lexing: thys/Re.thy@0fa636821349 (annotated)

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 *}
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	8
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
211 0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1030	section (* tryout with all-mkeps *)
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
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1041	fun
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1042	allvals :: "rexp \<Rightarrow> string \<Rightarrow> val set"
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1043	where
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1044	"allvals r [] = alleps r"
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1045	\| "allvals r (c#s) = {injval r c v \| v. v \<in> allvals (der c r) s}"
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1046
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1047
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1048	lemma q1:
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1049	assumes "v \<in> alleps r"
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1050	shows "\<turnstile> v : r \<and> flat v = []"
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1051	using assms
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1052	apply(induct r arbitrary: v)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1053	apply(auto intro: Prf.intros)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1054	done
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1055
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1056	lemma q11:
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1057	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	1058	shows "v \<in> alleps r"
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1059	using assms
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1060	apply(induct r arbitrary: v)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1061	apply(auto)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1062	apply(erule Prf.cases)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1063	apply(simp_all)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1064	apply(erule Prf.cases)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1065	apply(simp_all)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1066	apply(erule Prf.cases)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1067	apply(simp_all)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1068	apply(auto)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1069	apply(subgoal_tac "nullable r2a")
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1070	apply(auto)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1071	using not_nullable_flat apply auto[1]
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1072	apply(erule Prf.cases)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1073	apply(simp_all)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1074	apply(auto)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1075	apply(subgoal_tac "nullable r1a")
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1076	apply(auto)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1077	using not_nullable_flat apply auto[1]
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1078	done
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1079
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1080	lemma q33:
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1081	assumes "nullable r"
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1082	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	1083	using assms
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1084	apply(auto)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1085	apply (simp_all add: q1)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1086	by (simp add: q11)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1087
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1088	lemma q22:
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1089	assumes "v \<in> allvals r s"
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1090	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	1091	using assms
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1092	apply(induct s arbitrary: v r)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1093	apply(auto)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1094	apply(simp_all add: q1)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1095	using Prf_flat_L q1 apply fastforce
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1096	apply (simp add: v3)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1097	apply (metis Prf_flat_L v3 v4)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1098	by (simp add: v4)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1099
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1100
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1101	lemma qa_oops:
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1102	assumes "\<turnstile> v : r" "\<exists>s. flat v = a # s \<and> a # s \<in> L r" "\<turnstile> projval r a v : der a r"
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1103	shows "injval r a (projval r a v) = v"
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1104	using assms
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1105	apply(induct v r arbitrary: )
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1106	defer
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1107	defer
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1108	apply(simp)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1109	apply(erule Prf.cases)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1110	apply(simp_all)[4]
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1111	apply(auto simp only:)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1112	apply(simp)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1113	apply(erule Prf.cases)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1114	apply(simp_all)[5]
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1115
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1116	done
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1117	apply(erule Prf.cases)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1118	apply(simp_all)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1119	apply(erule Prf.cases)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1120	apply(simp_all)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1121	apply(erule Prf.cases)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1122	apply(simp_all)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1123	apply(erule Prf.cases)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1124	apply(simp_all)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1125	apply(erule Prf.cases)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1126	apply(simp_all)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1127	apply(auto)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1128	using Prf_flat_L apply fastforce
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1129	apply(erule Prf.cases)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1130	apply(simp_all)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1131	apply(erule Prf.cases)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1132	apply(simp_all)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1133	using Prf_flat_L apply force
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1134	apply(erule Prf.cases)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1135	apply(simp_all)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1136	apply(erule Prf.cases)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1137	apply(simp_all)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1138	apply(erule Prf.cases)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1139	apply(simp_all)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1140	apply(auto)[1]
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1141	apply(case_tac "nullable r1a")
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1142	apply(simp)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1143	apply(erule Prf.cases)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1144	apply(simp_all)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1145	apply(auto simp add: Sequ_def)[1]
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1146	apply(simp add: Cons_eq_append_conv)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1147	apply(auto)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1148
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1149
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1150	using Prf_flat_L
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1151	apply(erule Prf.cases)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1152	apply(simp_all)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1153	using Prf_flat_L
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(simp add: Sequ_def)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1159	apply(auto)[1]
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1160	apply(simp add: Cons_eq_append_conv)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1161	apply(auto)[1]
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1162	sorry
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1163
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1164
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1165	lemma q2:
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1166	assumes "s \<in> L(r)"
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1167	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	1168	using assms
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1169	apply(induct s arbitrary: r)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1170	apply(simp)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1171	apply(subst q33)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1172	using nullable_correctness apply blast
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1173	apply(simp)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1174	apply(simp)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1175	apply(drule_tac x="der a r" in meta_spec)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1176	apply(drule meta_mp)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1177	using lex_correct1 lex_correct3 apply fastforce
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1178	apply(simp)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1179	apply(auto)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1180	using v3 apply blast
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1181	apply (simp add: v4)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1182	apply(rule_tac x="projval r a x" in exI)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1183	apply(rule conjI)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1184	defer
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1185	apply(rule conjI)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1186	apply (simp add: v3_proj)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1187	apply (simp add: v4_proj2)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1188	apply(subgoal_tac "projval r a x \<in> allvals (der a r) s")
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1189	apply(simp)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1190	apply(subst qa_oops)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1191	apply(simp)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1192	apply(blast)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1193
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1194	lemma q2:
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1195	assumes "\<turnstile> v : r" "s \<in> L (r)" "flat v = s"
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1196	shows "v \<in> allvals r s"
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1197	using assms
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1198	apply(induct s arbitrary: r v)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1199	apply(auto)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1200	apply(subgoal_tac "nullable r")
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1201	apply(simp add: q11)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1202	using not_nullable_flat apply fastforce
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1203	apply(drule sym)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1204	apply(simp)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1205	apply(case_tac s)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1206	apply(simp)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1207
