187 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
188 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
189 |
defer |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
190 |
(* ALT case *) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
191 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
192 |
apply(simp_all (no_asm_use))[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
193 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
194 |
apply(simp_all (no_asm_use))[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
195 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
196 |
apply(simp_all (no_asm_use))[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
197 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
198 |
apply(erule GrOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
199 |
apply(simp_all (no_asm_use))[7] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
200 |
apply(erule GrOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
201 |
apply(simp_all (no_asm_use))[7] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
202 |
apply (metis GrOrd.intros(3)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
203 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
204 |
apply(erule GrOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
205 |
apply(simp_all (no_asm_use))[7] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
206 |
apply(erule GrOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
207 |
apply(simp_all (no_asm_use))[7] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
208 |
apply (metis GrOrd.intros(5)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
209 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
210 |
apply(simp_all (no_asm_use))[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
211 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
212 |
apply(erule GrOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
213 |
apply(simp_all (no_asm_use))[7] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
214 |
apply(erule GrOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
215 |
apply(simp_all (no_asm_use))[7] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
216 |
apply (metis GrOrd.intros(5)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
217 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
218 |
apply(simp_all (no_asm_use))[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
219 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
220 |
apply(simp_all (no_asm_use))[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
221 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
222 |
apply(erule GrOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
223 |
apply(simp_all (no_asm_use))[7] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
224 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
225 |
apply(erule GrOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
226 |
apply(simp_all (no_asm_use))[7] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
227 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
228 |
apply(simp_all (no_asm_use))[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
229 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
230 |
apply(erule GrOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
231 |
apply(simp_all (no_asm_use))[7] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
232 |
apply(erule GrOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
233 |
apply(simp_all (no_asm_use))[7] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
234 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
235 |
apply(erule GrOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
236 |
apply(simp_all (no_asm_use))[7] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
237 |
apply(erule GrOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
238 |
apply(simp_all (no_asm_use))[7] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
239 |
apply (metis GrOrd.intros(4)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
240 |
(* SEQ case *) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
241 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
242 |
apply(simp_all (no_asm_use))[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
243 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
244 |
apply(simp_all (no_asm_use))[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
245 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
246 |
apply(simp_all (no_asm_use))[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
247 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
248 |
apply(erule GrOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
249 |
apply(simp_all (no_asm_use))[7] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
250 |
apply(erule GrOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
251 |
apply(simp_all (no_asm_use))[7] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
252 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
253 |
apply (metis GrOrd.intros(1)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
254 |
apply (metis GrOrd.intros(1)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
255 |
apply(erule GrOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
256 |
apply(simp_all (no_asm_use))[7] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
257 |
apply (metis GrOrd.intros(1)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
258 |
by (metis GrOrd.intros(1) Gr_refl) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
259 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
260 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
261 |
section {* Values Sets *} |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
262 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
263 |
definition prefix :: "string \<Rightarrow> string \<Rightarrow> bool" ("_ \<sqsubseteq> _" [100, 100] 100) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
264 |
where |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
265 |
"s1 \<sqsubseteq> s2 \<equiv> \<exists>s3. s1 @ s3 = s2" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
266 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
267 |
definition sprefix :: "string \<Rightarrow> string \<Rightarrow> bool" ("_ \<sqsubset> _" [100, 100] 100) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
268 |
where |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
269 |
"s1 \<sqsubset> s2 \<equiv> (s1 \<sqsubseteq> s2 \<and> s1 \<noteq> s2)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
270 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
271 |
lemma length_sprefix: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
272 |
"s1 \<sqsubset> s2 \<Longrightarrow> length s1 < length s2" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
273 |
unfolding sprefix_def prefix_def |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
274 |
by (auto) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
275 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
276 |
definition Prefixes :: "string \<Rightarrow> string set" where |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
277 |
"Prefixes s \<equiv> {sp. sp \<sqsubseteq> s}" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
278 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
279 |
definition Suffixes :: "string \<Rightarrow> string set" where |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
280 |
"Suffixes s \<equiv> rev ` (Prefixes (rev s))" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
281 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
282 |
lemma Suffixes_in: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
283 |
"\<exists>s1. s1 @ s2 = s3 \<Longrightarrow> s2 \<in> Suffixes s3" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
284 |
unfolding Suffixes_def Prefixes_def prefix_def image_def |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
285 |
apply(auto) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
286 |
by (metis rev_rev_ident) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
287 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
288 |
lemma Prefixes_Cons: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
289 |
"Prefixes (c # s) = {[]} \<union> {c # sp | sp. sp \<in> Prefixes s}" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
290 |
unfolding Prefixes_def prefix_def |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
291 |
apply(auto simp add: append_eq_Cons_conv) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
292 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
293 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
294 |
lemma finite_Prefixes: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
295 |
"finite (Prefixes s)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
296 |
apply(induct s) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
297 |
apply(auto simp add: Prefixes_def prefix_def)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
298 |
apply(simp add: Prefixes_Cons) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
299 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
300 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
301 |
lemma finite_Suffixes: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
302 |
"finite (Suffixes s)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
303 |
unfolding Suffixes_def |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
304 |
apply(rule finite_imageI) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
305 |
apply(rule finite_Prefixes) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
306 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
307 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
308 |
lemma prefix_Cons: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
309 |
"((c # s1) \<sqsubseteq> (c # s2)) = (s1 \<sqsubseteq> s2)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
310 |
apply(auto simp add: prefix_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
311 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
312 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
313 |
lemma prefix_append: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
314 |
"((s @ s1) \<sqsubseteq> (s @ s2)) = (s1 \<sqsubseteq> s2)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
315 |
apply(induct s) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
316 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
317 |
apply(simp add: prefix_Cons) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
318 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
319 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
320 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
321 |
definition Values :: "rexp \<Rightarrow> string \<Rightarrow> val set" where |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
322 |
"Values r s \<equiv> {v. \<turnstile> v : r \<and> flat v \<sqsubseteq> s}" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
323 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
324 |
definition rest :: "val \<Rightarrow> string \<Rightarrow> string" where |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
325 |
"rest v s \<equiv> drop (length (flat v)) s" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
326 |
|
88
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
327 |
lemma rest_flat: |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
328 |
assumes "flat v1 \<sqsubseteq> s" |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
329 |
shows "flat v1 @ rest v1 s = s" |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
330 |
using assms |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
331 |
apply(auto simp add: rest_def prefix_def) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
332 |
done |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
333 |
|
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
334 |
|
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
335 |
lemma rest_Suffixes: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
336 |
"rest v s \<in> Suffixes s" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
337 |
unfolding rest_def |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
338 |
by (metis Suffixes_in append_take_drop_id) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
339 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
340 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
341 |
lemma Values_recs: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
342 |
"Values (NULL) s = {}" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
343 |
"Values (EMPTY) s = {Void}" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
344 |
"Values (CHAR c) s = (if [c] \<sqsubseteq> s then {Char c} else {})" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
345 |
"Values (ALT r1 r2) s = {Left v | v. v \<in> Values r1 s} \<union> {Right v | v. v \<in> Values r2 s}" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
346 |
"Values (SEQ r1 r2) s = {Seq v1 v2 | v1 v2. v1 \<in> Values r1 s \<and> v2 \<in> Values r2 (rest v1 s)}" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
347 |
unfolding Values_def |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
348 |
apply(auto) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
349 |
(*NULL*) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
350 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
351 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
352 |
(*EMPTY*) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
353 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
354 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
355 |
apply(rule Prf.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
356 |
apply (metis append_Nil prefix_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
357 |
(*CHAR*) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
358 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
359 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
360 |
apply(rule Prf.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
361 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
362 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
363 |
(*ALT*) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
364 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
365 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
366 |
apply (metis Prf.intros(2)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
367 |
apply (metis Prf.intros(3)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
368 |
(*SEQ*) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
369 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
370 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
371 |
apply (simp add: append_eq_conv_conj prefix_def rest_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
372 |
apply (metis Prf.intros(1)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
373 |
apply (simp add: append_eq_conv_conj prefix_def rest_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
374 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
375 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
376 |
lemma Values_finite: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
377 |
"finite (Values r s)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
378 |
apply(induct r arbitrary: s) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
379 |
apply(simp_all add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
380 |
thm finite_surj |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
381 |
apply(rule_tac f="\<lambda>(x, y). Seq x y" and |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
382 |
A="{(v1, v2) | v1 v2. v1 \<in> Values r1 s \<and> v2 \<in> Values r2 (rest v1 s)}" in finite_surj) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
383 |
prefer 2 |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
384 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
385 |
apply(rule_tac B="\<Union>sp \<in> Suffixes s. {(v1, v2). v1 \<in> Values r1 s \<and> v2 \<in> Values r2 sp}" in finite_subset) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
386 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
387 |
apply (metis rest_Suffixes) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
388 |
apply(rule finite_UN_I) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
389 |
apply(rule finite_Suffixes) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
390 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
391 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
392 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
393 |
section {* Sulzmann functions *} |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
394 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
395 |
fun |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
396 |
mkeps :: "rexp \<Rightarrow> val" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
397 |
where |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
398 |
"mkeps(EMPTY) = Void" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
399 |
| "mkeps(SEQ r1 r2) = Seq (mkeps r1) (mkeps r2)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
400 |
| "mkeps(ALT r1 r2) = (if nullable(r1) then Left (mkeps r1) else Right (mkeps r2))" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
401 |
|
83
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
402 |
section {* Derivatives *} |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
403 |
|
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
404 |
fun |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
405 |
der :: "char \<Rightarrow> rexp \<Rightarrow> rexp" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
406 |
where |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
407 |
"der c (NULL) = NULL" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
408 |
| "der c (EMPTY) = NULL" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
409 |
| "der c (CHAR c') = (if c = c' then EMPTY else NULL)" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
410 |
| "der c (ALT r1 r2) = ALT (der c r1) (der c r2)" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
411 |
| "der c (SEQ r1 r2) = |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
412 |
(if nullable r1 |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
413 |
then ALT (SEQ (der c r1) r2) (der c r2) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
414 |
else SEQ (der c r1) r2)" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
415 |
|
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
416 |
fun |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
417 |
ders :: "string \<Rightarrow> rexp \<Rightarrow> rexp" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
418 |
where |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
419 |
"ders [] r = r" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
420 |
| "ders (c # s) r = ders s (der c r)" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
421 |
|
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
422 |
|
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
423 |
section {* Injection function *} |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
424 |
|
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
425 |
fun injval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
426 |
where |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
427 |
"injval (EMPTY) c Void = Char c" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
428 |
| "injval (CHAR d) c Void = Char d" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
429 |
| "injval (CHAR d) c (Char c') = Seq (Char d) (Char c')" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
430 |
| "injval (ALT r1 r2) c (Left v1) = Left(injval r1 c v1)" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
431 |
| "injval (ALT r1 r2) c (Right v2) = Right(injval r2 c v2)" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
432 |
| "injval (SEQ r1 r2) c (Char c') = Seq (Char c) (Char c')" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
433 |
| "injval (SEQ r1 r2) c (Seq v1 v2) = Seq (injval r1 c v1) v2" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
434 |
| "injval (SEQ r1 r2) c (Left (Seq v1 v2)) = Seq (injval r1 c v1) v2" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
435 |
| "injval (SEQ r1 r2) c (Right v2) = Seq (mkeps r1) (injval r2 c v2)" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
436 |
|
85
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
437 |
fun |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
438 |
lex :: "rexp \<Rightarrow> string \<Rightarrow> val option" |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
439 |
where |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
440 |
"lex r [] = (if nullable r then Some(mkeps r) else None)" |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
441 |
| "lex r (c#s) = (case (lex (der c r) s) of |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
442 |
None \<Rightarrow> None |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
443 |
| Some(v) \<Rightarrow> Some(injval r c v))" |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
444 |
|
86
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
445 |
fun |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
446 |
lex2 :: "rexp \<Rightarrow> string \<Rightarrow> val" |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
447 |
where |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
448 |
"lex2 r [] = mkeps r" |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
449 |
| "lex2 r (c#s) = injval r c (lex2 (der c r) s)" |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
450 |
|
83
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
451 |
|
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
452 |
section {* Projection function *} |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
453 |
|
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
454 |
fun projval :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
455 |
where |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
456 |
"projval (CHAR d) c _ = Void" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
457 |
| "projval (ALT r1 r2) c (Left v1) = Left (projval r1 c v1)" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
458 |
| "projval (ALT r1 r2) c (Right v2) = Right (projval r2 c v2)" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
459 |
| "projval (SEQ r1 r2) c (Seq v1 v2) = |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
460 |
(if flat v1 = [] then Right(projval r2 c v2) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
461 |
else if nullable r1 then Left (Seq (projval r1 c v1) v2) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
462 |
else Seq (projval r1 c v1) v2)" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
463 |
|
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
464 |
|
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
465 |
|
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
466 |
lemma mkeps_nullable: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
467 |
assumes "nullable(r)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
468 |
shows "\<turnstile> mkeps r : r" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
469 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
470 |
apply(induct rule: nullable.induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
471 |
apply(auto intro: Prf.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
472 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
473 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
474 |
lemma mkeps_flat: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
475 |
assumes "nullable(r)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
476 |
shows "flat (mkeps r) = []" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
477 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
478 |
apply(induct rule: nullable.induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
479 |
apply(auto) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
480 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
481 |
|
83
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
482 |
lemma v3: |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
483 |
assumes "\<turnstile> v : der c r" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
484 |
shows "\<turnstile> (injval r c v) : r" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
485 |
using assms |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
486 |
apply(induct arbitrary: v rule: der.induct) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
487 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
488 |
apply(erule Prf.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
489 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
490 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
491 |
apply(erule Prf.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
492 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
493 |
apply(case_tac "c = c'") |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
494 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
495 |
apply(erule Prf.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
496 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
497 |
apply (metis Prf.intros(5)) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
498 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
499 |
apply(erule Prf.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
500 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
501 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
502 |
apply(erule Prf.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
503 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
504 |
apply (metis Prf.intros(2)) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
505 |
apply (metis Prf.intros(3)) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
506 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
507 |
apply(case_tac "nullable r1") |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
508 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
509 |
apply(erule Prf.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
510 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
511 |
apply(auto)[1] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
512 |
apply(erule Prf.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
513 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
514 |
apply(auto)[1] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
515 |
apply (metis Prf.intros(1)) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
516 |
apply(auto)[1] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
517 |
apply (metis Prf.intros(1) mkeps_nullable) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
518 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
519 |
apply(erule Prf.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
520 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
521 |
apply(auto)[1] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
522 |
apply(rule Prf.intros) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
523 |
apply(auto)[2] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
524 |
done |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
525 |
|
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
526 |
lemma v3_proj: |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
527 |
assumes "\<turnstile> v : r" and "\<exists>s. (flat v) = c # s" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
528 |
shows "\<turnstile> (projval r c v) : der c r" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
529 |
using assms |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
530 |
apply(induct rule: Prf.induct) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
531 |
prefer 4 |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
532 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
533 |
prefer 4 |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
534 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
535 |
apply (metis Prf.intros(4)) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
536 |
prefer 2 |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
537 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
538 |
apply (metis Prf.intros(2)) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
539 |
prefer 2 |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
540 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
541 |
apply (metis Prf.intros(3)) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
542 |
apply(auto) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
543 |
apply(rule Prf.intros) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
544 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
545 |
apply (metis Prf_flat_L nullable_correctness) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
546 |
apply(rule Prf.intros) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
547 |
apply(rule Prf.intros) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
548 |
apply (metis Cons_eq_append_conv) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
549 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
550 |
apply(rule Prf.intros) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
551 |
apply (metis Cons_eq_append_conv) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
552 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
553 |
done |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
554 |
|
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
555 |
lemma v4: |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
556 |
assumes "\<turnstile> v : der c r" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
557 |
shows "flat (injval r c v) = c # (flat v)" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
558 |
using assms |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
559 |
apply(induct arbitrary: v rule: der.induct) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
560 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
561 |
apply(erule Prf.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
562 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
563 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
564 |
apply(erule Prf.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
565 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
566 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
567 |
apply(case_tac "c = c'") |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
568 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
569 |
apply(auto)[1] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
570 |
apply(erule Prf.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
571 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
572 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
573 |
apply(erule Prf.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
574 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
575 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
576 |
apply(erule Prf.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
577 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
578 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
579 |
apply(case_tac "nullable r1") |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
580 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
581 |
apply(erule Prf.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
582 |
apply(simp_all (no_asm_use))[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
583 |
apply(auto)[1] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
584 |
apply(erule Prf.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
585 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
586 |
apply(clarify) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
587 |
apply(simp only: injval.simps flat.simps) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
588 |
apply(auto)[1] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
589 |
apply (metis mkeps_flat) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
590 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
591 |
apply(erule Prf.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
