author | kuncar |
Tue, 13 Aug 2013 15:59:22 +0200 | |
changeset 53012 | cb82606b8215 |
child 53026 | e1a548c11845 |
permissions | -rw-r--r-- |
53012
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
1 |
(* Title: HOL/Lifting_Sum.thy |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
2 |
Author: Brian Huffman and Ondrej Kuncar |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
3 |
*) |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
4 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
5 |
header {* Setup for Lifting/Transfer for the sum type *} |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
6 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
7 |
theory Lifting_Sum |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
8 |
imports Lifting FunDef |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
9 |
begin |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
10 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
11 |
subsection {* Relator and predicator properties *} |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
12 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
13 |
fun |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
14 |
sum_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> 'c + 'd \<Rightarrow> bool" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
15 |
where |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
16 |
"sum_rel R1 R2 (Inl a1) (Inl b1) = R1 a1 b1" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
17 |
| "sum_rel R1 R2 (Inl a1) (Inr b2) = False" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
18 |
| "sum_rel R1 R2 (Inr a2) (Inl b1) = False" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
19 |
| "sum_rel R1 R2 (Inr a2) (Inr b2) = R2 a2 b2" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
20 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
21 |
lemma sum_rel_unfold: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
22 |
"sum_rel R1 R2 x y = (case (x, y) of (Inl x, Inl y) \<Rightarrow> R1 x y |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
23 |
| (Inr x, Inr y) \<Rightarrow> R2 x y |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
24 |
| _ \<Rightarrow> False)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
25 |
by (cases x) (cases y, simp_all)+ |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
26 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
27 |
fun sum_pred :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> bool" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
28 |
where |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
29 |
"sum_pred P1 P2 (Inl a) = P1 a" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
30 |
| "sum_pred P1 P2 (Inr a) = P2 a" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
31 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
32 |
lemma sum_pred_unfold: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
33 |
"sum_pred P1 P2 x = (case x of Inl x \<Rightarrow> P1 x |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
34 |
| Inr x \<Rightarrow> P2 x)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
35 |
by (cases x) simp_all |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
36 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
37 |
lemma sum_rel_eq [relator_eq]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
38 |
"sum_rel (op =) (op =) = (op =)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
39 |
by (simp add: sum_rel_unfold fun_eq_iff split: sum.split) |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
40 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
41 |
lemma sum_rel_mono[relator_mono]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
42 |
assumes "A \<le> C" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
43 |
assumes "B \<le> D" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
44 |
shows "(sum_rel A B) \<le> (sum_rel C D)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
45 |
using assms by (auto simp: sum_rel_unfold split: sum.splits) |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
46 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
47 |
lemma sum_rel_OO[relator_distr]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
48 |
"(sum_rel A B) OO (sum_rel C D) = sum_rel (A OO C) (B OO D)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
49 |
by (rule ext)+ (auto simp add: sum_rel_unfold OO_def split_sum_ex split: sum.split) |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
50 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
51 |
lemma Domainp_sum[relator_domain]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
52 |
assumes "Domainp R1 = P1" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
53 |
assumes "Domainp R2 = P2" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
54 |
shows "Domainp (sum_rel R1 R2) = (sum_pred P1 P2)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
55 |
using assms |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
56 |
by (auto simp add: Domainp_iff split_sum_ex sum_pred_unfold iff: fun_eq_iff split: sum.split) |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
57 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
58 |
lemma reflp_sum_rel[reflexivity_rule]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
59 |
"reflp R1 \<Longrightarrow> reflp R2 \<Longrightarrow> reflp (sum_rel R1 R2)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
60 |
unfolding reflp_def split_sum_all sum_rel.simps by fast |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
61 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
62 |
lemma left_total_sum_rel[reflexivity_rule]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
63 |
"left_total R1 \<Longrightarrow> left_total R2 \<Longrightarrow> left_total (sum_rel R1 R2)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
64 |
using assms unfolding left_total_def split_sum_all split_sum_ex by simp |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
65 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
66 |
lemma left_unique_sum_rel [reflexivity_rule]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
67 |
"left_unique R1 \<Longrightarrow> left_unique R2 \<Longrightarrow> left_unique (sum_rel R1 R2)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
68 |
using assms unfolding left_unique_def split_sum_all by simp |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
69 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
70 |
lemma right_total_sum_rel [transfer_rule]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
71 |
"right_total R1 \<Longrightarrow> right_total R2 \<Longrightarrow> right_total (sum_rel R1 R2)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
72 |
unfolding right_total_def split_sum_all split_sum_ex by simp |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
73 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
74 |
lemma right_unique_sum_rel [transfer_rule]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
75 |
"right_unique R1 \<Longrightarrow> right_unique R2 \<Longrightarrow> right_unique (sum_rel R1 R2)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
76 |
unfolding right_unique_def split_sum_all by simp |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
77 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
78 |
lemma bi_total_sum_rel [transfer_rule]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
79 |
"bi_total R1 \<Longrightarrow> bi_total R2 \<Longrightarrow> bi_total (sum_rel R1 R2)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
80 |
using assms unfolding bi_total_def split_sum_all split_sum_ex by simp |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
81 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
82 |
lemma bi_unique_sum_rel [transfer_rule]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
83 |
"bi_unique R1 \<Longrightarrow> bi_unique R2 \<Longrightarrow> bi_unique (sum_rel R1 R2)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
84 |
using assms unfolding bi_unique_def split_sum_all by simp |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
85 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
86 |
lemma sum_invariant_commute [invariant_commute]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
87 |
"sum_rel (Lifting.invariant P1) (Lifting.invariant P2) = Lifting.invariant (sum_pred P1 P2)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
88 |
by (auto simp add: fun_eq_iff Lifting.invariant_def sum_rel_unfold sum_pred_unfold split: sum.split) |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
89 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
90 |
subsection {* Quotient theorem for the Lifting package *} |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
91 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
92 |
lemma Quotient_sum[quot_map]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
93 |
assumes "Quotient R1 Abs1 Rep1 T1" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
94 |
assumes "Quotient R2 Abs2 Rep2 T2" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
95 |
shows "Quotient (sum_rel R1 R2) (sum_map Abs1 Abs2) |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
96 |
(sum_map Rep1 Rep2) (sum_rel T1 T2)" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
97 |
using assms unfolding Quotient_alt_def |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
98 |
by (simp add: split_sum_all) |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
99 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
100 |
subsection {* Transfer rules for the Transfer package *} |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
101 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
102 |
context |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
103 |
begin |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
104 |
interpretation lifting_syntax . |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
105 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
106 |
lemma Inl_transfer [transfer_rule]: "(A ===> sum_rel A B) Inl Inl" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
107 |
unfolding fun_rel_def by simp |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
108 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
109 |
lemma Inr_transfer [transfer_rule]: "(B ===> sum_rel A B) Inr Inr" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
110 |
unfolding fun_rel_def by simp |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
111 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
112 |
lemma sum_case_transfer [transfer_rule]: |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
113 |
"((A ===> C) ===> (B ===> C) ===> sum_rel A B ===> C) sum_case sum_case" |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
114 |
unfolding fun_rel_def sum_rel_unfold by (simp split: sum.split) |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
115 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
116 |
end |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
117 |
|
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
118 |
end |
cb82606b8215
move Lifting/Transfer relevant parts of Library/Quotient_* to Main
kuncar
parents:
diff
changeset
|
119 |