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1208	apply(drule_tac x="projval r a v" in meta_spec)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1209	apply(drule_tac x="der a r" in meta_spec)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1210	apply(drule meta_mp)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1211	defer
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1212	apply(drule meta_mp)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1213	using v3_proj apply blast
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1214	apply(rule_tac x="projval r a v" in exI)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1215	apply(auto)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1216	defer
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1217	apply(subst (asm) v4_proj2)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1218	apply(assumption)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1219	apply(assumption)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1220	apply(simp)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1221	apply(subst v4_proj2)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1222	apply(assumption)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1223	apply(assumption)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1224	apply(simp)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1225	apply(subst (asm) v4_proj2)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1226	apply(assumption)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1227	apply(assumption)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1228	sorry
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1229
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1230
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1231
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1232	lemma
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1233	"{v. \<turnstile> v : r \<and> flat v = s} = allvals r s"
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1234	apply(auto)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1235	apply(rule q2)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1236	apply(simp)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1237
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1238
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1239	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	1240	where
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1241	"POSIX2 v r s \<equiv> (\<turnstile> v : r \<and> (\<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	1242
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1243	lemma k1:
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1244	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	1245	using assms
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1246	apply(induct r arbitrary: v)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1247	apply(simp_all)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1248	apply(simp add: POSIX2_def)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1249	apply(auto)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1250	apply(simp add: POSIX2_def)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1251	apply(simp add: POSIX2_def)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1252	apply(simp add: POSIX2_def)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1253	apply(simp add: POSIX2_def)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1254	apply(simp add: POSIX2_def)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1255	done
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1256
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1257	lemma k2:
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1258	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	1259	using assms
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1260	apply(induct s arbitrary: r v)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1261	apply(simp add: k1)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1262	apply(auto)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1263	apply(simp only: POSIX2_def)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1264	apply(simp)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1265	apply(clarify)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1266	apply(drule_tac x="injval r a va" in spec)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1267	apply(drule mp)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1268	defer
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1269	apply(auto)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1270	done
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1271	done
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1272
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1273	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	1274	apply(induct r arbitrary: s)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1275	apply(auto)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1276	apply(rule_tac x="Void" in exI)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1277	apply(simp add: POSIX2_def)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1278	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	1279	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	1280	apply(simp add: POSIX2_def)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1281	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	1282	defer
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1283	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	1284	apply(auto)[1]
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1285	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	1286	apply(simp add: POSIX2_def)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1287	apply(auto)[1]
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1288	using Prf.intros(2) apply blast
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1289	apply(case_tac s)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1290	apply(simp)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1291	apply(auto)[1]
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1292	apply (simp add: ValOrd.intros(6))
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1293	apply(rule ValOrd.intros)
0fa636821349 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 88 diff changeset	1294
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	1295	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	1296
030939b7d475 added type inference paper and updated Re.thy Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 86 diff changeset	1297	lemma POSIX_PMatch:
88 532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1298	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	1299	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	1300	using assms
88 532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1301	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	1302	apply(simp_all add: Values_recs)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1303	apply(erule PMatch.cases)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1304	apply(simp_all)[5]
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1305	apply (metis ValOrd.intros(7))
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1306	apply(erule PMatch.cases)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1307	apply(simp_all)[5]
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1308	apply(simp add: prefix_def)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1309	apply (metis ValOrd.intros(8))
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1310	apply(auto)[1]
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1311	defer
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1312	apply(auto)[1]
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1313	apply(erule PMatch.cases)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1314	apply(simp_all)[5]
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1315	apply (metis ValOrd.intros(6))
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1316	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	1317	apply(erule PMatch.cases)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1318	apply(simp_all)[5]
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1319	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	1320	apply (metis ValOrd.intros(5))
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1321	apply(erule PMatch.cases)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1322	apply(simp_all)[5]
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1323	apply(auto)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1324	apply(case_tac "v1a = v1")
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1325	apply(simp)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1326	apply(rule ValOrd.intros(1))
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1327	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	1328	apply(rule ValOrd.intros(2))
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1329	apply(auto)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1330	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	1331	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	1332	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	1333	apply(auto)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1334	apply(drule meta_mp)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1335	defer
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1336	apply(auto)[1]
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1337	apply(frule PMatch1)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1338	apply(drule PMatch1(2))
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1339	apply(frule PMatch1)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1340	apply(drule PMatch1(2))
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1341	apply(auto)[1]
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1342	apply(simp add: Values_def)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1343	apply(simp add: prefix_def)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1344	apply(auto)[1]
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1345	apply(simp add: append_eq_append_conv2)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1346	apply(auto)[1]
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1347	apply(rotate_tac 10)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1348	apply(drule sym)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1349	apply(simp)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1350	apply(simp add: rest_def)
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1351	apply(case_tac "s3a = []")
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1352	apply(auto)[1]
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1353
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1354
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1355	apply (metis ValOrd.intros(6))
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1356	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	1357	apply(auto)[1]
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1358	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	1359	apply (metis ValOrd.intros(5))
532bb9df225d updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 87 diff changeset	1360	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	1361	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	1362	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	1363	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	1364	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	1365	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	1366	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	1367	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	1368	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	1369	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	1370	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	1371	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	1372	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	1373	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	1374	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	1375	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	1376
030939b7d475 added type inference paper and updated Re.thy Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 86 diff changeset	1377	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	1378	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	1379	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	1380	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	1381	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	1382	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	1383	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	1384	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	1385	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	1386	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	1387