592 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
593 |
done |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
594 |
|
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
595 |
lemma v4_proj: |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
596 |
assumes "\<turnstile> v : r" and "\<exists>s. (flat v) = c # s" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
597 |
shows "c # flat (projval r c v) = flat v" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
598 |
using assms |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
599 |
apply(induct rule: Prf.induct) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
600 |
prefer 4 |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
601 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
602 |
prefer 4 |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
603 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
604 |
prefer 2 |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
605 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
606 |
prefer 2 |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
607 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
608 |
apply(auto) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
609 |
by (metis Cons_eq_append_conv) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
610 |
|
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
611 |
lemma v4_proj2: |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
612 |
assumes "\<turnstile> v : r" and "(flat v) = c # s" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
613 |
shows "flat (projval r c v) = s" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
614 |
using assms |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
615 |
by (metis list.inject v4_proj) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
616 |
|
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
617 |
|
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
618 |
section {* Alternative Posix definition *} |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
619 |
|
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
620 |
inductive |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
621 |
PMatch :: "string \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<in> _ \<rightarrow> _" [100, 100, 100] 100) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
622 |
where |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
623 |
"[] \<in> EMPTY \<rightarrow> Void" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
624 |
| "[c] \<in> (CHAR c) \<rightarrow> (Char c)" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
625 |
| "s \<in> r1 \<rightarrow> v \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Left v)" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
626 |
| "\<lbrakk>s \<in> r2 \<rightarrow> v; s \<notin> L(r1)\<rbrakk> \<Longrightarrow> s \<in> (ALT r1 r2) \<rightarrow> (Right v)" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
627 |
| "\<lbrakk>s1 \<in> r1 \<rightarrow> v1; s2 \<in> r2 \<rightarrow> v2; |
84
f89372781a4c
the algorithm is correct according to the Type Inference definition
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
83
diff
changeset
|
628 |
\<not>(\<exists>s3 s4. s3 \<noteq> [] \<and> s3 @ s4 = s2 \<and> (s1 @ s3) \<in> L r1 \<and> s4 \<in> L r2)\<rbrakk> \<Longrightarrow> |
83
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
629 |
(s1 @ s2) \<in> (SEQ r1 r2) \<rightarrow> (Seq v1 v2)" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
630 |
|
86
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
631 |
|
83
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
632 |
lemma PMatch_mkeps: |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
633 |
assumes "nullable r" |
85
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
634 |
shows "[] \<in> r \<rightarrow> mkeps r" |
83
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
635 |
using assms |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
636 |
apply(induct r) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
637 |
apply(auto) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
638 |
apply (metis PMatch.intros(1)) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
639 |
apply(subst append.simps(1)[symmetric]) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
640 |
apply (rule PMatch.intros) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
641 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
642 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
643 |
apply(auto)[1] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
644 |
apply (rule PMatch.intros) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
645 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
646 |
apply (rule PMatch.intros) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
647 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
648 |
apply (rule PMatch.intros) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
649 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
650 |
by (metis nullable_correctness) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
651 |
|
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
652 |
|
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
653 |
lemma PMatch1: |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
654 |
assumes "s \<in> r \<rightarrow> v" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
655 |
shows "\<turnstile> v : r" "flat v = s" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
656 |
using assms |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
657 |
apply(induct s r v rule: PMatch.induct) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
658 |
apply(auto) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
659 |
apply (metis Prf.intros(4)) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
660 |
apply (metis Prf.intros(5)) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
661 |
apply (metis Prf.intros(2)) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
662 |
apply (metis Prf.intros(3)) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
663 |
apply (metis Prf.intros(1)) |
84
f89372781a4c
the algorithm is correct according to the Type Inference definition
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
83
diff
changeset
|
664 |
done |
83
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
665 |
|
86
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
666 |
lemma PMatch_Values: |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
667 |
assumes "s \<in> r \<rightarrow> v" |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
668 |
shows "v \<in> Values r s" |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
669 |
using assms |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
670 |
apply(simp add: Values_def PMatch1) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
671 |
by (metis append_Nil2 prefix_def) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
672 |
|
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
673 |
lemma PMatch2: |
83
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
674 |
assumes "s \<in> (der c r) \<rightarrow> v" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
675 |
shows "(c#s) \<in> r \<rightarrow> (injval r c v)" |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
676 |
using assms |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
677 |
apply(induct c r arbitrary: s v rule: der.induct) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
678 |
apply(auto) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
679 |
apply(erule PMatch.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
680 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
681 |
apply(erule PMatch.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
682 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
683 |
apply(case_tac "c = c'") |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
684 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
685 |
apply(erule PMatch.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
686 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
687 |
apply (metis PMatch.intros(2)) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
688 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
689 |
apply(erule PMatch.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
690 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
691 |
apply(erule PMatch.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
692 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
693 |
apply (metis PMatch.intros(3)) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
694 |
apply(clarify) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
695 |
apply(rule PMatch.intros) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
696 |
apply metis |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
697 |
apply(simp add: L_flat_Prf) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
698 |
apply(auto)[1] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
699 |
thm v3_proj |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
700 |
apply(frule_tac c="c" in v3_proj) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
701 |
apply metis |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
702 |
apply(drule_tac x="projval r1 c v" in spec) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
703 |
apply(drule mp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
704 |
apply (metis v4_proj2) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
705 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
706 |
apply(case_tac "nullable r1") |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
707 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
708 |
defer |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
709 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
710 |
apply(erule PMatch.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
711 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
712 |
apply(clarify) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
713 |
apply(subst append.simps(2)[symmetric]) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
714 |
apply(rule PMatch.intros) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
715 |
apply metis |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
716 |
apply metis |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
717 |
apply(auto)[1] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
718 |
apply(simp add: L_flat_Prf) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
719 |
apply(auto)[1] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
720 |
apply(frule_tac c="c" in v3_proj) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
721 |
apply metis |
84
f89372781a4c
the algorithm is correct according to the Type Inference definition
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
83
diff
changeset
|
722 |
apply(drule_tac x="s3" in spec) |
83
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
723 |
apply(drule mp) |
84
f89372781a4c
the algorithm is correct according to the Type Inference definition
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
83
diff
changeset
|
724 |
apply(rule_tac x="projval r1 c v" in exI) |
f89372781a4c
the algorithm is correct according to the Type Inference definition
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
83
diff
changeset
|
725 |
apply(rule conjI) |
83
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
726 |
apply (metis v4_proj2) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
727 |
apply(simp) |
84
f89372781a4c
the algorithm is correct according to the Type Inference definition
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
83
diff
changeset
|
728 |
apply metis |
83
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
729 |
(* nullable case *) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
730 |
apply(erule PMatch.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
731 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
732 |
apply(clarify) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
733 |
apply(erule PMatch.cases) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
734 |
apply(simp_all)[5] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
735 |
apply(clarify) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
736 |
apply(subst append.simps(2)[symmetric]) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
737 |
apply(rule PMatch.intros) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
738 |
apply metis |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
739 |
apply metis |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
740 |
apply(auto)[1] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
741 |
apply(simp add: L_flat_Prf) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
742 |
apply(auto)[1] |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
743 |
apply(frule_tac c="c" in v3_proj) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
744 |
apply metis |
84
f89372781a4c
the algorithm is correct according to the Type Inference definition
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
83
diff
changeset
|
745 |
apply(drule_tac x="s3" in spec) |
83
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
746 |
apply(drule mp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
747 |
apply (metis v4_proj2) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
748 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
749 |
(* interesting case *) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
750 |
apply(clarify) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
751 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
752 |
thm L.simps |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
753 |
apply(subst (asm) L.simps(4)[symmetric]) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
754 |
apply(simp only: L_flat_Prf) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
755 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
756 |
apply(subst append.simps(1)[symmetric]) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
757 |
apply(rule PMatch.intros) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
758 |
apply (metis PMatch_mkeps) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
759 |
apply metis |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
760 |
apply(auto) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
761 |
apply(simp only: L_flat_Prf) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
762 |
apply(simp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
763 |
apply(auto) |
84
f89372781a4c
the algorithm is correct according to the Type Inference definition
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
83
diff
changeset
|
764 |
apply(drule_tac x="Seq (projval r1 c v) vb" in spec) |
83
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
765 |
apply(drule mp) |
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
766 |
apply(simp) |
84
f89372781a4c
the algorithm is correct according to the Type Inference definition
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
83
diff
changeset
|
767 |
apply (metis append_Cons butlast_snoc last_snoc neq_Nil_conv rotate1.simps(2) v4_proj2) |
f89372781a4c
the algorithm is correct according to the Type Inference definition
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
83
diff
changeset
|
768 |
apply(subgoal_tac "\<turnstile> projval r1 c v : der c r1") |
f89372781a4c
the algorithm is correct according to the Type Inference definition
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
83
diff
changeset
|
769 |
apply (metis Prf.intros(1)) |
83
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
770 |
apply(rule v3_proj) |
84
f89372781a4c
the algorithm is correct according to the Type Inference definition
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
83
diff
changeset
|
771 |
apply(simp) |
f89372781a4c
the algorithm is correct according to the Type Inference definition
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
83
diff
changeset
|
772 |
by (metis Cons_eq_append_conv) |
83
a8bcb5a0f9b9
added POSIX relation from the Type-Inference paper
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
82
diff
changeset
|
773 |
|
86
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
774 |
lemma lex_correct1: |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
775 |
assumes "s \<notin> L r" |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
776 |
shows "lex r s = None" |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
777 |
using assms |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
778 |
apply(induct s arbitrary: r) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
779 |
apply(simp) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
780 |
apply (metis nullable_correctness) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
781 |
apply(auto) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
782 |
apply(drule_tac x="der a r" in meta_spec) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
783 |
apply(drule meta_mp) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
784 |
apply(auto) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
785 |
apply(simp add: L_flat_Prf) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
786 |
by (metis v3 v4) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
787 |
|
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
788 |
|
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
789 |
lemma lex_correct2: |
85
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
790 |
assumes "s \<in> L r" |
86
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
791 |
shows "\<exists>v. lex r s = Some(v) \<and> \<turnstile> v : r \<and> flat v = s" |
85
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
792 |
using assms |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
793 |
apply(induct s arbitrary: r) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
794 |
apply(simp) |
86
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
795 |
apply (metis mkeps_flat mkeps_nullable nullable_correctness) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
796 |
apply(drule_tac x="der a r" in meta_spec) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
797 |
apply(drule meta_mp) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
798 |
apply(simp add: L_flat_Prf) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
799 |
apply(auto) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
800 |
apply (metis v3_proj v4_proj2) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
801 |
apply (metis v3) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
802 |
apply(rule v4) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
803 |
by metis |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
804 |
|
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
805 |
lemma lex_correct3: |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
806 |
assumes "s \<in> L r" |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
807 |
shows "\<exists>v. lex r s = Some(v) \<and> s \<in> r \<rightarrow> v" |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
808 |
using assms |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
809 |
apply(induct s arbitrary: r) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
810 |
apply(simp) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
811 |
apply (metis PMatch_mkeps nullable_correctness) |
85
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
812 |
apply(drule_tac x="der a r" in meta_spec) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
813 |
apply(drule meta_mp) |
86
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
814 |
apply(simp add: L_flat_Prf) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
815 |
apply(auto) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
816 |
apply (metis v3_proj v4_proj2) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
817 |
apply(rule PMatch2) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
818 |
apply(simp) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
819 |
done |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
820 |
|
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
821 |
lemma lex_correct4: |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
822 |
assumes "s \<in> L r" |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
823 |
shows "s \<in> r \<rightarrow> (lex2 r s)" |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
824 |
using assms |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
825 |
apply(induct s arbitrary: r) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
826 |
apply(simp) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
827 |
apply (metis PMatch_mkeps nullable_correctness) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
828 |
apply(simp) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
829 |
apply(rule PMatch2) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
830 |
apply(drule_tac x="der a r" in meta_spec) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
831 |
apply(drule meta_mp) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
832 |
apply(simp add: L_flat_Prf) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
833 |
apply(auto) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
834 |
apply (metis v3_proj v4_proj2) |
56dd3d1d479b
added a proof about Values and PMatch
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
85
diff
changeset
|
835 |
done |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
836 |
|
87
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
837 |
lemma |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
838 |
"lex2 (ALT (CHAR a) (ALT (CHAR b) (SEQ (CHAR a) (CHAR b)))) [a,b] = Right (Right (Seq (Char a) (Char b)))" |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
839 |
apply(simp) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
840 |
done |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
841 |
|
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
842 |
|
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
843 |
section {* Sulzmann's Ordering of values *} |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
844 |
|
88
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
845 |
|
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
846 |
inductive ValOrd :: "val \<Rightarrow> rexp \<Rightarrow> val \<Rightarrow> bool" ("_ \<succ>_ _" [100, 100, 100] 100) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
847 |
where |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
848 |
"v2 \<succ>r2 v2' \<Longrightarrow> (Seq v1 v2) \<succ>(SEQ r1 r2) (Seq v1 v2')" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
849 |
| "\<lbrakk>v1 \<succ>r1 v1'; v1 \<noteq> v1'\<rbrakk> \<Longrightarrow> (Seq v1 v2) \<succ>(SEQ r1 r2) (Seq v1' v2')" |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
850 |
| "length (flat v1) \<ge> length (flat v2) \<Longrightarrow> (Left v1) \<succ>(ALT r1 r2) (Right v2)" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
851 |
| "length (flat v2) > length (flat v1) \<Longrightarrow> (Right v2) \<succ>(ALT r1 r2) (Left v1)" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
852 |
| "v2 \<succ>r2 v2' \<Longrightarrow> (Right v2) \<succ>(ALT r1 r2) (Right v2')" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
853 |
| "v1 \<succ>r1 v1' \<Longrightarrow> (Left v1) \<succ>(ALT r1 r2) (Left v1')" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
854 |
| "Void \<succ>EMPTY Void" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
855 |
| "(Char c) \<succ>(CHAR c) (Char c)" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
856 |
|
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
857 |
inductive ValOrd2 :: "val \<Rightarrow> val \<Rightarrow> bool" ("_ 2\<succ> _" [100, 100] 100) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
858 |
where |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
859 |
"v2 2\<succ> v2' \<Longrightarrow> (Seq v1 v2) 2\<succ> (Seq v1 v2')" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
860 |
| "\<lbrakk>v1 2\<succ> v1'; v1 \<noteq> v1'\<rbrakk> \<Longrightarrow> (Seq v1 v2) 2\<succ> (Seq v1' v2')" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
861 |
| "length (flat v1) \<ge> length (flat v2) \<Longrightarrow> (Left v1) 2\<succ> (Right v2)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
862 |
| "length (flat v2) > length (flat v1) \<Longrightarrow> (Right v2) 2\<succ> (Left v1)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
863 |
| "v2 2\<succ> v2' \<Longrightarrow> (Right v2) 2\<succ> (Right v2')" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
864 |
| "v1 2\<succ> v1' \<Longrightarrow> (Left v1) 2\<succ> (Left v1')" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
865 |
| "Void 2\<succ> Void" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
866 |
| "(Char c) 2\<succ> (Char c)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
867 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
868 |
lemma Ord1: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
869 |
"v1 \<succ>r v2 \<Longrightarrow> v1 2\<succ> v2" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
870 |
apply(induct rule: ValOrd.induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
871 |
apply(auto intro: ValOrd2.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
872 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
873 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
874 |
lemma Ord2: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
875 |
"v1 2\<succ> v2 \<Longrightarrow> \<exists>r. v1 \<succ>r v2" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
876 |
apply(induct v1 v2 rule: ValOrd2.induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
877 |
apply(auto intro: ValOrd.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
878 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
879 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
880 |
lemma Ord3: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
881 |
"\<lbrakk>v1 2\<succ> v2; \<turnstile> v1 : r\<rbrakk> \<Longrightarrow> v1 \<succ>r v2" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
882 |
apply(induct v1 v2 arbitrary: r rule: ValOrd2.induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
883 |
apply(auto intro: ValOrd.intros elim: Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
884 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
885 |
|
87
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
886 |
section {* Posix definition *} |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
887 |
|
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
888 |
definition POSIX :: "val \<Rightarrow> rexp \<Rightarrow> bool" |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
889 |
where |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
890 |
"POSIX v r \<equiv> (\<turnstile> v : r \<and> (\<forall>v'. (\<turnstile> v' : r \<and> flat v' \<sqsubseteq> flat v) \<longrightarrow> v \<succ>r v'))" |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
891 |
|
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
892 |
lemma ValOrd_refl: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
893 |
assumes "\<turnstile> v : r" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
894 |
shows "v \<succ>r v" |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
895 |
using assms |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
896 |
apply(induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
897 |
apply(auto intro: ValOrd.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
898 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
899 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
900 |
lemma ValOrd_total: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
901 |
shows "\<lbrakk>\<turnstile> v1 : r; \<turnstile> v2 : r\<rbrakk> \<Longrightarrow> v1 \<succ>r v2 \<or> v2 \<succ>r v1" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
902 |
apply(induct r arbitrary: v1 v2) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
903 |
apply(auto) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
904 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
905 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
906 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
907 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
908 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
909 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
910 |
apply (metis ValOrd.intros(7)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
911 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
912 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
913 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
914 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
915 |
apply (metis ValOrd.intros(8)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
916 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
917 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
918 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
919 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
920 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
921 |
apply(case_tac "v1a = v1b") |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
922 |
apply(simp) |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
923 |
apply(rule ValOrd.intros(1)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
924 |
apply (metis ValOrd.intros(1)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
925 |
apply(rule ValOrd.intros(2)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
926 |
apply(auto)[2] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
927 |
apply(erule contrapos_np) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
928 |
apply(rule ValOrd.intros(2)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
929 |
apply(auto) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
930 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
931 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
932 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
933 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
934 |
apply (metis Ord1 Ord3 Prf.intros(2) ValOrd2.intros(6)) |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
935 |
apply(rule ValOrd.intros) |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
936 |
apply(erule contrapos_np) |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
937 |
apply(rule ValOrd.intros) |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
938 |
apply (metis le_eq_less_or_eq neq_iff) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
939 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
940 |
apply(simp_all)[5] |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
941 |
apply(rule ValOrd.intros) |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
942 |
apply(erule contrapos_np) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
943 |
apply(rule ValOrd.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
944 |
apply (metis le_eq_less_or_eq neq_iff) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
945 |
apply(rule ValOrd.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
946 |
apply(erule contrapos_np) |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
947 |
apply(rule ValOrd.intros) |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
948 |
by metis |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
949 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
950 |
lemma ValOrd_anti: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
951 |
shows "\<lbrakk>\<turnstile> v1 : r; \<turnstile> v2 : r; v1 \<succ>r v2; v2 \<succ>r v1\<rbrakk> \<Longrightarrow> v1 = v2" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
952 |
apply(induct r arbitrary: v1 v2) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
953 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
954 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
955 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
956 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
957 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
958 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
959 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
960 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
