| author | Cezary Kaliszyk <kaliszyk@in.tum.de> |
| Fri, 08 Jan 2010 15:02:12 +0100 | |
| changeset 829 | 42b90994ac77 |
| parent 779 | 3b21b24a5fb6 |
| child 924 | 5455b19ef138 |
| permissions | -rw-r--r-- |
|
698
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
1 |
theory QuotSum |
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ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
2 |
imports QuotMain |
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begin |
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||
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fun |
|
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545
95371a8b17e1
Cleaning the Quotients file
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
538
diff
changeset
|
6 |
sum_rel |
| 0 | 7 |
where |
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545
95371a8b17e1
Cleaning the Quotients file
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
538
diff
changeset
|
8 |
"sum_rel R1 R2 (Inl a1) (Inl b1) = R1 a1 b1" |
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95371a8b17e1
Cleaning the Quotients file
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
538
diff
changeset
|
9 |
| "sum_rel R1 R2 (Inl a1) (Inr b2) = False" |
|
95371a8b17e1
Cleaning the Quotients file
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
538
diff
changeset
|
10 |
| "sum_rel R1 R2 (Inr a2) (Inl b1) = False" |
|
95371a8b17e1
Cleaning the Quotients file
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
538
diff
changeset
|
11 |
| "sum_rel R1 R2 (Inr a2) (Inr b2) = R2 a2 b2" |
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|
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fun |
|
14 |
sum_map |
|
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where |
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"sum_map f1 f2 (Inl a) = Inl (f1 a)" |
|
17 |
| "sum_map f1 f2 (Inr a) = Inr (f2 a)" |
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18 |
||
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779
3b21b24a5fb6
corrected map declarations for Sum and Prod; moved absrep_fun examples in separate file
Christian Urban <urbanc@in.tum.de>
parents:
698
diff
changeset
|
19 |
declare [[map "+" = (sum_map, sum_rel)]] |
|
3b21b24a5fb6
corrected map declarations for Sum and Prod; moved absrep_fun examples in separate file
Christian Urban <urbanc@in.tum.de>
parents:
698
diff
changeset
|
20 |
|
|
545
95371a8b17e1
Cleaning the Quotients file
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
538
diff
changeset
|
21 |
|
|
698
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
22 |
lemma sum_equivp[quot_equiv]: |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
23 |
assumes a: "equivp R1" |
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ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
24 |
assumes b: "equivp R2" |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
25 |
shows "equivp (sum_rel R1 R2)" |
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ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
26 |
unfolding equivp_reflp_symp_transp reflp_def symp_def transp_def |
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ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
27 |
apply(auto) |
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ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
28 |
apply(case_tac x) |
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ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
29 |
apply(auto simp add: equivp_reflp[OF a] equivp_reflp[OF b]) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
30 |
apply(case_tac x) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
31 |
apply(case_tac y) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
32 |
prefer 3 |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
33 |
apply(case_tac y) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
34 |
apply(auto simp add: equivp_symp[OF a] equivp_symp[OF b]) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
35 |
apply(case_tac x) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
36 |
apply(case_tac y) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
37 |
apply(case_tac z) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
38 |
prefer 3 |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
39 |
apply(case_tac z) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
40 |
prefer 5 |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
41 |
apply(case_tac y) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
42 |
apply(case_tac z) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
43 |
prefer 3 |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
44 |
apply(case_tac z) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
45 |
apply(auto) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
46 |
apply(metis equivp_transp[OF b]) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
47 |
apply(metis equivp_transp[OF a]) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
48 |
done |
| 0 | 49 |
|
|
698
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
50 |
lemma sum_fun_fun: |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
51 |
assumes q1: "Quotient R1 Abs1 Rep1" |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
52 |
assumes q2: "Quotient R2 Abs2 Rep2" |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
53 |
shows "sum_rel R1 R2 r s = |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
54 |
(sum_rel R1 R2 r r \<and> sum_rel R1 R2 s s \<and> sum_map Abs1 Abs2 r = sum_map Abs1 Abs2 s)" |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
55 |
using q1 q2 |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
56 |
apply(case_tac r) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
57 |
apply(case_tac s) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
58 |
apply(simp_all) |
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ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
59 |
prefer 2 |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
60 |
apply(case_tac s) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
61 |
apply(auto) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
62 |
unfolding Quotient_def |
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ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
63 |
apply metis+ |
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ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
64 |
done |
| 0 | 65 |
|
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698
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
66 |
lemma sum_quotient[quot_thm]: |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
67 |
assumes q1: "Quotient R1 Abs1 Rep1" |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
68 |
assumes q2: "Quotient R2 Abs2 Rep2" |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
69 |
shows "Quotient (sum_rel R1 R2) (sum_map Abs1 Abs2) (sum_map Rep1 Rep2)" |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
70 |
unfolding Quotient_def |
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ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
71 |
apply(rule conjI) |
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ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
72 |
apply(rule allI) |
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ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
73 |
apply(case_tac a) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
74 |
apply(simp add: Quotient_abs_rep[OF q1]) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
75 |
apply(simp add: Quotient_abs_rep[OF q2]) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
76 |
apply(rule conjI) |
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ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
77 |
apply(rule allI) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
78 |
apply(case_tac a) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
79 |
apply(simp add: Quotient_rel_rep[OF q1]) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
80 |
apply(simp add: Quotient_rel_rep[OF q2]) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
81 |
apply(rule allI)+ |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
82 |
apply(rule sum_fun_fun[OF q1 q2]) |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
83 |
done |
|
ed44eaaea63e
Option and Sum quotients.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
613
diff
changeset
|
84 |
|
|
829
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
779
diff
changeset
|
85 |
lemma sum_map_id[id_simps]: "sum_map id id \<equiv> id" |
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42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
779
diff
changeset
|
86 |
apply (rule eq_reflection) |
|
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
779
diff
changeset
|
87 |
apply (rule ext) |
|
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
779
diff
changeset
|
88 |
apply (case_tac x) |
|
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
779
diff
changeset
|
89 |
apply (auto) |
|
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
779
diff
changeset
|
90 |
done |
|
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
779
diff
changeset
|
91 |
|
|
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
779
diff
changeset
|
92 |
lemma sum_rel_eq[id_simps]: "sum_rel op = op = \<equiv> op =" |
|
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
779
diff
changeset
|
93 |
apply (rule eq_reflection) |
|
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
779
diff
changeset
|
94 |
apply (rule ext)+ |
|
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
779
diff
changeset
|
95 |
apply (case_tac x) |
|
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
779
diff
changeset
|
96 |
apply auto |
|
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
779
diff
changeset
|
97 |
apply (case_tac xa) |
|
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
779
diff
changeset
|
98 |
apply auto |
|
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
779
diff
changeset
|
99 |
apply (case_tac xa) |
|
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
779
diff
changeset
|
100 |
apply auto |
|
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
779
diff
changeset
|
101 |
done |
|
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
779
diff
changeset
|
102 |
|
|
42b90994ac77
map and rel simps for all quotients; needed when changing the relations to aggregate ones.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
779
diff
changeset
|
103 |
end |