030939b7d475 added type inference paper and updated Re.thy Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 86 diff changeset	1388
030939b7d475 added type inference paper and updated Re.thy Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 86 diff changeset	1389	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	1390	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	1391	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	1392	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	1393	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	1394	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	1395	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	1396	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	1397	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	1398	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	1399	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	1400	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	1401	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	1402	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	1403	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	1404	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	1405	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	1406	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	1407	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	1408	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	1409	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	1410	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	1411	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	1412	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	1413	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	1414	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	1415
030939b7d475 added type inference paper and updated Re.thy Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 86 diff changeset	1416
85 53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1417
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1418	lemma ValOrd_PMatch:
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1419	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	1420	shows "v1 \<succ>r v2"
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1421	using assms
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1422	apply(induct arbitrary: v2 rule: PMatch.induct)
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1423	apply(erule Prf.cases)
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1424	apply(simp_all)[5]
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1425	apply (metis ValOrd.intros(7))
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1426	apply(erule Prf.cases)
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1427	apply(simp_all)[5]
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1428	apply (metis ValOrd.intros(8))
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1429	apply(erule Prf.cases)
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1430	apply(simp_all)[5]
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1431	apply(clarify)
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1432	apply (metis ValOrd.intros(6))
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1433	apply(clarify)
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1434	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	1435	apply(erule Prf.cases)
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1436	apply(simp_all)[5]
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1437	apply(clarify)
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1438	apply (metis Prf_flat_L)
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1439	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	1440	(* Seq case *)
85 53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1441	apply(erule Prf.cases)
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1442	apply(simp_all)[5]
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1443	apply(auto)
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1444	apply(case_tac "v1 = v1a")
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1445	apply(auto)
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1446	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	1447	apply(rule ValOrd.intros(2))
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1448	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	1449	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	1450	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	1451	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	1452	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	1453	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	1454	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	1455	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	1456	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	1457	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	1458	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	1459	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	1460	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	1461	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	1462
85 53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1463	thm L_flat_Prf
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1464	apply(simp add: L_flat_Prf)
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1465	thm append_eq_append_conv2
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1466	apply(simp add: append_eq_append_conv2)
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1467	apply(auto)
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1468	apply(drule_tac x="us" in spec)
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1469	apply(drule mp)
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1470	apply metis
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1471	apply (metis append_Nil2)
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1472	apply(case_tac "us = []")
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1473	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	1474	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	1475	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	1476
030939b7d475 added type inference paper and updated Re.thy Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 86 diff changeset	1477	apply(auto)[1]
85 53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1478	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	1479	apply(simp)
53d5f9a5bbd3 updated Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 84 diff changeset	1480
82 26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1481	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	1482	"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	1483	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	1484	apply(auto)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1485	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	1486	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	1487	done
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1488
5 fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1489
7 b409ecf47f64 cleaned up the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 6 diff changeset	1490	section {* Posix definition *}
b409ecf47f64 cleaned up the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 6 diff changeset	1491
5 fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1492	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	1493	where
82 26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1494	"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	1495
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1496	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	1497	where
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1498	"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	1499
82 26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1500	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	1501	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	1502	apply(auto)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1503	apply(rule Ord1)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1504	apply(auto)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1505	apply(rule Ord3)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1506	apply(auto)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1507	done
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1508
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1509	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	1510
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1511	lemma POSIX_SEQ1:
5 fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1512	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	1513	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	1514	using assms
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1515	unfolding POSIX_def
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1516	apply(auto)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1517	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	1518	apply(simp)
82 26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1519	apply(erule impE)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1520	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	1521	apply(simp)
5 fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1522	apply(simp)
82 26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1523	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	1524	apply(simp_all)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1525	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1526	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	1527
82 26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1528	lemma POSIX_SEQ2:
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1529	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	1530	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	1531	using assms
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1532	unfolding POSIX_def
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1533	apply(auto)
82 26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1534	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	1535	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1536	apply(erule impE)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1537	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	1538	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1539	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1540	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	1541	apply(simp_all)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1542	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	1543
82 26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1544	lemma POSIX_ALT2:
5 fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1545	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	1546	shows "POSIX v1 r1"
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1547	using assms
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1548	unfolding POSIX_def
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1549	apply(auto)
82 26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1550	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	1551	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	1552	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	1553	apply(simp)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1554	apply(drule mp)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1555	apply(rule Prf.intros)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1556	apply(auto)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1557	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	1558	apply(simp_all)
5 fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1559	done
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1560
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1561	lemma POSIX_ALT1a:
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1562	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	1563	shows "POSIX v2 r2"
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1564	using assms
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1565	unfolding POSIX_def
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1566	apply(auto)
82 26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1567	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	1568	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	1569	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	1570	apply(simp)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1571	apply(drule mp)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1572	apply(rule Prf.intros)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1573	apply(auto)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1574	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	1575	apply(simp_all)
