author | wenzelm |
Sat, 07 Apr 2012 16:41:59 +0200 | |
changeset 47389 | e8552cba702d |
parent 47308 | 9caab698dbe4 |
child 47455 | 26315a545e26 |
permissions | -rw-r--r-- |
47308 | 1 |
(* Title: HOL/Library/Quotient3_Sum.thy |
35222
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
2 |
Author: Cezary Kaliszyk and Christian Urban |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
3 |
*) |
35788 | 4 |
|
5 |
header {* Quotient infrastructure for the sum type *} |
|
6 |
||
35222
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
7 |
theory Quotient_Sum |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
8 |
imports Main Quotient_Syntax |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
9 |
begin |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
10 |
|
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
11 |
fun |
40542
9a173a22771c
re-generalized type of option_rel and sum_rel (accident from 2989f9f3aa10)
haftmann
parents:
40465
diff
changeset
|
12 |
sum_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> 'c + 'd \<Rightarrow> bool" |
35222
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
13 |
where |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
14 |
"sum_rel R1 R2 (Inl a1) (Inl b1) = R1 a1 b1" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
15 |
| "sum_rel R1 R2 (Inl a1) (Inr b2) = False" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
16 |
| "sum_rel R1 R2 (Inr a2) (Inl b1) = False" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
17 |
| "sum_rel R1 R2 (Inr a2) (Inr b2) = R2 a2 b2" |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
18 |
|
40820
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
19 |
lemma sum_rel_unfold: |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
20 |
"sum_rel R1 R2 x y = (case (x, y) of (Inl x, Inl y) \<Rightarrow> R1 x y |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
21 |
| (Inr x, Inr y) \<Rightarrow> R2 x y |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
22 |
| _ \<Rightarrow> False)" |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
23 |
by (cases x) (cases y, simp_all)+ |
35222
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
24 |
|
40820
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
25 |
lemma sum_rel_map1: |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
26 |
"sum_rel R1 R2 (sum_map f1 f2 x) y \<longleftrightarrow> sum_rel (\<lambda>x. R1 (f1 x)) (\<lambda>x. R2 (f2 x)) x y" |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
27 |
by (simp add: sum_rel_unfold split: sum.split) |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
28 |
|
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
29 |
lemma sum_rel_map2: |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
30 |
"sum_rel R1 R2 x (sum_map f1 f2 y) \<longleftrightarrow> sum_rel (\<lambda>x y. R1 x (f1 y)) (\<lambda>x y. R2 x (f2 y)) x y" |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
31 |
by (simp add: sum_rel_unfold split: sum.split) |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
32 |
|
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
33 |
lemma sum_map_id [id_simps]: |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
34 |
"sum_map id id = id" |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
35 |
by (simp add: id_def sum_map.identity fun_eq_iff) |
35222
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
36 |
|
40820
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
37 |
lemma sum_rel_eq [id_simps]: |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
38 |
"sum_rel (op =) (op =) = (op =)" |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
39 |
by (simp add: sum_rel_unfold fun_eq_iff split: sum.split) |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
40 |
|
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
41 |
lemma sum_reflp: |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
42 |
"reflp R1 \<Longrightarrow> reflp R2 \<Longrightarrow> reflp (sum_rel R1 R2)" |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
43 |
by (auto simp add: sum_rel_unfold split: sum.splits intro!: reflpI elim: reflpE) |
35222
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
44 |
|
40820
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
45 |
lemma sum_symp: |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
46 |
"symp R1 \<Longrightarrow> symp R2 \<Longrightarrow> symp (sum_rel R1 R2)" |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