961 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
962 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
963 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
964 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
965 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
966 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
967 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
968 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
969 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
970 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
971 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
972 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
973 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
974 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
975 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
976 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
977 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
978 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
979 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
980 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
981 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
982 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
983 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
984 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
985 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
986 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
987 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
988 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
989 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
990 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
991 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
992 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
993 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
994 |
apply(simp_all)[8] |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
995 |
done |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
996 |
|
87
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
997 |
lemma POSIX_ALT_I1: |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
998 |
assumes "POSIX v1 r1" |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
999 |
shows "POSIX (Left v1) (ALT r1 r2)" |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1000 |
using assms |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1001 |
unfolding POSIX_def |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1002 |
apply(auto) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1003 |
apply (metis Prf.intros(2)) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1004 |
apply(rotate_tac 2) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1005 |
apply(erule Prf.cases) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1006 |
apply(simp_all)[5] |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1007 |
apply(auto) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1008 |
apply(rule ValOrd.intros) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1009 |
apply(auto) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1010 |
apply(rule ValOrd.intros) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1011 |
by (metis le_eq_less_or_eq length_sprefix sprefix_def) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1012 |
|
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1013 |
lemma POSIX_ALT_I2: |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1014 |
assumes "POSIX v2 r2" "\<forall>v'. \<turnstile> v' : r1 \<longrightarrow> length (flat v2) > length (flat v')" |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1015 |
shows "POSIX (Right v2) (ALT r1 r2)" |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1016 |
using assms |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1017 |
unfolding POSIX_def |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1018 |
apply(auto) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1019 |
apply (metis Prf.intros) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1020 |
apply(rotate_tac 3) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1021 |
apply(erule Prf.cases) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1022 |
apply(simp_all)[5] |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1023 |
apply(auto) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1024 |
apply(rule ValOrd.intros) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1025 |
apply metis |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1026 |
apply(rule ValOrd.intros) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1027 |
apply metis |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1028 |
done |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
1029 |
|
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1030 |
section {* tryout with all-mkeps *} |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1031 |
|
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1032 |
fun |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1033 |
alleps :: "rexp \<Rightarrow> val set" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1034 |
where |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1035 |
"alleps(NULL) = {}" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1036 |
| "alleps(EMPTY) = {Void}" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1037 |
| "alleps(CHAR c) = {}" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1038 |
| "alleps(SEQ r1 r2) = {Seq v1 v2 | v1 v2. v1 \<in> alleps r1 \<and> v2 \<in> alleps r2}" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1039 |
| "alleps(ALT r1 r2) = {Left v1 | v1. v1 \<in> alleps r1} \<union> {Right v2 | v2. v2 \<in> alleps r2}" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1040 |
|
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1041 |
fun injall :: "rexp \<Rightarrow> char \<Rightarrow> val \<Rightarrow> val set" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1042 |
where |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1043 |
"injall (EMPTY) c Void = {}" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1044 |
| "injall (CHAR d) c Void = (if c = d then {Char d} else {})" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1045 |
| "injall (ALT r1 r2) c (Left v1) = {Left v | v. v \<in> injall r1 c v1}" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1046 |
| "injall (ALT r1 r2) c (Right v2) = {Right v | v. v \<in> injall r2 c v2}" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1047 |
| "injall (SEQ r1 r2) c (Seq v1 v2) = {Seq v v2 | v. v \<in> injall r1 c v1}" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1048 |
| "injall (SEQ r1 r2) c (Left (Seq v1 v2)) = {Seq v v2 | v. v \<in> injall r1 c v1}" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1049 |
| "injall (SEQ r1 r2) c (Right v2) = {Seq v v' | v v'. v \<in> alleps r1 \<and> v' \<in> injall r2 c v2}" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1050 |
|
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1051 |
fun |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1052 |
allvals :: "rexp \<Rightarrow> string \<Rightarrow> val set" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1053 |
where |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1054 |
"allvals r [] = alleps r" |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1055 |
| "allvals r (c#s) = {v | v v'. v \<in> injall r c v' \<and> v' \<in> allvals (der c r) s}" |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1056 |
|
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1057 |
lemma q1: |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1058 |
assumes "v \<in> alleps r" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1059 |
shows "\<turnstile> v : r \<and> flat v = []" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1060 |
using assms |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1061 |
apply(induct r arbitrary: v) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1062 |
apply(auto intro: Prf.intros) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1063 |
done |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1064 |
|
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1065 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1066 |
lemma allvals_NULL: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1067 |
shows "allvals (NULL) s = {}" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1068 |
apply(induct_tac s) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1069 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1070 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1071 |
done |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1072 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1073 |
lemma allvals_EMPTY: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1074 |
shows "allvals (EMPTY) [] = {Void}" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1075 |
and "s \<noteq> [] \<Longrightarrow> allvals (EMPTY) s = {}" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1076 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1077 |
apply(case_tac s) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1078 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1079 |
apply(simp add: allvals_NULL) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1080 |
done |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1081 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1082 |
lemma allvals_CHAR: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1083 |
shows "allvals (CHAR c) [c] = {Char c}" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1084 |
and "s \<noteq> [c] \<Longrightarrow> allvals (CHAR c) s = {}" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1085 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1086 |
apply(case_tac s) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1087 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1088 |
apply(case_tac "c = a") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1089 |
apply(simp add: allvals_EMPTY) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1090 |
apply(simp add: allvals_NULL) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1091 |
done |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1092 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1093 |
lemma allvals_ALT: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1094 |
shows "allvals (ALT r1 r2) s = {Left v1 | v1. v1 \<in> allvals r1 s} \<union> |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1095 |
{Right v2 | v2. v2 \<in> allvals r2 s}" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1096 |
apply(induct s arbitrary: r1 r2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1097 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1098 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1099 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1100 |
apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1101 |
apply(rule_tac x="Left v'" in exI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1102 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1103 |
apply(rule_tac x="Right v'" in exI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1104 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1105 |
done |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1106 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1107 |
lemma allvals_SEQ_Nil: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1108 |
"allvals (SEQ r1 r2) [] = {Seq v1 v2 | v1 v2. v1 \<in> allvals r1 [] \<and> v2 \<in> allvals r2 []}" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1109 |
by simp |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1110 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1111 |
lemma allvals_SEQ: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1112 |
shows "allvals (SEQ r1 r2) s = {Seq v1 v2 | v1 v2 s1 s2. |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1113 |
s = s1 @ s2 \<and> v1 \<in> allvals r1 s1 \<and> v2 \<in> allvals r2 s2}" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1114 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1115 |
apply(induct s arbitrary: r1 r2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1116 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1117 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1118 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1119 |
apply(simp_all add: allvals_ALT) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1120 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1121 |
apply (metis (mono_tags, lifting) allvals.simps(2) append_Cons mem_Collect_eq) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1122 |
apply fastforce |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1123 |
prefer 2 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1124 |
apply(rule_tac x="a#s1" in exI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1125 |
apply(rule_tac x="s2" in exI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1126 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1127 |
apply(fastforce) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1128 |
prefer 2 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1129 |
apply(subst (asm) Cons_eq_append_conv) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1130 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1131 |
using Prf_flat_L nullable_correctness q1 apply fastforce |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1132 |
apply(rule_tac x="Seq v' v2" in exI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1133 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1134 |
apply(rule_tac x="ys'" in exI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1135 |
apply(rule_tac x="s2" in exI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1136 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1137 |
apply(subst (asm) Cons_eq_append_conv) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1138 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1139 |
apply(rule_tac x="Right v'" in exI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1140 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1141 |
apply(rule_tac x="Left (Seq v' v2)" in exI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1142 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1143 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1144 |
done |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1145 |
|
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1146 |
lemma q11: |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1147 |
assumes "nullable r" "\<turnstile> v : r" "flat v = []" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1148 |
shows "v \<in> alleps r" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1149 |
using assms |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1150 |
apply(induct r arbitrary: v) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1151 |
apply(auto) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1152 |
apply(erule Prf.cases) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1153 |
apply(simp_all) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1154 |
apply(erule Prf.cases) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1155 |
apply(simp_all) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1156 |
apply(erule Prf.cases) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1157 |
apply(simp_all) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1158 |
apply(auto) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1159 |
apply(subgoal_tac "nullable r2a") |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1160 |
apply(auto) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1161 |
using not_nullable_flat apply auto[1] |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1162 |
apply(erule Prf.cases) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1163 |
apply(simp_all) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1164 |
apply(auto) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1165 |
apply(subgoal_tac "nullable r1a") |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1166 |
apply(auto) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1167 |
using not_nullable_flat apply auto[1] |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1168 |
done |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1169 |
|
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1170 |
lemma q33: |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1171 |
assumes "nullable r" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1172 |
shows "alleps r = {v. \<turnstile> v : r \<and> flat v = []}" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1173 |
using assms |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1174 |
apply(auto) |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1175 |
apply (simp_all add: q1) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1176 |
by (simp add: q11) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1177 |
|
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1178 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1179 |
lemma k0: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1180 |
assumes "\<turnstile> v : der a r" "v' \<in> injall r a v" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1181 |
shows "flat v' = a # flat v" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1182 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1183 |
apply(induct a r arbitrary: v v' rule: der.induct) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1184 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1185 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1186 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1187 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1188 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1189 |
apply(case_tac "c = c'") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1190 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1191 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1192 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1193 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1194 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1195 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1196 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1197 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1198 |
apply(case_tac "nullable r1") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1199 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1200 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1201 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1202 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1203 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1204 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1205 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1206 |
using q1 apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1207 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1208 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1209 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1210 |
done |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1211 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1212 |
lemma k: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1213 |
assumes "\<turnstile> v' : der a r" "v \<in> injall r a v'" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1214 |
shows "\<turnstile> v : r" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1215 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1216 |
apply(induct a r arbitrary: v v' rule: der.induct) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1217 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1218 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1219 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1220 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1221 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1222 |
apply(case_tac "c = c'") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1223 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1224 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1225 |
apply(rule Prf.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1226 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1227 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1228 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1229 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1230 |
apply(auto intro: Prf.intros)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1231 |
apply(auto intro: Prf.intros)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1232 |
apply(case_tac "nullable r1") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1233 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1234 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1235 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1236 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1237 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1238 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1239 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1240 |
apply(auto intro: Prf.intros)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1241 |
using Prf.intros(1) q1 apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1242 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1243 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1244 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1245 |
using Prf.intros(1) by auto |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1246 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1247 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1248 |
|
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1249 |
lemma q22: |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1250 |
assumes "v \<in> allvals r s" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1251 |
shows "\<turnstile> v : r \<and> s \<in> L (r) \<and> flat v = s" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1252 |
using assms |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1253 |
apply(induct s arbitrary: v r) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1254 |
apply(auto) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1255 |
apply(simp_all add: q1) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1256 |
using Prf_flat_L q1 apply fastforce |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1257 |
apply(drule_tac x="v'" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1258 |
apply(drule_tac x="der a r" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1259 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1260 |
apply(clarify) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1261 |
apply(rule k) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1262 |
apply(assumption) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1263 |
apply(assumption) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1264 |
apply(drule_tac x="v'" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1265 |
apply(drule_tac x="der a r" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1266 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1267 |
apply(clarify) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1268 |
using Prf_flat_L v3 v4 apply fastforce |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1269 |
apply(drule_tac x="v'" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1270 |
apply(drule_tac x="der a r" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1271 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1272 |
apply(clarify) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1273 |
using k0 by blast |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1274 |
|
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1275 |
lemma ra: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1276 |
assumes "v \<in> allvals r1 s" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1277 |
shows "Left v \<in> allvals (ALT r1 r2) s" |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1278 |
using assms |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1279 |
apply(induct s arbitrary: r1 r2 v) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1280 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1281 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1282 |
apply(rule_tac x="Left v'" in exI) |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1283 |
apply(simp) |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1284 |
done |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1285 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1286 |
lemma rb: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1287 |
assumes "v \<in> allvals r2 s" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1288 |
shows "Right v \<in> allvals (ALT r1 r2) s" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1289 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1290 |
apply(induct s arbitrary: r1 r2 v) |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1291 |
apply(simp) |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1292 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1293 |
apply(rule_tac x="Right v'" in exI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1294 |
apply(simp) |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1295 |
done |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1296 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1297 |
lemma r1: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1298 |
assumes "v1 \<in> alleps r1" "v2 \<in> allvals r2 s2" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1299 |
shows "Seq v1 v2 \<in> allvals (SEQ r1 r2) s2" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1300 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1301 |
apply(induct s2 arbitrary: r1 r2 v1 v2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1302 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1303 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1304 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1305 |
apply(rule_tac x="Right v'" in exI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1306 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1307 |
apply(rule rb) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1308 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1309 |
using not_nullable_flat q1 by blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1310 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1311 |
lemma r2: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1312 |
assumes "v1 \<in> allvals r1 s1" "v2 \<in> allvals r2 s2" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1313 |
shows "Seq v1 v2 \<in> allvals (SEQ r1 r2) (s1 @ s2)" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1314 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1315 |
apply(induct s1 arbitrary: r1 r2 v1 v2 s2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1316 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1317 |
apply(rule r1) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1318 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1319 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1320 |
apply(simp) |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1321 |
apply(auto) |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1322 |
apply(drule_tac x="der a r1" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1323 |
apply(drule_tac x="r2" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1324 |
apply(drule_tac x="v'" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1325 |
apply(drule_tac x="v2" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1326 |
apply(drule_tac x="s2" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1327 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1328 |
apply(rule_tac x="Left (Seq v' v2)" in exI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1329 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1330 |
apply(rule ra) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1331 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1332 |
apply(drule_tac x="der a r1" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1333 |
apply(drule_tac x="r2" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1334 |
apply(drule_tac x="v'" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1335 |
apply(drule_tac x="v2" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1336 |
apply(drule_tac x="s2" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1337 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1338 |
apply(rule_tac x="Seq v' v2" in exI) |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1339 |
apply(simp) |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1340 |
done |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1341 |
|
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1342 |
lemma q22a: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1343 |
assumes " s \<in> L (r)" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1344 |
shows "\<exists>v. v \<in> allvals r s \<and> flat v = s" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1345 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1346 |
apply(induct r arbitrary: s) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1347 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1348 |
apply(simp add: Sequ_def) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1349 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1350 |
apply(drule_tac x="s1" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1351 |
apply(drule_tac x="s2" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1352 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1353 |
apply(rule_tac x="Seq v va" in exI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1354 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1355 |
apply(rule r2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1356 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1357 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1358 |
apply(drule_tac x="s" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1359 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1360 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1361 |
apply(rule_tac x="Left v" in exI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1362 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1363 |
apply(rule ra) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1364 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1365 |
apply(drule_tac x="s" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1366 |
apply(drule_tac x="s" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1367 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1368 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1369 |
apply(rule_tac x="Right v" in exI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1370 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1371 |
apply(rule rb) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1372 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1373 |
done |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1374 |
|
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1375 |
lemma q22b: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1376 |
assumes " s \<in> L (r)" "\<turnstile> v : r" "flat v = s" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1377 |
shows "v \<in> allvals r s" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1378 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1379 |
apply(induct r arbitrary: s v) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1380 |
apply(auto) |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1381 |
apply(erule Prf.cases) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1382 |
apply(simp_all) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1383 |
apply(erule Prf.cases) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1384 |
apply(simp_all) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1385 |