5 fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1576	done
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1577
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1578	lemma POSIX_ALT1b:
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1579	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	1580	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	1581	using assms
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1582	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	1583	unfolding POSIX_def
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1584	apply(auto)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1585	done
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1586
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1587	lemma POSIX_ALT_I1:
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1588	assumes "POSIX v1 r1"
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1589	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	1590	using assms
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1591	unfolding POSIX_def
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1592	apply(auto)
82 26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1593	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	1594	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	1595	apply(erule Prf.cases)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1596	apply(simp_all)[5]
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1597	apply(auto)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1598	apply(rule ValOrd.intros)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1599	apply(auto)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1600	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	1601	by simp
5 fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1602
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1603	lemma POSIX_ALT_I2:
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1604	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	1605	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	1606	using assms
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1607	unfolding POSIX_def
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1608	apply(auto)
82 26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1609	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	1610	apply(rotate_tac 3)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1611	apply(erule Prf.cases)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1612	apply(simp_all)[5]
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1613	apply(auto)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1614	apply(rule ValOrd.intros)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1615	apply metis
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1616	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	1617
5 fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1618	lemma mkeps_POSIX:
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1619	assumes "nullable r"
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1620	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	1621	using assms
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1622	apply(induct r)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1623	apply(auto)[1]
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1624	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	1625	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	1626	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	1627	apply(erule Prf.cases)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1628	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	1629	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	1630	apply(simp)
5 fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1631	apply(auto)[1]
7 b409ecf47f64 cleaned up the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 6 diff changeset	1632	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	1633	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	1634	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	1635	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	1636	apply(erule Prf.cases)
b409ecf47f64 cleaned up the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 6 diff changeset	1637	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	1638	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	1639	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	1640	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1641	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	1642	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	1643	apply metis
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1644	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1645	(* ALT case *)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1646	thm mkeps.simps
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1647	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1648	apply(erule disjE)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1649	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1650	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	1651	(* *)
7 b409ecf47f64 cleaned up the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 6 diff changeset	1652	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	1653	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	1654	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	1655	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	1656	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1657	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	1658	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	1659	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	1660	apply(erule Prf.cases)
b409ecf47f64 cleaned up the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 6 diff changeset	1661	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	1662	thm mkeps_flat
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1663	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	1664	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	1665	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	1666	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	1667	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	1668	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	1669	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1670	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	1671	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	1672
5 fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1673
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1674
7 b409ecf47f64 cleaned up the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 6 diff changeset	1675	text {*
b409ecf47f64 cleaned up the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 6 diff changeset	1676	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	1677	*}
b409ecf47f64 cleaned up the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 6 diff changeset	1678
5 fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1679
82 26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1680
7 b409ecf47f64 cleaned up the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 6 diff changeset	1681	text {*
82 26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1682	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	1683	*}
5 fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	1684
82 26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1685
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1686	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	1687	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	1688	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	1689	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1690	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	1691	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	1692	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1693	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	1694	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	1695	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1696	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	1697	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	1698	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	1699	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	1700	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	1701	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	1702	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1703	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	1704	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	1705	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	1706	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	1707	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	1708	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	1709	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1710	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	1711	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	1712	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	1713	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	1714	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1715	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	1716	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	1717	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	1718	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	1719	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1720	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	1721	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	1722	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1723	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	1724	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	1725	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	1726	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1727	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	1728	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	1729	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	1730	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	1731	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	1732	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	1733	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	1734	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	1735	done
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1736
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1737	lemma Values_nullable:
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1738	assumes "nullable r1"
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1739	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	1740	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1741	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	1742	apply(simp_all)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1743	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	1744	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	1745	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	1746	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1747	done
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1748
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1749	lemma Values_injval:
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1750	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	1751	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	1752	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1753	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	1754	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	1755	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	1756	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	1757	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1758	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	1759	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	1760	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1761	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	1762	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1763	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	1764	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1765	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	1766	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1767	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	1768	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1769	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	1770	apply(subst v4)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1771	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	1772	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	1773	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	1774	apply(assumption)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1775	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	1776	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	1777	apply(assumption)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1778	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1779	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1780	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	1781	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1782	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	1783	apply(subst v4)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1784	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	1785	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	1786	done
