1 (* Title: HOL/Lifting_Sum.thy
2 Author: Brian Huffman and Ondrej Kuncar
5 header {* Setup for Lifting/Transfer for the sum type *}
8 imports Lifting Basic_BNFs
11 subsection {* Relator and predicator properties *}
13 abbreviation (input) "sum_pred \<equiv> sum_case"
15 lemmas sum_rel_eq[relator_eq] = sum.rel_eq
16 lemmas sum_rel_mono[relator_mono] = sum.rel_mono
18 lemma sum_rel_OO[relator_distr]:
19 "(sum_rel A B) OO (sum_rel C D) = sum_rel (A OO C) (B OO D)"
20 by (rule ext)+ (auto simp add: sum_rel_def OO_def split_sum_ex split: sum.split)
22 lemma Domainp_sum[relator_domain]:
23 assumes "Domainp R1 = P1"
24 assumes "Domainp R2 = P2"
25 shows "Domainp (sum_rel R1 R2) = (sum_pred P1 P2)"
27 by (auto simp add: Domainp_iff split_sum_ex iff: fun_eq_iff split: sum.split)
29 lemma reflp_sum_rel[reflexivity_rule]:
30 "reflp R1 \<Longrightarrow> reflp R2 \<Longrightarrow> reflp (sum_rel R1 R2)"
31 unfolding reflp_def split_sum_all sum_rel_simps by fast
33 lemma left_total_sum_rel[reflexivity_rule]:
34 "left_total R1 \<Longrightarrow> left_total R2 \<Longrightarrow> left_total (sum_rel R1 R2)"
35 using assms unfolding left_total_def split_sum_all split_sum_ex by simp
37 lemma left_unique_sum_rel [reflexivity_rule]:
38 "left_unique R1 \<Longrightarrow> left_unique R2 \<Longrightarrow> left_unique (sum_rel R1 R2)"
39 using assms unfolding left_unique_def split_sum_all by simp
41 lemma right_total_sum_rel [transfer_rule]:
42 "right_total R1 \<Longrightarrow> right_total R2 \<Longrightarrow> right_total (sum_rel R1 R2)"
43 unfolding right_total_def split_sum_all split_sum_ex by simp
45 lemma right_unique_sum_rel [transfer_rule]:
46 "right_unique R1 \<Longrightarrow> right_unique R2 \<Longrightarrow> right_unique (sum_rel R1 R2)"
47 unfolding right_unique_def split_sum_all by simp
49 lemma bi_total_sum_rel [transfer_rule]:
50 "bi_total R1 \<Longrightarrow> bi_total R2 \<Longrightarrow> bi_total (sum_rel R1 R2)"
51 using assms unfolding bi_total_def split_sum_all split_sum_ex by simp
53 lemma bi_unique_sum_rel [transfer_rule]:
54 "bi_unique R1 \<Longrightarrow> bi_unique R2 \<Longrightarrow> bi_unique (sum_rel R1 R2)"
55 using assms unfolding bi_unique_def split_sum_all by simp
57 lemma sum_invariant_commute [invariant_commute]:
58 "sum_rel (Lifting.invariant P1) (Lifting.invariant P2) = Lifting.invariant (sum_pred P1 P2)"
59 by (auto simp add: fun_eq_iff Lifting.invariant_def sum_rel_def split: sum.split)
61 subsection {* Quotient theorem for the Lifting package *}
63 lemma Quotient_sum[quot_map]:
64 assumes "Quotient R1 Abs1 Rep1 T1"
65 assumes "Quotient R2 Abs2 Rep2 T2"
66 shows "Quotient (sum_rel R1 R2) (sum_map Abs1 Abs2)
67 (sum_map Rep1 Rep2) (sum_rel T1 T2)"
68 using assms unfolding Quotient_alt_def
69 by (simp add: split_sum_all)
71 subsection {* Transfer rules for the Transfer package *}
75 interpretation lifting_syntax .
77 lemma Inl_transfer [transfer_rule]: "(A ===> sum_rel A B) Inl Inl"
78 unfolding fun_rel_def by simp
80 lemma Inr_transfer [transfer_rule]: "(B ===> sum_rel A B) Inr Inr"
81 unfolding fun_rel_def by simp
83 lemma sum_case_transfer [transfer_rule]:
84 "((A ===> C) ===> (B ===> C) ===> sum_rel A B ===> C) sum_case sum_case"
85 unfolding fun_rel_def sum_rel_def by (simp split: sum.split)