apply(simp add: Sequ_def) |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1386 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1387 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1388 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1389 |
apply(simp add: append_eq_append_conv2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1390 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1391 |
apply (metis Prf_flat_L append_assoc r2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1392 |
apply (metis Prf_flat_L r2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1393 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1394 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1395 |
apply(auto intro: ra rb)[2] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1396 |
apply(rule rb) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1397 |
apply (simp add: Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1398 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1399 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1400 |
apply(auto intro: ra rb)[2] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1401 |
apply(rule ra) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1402 |
by (simp add: Prf_flat_L) |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1403 |
|
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1404 |
|
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1405 |
lemma q2: |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1406 |
assumes "s \<in> L(r)" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1407 |
shows "allvals r s = {v. \<turnstile> v : r \<and> s \<in> L (r) \<and> flat v = s}" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1408 |
using assms |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1409 |
apply(auto) |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1410 |
apply (simp add: q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1411 |
apply (simp add: q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1412 |
by (simp add: q22b) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1413 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1414 |
lemma r3a: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1415 |
assumes "v' \<in> allvals (SEQ r1 r2) (s1 @ s2)" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1416 |
"(s1 @ s2) \<in> L (SEQ r1 r2)" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1417 |
shows "\<exists>v1 v2. v' = Seq v1 v2 \<and> v1 \<in> allvals r1 s1 \<and> v2 \<in> allvals r2 s2" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1418 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1419 |
apply(subst (asm) q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1420 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1421 |
apply(erule_tac Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1422 |
apply(simp_all) |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1423 |
apply(rule conjI) |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1424 |
apply(simp add: append_eq_append_conv2) |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1425 |
apply(auto) |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1426 |
apply(subst q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1427 |
oops |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1428 |
|
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1429 |
lemma r3: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1430 |
assumes "Seq v1 v2 \<in> allvals (SEQ r1 r2) (s1 @ s2)" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1431 |
"flat v1 = s1" "flat v2 = s2" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1432 |
"(s1 @ s2) \<in> L (SEQ r1 r2)" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1433 |
shows "v1 \<in> allvals r1 s1" "v2 \<in> allvals r2 s2" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1434 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1435 |
apply(subst (asm) q2) |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1436 |
apply(auto) |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1437 |
apply(erule_tac Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1438 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1439 |
apply(subst q2) |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1440 |
apply(auto) |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1441 |
using Prf_flat_L apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1442 |
using Prf_flat_L apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1443 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1444 |
apply(subst (asm) q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1445 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1446 |
apply(erule_tac Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1447 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1448 |
apply(subst q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1449 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1450 |
using Prf_flat_L apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1451 |
using Prf_flat_L apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1452 |
done |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1453 |
|
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1454 |
|
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1455 |
definition POSIX2 :: "val \<Rightarrow> rexp \<Rightarrow> string \<Rightarrow> bool" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1456 |
where |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1457 |
"POSIX2 v r s \<equiv> (\<turnstile> v : r \<and> flat v = s \<and> (\<forall>v'\<in>allvals r s. v \<succ>r v'))" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1458 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1459 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1460 |
|
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1461 |
|
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1462 |
lemma k1: |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1463 |
assumes "nullable r" |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1464 |
shows "POSIX2 v r [] \<Longrightarrow> \<forall>v' \<in> alleps r. v \<succ>r v'" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1465 |
using assms |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1466 |
apply(induct r arbitrary: v) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1467 |
apply(simp_all) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1468 |
apply(simp add: POSIX2_def) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1469 |
apply(auto) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1470 |
apply(simp add: POSIX2_def) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1471 |
apply(simp add: POSIX2_def) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1472 |
apply(simp add: POSIX2_def) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1473 |
apply(simp add: POSIX2_def) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1474 |
apply(simp add: POSIX2_def) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1475 |
done |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1476 |
|
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1477 |
lemma k2: |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1478 |
assumes "s \<in> L r" |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1479 |
shows "POSIX2 v r s \<Longrightarrow> \<forall>v' \<in> allvals r s. v \<succ>r v'" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1480 |
using assms |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1481 |
apply(induct s arbitrary: r v) |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1482 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1483 |
apply(rule k1) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1484 |
apply (simp add: nullable_correctness) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1485 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1486 |
apply(simp) |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1487 |
apply(auto) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1488 |
apply(simp only: POSIX2_def) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1489 |
apply(simp) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1490 |
apply(clarify) |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1491 |
apply(drule_tac x="x" in spec) |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1492 |
apply(drule mp) |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1493 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1494 |
done |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1495 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1496 |
lemma pos: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1497 |
assumes "s \<in> r \<rightarrow> v" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1498 |
shows "v \<in> allvals r s" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1499 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1500 |
apply(subst q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1501 |
using PMatch1(1) PMatch1(2) Prf_flat_L apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1502 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1503 |
using PMatch1(1) PMatch1(2) Prf_flat_L by blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1504 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1505 |
lemma mkeps_val: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1506 |
assumes "nullable r" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1507 |
shows "mkeps r \<in> alleps r" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1508 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1509 |
apply(induct r) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1510 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1511 |
done |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1512 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1513 |
lemma injval_injall: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1514 |
assumes "\<turnstile> v : der a r" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1515 |
shows "injval r a v \<in> injall r a v" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1516 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1517 |
apply(induct a r arbitrary: v rule: der.induct) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1518 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1519 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1520 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1521 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1522 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1523 |
apply(case_tac "c = c'") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1524 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1525 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1526 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1527 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1528 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1529 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1530 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1531 |
apply(case_tac "nullable r1") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1532 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1533 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1534 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1535 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1536 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1537 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1538 |
using mkeps_val apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1539 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1540 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1541 |
done |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1542 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1543 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1544 |
lemma mkeps_val1: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1545 |
assumes "nullable r" "v \<succ>r mkeps r" "flat v = []" "\<turnstile> v : r" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1546 |
shows "v = mkeps r" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1547 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1548 |
apply(induct r arbitrary: v) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1549 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1550 |
apply(erule ValOrd.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1551 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1552 |
apply(erule ValOrd.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1553 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1554 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1555 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1556 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1557 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1558 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1559 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1560 |
apply(erule ValOrd.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1561 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1562 |
apply(erule ValOrd.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1563 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1564 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1565 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1566 |
apply(erule ValOrd.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1567 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1568 |
apply(erule ValOrd.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1569 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1570 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1571 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1572 |
apply(erule ValOrd.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1573 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1574 |
apply (simp add: not_nullable_flat) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1575 |
apply(erule ValOrd.cases) |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1576 |
apply(auto) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1577 |
done |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1578 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1579 |
lemma sulzmann_our: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1580 |
assumes "v \<in> alleps r" "nullable r" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1581 |
shows "mkeps r \<succ>r v" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1582 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1583 |
apply(induct r arbitrary: v) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1584 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1585 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1586 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1587 |
apply(case_tac "mkeps r1 = v1") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1588 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1589 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1590 |
apply(blast) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1591 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1592 |
apply(blast) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1593 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1594 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1595 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1596 |
apply(blast) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1597 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1598 |
apply(blast) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1599 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1600 |
using not_nullable_flat q1 apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1601 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1602 |
using q1 apply auto[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1603 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1604 |
apply (simp add: q1) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1605 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1606 |
apply(blast) |
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1607 |
done |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
1608 |
|
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1609 |
lemma destruct: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1610 |
assumes "\<forall>s3. s1 @ s3 \<in> L r1 \<longrightarrow> s3 = [] \<or> (\<forall>s4. s3 @ s4 = s2 \<longrightarrow> s4 \<notin> L r2)" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1611 |
and "s1 \<in> L r1" "s2 \<in> L r2" "(s1' @ s2') \<sqsubseteq> (s1 @ s2)" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1612 |
and "s1'@ s2' \<in> L (SEQ r1 r2)" "s1' \<in> L r1" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1613 |
shows "s1' \<sqsubseteq> s1" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1614 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1615 |
apply(simp add: prefix_def) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1616 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1617 |
apply(simp add: append_eq_append_conv2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1618 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1619 |
apply(simp add: Sequ_def) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1620 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1621 |
apply(drule_tac x="us" in spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1622 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1623 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1624 |
apply(simp add: append_eq_append_conv2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1625 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1626 |
oops |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1627 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1628 |
lemma inj_ord: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1629 |
assumes "v1 \<succ>(der a r) v2" "s \<in> (der a r) \<rightarrow> v1" "s' \<in> L (der a r)" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1630 |
"v1 \<in> allvals (der a r) s" "v2 \<in> allvals (der a r) s'" "s' \<sqsubseteq> s" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1631 |
shows "injval r a v1 \<succ>r injval r a v2" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1632 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1633 |
apply(induct a r arbitrary: s s' v1 v2 rule: der.induct) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1634 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1635 |
(*apply(simp add: allvals_NULL) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1636 |
apply(simp add: allvals_NULL)*) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1637 |
apply(case_tac "c = c'") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1638 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1639 |
apply(case_tac "s = []") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1640 |
apply(subgoal_tac "s' = []") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1641 |
prefer 2 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1642 |
using allvals_EMPTY(2) apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1643 |
apply(simp add: allvals_EMPTY) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1644 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1645 |
apply(simp add: allvals_EMPTY) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1646 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1647 |
(*apply(simp add: allvals_NULL)*) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1648 |
(* ALT case *) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1649 |
apply(simp add: allvals_ALT) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1650 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1651 |
apply(erule ValOrd.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1652 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1653 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1654 |
apply(erule PMatch.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1655 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1656 |
apply(erule ValOrd.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1657 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1658 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1659 |
apply(subst v4) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1660 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1661 |
apply (simp add: q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1662 |
apply(subst v4) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1663 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1664 |
apply (simp add: q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1665 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1666 |
apply(erule ValOrd.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1667 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1668 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1669 |
apply(subst v4) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1670 |
apply (simp add: q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1671 |
apply(subst v4) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1672 |
apply (simp add: q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1673 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1674 |
apply(erule ValOrd.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1675 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1676 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1677 |
apply(erule PMatch.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1678 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1679 |
using q22 apply auto[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1680 |
apply(erule PMatch.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1681 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1682 |
apply(erule ValOrd.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1683 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1684 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1685 |
using q22 apply auto[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1686 |
apply(erule PMatch.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1687 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1688 |
apply(erule ValOrd.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1689 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1690 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1691 |
apply(subst v4) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1692 |
apply (simp add: q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1693 |
apply(subst v4) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1694 |
apply (simp add: q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1695 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1696 |
apply(erule PMatch.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1697 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1698 |
apply(erule ValOrd.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1699 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1700 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1701 |
apply(subst v4) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1702 |
apply (simp add: q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1703 |
apply(subst v4) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1704 |
apply (simp add: q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1705 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1706 |
using q22 apply auto[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1707 |
apply(erule PMatch.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1708 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1709 |
apply(erule ValOrd.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1710 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1711 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1712 |
using q22 apply auto[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1713 |
(* seq case *) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1714 |
apply(case_tac "nullable r1") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1715 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1716 |
prefer 2 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1717 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1718 |
apply(simp add: allvals_SEQ) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1719 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1720 |
apply(erule ValOrd.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1721 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1722 |
apply(clarify) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1723 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1724 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1725 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1726 |
apply(erule PMatch.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1727 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1728 |
apply(clarify) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1729 |
apply(rotate_tac 1) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1730 |
apply(drule_tac x="s1b" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1731 |
apply(rotate_tac 13) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1732 |
apply(drule_tac x="s1a" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1733 |
apply(drule_tac x="v1c" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1734 |
apply(drule_tac x="v1'" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1735 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1736 |
apply(subgoal_tac "s1 = s1b") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1737 |
prefer 2 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1738 |
apply (metis PMatch1(2) q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1739 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1740 |
apply(clarify) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1741 |
apply(drule destruct) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1742 |
apply (metis PMatch1(2) q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1743 |
apply (metis PMatch1(2) q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1744 |
apply(assumption) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1745 |
apply (metis PMatch1(2) q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1746 |
apply (metis PMatch1(2) q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1747 |
apply(subgoal_tac "s2a = s2b") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1748 |
prefer 2 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1749 |
apply (metis PMatch1(2) q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1750 |
apply(drule destruct) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1751 |
apply (metis PMatch1(2) q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1752 |
apply (metis PMatch1(2) q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1753 |
apply(assumption) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1754 |
back |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1755 |
apply (metis PMatch1(2) q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1756 |
apply (metis PMatch1(2) q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1757 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1758 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1759 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1760 |
apply(simp add: allvals_ALT) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1761 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1762 |
apply(erule ValOrd.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1763 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1764 |
apply(clarify) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1765 |
apply(erule ValOrd.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1766 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1767 |
apply(clarify) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1768 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1769 |
apply(blast) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1770 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1771 |
apply(clarify) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1772 |
apply(simp add: allvals_SEQ) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1773 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1774 |
apply(erule PMatch.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1775 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1776 |
apply(clarify) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1777 |
apply(erule PMatch.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1778 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1779 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1780 |
apply(drule_tac x="s1b" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1781 |
apply(drule_tac x="v1" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1782 |
apply(drule_tac x="v1'a" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1783 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1784 |
apply(drule_tac meta_mp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1785 |
apply(subgoal_tac "s1 = s1b") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1786 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1787 |
apply (metis PMatch1(2) q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1788 |
apply(drule_tac meta_mp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1789 |
apply(subgoal_tac "s1a = s1b") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1790 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1791 |
apply(simp add: append_eq_append_conv2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1792 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1793 |
apply(subgoal_tac "s2 = s2a") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1794 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1795 |
apply(clarify) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1796 |
prefer 2 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1797 |
using q22 apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1798 |
using q22 apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1799 |
using q22 apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1800 |
apply(subgoal_tac "usa = []") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1801 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1802 |
prefer 2 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1803 |
using q22 apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1804 |
prefer 3 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1805 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1806 |
prefer 4 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1807 |
apply(erule PMatch.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1808 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1809 |
apply(clarify) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1810 |
apply(erule PMatch.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1811 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1812 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1813 |
apply(erule ValOrd.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1814 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1815 |
apply(clarify) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1816 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1817 |
prefer 5 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1818 |
apply(erule PMatch.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1819 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1820 |
apply(clarify) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1821 |
apply(erule ValOrd.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1822 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1823 |
apply(clarify) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1824 |
apply(simp add: allvals_SEQ) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1825 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1826 |