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1787
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1788	lemma Values_projval:
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1789	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	1790	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	1791	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1792	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	1793	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	1794	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	1795	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	1796	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1797	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	1798	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1799	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	1800	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	1801	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	1802	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1803	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	1804	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1805	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	1806	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	1807	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	1808	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	1809	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1810	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	1811	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	1812	apply(assumption)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1813	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1814	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1815	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	1816	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1817	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	1818	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	1819	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1820	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	1821	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1822	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	1823	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	1824	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	1825	apply(assumption)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1826	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1827	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	1828	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1829	done
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1830
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1831
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1832	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	1833
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1834	lemma MValue_ALTE:
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1835	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	1836	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	1837	(\<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	1838	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1839	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	1840	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	1841	apply(auto)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1842	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	1843	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1844	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	1845	apply(simp_all)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1846	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	1847	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1848	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	1849	apply(simp_all)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1850	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	1851	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1852	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	1853	apply(simp_all)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1854	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	1855	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1856	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	1857	apply(simp_all)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1858	done
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1859
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1860	lemma MValue_ALTI1:
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1861	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	1862	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	1863	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1864	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	1865	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	1866	apply(auto)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1867	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	1868	apply metis
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1869	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	1870	apply metis
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1871	done
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1872
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1873	lemma MValue_ALTI2:
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1874	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	1875	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	1876	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1877	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	1878	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	1879	apply(auto)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1880	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	1881	apply metis
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1882	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	1883	apply metis
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1884	done
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1885
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1886	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	1887	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	1888
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1889	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	1890	by (metis)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1891
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1892	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	1893	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	1894
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1895
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1896	lemma LeftRight:
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1897	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	1898	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	1899	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	1900	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1901	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1902	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	1903	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	1904	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	1905	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1906	apply(subst v4)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1907	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1908	apply(subst v4)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1909	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1910	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1911	done
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1912
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1913	lemma RightLeft:
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1914	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	1915	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	1916	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	1917	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1918	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1919	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	1920	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	1921	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	1922	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1923	apply(subst v4)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1924	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1925	apply(subst v4)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1926	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1927	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1928	done
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1929
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1930	lemma h:
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1931	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	1932	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	1933	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1934	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	1935	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1936	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1937	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	1938	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	1939	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1940	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1941	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	1942	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	1943	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1944	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1945	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	1946	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	1947	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	1948	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1949	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	1950	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	1951	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	1952	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1953	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	1954	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	1955	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1956	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	1957	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	1958	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1959	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	1960	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1961	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	1962
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1963	lemma LeftRightSeq:
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1964	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	1965	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	1966	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	1967	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1968	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1969	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	1970	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	1971	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1972	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1973	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	1974	prefer 2
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1975	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	1976	by (metis h)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1977
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1978	lemma rr1:
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1979	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	1980	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	1981	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1982	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	1983