47 |
by (auto simp add: sum_rel_unfold split: sum.splits intro!: sympI elim: sympE) |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
48 |
|
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
49 |
lemma sum_transp: |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
50 |
"transp R1 \<Longrightarrow> transp R2 \<Longrightarrow> transp (sum_rel R1 R2)" |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
51 |
by (auto simp add: sum_rel_unfold split: sum.splits intro!: transpI elim: transpE) |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
52 |
|
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
53 |
lemma sum_equivp [quot_equiv]: |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
54 |
"equivp R1 \<Longrightarrow> equivp R2 \<Longrightarrow> equivp (sum_rel R1 R2)" |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
55 |
by (blast intro: equivpI sum_reflp sum_symp sum_transp elim: equivpE) |
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
56 |
|
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
57 |
lemma sum_quotient [quot_thm]: |
47308 | 58 |
assumes q1: "Quotient3 R1 Abs1 Rep1" |
59 |
assumes q2: "Quotient3 R2 Abs2 Rep2" |
|
60 |
shows "Quotient3 (sum_rel R1 R2) (sum_map Abs1 Abs2) (sum_map Rep1 Rep2)" |
|
61 |
apply (rule Quotient3I) |
|
41372 | 62 |
apply (simp_all add: sum_map.compositionality comp_def sum_map.identity sum_rel_eq sum_rel_map1 sum_rel_map2 |
47308 | 63 |
Quotient3_abs_rep [OF q1] Quotient3_rel_rep [OF q1] Quotient3_abs_rep [OF q2] Quotient3_rel_rep [OF q2]) |
64 |
using Quotient3_rel [OF q1] Quotient3_rel [OF q2] |
|
41372 | 65 |
apply (simp add: sum_rel_unfold comp_def split: sum.split) |
35222
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
66 |
done |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
67 |
|
47308 | 68 |
declare [[mapQ3 sum = (sum_rel, sum_quotient)]] |
47094 | 69 |
|
40820
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
70 |
lemma sum_Inl_rsp [quot_respect]: |
47308 | 71 |
assumes q1: "Quotient3 R1 Abs1 Rep1" |
72 |
assumes q2: "Quotient3 R2 Abs2 Rep2" |
|
35222
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
73 |
shows "(R1 ===> sum_rel R1 R2) Inl Inl" |
40465
2989f9f3aa10
more appropriate specification packages; fun_rel_def is no simp rule by default
haftmann
parents:
39302
diff
changeset
|
74 |
by auto |
35222
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
75 |
|
40820
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
76 |
lemma sum_Inr_rsp [quot_respect]: |
47308 | 77 |
assumes q1: "Quotient3 R1 Abs1 Rep1" |
78 |
assumes q2: "Quotient3 R2 Abs2 Rep2" |
|
35222
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
79 |
shows "(R2 ===> sum_rel R1 R2) Inr Inr" |
40465
2989f9f3aa10
more appropriate specification packages; fun_rel_def is no simp rule by default
haftmann
parents:
39302
diff
changeset
|
80 |
by auto |
35222
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
81 |
|
40820
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
82 |
lemma sum_Inl_prs [quot_preserve]: |
47308 | 83 |
assumes q1: "Quotient3 R1 Abs1 Rep1" |
84 |
assumes q2: "Quotient3 R2 Abs2 Rep2" |
|
35222
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
85 |
shows "(Rep1 ---> sum_map Abs1 Abs2) Inl = Inl" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
86 |
apply(simp add: fun_eq_iff) |
47308 | 87 |
apply(simp add: Quotient3_abs_rep[OF q1]) |
35222
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
88 |
done |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
89 |
|
40820
fd9c98ead9a9
more systematic and compact proofs on type relation operators using natural deduction rules
haftmann
parents:
40610
diff
changeset
|
90 |
lemma sum_Inr_prs [quot_preserve]: |
47308 | 91 |
assumes q1: "Quotient3 R1 Abs1 Rep1" |
92 |
assumes q2: "Quotient3 R2 Abs2 Rep2" |
|
35222
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
93 |
shows "(Rep2 ---> sum_map Abs1 Abs2) Inr = Inr" |
39302
d7728f65b353
renamed lemmas: ext_iff -> fun_eq_iff, set_ext_iff -> set_eq_iff, set_ext -> set_eqI
nipkow
parents:
39198
diff
changeset
|
94 |
apply(simp add: fun_eq_iff) |
47308 | 95 |
apply(simp add: Quotient3_abs_rep[OF q2]) |
35222
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
96 |
done |
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
97 |
|
4f1fba00f66d
Initial version of HOL quotient package.
Cezary Kaliszyk <kaliszyk@in.tum.de>
parents:
diff
changeset
|
98 |
end |