apply (simp add: q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1827 |
apply(simp add: allvals_SEQ) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1828 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1829 |
apply(simp add: append_eq_append_conv2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1830 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1831 |
apply (simp add: q22) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1832 |
thm PMatch2 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1833 |
apply(drule PMatch2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1834 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1835 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1836 |
lemma sulzmann_our: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1837 |
assumes "\<forall>v' \<in> allvals r s. v \<succ>r v'" "s \<in> L r" "\<turnstile> v : r" "flat v = s" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1838 |
shows "s \<in> r \<rightarrow> v" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1839 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1840 |
apply(induct s arbitrary: r v) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1841 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1842 |
apply(subst (asm) q33) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1843 |
apply (simp add: nullable_correctness) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1844 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1845 |
apply(drule_tac x="mkeps r" in spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1846 |
apply(drule mp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1847 |
apply(rule conjI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1848 |
using mkeps_val not_nullable_flat q1 apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1849 |
using mkeps_flat not_nullable_flat apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1850 |
apply(subgoal_tac "nullable r") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1851 |
apply(drule mkeps_val1) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1852 |
apply(assumption) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1853 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1854 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1855 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1856 |
using PMatch_mkeps not_nullable_flat apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1857 |
using not_nullable_flat apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1858 |
apply(drule_tac x="der a r" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1859 |
apply(drule_tac x="projval r a v" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1860 |
apply(drule meta_mp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1861 |
defer |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1862 |
apply(drule meta_mp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1863 |
using Prf_flat_L v3_proj v4_proj2 apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1864 |
apply(drule meta_mp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1865 |
using v3_proj apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1866 |
apply(drule meta_mp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1867 |
apply (simp add: v4_proj2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1868 |
defer |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1869 |
apply(rule ballI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1870 |
apply(drule_tac x="injval r a x" in spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1871 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1872 |
apply(drule_tac x="x" in spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1873 |
apply(drule mp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1874 |
apply(rule injval_injall) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1875 |
using q22 apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1876 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1877 |
(* HERE *) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1878 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1879 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1880 |
lemma our_sulzmann: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1881 |
assumes "s \<in> r \<rightarrow> v" "v' \<in> allvals r s" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1882 |
shows "v \<succ>r v'" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1883 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1884 |
apply(induct r arbitrary: s v v') |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1885 |
apply(erule PMatch.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1886 |
apply(simp_all)[5] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1887 |
apply(erule PMatch.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1888 |
apply(simp_all)[5] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1889 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1890 |
apply(erule PMatch.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1891 |
apply(simp_all)[5] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1892 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1893 |
(* SEQ - case *) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1894 |
apply(erule PMatch.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1895 |
apply(simp_all)[5] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1896 |
apply(clarify) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1897 |
apply(subst (asm) (3) q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1898 |
apply(simp add: Sequ_def) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1899 |
apply(rule_tac x="s1" in exI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1900 |
apply(rule_tac x="s2" in exI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1901 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1902 |
apply(rule conjI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1903 |
using PMatch1(1) PMatch1(2) Prf_flat_L apply fastforce |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1904 |
apply (metis PMatch1(1) PMatch1(2) Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1905 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1906 |
apply(clarify) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1907 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1908 |
apply(simp_all)[5] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1909 |
apply(case_tac "v1 = v1a") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1910 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1911 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1912 |
apply(rotate_tac 1) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1913 |
apply(drule_tac x="s2" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1914 |
apply(drule_tac x="v2" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1915 |
apply(drule_tac x="v2a" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1916 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1917 |
apply(drule_tac meta_mp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1918 |
apply(subst q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1919 |
using PMatch1(1) PMatch1(2) Prf_flat_L apply fastforce |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1920 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1921 |
apply(rule conjI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1922 |
using PMatch1(1) PMatch1(2) Prf_flat_L apply fastforce |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1923 |
apply (simp add: PMatch1(2)) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1924 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1925 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1926 |
prefer 2 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1927 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1928 |
apply(drule_tac x="s1" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1929 |
apply(drule_tac x="v1" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1930 |
apply(drule_tac x="v1a" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1931 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1932 |
apply(drule_tac meta_mp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1933 |
apply(subst q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1934 |
apply (metis PMatch1(1) PMatch1(2) Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1935 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1936 |
apply(rule conjI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1937 |
apply (metis PMatch1(1) PMatch1(2) Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1938 |
apply(subst (asm) append_eq_append_conv2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1939 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1940 |
using Prf_flat_L apply fastforce |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1941 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1942 |
apply(drule_tac x="us" in spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1943 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1944 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1945 |
using Prf_flat_L apply fastforce |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1946 |
using Prf_flat_L apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1947 |
apply(drule_tac meta_mp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1948 |
apply(subst q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1949 |
using Prf_flat_L apply fastforce |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1950 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1951 |
using Prf_flat_L apply fastforce |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1952 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1953 |
apply(drule_tac x="flat v1a" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1954 |
apply(drule_tac x="v1" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1955 |
apply(drule_tac x="v1a" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1956 |
apply(drule_tac meta_mp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1957 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1958 |
apply(drule meta_mp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1959 |
apply(subst q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1960 |
apply (metis PMatch1(1) PMatch1(2) Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1961 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1962 |
apply(rule conjI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1963 |
apply (metis PMatch1(1) PMatch1(2) Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1964 |
apply(drule_tac x="[]" in spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1965 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1966 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1967 |
using Prf_flat_L apply fast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1968 |
apply(drule_tac x="us" in spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1969 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1970 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1971 |
apply (simp add: Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1972 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1973 |
thm PMatch1 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1974 |
qed |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1975 |
done |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1976 |
using PMatch1(1) PMatch1(2) Prf_flat_L apply fastforce |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1977 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1978 |
apply(clarify) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1979 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1980 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1981 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1982 |
apply(drule_tac x="v1" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1983 |
apply(drule meta_mp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1984 |
apply(subst q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1985 |
apply (simp add: Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1986 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1987 |
apply (simp add: Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1988 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1989 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1990 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1991 |
apply (simp add: PMatch1(2)) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1992 |
apply (simp add: PMatch1(2)) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1993 |
apply(subst (asm) (2) q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1994 |
using PMatch1(1) PMatch1(2) Prf_flat_L apply fastforce |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1995 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1996 |
apply(clarify) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1997 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1998 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
1999 |
prefer 2 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2000 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2001 |
using q22b apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2002 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2003 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2004 |
using Prf_flat_L apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2005 |
apply(subst (asm) (3) q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2006 |
apply(simp add: Sequ_def) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2007 |
apply(rule_tac x="s1" in exI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2008 |
apply(rule_tac x="s2" in exI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2009 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2010 |
using PMatch1(1) PMatch1(2) Prf_flat_L apply fastforce |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2011 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2012 |
apply(clarify) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2013 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2014 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2015 |
apply(auto simp add: Sequ_def)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2016 |
apply(case_tac "v1 = v1a") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2017 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2018 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2019 |
apply(rotate_tac 3) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2020 |
apply(drule_tac x="v2a" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2021 |
apply(drule_tac meta_mp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2022 |
apply(subst q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2023 |
using PMatch1(1) PMatch1(2) Prf_flat_L apply fastforce |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2024 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2025 |
apply(rule conjI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2026 |
using PMatch1(1) PMatch1(2) Prf_flat_L apply fastforce |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2027 |
apply (metis PMatch1(2) same_append_eq) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2028 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2029 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2030 |
apply(drule_tac x="v1a" in meta_spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2031 |
apply(drule_tac meta_mp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2032 |
apply(subst q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2033 |
using PMatch1(1) PMatch1(2) Prf_flat_L apply fastforce |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2034 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2035 |
apply(rule conjI) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2036 |
using PMatch1(1) PMatch1(2) Prf_flat_L apply fastforce |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2037 |
prefer 2 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2038 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2039 |
prefer 2 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2040 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2041 |
apply(rotate_tac 7) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2042 |
apply(drule sym) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2043 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2044 |
apply(subst (asm) (2) append_eq_append_conv2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2045 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2046 |
apply(frule_tac x="us" in spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2047 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2048 |
prefer 2 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2049 |
apply(drule_tac x="flat v2a" in spec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2050 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2051 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2052 |
apply(subgoal_tac "flat v2a = s2") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2053 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2054 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2055 |
apply(simp add: append_eq_append_conv2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2056 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2057 |
prefer 2 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2058 |
apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2059 |
prefer 2 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2060 |
apply (metis Prf_flat_L append_self_conv2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2061 |
prefer 4 |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2062 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2063 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2064 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2065 |
lemma our_sulzmann: |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2066 |
assumes "s \<in> r \<rightarrow> v" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2067 |
shows "POSIX2 v r s" |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2068 |
using assms |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2069 |
apply(induct s r v) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2070 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2071 |
apply(simp add: POSIX2_def) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2072 |
using Prf.intros(4) ValOrd.intros(7) apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2073 |
apply(simp add: POSIX2_def) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2074 |
apply (simp add: Prf.intros(5) ValOrd.intros(8)) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2075 |
apply(simp add: POSIX2_def) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2076 |
apply(auto) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2077 |
apply(rule Prf.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2078 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2079 |
apply(subgoal_tac "(\<exists>x1. x = Left x1) \<or> (\<exists>x1. x = Right x1)") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2080 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2081 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2082 |
apply(drule_tac x="x1" in bspec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2083 |
apply(subst (asm) q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2084 |
apply (simp add: Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2085 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2086 |
apply(subst q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2087 |
apply (simp add: Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2088 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2089 |
apply(rotate_tac 2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2090 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2091 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2092 |
apply(rotate_tac 2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2093 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2094 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2095 |
apply (simp add: Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2096 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2097 |
apply(subst (asm) (2) q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2098 |
apply (simp add: Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2099 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2100 |
defer |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2101 |
apply(simp add: POSIX2_def) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2102 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2103 |
apply(rule Prf.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2104 |
apply (simp add: Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2105 |
apply(subgoal_tac "(\<exists>x1. x = Left x1) \<or> (\<exists>x1. x = Right x1)") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2106 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2107 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2108 |
apply(subst (asm) (2) q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2109 |
apply (simp add: Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2110 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2111 |
apply(rotate_tac 4) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2112 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2113 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2114 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2115 |
using Prf_flat_L apply force |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2116 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2117 |
apply(drule_tac x="x1" in bspec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2118 |
apply(subst (asm) q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2119 |
apply (simp add: Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2120 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2121 |
apply(subst q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2122 |
apply(rotate_tac 3) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2123 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2124 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2125 |
apply(rotate_tac 3) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2126 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2127 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2128 |
defer |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2129 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2130 |
apply(simp add: POSIX2_def) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2131 |
apply(auto intro: Prf.intros)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2132 |
apply(subgoal_tac "(\<exists>x1 x2. x = Seq x1 x2 \<and> flat v1 @ flat v2 = flat x1 @ flat x2)") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2133 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2134 |
apply(case_tac "v1 = x1") |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2135 |
apply(simp) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2136 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2137 |
apply(rotate_tac 6) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2138 |
apply(drule_tac x="x2" in bspec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2139 |
apply(subst (asm) q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2140 |
apply (simp add: Sequ_def Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2141 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2142 |
using Prf_flat_L apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2143 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2144 |
apply(rotate_tac 6) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2145 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2146 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2147 |
apply(subst q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2148 |
apply (simp add: Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2149 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2150 |
apply(auto simp add: Sequ_def) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2151 |
using Prf_flat_L apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2152 |
apply(rule ValOrd.intros) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2153 |
apply(rotate_tac 5) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2154 |
apply(drule_tac x="x1" in bspec) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2155 |
apply(rotate_tac 1) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2156 |
apply(subst (asm) q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2157 |
apply (simp add: Sequ_def Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2158 |
using Prf_flat_L apply blast |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2159 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2160 |
apply(subst q2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2161 |
apply (simp add: Sequ_def Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2162 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2163 |
apply(rotate_tac 7) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2164 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2165 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2166 |
apply (simp add: Sequ_def Prf_flat_L) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2167 |
apply(rotate_tac 7) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2168 |
apply(erule Prf.cases) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2169 |
apply(simp_all) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2170 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2171 |
apply(simp add: append_eq_append_conv2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2172 |
apply(auto simp add: Sequ_def)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2173 |
using Prf_flat_L apply fastforce |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2174 |
apply(simp add: append_eq_append_conv2) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2175 |
apply(auto simp add: Sequ_def)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2176 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2177 |
apply(auto)[1] |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2178 |
|
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2179 |
apply(simp add: POSIX2_def) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2180 |
apply(simp add: POSIX2_def) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2181 |
apply(simp add: POSIX2_def) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2182 |
apply(simp add: POSIX2_def) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2183 |
apply(simp add: POSIX2_def) |
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2184 |
|
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2185 |
lemma "s \<in> L(r) \<Longrightarrow> \<exists>v. POSIX2 v r s" |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2186 |
apply(induct r arbitrary: s) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2187 |
apply(auto) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2188 |
apply(rule_tac x="Void" in exI) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2189 |
apply(simp add: POSIX2_def) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2190 |
apply (simp add: Prf.intros(4) ValOrd.intros(7)) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2191 |
apply(rule_tac x="Char x" in exI) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2192 |
apply(simp add: POSIX2_def) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2193 |
apply (simp add: Prf.intros(5) ValOrd.intros(8)) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2194 |
defer |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2195 |
apply(drule_tac x="s" in meta_spec) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2196 |
apply(auto)[1] |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2197 |
apply(rule_tac x="Left v" in exI) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2198 |
apply(simp add: POSIX2_def) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2199 |
apply(auto)[1] |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2200 |
using Prf.intros(2) apply blast |
212
9fd41f224e8d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
211
diff
changeset
|
2201 |
|
211
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2202 |
apply(case_tac s) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2203 |
apply(simp) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2204 |
apply(auto)[1] |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2205 |
apply (simp add: ValOrd.intros(6)) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2206 |
apply(rule ValOrd.intros) |
0fa636821349
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
88
diff
changeset
|
2207 |
|
87
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2208 |
thm PMatch.intros[no_vars] |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2209 |
|
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2210 |
lemma POSIX_PMatch: |
88
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2211 |
assumes "s \<in> r \<rightarrow> v" "v' \<in> Values r s" |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2212 |
shows "v \<succ>r v' \<or> length (flat v') < length (flat v)" |
87
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2213 |
using assms |
88
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2214 |
apply(induct r arbitrary: s v v' rule: rexp.induct) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2215 |
apply(simp_all add: Values_recs) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2216 |
apply(erule PMatch.cases) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2217 |
apply(simp_all)[5] |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2218 |
apply (metis ValOrd.intros(7)) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2219 |
apply(erule PMatch.cases) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2220 |
apply(simp_all)[5] |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2221 |
apply(simp add: prefix_def) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2222 |
apply (metis ValOrd.intros(8)) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2223 |
apply(auto)[1] |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2224 |
defer |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2225 |
apply(auto)[1] |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2226 |
apply(erule PMatch.cases) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2227 |
apply(simp_all)[5] |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2228 |
apply (metis ValOrd.intros(6)) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2229 |
apply (metis (no_types, lifting) PMatch1(2) Prf_flat_L Values_def length_sprefix mem_Collect_eq sprefix_def) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2230 |
apply(erule PMatch.cases) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2231 |
apply(simp_all)[5] |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2232 |
apply (metis (no_types, lifting) PMatch1(2) ValOrd.intros(3) Values_def length_sprefix mem_Collect_eq order_refl sprefix_def) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2233 |
apply (metis ValOrd.intros(5)) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2234 |
apply(erule PMatch.cases) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2235 |
apply(simp_all)[5] |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2236 |
apply(auto) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2237 |
apply(case_tac "v1a = v1") |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2238 |
apply(simp) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2239 |
apply(rule ValOrd.intros(1)) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2240 |
apply (metis PMatch1(2) append_Nil comm_monoid_diff_class.diff_cancel drop_0 drop_all drop_append order_refl rest_def) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2241 |