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1984	(* HERE *)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1985
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1986	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	1987	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	1988	"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	1989	"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	1990	"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	1991	"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	1992	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	1993	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1994	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	1995	(* NULL case *)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1996	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	1997	(* EMPTY case *)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	1998	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	1999	(* CHAR case *)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2000	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	2001	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2002	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	2003	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	2004	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	2005	(* ALT case *)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2006	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2007	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	2008	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2009	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	2010	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	2011	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	2012	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	2013	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	2014	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	2015	apply(subst v4)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2016	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	2017	apply(subst v4)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2018	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	2019	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2020	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	2021	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	2022	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	2023	apply(subst v4)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2024	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	2025	apply(subst v4)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2026	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	2027	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2028	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	2029	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	2030	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	2031	(* SEQ case*)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2032	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2033	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	2034	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2035	defer
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2036	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2037	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	2038	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2039	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	2040	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	2041	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2042	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	2043	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2044	apply (metis Ord1)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2045	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2046	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	2047	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	2048	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2049	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	2050	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2051	oops
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2052
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2053	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	2054	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	2055	"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	2056	"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	2057	"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	2058	"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	2059	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	2060	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2061	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	2062	(* NULL case *)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2063	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	2064	(* EMPTY case *)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2065	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	2066	(* CHAR case *)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2067	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	2068	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2069	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	2070	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	2071	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	2072	(* ALT case *)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2073	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2074	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	2075	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2076	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	2077	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	2078	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	2079	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	2080	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	2081	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	2082	apply(subst v4)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2083	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	2084	apply(subst v4)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2085	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	2086	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2087	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	2088	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	2089	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	2090	apply(subst v4)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2091	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	2092	apply(subst v4)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2093	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	2094	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2095	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	2096	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	2097	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	2098	(* SEQ case*)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2099	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2100	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	2101	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2102	defer
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2103	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2104	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	2105	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2106	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	2107	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	2108	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2109	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	2110	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2111	apply (metis Ord1)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2112	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2113	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	2114	apply metis
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2115	using injval_inj
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2116	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	2117	apply metis
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2118	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	2119	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2120	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	2121	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	2122	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2123	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	2124	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	2125	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2126	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	2127	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2128	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	2129	apply metis
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2130	using injval_inj
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2131	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	2132	apply metis
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2133	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	2134	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	2135	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	2136	thm h
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2137	apply(rule Ord1)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2138	apply(rule h)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2139	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2140	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	2141	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	2142	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	2143	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	2144	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	2145	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2146	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	2147	defer
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2148	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	2149	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	2150	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2151	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	2152	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	2153	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	2154	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	2155	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	2156	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	2157	oops
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2158
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2159	lemma POSIX_der:
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2160	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	2161	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	2162	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2163	unfolding POSIX_def
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2164	apply(auto)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2165	thm v3
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2166	apply (erule v3)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2167	thm v4
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2168	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	2169	apply(assumption)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2170	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	2171	apply(drule mp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2172	apply(rule conjI)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2173	thm v3_proj
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2174	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	2175	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2176	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	2177	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2178	thm t
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2179	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	2180	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2181	thm v4_proj
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2182	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	2183	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2184	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	2185	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2186	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2187	oops
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2188
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2189	lemma POSIX_der:
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2190	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	2191	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	2192	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2193	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	2194	(* null case*)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2195	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	2196	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2197	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	2198	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	2199	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	2200	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	2201	(* empty case *)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2202	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	2203	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2204	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	2205	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	2206	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	2207	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	2208	(* char case *)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2209	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	2210	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	2211	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2212	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	2213	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	2214	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	2215	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	2216	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	2217	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	2218	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	2219	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	2220	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2221	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	2222	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	2223	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	2224	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	2225	(* alt case *)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2226	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	2227	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	2228	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2229	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	2230	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2231	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	2232	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	2233	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	2234	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	2235	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	2236	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	2237	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	2238	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2239	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	2240	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	2241	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	2242	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	2243	defer
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2244	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	2245	prefer 2
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2246	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	2247	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2248	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	2249	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	2250	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	2251	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	2252	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	2253	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2254	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	2255	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2256	apply(drule mp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2257	apply(rule conjI)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2258	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	2259	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2260	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	2261	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	2262	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	2263	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	2264	(* seq case *)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2265	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	2266	defer
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2267	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	2268	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2269	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	2270	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	2271	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	2272	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	2273	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	2274	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	2275	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	2276	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2277	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	2278	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2279	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2280	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	2281	prefer 2
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2282	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	2283	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	2284	prefer 2
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2285	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	2286	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2287	oops
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2288
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2289
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2290	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	2291	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	2292	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	2293	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	2294	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	2295	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	2296	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	2297	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	2298	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2299	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	2300	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	2301	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	2302	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	2303	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	2304	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	2305	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	2306	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	2307	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2308	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	2309	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	2310	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	2311	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	2312	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	2313	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	2314	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2315	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	2316	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	2317	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2318	defer
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2319	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	2320	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	2321	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2322	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	2323	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	2324	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	2325	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	2326	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	2327	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	2328	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	2329	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	2330	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	2331	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	2332	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	2333	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2334	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	2335	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	2336	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	2337	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	2338	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	2339	oops
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2340
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2341	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	2342	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	2343	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	2344	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	2345	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	2346	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	2347	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	2348	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	2349	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2350	oops
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2351
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2352	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	2353	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	2354	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	2355	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2356	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	2357	apply(simp_all)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2358	unfolding POSIX_def
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2359	apply(auto)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2360	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	2361	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	2362
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2363	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	2364	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	2365	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	2366	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	2367
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2368	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	2369	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	2370	shows "nullable r"
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2371	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2372	apply(induct)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2373	apply(auto)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2374	done
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2375
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2376	lemma POSIX_proj:
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2377	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	2378	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	2379	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2380	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	2381	defer
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2382	defer
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2383	defer
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2384	defer
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2385	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	2386	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	2387	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	2388	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	2389	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	2390	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	2391	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	2392	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	2393	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	2394	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	2395	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	2396	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	2397	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	2398	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	2399	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	2400	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	2401	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	2402	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	2403	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	2404	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	2405	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	2406	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2407	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	2408	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	2409	oops
