(* Title: HOL/Lifting_Sum.thy
Author: Brian Huffman and Ondrej Kuncar
*)
header {* Setup for Lifting/Transfer for the sum type *}
theory Lifting_Sum
imports Lifting Basic_BNFs
begin
subsection {* Relator and predicator properties *}
abbreviation (input) "sum_pred \<equiv> case_sum"
lemmas rel_sum_eq[relator_eq] = sum.rel_eq
lemmas rel_sum_mono[relator_mono] = sum.rel_mono
lemma rel_sum_OO[relator_distr]:
"(rel_sum A B) OO (rel_sum C D) = rel_sum (A OO C) (B OO D)"
by (rule ext)+ (auto simp add: rel_sum_def OO_def split_sum_ex split: sum.split)
lemma Domainp_sum[relator_domain]:
assumes "Domainp R1 = P1"
assumes "Domainp R2 = P2"
shows "Domainp (rel_sum R1 R2) = (sum_pred P1 P2)"
using assms
by (auto simp add: Domainp_iff split_sum_ex iff: fun_eq_iff split: sum.split)
lemma left_total_rel_sum[reflexivity_rule]:
"left_total R1 \<Longrightarrow> left_total R2 \<Longrightarrow> left_total (rel_sum R1 R2)"
using assms unfolding left_total_def split_sum_all split_sum_ex by simp
lemma left_unique_rel_sum [reflexivity_rule]:
"left_unique R1 \<Longrightarrow> left_unique R2 \<Longrightarrow> left_unique (rel_sum R1 R2)"
using assms unfolding left_unique_def split_sum_all by simp
lemma right_total_rel_sum [transfer_rule]:
"right_total R1 \<Longrightarrow> right_total R2 \<Longrightarrow> right_total (rel_sum R1 R2)"
unfolding right_total_def split_sum_all split_sum_ex by simp
lemma right_unique_rel_sum [transfer_rule]:
"right_unique R1 \<Longrightarrow> right_unique R2 \<Longrightarrow> right_unique (rel_sum R1 R2)"
unfolding right_unique_def split_sum_all by simp
lemma bi_total_rel_sum [transfer_rule]:
"bi_total R1 \<Longrightarrow> bi_total R2 \<Longrightarrow> bi_total (rel_sum R1 R2)"
using assms unfolding bi_total_def split_sum_all split_sum_ex by simp
lemma bi_unique_rel_sum [transfer_rule]:
"bi_unique R1 \<Longrightarrow> bi_unique R2 \<Longrightarrow> bi_unique (rel_sum R1 R2)"
using assms unfolding bi_unique_def split_sum_all by simp
lemma sum_invariant_commute [invariant_commute]:
"rel_sum (Lifting.invariant P1) (Lifting.invariant P2) = Lifting.invariant (sum_pred P1 P2)"
by (auto simp add: fun_eq_iff Lifting.invariant_def rel_sum_def split: sum.split)
subsection {* Quotient theorem for the Lifting package *}
lemma Quotient_sum[quot_map]:
assumes "Quotient R1 Abs1 Rep1 T1"
assumes "Quotient R2 Abs2 Rep2 T2"
shows "Quotient (rel_sum R1 R2) (map_sum Abs1 Abs2) (map_sum Rep1 Rep2) (rel_sum T1 T2)"
using assms unfolding Quotient_alt_def
by (simp add: split_sum_all)
subsection {* Transfer rules for the Transfer package *}
context
begin
interpretation lifting_syntax .
lemma Inl_transfer [transfer_rule]: "(A ===> rel_sum A B) Inl Inl"
unfolding fun_rel_def by simp
lemma Inr_transfer [transfer_rule]: "(B ===> rel_sum A B) Inr Inr"
unfolding fun_rel_def by simp
lemma case_sum_transfer [transfer_rule]:
"((A ===> C) ===> (B ===> C) ===> rel_sum A B ===> C) case_sum case_sum"
unfolding fun_rel_def rel_sum_def by (simp split: sum.split)
end
end