apply(rule ValOrd.intros(2)) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2242 |
apply(auto) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2243 |
apply(drule_tac x="s1" in meta_spec) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2244 |
apply(drule_tac x="v1a" in meta_spec) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2245 |
apply(drule_tac x="v1" in meta_spec) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2246 |
apply(auto) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2247 |
apply(drule meta_mp) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2248 |
defer |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2249 |
apply(auto)[1] |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2250 |
apply(frule PMatch1) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2251 |
apply(drule PMatch1(2)) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2252 |
apply(frule PMatch1) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2253 |
apply(drule PMatch1(2)) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2254 |
apply(auto)[1] |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2255 |
apply(simp add: Values_def) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2256 |
apply(simp add: prefix_def) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2257 |
apply(auto)[1] |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2258 |
apply(simp add: append_eq_append_conv2) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2259 |
apply(auto)[1] |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2260 |
apply(rotate_tac 10) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2261 |
apply(drule sym) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2262 |
apply(simp) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2263 |
apply(simp add: rest_def) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2264 |
apply(case_tac "s3a = []") |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2265 |
apply(auto)[1] |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2266 |
|
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2267 |
|
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2268 |
apply (metis ValOrd.intros(6)) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2269 |
apply (metis (no_types, lifting) PMatch1(2) ValOrd.intros(3) Values_def length_sprefix mem_Collect_eq order_refl sprefix_def) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2270 |
apply(auto)[1] |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2271 |
apply (metis (no_types, lifting) PMatch1(2) Prf_flat_L Values_def length_sprefix mem_Collect_eq sprefix_def) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2272 |
apply (metis ValOrd.intros(5)) |
532bb9df225d
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
87
diff
changeset
|
2273 |
apply(auto)[1] |
87
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2274 |
apply(erule Prf.cases) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2275 |
apply(simp_all)[5] |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2276 |
apply(erule Prf.cases) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2277 |
apply(simp_all)[5] |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2278 |
apply(erule Prf.cases) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2279 |
apply(simp_all)[5] |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2280 |
apply(erule PMatch.cases) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2281 |
apply(simp_all)[5] |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2282 |
defer |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2283 |
apply(erule Prf.cases) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2284 |
apply(simp_all)[5] |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2285 |
apply(erule PMatch.cases) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2286 |
apply(simp_all)[5] |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2287 |
apply(clarify) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2288 |
apply(simp add: L_flat_Prf) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2289 |
|
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2290 |
apply(clarify) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2291 |
apply (metis ValOrd.intros(8)) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2292 |
apply (metis POSIX_ALT_I1) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2293 |
apply(rule POSIX_ALT_I2) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2294 |
apply(simp) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2295 |
apply(auto)[1] |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2296 |
apply(simp add: POSIX_def) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2297 |
apply(frule PMatch1(1)) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2298 |
apply(frule PMatch1(2)) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2299 |
apply(simp) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2300 |
|
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2301 |
|
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2302 |
lemma POSIX_PMatch: |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2303 |
assumes "s \<in> r \<rightarrow> v" |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2304 |
shows "POSIX v r" |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2305 |
using assms |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2306 |
apply(induct arbitrary: rule: PMatch.induct) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2307 |
apply(auto) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2308 |
apply(simp add: POSIX_def) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2309 |
apply(auto)[1] |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2310 |
apply (metis Prf.intros(4)) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2311 |
apply(erule Prf.cases) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2312 |
apply(simp_all)[5] |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2313 |
apply (metis ValOrd.intros(7)) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2314 |
apply(simp add: POSIX_def) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2315 |
apply(auto)[1] |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2316 |
apply (metis Prf.intros(5)) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2317 |
apply(erule Prf.cases) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2318 |
apply(simp_all)[5] |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2319 |
apply (metis ValOrd.intros(8)) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2320 |
apply (metis POSIX_ALT_I1) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2321 |
apply(rule POSIX_ALT_I2) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2322 |
apply(simp) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2323 |
apply(auto)[1] |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2324 |
apply(simp add: POSIX_def) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2325 |
apply(frule PMatch1(1)) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2326 |
apply(frule PMatch1(2)) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2327 |
apply(simp) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2328 |
|
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2329 |
|
85
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2330 |
|
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2331 |
lemma ValOrd_PMatch: |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2332 |
assumes "s \<in> r \<rightarrow> v1" "\<turnstile> v2 : r" "flat v2 = s" |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2333 |
shows "v1 \<succ>r v2" |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2334 |
using assms |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2335 |
apply(induct arbitrary: v2 rule: PMatch.induct) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2336 |
apply(erule Prf.cases) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2337 |
apply(simp_all)[5] |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2338 |
apply (metis ValOrd.intros(7)) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2339 |
apply(erule Prf.cases) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2340 |
apply(simp_all)[5] |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2341 |
apply (metis ValOrd.intros(8)) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2342 |
apply(erule Prf.cases) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2343 |
apply(simp_all)[5] |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2344 |
apply(clarify) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2345 |
apply (metis ValOrd.intros(6)) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2346 |
apply(clarify) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2347 |
apply (metis PMatch1(2) ValOrd.intros(3) order_refl) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2348 |
apply(erule Prf.cases) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2349 |
apply(simp_all)[5] |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2350 |
apply(clarify) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2351 |
apply (metis Prf_flat_L) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2352 |
apply (metis ValOrd.intros(5)) |
87
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2353 |
(* Seq case *) |
85
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2354 |
apply(erule Prf.cases) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2355 |
apply(simp_all)[5] |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2356 |
apply(auto) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2357 |
apply(case_tac "v1 = v1a") |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2358 |
apply(auto) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2359 |
apply (metis PMatch1(2) ValOrd.intros(1) same_append_eq) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2360 |
apply(rule ValOrd.intros(2)) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2361 |
apply(auto) |
87
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2362 |
apply(drule_tac x="v1a" in meta_spec) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2363 |
apply(drule_tac meta_mp) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2364 |
apply(simp) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2365 |
apply(drule_tac meta_mp) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2366 |
prefer 2 |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2367 |
apply(simp) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2368 |
apply(simp add: append_eq_append_conv2) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2369 |
apply(auto) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2370 |
apply (metis Prf_flat_L) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2371 |
apply(case_tac "us = []") |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2372 |
apply(simp) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2373 |
apply(drule_tac x="us" in spec) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2374 |
apply(drule mp) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2375 |
|
85
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2376 |
thm L_flat_Prf |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2377 |
apply(simp add: L_flat_Prf) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2378 |
thm append_eq_append_conv2 |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2379 |
apply(simp add: append_eq_append_conv2) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2380 |
apply(auto) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2381 |
apply(drule_tac x="us" in spec) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2382 |
apply(drule mp) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2383 |
apply metis |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2384 |
apply (metis append_Nil2) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2385 |
apply(case_tac "us = []") |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2386 |
apply(auto) |
87
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2387 |
apply(drule_tac x="s2" in spec) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2388 |
apply(drule mp) |
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2389 |
|
030939b7d475
added type inference paper and updated Re.thy
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
86
diff
changeset
|
2390 |
apply(auto)[1] |
85
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2391 |
apply(drule_tac x="v1a" in meta_spec) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2392 |
apply(simp) |
53d5f9a5bbd3
updated
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
84
diff
changeset
|
2393 |
|
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2394 |
lemma refl_on_ValOrd: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2395 |
"refl_on (Values r s) {(v1, v2). v1 \<succ>r v2 \<and> v1 \<in> Values r s \<and> v2 \<in> Values r s}" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2396 |
unfolding refl_on_def |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2397 |
apply(auto) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2398 |
apply(rule ValOrd_refl) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2399 |
apply(simp add: Values_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2400 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2401 |
|
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2402 |
|
7
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
2403 |
section {* Posix definition *} |
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
2404 |
|
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2405 |
definition POSIX :: "val \<Rightarrow> rexp \<Rightarrow> bool" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2406 |
where |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2407 |
"POSIX v r \<equiv> (\<turnstile> v : r \<and> (\<forall>v'. (\<turnstile> v' : r \<and> flat v = flat v') \<longrightarrow> v \<succ>r v'))" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2408 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2409 |
definition POSIX2 :: "val \<Rightarrow> rexp \<Rightarrow> bool" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2410 |
where |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2411 |
"POSIX2 v r \<equiv> (\<turnstile> v : r \<and> (\<forall>v'. (\<turnstile> v' : r \<and> flat v = flat v') \<longrightarrow> v 2\<succ> v'))" |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2412 |
|
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2413 |
lemma "POSIX v r = POSIX2 v r" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2414 |
unfolding POSIX_def POSIX2_def |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2415 |
apply(auto) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2416 |
apply(rule Ord1) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2417 |
apply(auto) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2418 |
apply(rule Ord3) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2419 |
apply(auto) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2420 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2421 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2422 |
section {* POSIX for some constructors *} |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2423 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2424 |
lemma POSIX_SEQ1: |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2425 |
assumes "POSIX (Seq v1 v2) (SEQ r1 r2)" "\<turnstile> v1 : r1" "\<turnstile> v2 : r2" |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2426 |
shows "POSIX v1 r1" |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2427 |
using assms |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2428 |
unfolding POSIX_def |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2429 |
apply(auto) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2430 |
apply(drule_tac x="Seq v' v2" in spec) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2431 |
apply(simp) |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2432 |
apply(erule impE) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2433 |
apply(rule Prf.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2434 |
apply(simp) |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2435 |
apply(simp) |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2436 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2437 |
apply(simp_all) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2438 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2439 |
by (metis ValOrd_refl) |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2440 |
|
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2441 |
lemma POSIX_SEQ2: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2442 |
assumes "POSIX (Seq v1 v2) (SEQ r1 r2)" "\<turnstile> v1 : r1" "\<turnstile> v2 : r2" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2443 |
shows "POSIX v2 r2" |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2444 |
using assms |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2445 |
unfolding POSIX_def |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2446 |
apply(auto) |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2447 |
apply(drule_tac x="Seq v1 v'" in spec) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2448 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2449 |
apply(erule impE) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2450 |
apply(rule Prf.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2451 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2452 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2453 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2454 |
apply(simp_all) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2455 |
done |
8
a605dda64267
started a few arguments for the ALT case
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
7
diff
changeset
|
2456 |
|
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2457 |
lemma POSIX_ALT2: |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2458 |
assumes "POSIX (Left v1) (ALT r1 r2)" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2459 |
shows "POSIX v1 r1" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2460 |
using assms |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2461 |
unfolding POSIX_def |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2462 |
apply(auto) |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2463 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2464 |
apply(simp_all)[5] |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2465 |
apply(drule_tac x="Left v'" in spec) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2466 |
apply(simp) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2467 |
apply(drule mp) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2468 |
apply(rule Prf.intros) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2469 |
apply(auto) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2470 |
apply(erule ValOrd.cases) |
8
a605dda64267
started a few arguments for the ALT case
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
7
diff
changeset
|
2471 |
apply(simp_all) |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2472 |
done |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2473 |
|
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2474 |
lemma POSIX_ALT1a: |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2475 |
assumes "POSIX (Right v2) (ALT r1 r2)" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2476 |
shows "POSIX v2 r2" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2477 |
using assms |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2478 |
unfolding POSIX_def |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2479 |
apply(auto) |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2480 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2481 |
apply(simp_all)[5] |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2482 |
apply(drule_tac x="Right v'" in spec) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2483 |
apply(simp) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2484 |
apply(drule mp) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2485 |
apply(rule Prf.intros) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2486 |
apply(auto) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2487 |
apply(erule ValOrd.cases) |
8
a605dda64267
started a few arguments for the ALT case
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
7
diff
changeset
|
2488 |
apply(simp_all) |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2489 |
done |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2490 |
|
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2491 |
lemma POSIX_ALT1b: |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2492 |
assumes "POSIX (Right v2) (ALT r1 r2)" |
8
a605dda64267
started a few arguments for the ALT case
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
7
diff
changeset
|
2493 |
shows "(\<forall>v'. (\<turnstile> v' : r2 \<and> flat v' = flat v2) \<longrightarrow> v2 \<succ>r2 v')" |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2494 |
using assms |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2495 |
apply(drule_tac POSIX_ALT1a) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2496 |
unfolding POSIX_def |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2497 |
apply(auto) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2498 |
done |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2499 |
|
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2500 |
lemma POSIX_ALT_I1: |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2501 |
assumes "POSIX v1 r1" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2502 |
shows "POSIX (Left v1) (ALT r1 r2)" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2503 |
using assms |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2504 |
unfolding POSIX_def |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2505 |
apply(auto) |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2506 |
apply (metis Prf.intros(2)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2507 |
apply(rotate_tac 2) |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2508 |
apply(erule Prf.cases) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2509 |
apply(simp_all)[5] |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2510 |
apply(auto) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2511 |
apply(rule ValOrd.intros) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2512 |
apply(auto) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2513 |
apply(rule ValOrd.intros) |
8
a605dda64267
started a few arguments for the ALT case
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
7
diff
changeset
|
2514 |
by simp |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2515 |
|
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2516 |
lemma POSIX_ALT_I2: |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2517 |
assumes "POSIX v2 r2" "\<forall>v'. \<turnstile> v' : r1 \<longrightarrow> length (flat v2) > length (flat v')" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2518 |
shows "POSIX (Right v2) (ALT r1 r2)" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2519 |
using assms |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2520 |
unfolding POSIX_def |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2521 |
apply(auto) |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2522 |
apply (metis Prf.intros) |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2523 |
apply(rotate_tac 3) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2524 |
apply(erule Prf.cases) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2525 |
apply(simp_all)[5] |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2526 |
apply(auto) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2527 |
apply(rule ValOrd.intros) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2528 |
apply metis |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2529 |
done |
8
a605dda64267
started a few arguments for the ALT case
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
7
diff
changeset
|
2530 |
|
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2531 |
lemma mkeps_POSIX: |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2532 |
assumes "nullable r" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2533 |
shows "POSIX (mkeps r) r" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2534 |
using assms |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2535 |
apply(induct r) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2536 |
apply(auto)[1] |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2537 |
apply(simp add: POSIX_def) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2538 |
apply(auto)[1] |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2539 |
apply (metis Prf.intros(4)) |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2540 |
apply(erule Prf.cases) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2541 |
apply(simp_all)[5] |
6
87618dae0e04
getting back the original version by Sulzmann
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
5
diff
changeset
|
2542 |
apply (metis ValOrd.intros) |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2543 |
apply(simp) |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2544 |
apply(auto)[1] |
7
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
2545 |
apply(simp add: POSIX_def) |
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
2546 |
apply(auto)[1] |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2547 |
apply (metis mkeps.simps(2) mkeps_nullable nullable.simps(5)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2548 |
apply(rotate_tac 6) |
7
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
2549 |
apply(erule Prf.cases) |
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
2550 |
apply(simp_all)[5] |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2551 |
apply (simp add: mkeps_flat) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2552 |
apply(case_tac "mkeps r1a = v1") |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2553 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2554 |
apply (metis ValOrd.intros(1)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2555 |
apply (rule ValOrd.intros(2)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2556 |
apply metis |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2557 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2558 |
(* ALT case *) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2559 |
thm mkeps.simps |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2560 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2561 |
apply(erule disjE) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2562 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2563 |
apply (metis POSIX_ALT_I1) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2564 |
(* *) |
7
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
2565 |
apply(auto)[1] |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2566 |
thm POSIX_ALT_I1 |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2567 |
apply (metis POSIX_ALT_I1) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2568 |
apply(simp (no_asm) add: POSIX_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2569 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2570 |
apply(rule Prf.intros(3)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2571 |
apply(simp only: POSIX_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2572 |
apply(rotate_tac 4) |
7
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
2573 |
apply(erule Prf.cases) |
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
2574 |
apply(simp_all)[5] |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2575 |
thm mkeps_flat |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2576 |
apply(simp add: mkeps_flat) |
7
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
2577 |
apply(auto)[1] |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2578 |
thm Prf_flat_L nullable_correctness |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2579 |
apply (metis Prf_flat_L nullable_correctness) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2580 |
apply(rule ValOrd.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2581 |
apply(subst (asm) POSIX_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2582 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2583 |
apply(drule_tac x="v2" in spec) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2584 |
by simp |
6
87618dae0e04
getting back the original version by Sulzmann
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
5
diff
changeset
|
2585 |
|
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2586 |
|
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2587 |
|
7
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
2588 |
text {* |
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
2589 |
Injection value is related to r |
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
2590 |
*} |
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
2591 |
|
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2592 |
|
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2593 |
|
7
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
2594 |
text {* |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2595 |
The string behind the injection value is an added c |
7
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
2596 |
*} |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
2597 |
|
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2598 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2599 |
lemma injval_inj: "inj_on (injval r c) {v. \<turnstile> v : der c r}" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2600 |
apply(induct c r rule: der.induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2601 |
unfolding inj_on_def |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2602 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2603 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2604 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2605 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2606 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2607 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2608 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2609 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2610 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2611 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2612 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2613 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2614 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2615 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2616 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2617 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2618 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2619 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2620 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2621 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2622 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2623 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2624 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2625 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2626 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2627 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2628 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2629 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2630 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2631 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2632 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2633 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2634 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2635 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2636 |
apply (metis list.distinct(1) mkeps_flat v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2637 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2638 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2639 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2640 |