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2410
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2411	lemma POSIX_proj:
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2412	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	2413	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	2414	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2415	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	2416	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	2417	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	2418	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	2419	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	2420	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	2421	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	2422	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	2423	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2424	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	2425	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	2426	oops
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2427
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2428	lemma POSIX_proj:
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2429	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	2430	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	2431	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2432	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	2433	defer
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2434	defer
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2435	defer
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2436	defer
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2437	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	2438	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	2439	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	2440	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	2441	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	2442	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	2443	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	2444	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	2445	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	2446	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	2447	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	2448	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	2449	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	2450	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	2451	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	2452	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	2453	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	2454	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	2455	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	2456	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	2457	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	2458	apply(auto)[1]
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2459	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	2460	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	2461	oops
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2462
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2463	lemma Prf_inj:
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2464	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	2465	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	2466	using assms
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2467	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	2468	(* NULL case *)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2469	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2470	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	2471	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	2472	(* EMPTY case *)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2473	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	2474	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	2475	(* CHAR case *)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2476	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	2477	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2478	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	2479	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	2480	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	2481	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2482	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	2483	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	2484	(* ALT case *)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2485	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2486	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	2487	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	2488	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	2489	apply(subst v4)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2490	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2491	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	2492	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	2493	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	2494	apply(subst v4)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2495	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2496	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	2497	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	2498	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	2499	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2500	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	2501	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2502	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	2503	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	2504	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	2505	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2506	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	2507	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	2508	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	2509	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2510	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	2511	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	2512	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	2513	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	2514	(* SEQ case*)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2515	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2516	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	2517	defer
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2518	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2519	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	2520	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	2521	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2522	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	2523	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	2524	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	2525	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	2526	apply(clarify)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2527	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	2528	apply(simp)
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2529	oops
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2530
26202889f829 cleaned up version of Re1 Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 10 diff changeset	2531
7 b409ecf47f64 cleaned up the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 6 diff changeset	2532	text {*
b409ecf47f64 cleaned up the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 6 diff changeset	2533	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	2534	*}
5 fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2535
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2536	lemma proj_inj_id:
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2537	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	2538	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	2539	using assms
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2540	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	2541	apply(simp)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2542	apply(erule Prf.cases)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2543	apply(simp_all)[5]
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2544	apply(simp)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2545	apply(erule Prf.cases)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2546	apply(simp_all)[5]
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2547	apply(simp)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2548	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	2549	apply(simp)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2550	apply(erule Prf.cases)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2551	apply(simp_all)[5]
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2552	apply(simp)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2553	apply(erule Prf.cases)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2554	apply(simp_all)[5]
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2555	defer
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2556	apply(simp)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2557	apply(erule Prf.cases)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2558	apply(simp_all)[5]
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2559	apply(simp)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2560	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	2561	apply(simp)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2562	apply(erule Prf.cases)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2563	apply(simp_all)[5]
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2564	apply(auto)[1]
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2565	apply(erule Prf.cases)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2566	apply(simp_all)[5]
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2567	apply(auto)[1]
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2568	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	2569	apply(auto)[1]
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2570	apply (metis mkeps_flat)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2571	apply(auto)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2572	apply(erule Prf.cases)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2573	apply(simp_all)[5]
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2574	apply(auto)[1]
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2575	apply(simp add: v4)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2576	done
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2577
7 b409ecf47f64 cleaned up the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 6 diff changeset	2578	text {*
5 fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2579
7 b409ecf47f64 cleaned up the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 6 diff changeset	2580	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	2581
b409ecf47f64 cleaned up the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: 6 diff changeset	2582	*}
5 fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2583	lemma v5:
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2584	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	2585	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	2586	using assms
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2587	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	2588	(* NULL case *)
5 fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2589	apply(simp)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2590	apply(erule Prf.cases)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2591	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	2592	(* EMPTY case *)
5 fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2593	apply(simp)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2594	apply(erule Prf.cases)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2595	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	2596	(* CHAR case *)
5 fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2597	apply(simp)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2598	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	2599	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	2600	apply(erule Prf.cases)
fe177dfc4697 initial version of the theory Christian Urban <christian dot urban at kcl dot ac dot uk> parents: diff changeset	2601	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	2602	oops

author	Christian Urban <christian dot urban at kcl dot ac dot uk>
	Thu, 22 Sep 2016 00:40:03 +0100
changeset 211	0fa636821349
parent 88	532bb9df225d
child 212	9fd41f224e8d
permissions	-rw-r--r--