(*  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

 begin

 subsection {* Relator and predicator properties *}

 definition

    sum_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> 'c + 'd \<Rightarrow> bool"

 where

    "sum_rel R1 R2 x y =

      (case (x, y) of (Inl x, Inl y) \<Rightarrow> R1 x y

      | (Inr x, Inr y) \<Rightarrow> R2 x y

      | _ \<Rightarrow> False)"

 lemma sum_rel_simps[simp]:

   "sum_rel R1 R2 (Inl a1) (Inl b1) = R1 a1 b1"

   "sum_rel R1 R2 (Inl a1) (Inr b2) = False"

   "sum_rel R1 R2 (Inr a2) (Inl b1) = False"

   "sum_rel R1 R2 (Inr a2) (Inr b2) = R2 a2 b2"

   unfolding sum_rel_def by simp_all

 abbreviation (input) "sum_pred \<equiv> sum_case"

 lemma sum_rel_eq [relator_eq]:

   "sum_rel (op =) (op =) = (op =)"

   by (simp add: sum_rel_def fun_eq_iff split: sum.split)

 lemma sum_rel_mono[relator_mono]:

   assumes "A \<le> C"

   assumes "B \<le> D"

   shows "(sum_rel A B) \<le> (sum_rel C D)"

 using assms by (auto simp: sum_rel_def split: sum.splits)

 lemma sum_rel_OO[relator_distr]:

   "(sum_rel A B) OO (sum_rel C D) = sum_rel (A OO C) (B OO D)"

 by (rule ext)+ (auto simp add: sum_rel_def OO_def split_sum_ex split: sum.split)

 lemma Domainp_sum[relator_domain]:

   assumes "Domainp R1 = P1"

   assumes "Domainp R2 = P2"

   shows "Domainp (sum_rel 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 reflp_sum_rel[reflexivity_rule]:

   "reflp R1 \<Longrightarrow> reflp R2 \<Longrightarrow> reflp (sum_rel R1 R2)"

   unfolding reflp_def split_sum_all sum_rel_simps by fast

 lemma left_total_sum_rel[reflexivity_rule]:

   "left_total R1 \<Longrightarrow> left_total R2 \<Longrightarrow> left_total (sum_rel R1 R2)"

   using assms unfolding left_total_def split_sum_all split_sum_ex by simp

 lemma left_unique_sum_rel [reflexivity_rule]:

   "left_unique R1 \<Longrightarrow> left_unique R2 \<Longrightarrow> left_unique (sum_rel R1 R2)"

   using assms unfolding left_unique_def split_sum_all by simp

 lemma right_total_sum_rel [transfer_rule]:

   "right_total R1 \<Longrightarrow> right_total R2 \<Longrightarrow> right_total (sum_rel R1 R2)"

   unfolding right_total_def split_sum_all split_sum_ex by simp

 lemma right_unique_sum_rel [transfer_rule]:

   "right_unique R1 \<Longrightarrow> right_unique R2 \<Longrightarrow> right_unique (sum_rel R1 R2)"

   unfolding right_unique_def split_sum_all by simp

 lemma bi_total_sum_rel [transfer_rule]:

   "bi_total R1 \<Longrightarrow> bi_total R2 \<Longrightarrow> bi_total (sum_rel R1 R2)"

   using assms unfolding bi_total_def split_sum_all split_sum_ex by simp

 lemma bi_unique_sum_rel [transfer_rule]:

   "bi_unique R1 \<Longrightarrow> bi_unique R2 \<Longrightarrow> bi_unique (sum_rel R1 R2)"

   using assms unfolding bi_unique_def split_sum_all by simp

 lemma sum_invariant_commute [invariant_commute]: 

   "sum_rel (Lifting.invariant P1) (Lifting.invariant P2) = Lifting.invariant (sum_pred P1 P2)"

   by (auto simp add: fun_eq_iff Lifting.invariant_def sum_rel_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 (sum_rel R1 R2) (sum_map Abs1 Abs2)

     (sum_map Rep1 Rep2) (sum_rel 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 ===> sum_rel A B) Inl Inl"

   unfolding fun_rel_def by simp

 lemma Inr_transfer [transfer_rule]: "(B ===> sum_rel A B) Inr Inr"

   unfolding fun_rel_def by simp

 lemma sum_case_transfer [transfer_rule]:

   "((A ===> C) ===> (B ===> C) ===> sum_rel A B ===> C) sum_case sum_case"

   unfolding fun_rel_def sum_rel_def by (simp split: sum.split)

 end

 end