apply(rotate_tac 6) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2641 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2642 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2643 |
apply (metis list.distinct(1) mkeps_flat v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2644 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2645 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2646 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2647 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2648 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2649 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2650 |
lemma Values_nullable: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2651 |
assumes "nullable r1" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2652 |
shows "mkeps r1 \<in> Values r1 s" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2653 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2654 |
apply(induct r1 arbitrary: s) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2655 |
apply(simp_all) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2656 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2657 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2658 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2659 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2660 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2661 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2662 |
lemma Values_injval: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2663 |
assumes "v \<in> Values (der c r) s" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2664 |
shows "injval r c v \<in> Values r (c#s)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2665 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2666 |
apply(induct c r arbitrary: v s rule: der.induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2667 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2668 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2669 |
apply(case_tac "c = c'") |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2670 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2671 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2672 |
apply(simp add: prefix_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2673 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2674 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2675 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2676 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2677 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2678 |
apply(case_tac "nullable r1") |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2679 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2680 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2681 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2682 |
apply(simp add: rest_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2683 |
apply(subst v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2684 |
apply(simp add: Values_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2685 |
apply(simp add: Values_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2686 |
apply(rule Values_nullable) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2687 |
apply(assumption) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2688 |
apply(simp add: rest_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2689 |
apply(subst mkeps_flat) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2690 |
apply(assumption) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2691 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2692 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2693 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2694 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2695 |
apply(simp add: rest_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2696 |
apply(subst v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2697 |
apply(simp add: Values_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2698 |
apply(simp add: Values_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2699 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2700 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2701 |
lemma Values_projval: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2702 |
assumes "v \<in> Values r (c#s)" "\<exists>s. flat v = c # s" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2703 |
shows "projval r c v \<in> Values (der c r) s" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2704 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2705 |
apply(induct r arbitrary: v s c rule: rexp.induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2706 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2707 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2708 |
apply(case_tac "c = char") |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2709 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2710 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2711 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2712 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2713 |
apply(simp add: prefix_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2714 |
apply(case_tac "nullable rexp1") |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2715 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2716 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2717 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2718 |
apply(simp add: rest_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2719 |
apply (metis hd_Cons_tl hd_append2 list.sel(1)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2720 |
apply(simp add: rest_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2721 |
apply(simp add: append_eq_Cons_conv) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2722 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2723 |
apply(subst v4_proj2) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2724 |
apply(simp add: Values_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2725 |
apply(assumption) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2726 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2727 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2728 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2729 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2730 |
apply(auto simp add: Values_def not_nullable_flat)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2731 |
apply(simp add: append_eq_Cons_conv) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2732 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2733 |
apply(simp add: append_eq_Cons_conv) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2734 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2735 |
apply(simp add: rest_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2736 |
apply(subst v4_proj2) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2737 |
apply(simp add: Values_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2738 |
apply(assumption) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2739 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2740 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2741 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2742 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2743 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2744 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2745 |
definition "MValue v r s \<equiv> (v \<in> Values r s \<and> (\<forall>v' \<in> Values r s. v 2\<succ> v'))" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2746 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2747 |
lemma MValue_ALTE: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2748 |
assumes "MValue v (ALT r1 r2) s" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2749 |
shows "(\<exists>vl. v = Left vl \<and> MValue vl r1 s \<and> (\<forall>vr \<in> Values r2 s. length (flat vr) \<le> length (flat vl))) \<or> |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2750 |
(\<exists>vr. v = Right vr \<and> MValue vr r2 s \<and> (\<forall>vl \<in> Values r1 s. length (flat vl) < length (flat vr)))" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2751 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2752 |
apply(simp add: MValue_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2753 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2754 |
apply(auto) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2755 |
apply(drule_tac x="Left x" in bspec) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2756 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2757 |
apply(erule ValOrd2.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2758 |
apply(simp_all) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2759 |
apply(drule_tac x="Right vr" in bspec) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2760 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2761 |
apply(erule ValOrd2.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2762 |
apply(simp_all) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2763 |
apply(drule_tac x="Right x" in bspec) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2764 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2765 |
apply(erule ValOrd2.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2766 |
apply(simp_all) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2767 |
apply(drule_tac x="Left vl" in bspec) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2768 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2769 |
apply(erule ValOrd2.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2770 |
apply(simp_all) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2771 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2772 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2773 |
lemma MValue_ALTI1: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2774 |
assumes "MValue vl r1 s" "\<forall>vr \<in> Values r2 s. length (flat vr) \<le> length (flat vl)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2775 |
shows "MValue (Left vl) (ALT r1 r2) s" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2776 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2777 |
apply(simp add: MValue_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2778 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2779 |
apply(auto) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2780 |
apply(rule ValOrd2.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2781 |
apply metis |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2782 |
apply(rule ValOrd2.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2783 |
apply metis |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2784 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2785 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2786 |
lemma MValue_ALTI2: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2787 |
assumes "MValue vr r2 s" "\<forall>vl \<in> Values r1 s. length (flat vl) < length (flat vr)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2788 |
shows "MValue (Right vr) (ALT r1 r2) s" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2789 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2790 |
apply(simp add: MValue_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2791 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2792 |
apply(auto) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2793 |
apply(rule ValOrd2.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2794 |
apply metis |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2795 |
apply(rule ValOrd2.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2796 |
apply metis |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2797 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2798 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2799 |
lemma t: "(c#xs = c#ys) \<Longrightarrow> xs = ys" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2800 |
by (metis list.sel(3)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2801 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2802 |
lemma t2: "(xs = ys) \<Longrightarrow> (c#xs) = (c#ys)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2803 |
by (metis) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2804 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2805 |
lemma "\<not>(nullable r) \<Longrightarrow> \<not>(\<exists>v. \<turnstile> v : r \<and> flat v = [])" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2806 |
by (metis Prf_flat_L nullable_correctness) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2807 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2808 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2809 |
lemma LeftRight: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2810 |
assumes "(Left v1) \<succ>(der c (ALT r1 r2)) (Right v2)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2811 |
and "\<turnstile> v1 : der c r1" "\<turnstile> v2 : der c r2" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2812 |
shows "(injval (ALT r1 r2) c (Left v1)) \<succ>(ALT r1 r2) (injval (ALT r1 r2) c (Right v2))" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2813 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2814 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2815 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2816 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2817 |
apply(rule ValOrd.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2818 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2819 |
apply(subst v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2820 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2821 |
apply(subst v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2822 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2823 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2824 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2825 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2826 |
lemma RightLeft: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2827 |
assumes "(Right v1) \<succ>(der c (ALT r1 r2)) (Left v2)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2828 |
and "\<turnstile> v1 : der c r2" "\<turnstile> v2 : der c r1" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2829 |
shows "(injval (ALT r1 r2) c (Right v1)) \<succ>(ALT r1 r2) (injval (ALT r1 r2) c (Left v2))" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2830 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2831 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2832 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2833 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2834 |
apply(rule ValOrd.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2835 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2836 |
apply(subst v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2837 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2838 |
apply(subst v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2839 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2840 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2841 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2842 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2843 |
lemma h: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2844 |
assumes "nullable r1" "\<turnstile> v1 : der c r1" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2845 |
shows "injval r1 c v1 \<succ>r1 mkeps r1" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2846 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2847 |
apply(induct r1 arbitrary: v1 rule: der.induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2848 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2849 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2850 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2851 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2852 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2853 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2854 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2855 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2856 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2857 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2858 |
apply (metis ValOrd.intros(6)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2859 |
apply (metis ValOrd.intros(6)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2860 |
apply (metis ValOrd.intros(3) le_add2 list.size(3) mkeps_flat monoid_add_class.add.right_neutral) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2861 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2862 |
apply (metis ValOrd.intros(4) length_greater_0_conv list.distinct(1) list.size(3) mkeps_flat v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2863 |
apply (metis ValOrd.intros(4) length_greater_0_conv list.distinct(1) list.size(3) mkeps_flat v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2864 |
apply (metis ValOrd.intros(5)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2865 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2866 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2867 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2868 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2869 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2870 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2871 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2872 |
apply (metis ValOrd.intros(2) list.distinct(1) mkeps_flat v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2873 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2874 |
by (metis ValOrd.intros(1)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2875 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2876 |
lemma LeftRightSeq: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2877 |
assumes "(Left (Seq v1 v2)) \<succ>(der c (SEQ r1 r2)) (Right v3)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2878 |
and "nullable r1" "\<turnstile> v1 : der c r1" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2879 |
shows "(injval (SEQ r1 r2) c (Seq v1 v2)) \<succ>(SEQ r1 r2) (injval (SEQ r1 r2) c (Right v2))" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2880 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2881 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2882 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2883 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2884 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2885 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2886 |
apply(rule ValOrd.intros(2)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2887 |
prefer 2 |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2888 |
apply (metis list.distinct(1) mkeps_flat v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2889 |
by (metis h) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2890 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2891 |
lemma rr1: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2892 |
assumes "\<turnstile> v : r" "\<not>nullable r" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2893 |
shows "flat v \<noteq> []" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2894 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2895 |
by (metis Prf_flat_L nullable_correctness) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2896 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2897 |
(* HERE *) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2898 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2899 |
lemma Prf_inj_test: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2900 |
assumes "v1 \<succ>(der c r) v2" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2901 |
"v1 \<in> Values (der c r) s" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2902 |
"v2 \<in> Values (der c r) s" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2903 |
"injval r c v1 \<in> Values r (c#s)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2904 |
"injval r c v2 \<in> Values r (c#s)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2905 |
shows "(injval r c v1) 2\<succ> (injval r c v2)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2906 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2907 |
apply(induct c r arbitrary: v1 v2 s rule: der.induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2908 |
(* NULL case *) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2909 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2910 |
(* EMPTY case *) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2911 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2912 |
(* CHAR case *) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2913 |
apply(case_tac "c = c'") |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2914 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2915 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2916 |
apply (metis ValOrd2.intros(8)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2917 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2918 |
(* ALT case *) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2919 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2920 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2921 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2922 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2923 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2924 |
apply (metis ValOrd2.intros(6)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2925 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2926 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2927 |
apply(rule ValOrd2.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2928 |
apply(subst v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2929 |
apply(simp add: Values_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2930 |
apply(subst v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2931 |
apply(simp add: Values_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2932 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2933 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2934 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2935 |
apply(rule ValOrd2.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2936 |
apply(subst v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2937 |
apply(simp add: Values_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2938 |
apply(subst v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2939 |
apply(simp add: Values_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2940 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2941 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2942 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2943 |
apply (metis ValOrd2.intros(5)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2944 |
(* SEQ case*) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2945 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2946 |
apply(case_tac "nullable r1") |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2947 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2948 |
defer |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2949 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2950 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2951 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2952 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2953 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2954 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2955 |
apply(rule ValOrd2.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2956 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2957 |
apply (metis Ord1) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2958 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2959 |
apply(rule ValOrd2.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2960 |
apply(subgoal_tac "rest v1 (flat v1 @ flat v2) = flat v2") |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2961 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2962 |
apply(subgoal_tac "rest (injval r1 c v1) (c # flat v1 @ flat v2) = flat v2") |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2963 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2964 |
oops |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2965 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2966 |
lemma Prf_inj_test: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2967 |
assumes "v1 \<succ>(der c r) v2" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2968 |
"v1 \<in> Values (der c r) s" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2969 |
"v2 \<in> Values (der c r) s" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2970 |
"injval r c v1 \<in> Values r (c#s)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2971 |
"injval r c v2 \<in> Values r (c#s)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2972 |
shows "(injval r c v1) 2\<succ> (injval r c v2)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2973 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2974 |
apply(induct c r arbitrary: v1 v2 s rule: der.induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2975 |
(* NULL case *) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2976 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2977 |
(* EMPTY case *) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2978 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2979 |
(* CHAR case *) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2980 |
apply(case_tac "c = c'") |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2981 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2982 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2983 |
apply (metis ValOrd2.intros(8)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2984 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2985 |
(* ALT case *) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2986 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2987 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2988 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2989 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2990 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2991 |
apply (metis ValOrd2.intros(6)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2992 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2993 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2994 |
apply(rule ValOrd2.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2995 |
apply(subst v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2996 |
apply(simp add: Values_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2997 |
apply(subst v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2998 |
apply(simp add: Values_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
2999 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3000 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3001 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3002 |
apply(rule ValOrd2.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3003 |
apply(subst v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3004 |
apply(simp add: Values_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3005 |
apply(subst v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3006 |
apply(simp add: Values_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3007 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3008 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3009 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3010 |
apply (metis ValOrd2.intros(5)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3011 |
(* SEQ case*) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3012 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3013 |
apply(case_tac "nullable r1") |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3014 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3015 |
defer |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3016 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3017 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3018 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3019 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3020 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3021 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3022 |
apply(rule ValOrd2.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3023 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3024 |
apply (metis Ord1) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3025 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3026 |
apply(rule ValOrd2.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3027 |
apply metis |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3028 |
using injval_inj |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3029 |
apply(simp add: Values_def inj_on_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3030 |
apply metis |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3031 |
apply(simp add: Values_recs) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3032 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3033 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3034 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3035 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3036 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3037 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3038 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3039 |
apply (metis Ord1 ValOrd2.intros(1)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3040 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3041 |
apply(rule ValOrd2.intros(2)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3042 |
apply metis |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3043 |
using injval_inj |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3044 |
apply(simp add: Values_def inj_on_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3045 |
apply metis |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3046 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3047 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3048 |
apply(rule ValOrd2.intros(2)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3049 |
thm h |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3050 |
apply(rule Ord1) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3051 |
apply(rule h) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3052 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3053 |
apply(simp add: Values_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3054 |
apply(simp add: Values_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3055 |
apply (metis list.distinct(1) mkeps_flat v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3056 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3057 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3058 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3059 |
apply(simp add: Values_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3060 |
defer |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3061 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3062 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3063 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3064 |
apply(rule ValOrd2.intros(1)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3065 |
apply(rotate_tac 1) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3066 |
apply(drule_tac x="v2" in meta_spec) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3067 |
apply(rotate_tac 8) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3068 |
apply(drule_tac x="v2'" in meta_spec) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3069 |
apply(rotate_tac 8) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3070 |
oops |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3071 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3072 |
lemma POSIX_der: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3073 |
assumes "POSIX v (der c r)" "\<turnstile> v : der c r" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3074 |
shows "POSIX (injval r c v) r" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3075 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3076 |
unfolding POSIX_def |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3077 |
apply(auto) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3078 |
thm v3 |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3079 |
apply (erule v3) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3080 |
thm v4 |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3081 |
apply(subst (asm) v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3082 |
apply(assumption) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3083 |
apply(drule_tac x="projval r c v'" in spec) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3084 |
apply(drule mp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3085 |
apply(rule conjI) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3086 |
thm v3_proj |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3087 |
apply(rule v3_proj) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3088 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3089 |
apply(rule_tac x="flat v" in exI) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3090 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3091 |
thm t |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3092 |
apply(rule_tac c="c" in t) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3093 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3094 |
thm v4_proj |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3095 |
apply(subst v4_proj) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3096 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3097 |
apply(rule_tac x="flat v" in exI) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3098 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3099 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3100 |
oops |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3101 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3102 |
lemma POSIX_der: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3103 |
assumes "POSIX v (der c r)" "\<turnstile> v : der c r" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3104 |
shows "POSIX (injval r c v) r" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3105 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3106 |
apply(induct c r arbitrary: v rule: der.induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3107 |
(* null case*) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3108 |
apply(simp add: POSIX_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3109 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3110 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3111 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3112 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3113 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3114 |
(* empty case *) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3115 |
apply(simp add: POSIX_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3116 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3117 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3118 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3119 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3120 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3121 |
(* char case *) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3122 |
apply(simp add: POSIX_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3123 |
apply(case_tac "c = c'") |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3124 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3125 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3126 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3127 |
apply (metis Prf.intros(5)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3128 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3129 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3130 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3131 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3132 |
apply (metis ValOrd.intros(8)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3133 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3134 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3135 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3136 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3137 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3138 |
(* alt case *) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3139 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3140 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3141 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3142 |
apply(simp (no_asm) add: POSIX_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3143 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3144 |
apply (metis Prf.intros(2) v3) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3145 |
apply(rotate_tac 4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3146 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3147 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3148 |
apply (metis POSIX_ALT2 POSIX_def ValOrd.intros(6)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3149 |
apply (metis ValOrd.intros(3) order_refl) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3150 |
apply(simp (no_asm) add: POSIX_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3151 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3152 |
apply (metis Prf.intros(3) v3) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3153 |
apply(rotate_tac 4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3154 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3155 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3156 |
defer |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3157 |
apply (metis POSIX_ALT1a POSIX_def ValOrd.intros(5)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3158 |
prefer 2 |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3159 |
apply(subst (asm) (5) POSIX_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3160 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3161 |
apply(rotate_tac 5) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3162 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3163 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3164 |
apply(rule ValOrd.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3165 |
apply(subst (asm) v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3166 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3167 |
apply(drule_tac x="Left (projval r1a c v1)" in spec) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3168 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3169 |
apply(drule mp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3170 |
apply(rule conjI) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3171 |
apply (metis Prf.intros(2) v3_proj) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3172 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3173 |
apply (metis v4_proj2) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3174 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3175 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3176 |
apply (metis less_not_refl v4_proj2) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3177 |
(* seq case *) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3178 |
apply(case_tac "nullable r1") |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3179 |
defer |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3180 |
apply(simp add: POSIX_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3181 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3182 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3183 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3184 |
apply (metis Prf.intros(1) v3) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3185 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3186 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3187 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3188 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3189 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3190 |
apply(subst (asm) (3) v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3191 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3192 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3193 |
apply(subgoal_tac "flat v1a \<noteq> []") |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3194 |
prefer 2 |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3195 |
apply (metis Prf_flat_L nullable_correctness) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3196 |
apply(subgoal_tac "\<exists>s. flat v1a = c # s") |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3197 |
prefer 2 |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3198 |
apply (metis append_eq_Cons_conv) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3199 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3200 |
oops |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3201 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3202 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3203 |
lemma POSIX_ex: "\<turnstile> v : r \<Longrightarrow> \<exists>v. POSIX v r" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3204 |
apply(induct r arbitrary: v) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3205 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3206 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3207 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3208 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3209 |
apply(rule_tac x="Void" in exI) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3210 |
apply(simp add: POSIX_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3211 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3212 |
apply (metis Prf.intros(4)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3213 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3214 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3215 |
apply (metis ValOrd.intros(7)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3216 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3217 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3218 |
apply(rule_tac x="Char c" in exI) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3219 |
apply(simp add: POSIX_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3220 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3221 |
apply (metis Prf.intros(5)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3222 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3223 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3224 |
apply (metis ValOrd.intros(8)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3225 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3226 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3227 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3228 |
apply(drule_tac x="v1" in meta_spec) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3229 |
apply(drule_tac x="v2" in meta_spec) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3230 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3231 |
defer |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3232 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3233 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3234 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3235 |
apply (metis POSIX_ALT_I1) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3236 |
apply (metis POSIX_ALT_I1 POSIX_ALT_I2) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3237 |
apply(case_tac "nullable r1a") |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3238 |
apply(rule_tac x="Seq (mkeps r1a) va" in exI) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3239 |
apply(auto simp add: POSIX_def)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3240 |
apply (metis Prf.intros(1) mkeps_nullable) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3241 |
apply(simp add: mkeps_flat) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3242 |
apply(rotate_tac 7) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3243 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3244 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3245 |
apply(case_tac "mkeps r1 = v1a") |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3246 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3247 |
apply (rule ValOrd.intros(1)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3248 |
apply (metis append_Nil mkeps_flat) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3249 |
apply (rule ValOrd.intros(2)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3250 |
apply(drule mkeps_POSIX) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3251 |
apply(simp add: POSIX_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3252 |
oops |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3253 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3254 |
lemma POSIX_ex2: "\<turnstile> v : r \<Longrightarrow> \<exists>v. POSIX v r \<and> \<turnstile> v : r" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3255 |
apply(induct r arbitrary: v) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3256 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3257 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3258 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3259 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3260 |
apply(rule_tac x="Void" in exI) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3261 |
apply(simp add: POSIX_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3262 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3263 |
oops |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3264 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3265 |
lemma POSIX_ALT_cases: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3266 |
assumes "\<turnstile> v : (ALT r1 r2)" "POSIX v (ALT r1 r2)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3267 |
shows "(\<exists>v1. v = Left v1 \<and> POSIX v1 r1) \<or> (\<exists>v2. v = Right v2 \<and> POSIX v2 r2)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3268 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3269 |
apply(erule_tac Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3270 |
apply(simp_all) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3271 |
unfolding POSIX_def |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3272 |
apply(auto) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3273 |
apply (metis POSIX_ALT2 POSIX_def assms(2)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3274 |
by (metis POSIX_ALT1b assms(2)) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3275 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3276 |
lemma POSIX_ALT_cases2: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3277 |
assumes "POSIX v (ALT r1 r2)" "\<turnstile> v : (ALT r1 r2)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3278 |
shows "(\<exists>v1. v = Left v1 \<and> POSIX v1 r1) \<or> (\<exists>v2. v = Right v2 \<and> POSIX v2 r2)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3279 |
using assms POSIX_ALT_cases by auto |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3280 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3281 |
lemma Prf_flat_empty: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3282 |
assumes "\<turnstile> v : r" "flat v = []" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3283 |
shows "nullable r" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3284 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3285 |
apply(induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3286 |
apply(auto) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3287 |
done |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3288 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3289 |
lemma POSIX_proj: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3290 |
assumes "POSIX v r" "\<turnstile> v : r" "\<exists>s. flat v = c#s" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3291 |
shows "POSIX (projval r c v) (der c r)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3292 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3293 |
apply(induct r c v arbitrary: rule: projval.induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3294 |
defer |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3295 |
defer |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3296 |
defer |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3297 |
defer |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3298 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3299 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3300 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3301 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3302 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3303 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3304 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3305 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3306 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3307 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3308 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3309 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3310 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3311 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3312 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3313 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3314 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3315 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3316 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3317 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3318 |
apply(simp add: POSIX_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3319 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3320 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3321 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3322 |
oops |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3323 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3324 |
lemma POSIX_proj: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3325 |
assumes "POSIX v r" "\<turnstile> v : r" "\<exists>s. flat v = c#s" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3326 |
shows "POSIX (projval r c v) (der c r)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3327 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3328 |
apply(induct r arbitrary: c v rule: rexp.induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3329 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3330 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3331 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3332 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3333 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3334 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3335 |
apply(simp add: POSIX_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3336 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3337 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3338 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3339 |
oops |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3340 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3341 |
lemma POSIX_proj: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3342 |
assumes "POSIX v r" "\<turnstile> v : r" "\<exists>s. flat v = c#s" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3343 |
shows "POSIX (projval r c v) (der c r)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3344 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3345 |
apply(induct r c v arbitrary: rule: projval.induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3346 |
defer |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3347 |
defer |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3348 |
defer |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3349 |
defer |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3350 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3351 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3352 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3353 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3354 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3355 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3356 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3357 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3358 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3359 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3360 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3361 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3362 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3363 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3364 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3365 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3366 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3367 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3368 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3369 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3370 |
apply(simp add: POSIX_def) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3371 |
apply(auto)[1] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3372 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3373 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3374 |
oops |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3375 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3376 |
lemma Prf_inj: |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3377 |
assumes "v1 \<succ>(der c r) v2" "\<turnstile> v1 : der c r" "\<turnstile> v2 : der c r" "flat v1 = flat v2" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3378 |
shows "(injval r c v1) \<succ>r (injval r c v2)" |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3379 |
using assms |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3380 |
apply(induct arbitrary: v1 v2 rule: der.induct) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3381 |
(* NULL case *) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3382 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3383 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3384 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3385 |
(* EMPTY case *) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3386 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3387 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3388 |
(* CHAR case *) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3389 |
apply(case_tac "c = c'") |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3390 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3391 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3392 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3393 |
apply(rule ValOrd.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3394 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3395 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3396 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3397 |
(* ALT case *) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3398 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3399 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3400 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3401 |
apply(rule ValOrd.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3402 |
apply(subst v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3403 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3404 |
apply(rotate_tac 3) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3405 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3406 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3407 |
apply(subst v4) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3408 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3409 |
apply(rotate_tac 2) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3410 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3411 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3412 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3413 |
apply(rule ValOrd.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3414 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3415 |
apply(rotate_tac 3) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3416 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3417 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3418 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3419 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3420 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3421 |
apply(rule ValOrd.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3422 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3423 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3424 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3425 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3426 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3427 |
(* SEQ case*) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3428 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3429 |
apply(case_tac "nullable r1") |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3430 |
defer |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3431 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3432 |
apply(erule ValOrd.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3433 |
apply(simp_all)[8] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3434 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3435 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3436 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3437 |
apply(erule Prf.cases) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3438 |
apply(simp_all)[5] |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3439 |
apply(clarify) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3440 |
apply(rule ValOrd.intros) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3441 |
apply(simp) |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3442 |
oops |
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3443 |
|
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3444 |
|
7
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
3445 |
text {* |
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
3446 |
Injection followed by projection is the identity. |
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
3447 |
*} |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3448 |
|
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3449 |
lemma proj_inj_id: |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3450 |
assumes "\<turnstile> v : der c r" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3451 |
shows "projval r c (injval r c v) = v" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3452 |
using assms |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3453 |
apply(induct r arbitrary: c v rule: rexp.induct) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3454 |
apply(simp) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3455 |
apply(erule Prf.cases) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3456 |
apply(simp_all)[5] |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3457 |
apply(simp) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3458 |
apply(erule Prf.cases) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3459 |
apply(simp_all)[5] |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3460 |
apply(simp) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3461 |
apply(case_tac "c = char") |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3462 |
apply(simp) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3463 |
apply(erule Prf.cases) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3464 |
apply(simp_all)[5] |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3465 |
apply(simp) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3466 |
apply(erule Prf.cases) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3467 |
apply(simp_all)[5] |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3468 |
defer |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3469 |
apply(simp) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3470 |
apply(erule Prf.cases) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3471 |
apply(simp_all)[5] |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3472 |
apply(simp) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3473 |
apply(case_tac "nullable rexp1") |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3474 |
apply(simp) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3475 |
apply(erule Prf.cases) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3476 |
apply(simp_all)[5] |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3477 |
apply(auto)[1] |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3478 |
apply(erule Prf.cases) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3479 |
apply(simp_all)[5] |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3480 |
apply(auto)[1] |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3481 |
apply (metis list.distinct(1) v4) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3482 |
apply(auto)[1] |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3483 |
apply (metis mkeps_flat) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3484 |
apply(auto) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3485 |
apply(erule Prf.cases) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3486 |
apply(simp_all)[5] |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3487 |
apply(auto)[1] |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3488 |
apply(simp add: v4) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3489 |
done |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3490 |
|
7
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
3491 |
text {* |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3492 |
|
7
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
3493 |
HERE: Crucial lemma that does not go through in the sequence case. |
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
3494 |
|
b409ecf47f64
cleaned up the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
6
diff
changeset
|
3495 |
*} |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3496 |
lemma v5: |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3497 |
assumes "\<turnstile> v : der c r" "POSIX v (der c r)" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3498 |
shows "POSIX (injval r c v) r" |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3499 |
using assms |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3500 |
apply(induct arbitrary: v rule: der.induct) |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3501 |
(* NULL case *) |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3502 |
apply(simp) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3503 |
apply(erule Prf.cases) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3504 |
apply(simp_all)[5] |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3505 |
(* EMPTY case *) |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3506 |
apply(simp) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3507 |
apply(erule Prf.cases) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3508 |
apply(simp_all)[5] |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3509 |
(* CHAR case *) |
5
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3510 |
apply(simp) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3511 |
apply(case_tac "c = c'") |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3512 |
apply(auto simp add: POSIX_def)[1] |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3513 |
apply(erule Prf.cases) |
fe177dfc4697
initial version of the theory
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
diff
changeset
|
3514 |
apply(simp_all)[5] |
82
26202889f829
cleaned up version of Re1
Christian Urban <christian dot urban at kcl dot ac dot uk>
parents:
10
diff
changeset
|
3515 |
oops |