src/HOL/Library/Quotient_Sum.thy
 author kuncar Mon, 13 May 2013 13:59:04 +0200 changeset 51956 a4d81cdebf8b parent 51377 7da251a6c16e child 51994 82cc2aeb7d13 permissions -rw-r--r--
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
```
(*  Title:      HOL/Library/Quotient_Sum.thy
Author:     Cezary Kaliszyk, Christian Urban and Brian Huffman
*)

header {* Quotient infrastructure for the sum type *}

theory Quotient_Sum
imports Main Quotient_Syntax
begin

subsection {* Relator for sum type *}

fun
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 (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"

lemma sum_rel_unfold:
"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)"
by (cases x) (cases y, simp_all)+

fun sum_pred :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> bool"
where
"sum_pred P1 P2 (Inl a) = P1 a"
| "sum_pred P1 P2 (Inr a) = P2 a"

lemma sum_pred_unfold:
"sum_pred P1 P2 x = (case x of Inl x \<Rightarrow> P1 x
| Inr x \<Rightarrow> P2 x)"
by (cases x) simp_all

lemma sum_rel_map1:
"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"
by (simp add: sum_rel_unfold split: sum.split)

lemma sum_rel_map2:
"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"
by (simp add: sum_rel_unfold split: sum.split)

lemma sum_map_id [id_simps]:
"sum_map id id = id"
by (simp add: id_def sum_map.identity fun_eq_iff)

lemma sum_rel_eq [id_simps, relator_eq]:
"sum_rel (op =) (op =) = (op =)"
by (simp add: sum_rel_unfold fun_eq_iff split: sum.split)

lemma split_sum_all: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (Inl x)) \<and> (\<forall>x. P (Inr x))"
by (metis sum.exhaust) (* TODO: move to Sum_Type.thy *)

lemma split_sum_ex: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P (Inl x)) \<or> (\<exists>x. P (Inr x))"
by (metis sum.exhaust) (* TODO: move to Sum_Type.thy *)

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_unfold 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_unfold 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 sum_pred_unfold iff: fun_eq_iff split: sum.split)

lemma sum_reflp[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 sum_left_total[reflexivity_rule]:
"left_total R1 \<Longrightarrow> left_total R2 \<Longrightarrow> left_total (sum_rel R1 R2)"
apply (intro left_totalI)
unfolding split_sum_ex
by (case_tac x) (auto elim: left_totalE)

lemma sum_symp:
"symp R1 \<Longrightarrow> symp R2 \<Longrightarrow> symp (sum_rel R1 R2)"
unfolding symp_def split_sum_all sum_rel.simps by fast

lemma sum_transp:
"transp R1 \<Longrightarrow> transp R2 \<Longrightarrow> transp (sum_rel R1 R2)"
unfolding transp_def split_sum_all sum_rel.simps by fast

lemma sum_equivp [quot_equiv]:
"equivp R1 \<Longrightarrow> equivp R2 \<Longrightarrow> equivp (sum_rel R1 R2)"
by (blast intro: equivpI sum_reflp sum_symp sum_transp elim: equivpE)

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

subsection {* Transfer rules for transfer package *}

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_unfold by (simp split: sum.split)

subsection {* Setup for 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

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_unfold sum_pred_unfold split: sum.split)

subsection {* Rules for quotient package *}

lemma sum_quotient [quot_thm]:
assumes q1: "Quotient3 R1 Abs1 Rep1"
assumes q2: "Quotient3 R2 Abs2 Rep2"
shows "Quotient3 (sum_rel R1 R2) (sum_map Abs1 Abs2) (sum_map Rep1 Rep2)"
apply (rule Quotient3I)
apply (simp_all add: sum_map.compositionality comp_def sum_map.identity sum_rel_eq sum_rel_map1 sum_rel_map2
Quotient3_abs_rep [OF q1] Quotient3_rel_rep [OF q1] Quotient3_abs_rep [OF q2] Quotient3_rel_rep [OF q2])
using Quotient3_rel [OF q1] Quotient3_rel [OF q2]
apply (simp add: sum_rel_unfold comp_def split: sum.split)
done

declare [[mapQ3 sum = (sum_rel, sum_quotient)]]

lemma sum_Inl_rsp [quot_respect]:
assumes q1: "Quotient3 R1 Abs1 Rep1"
assumes q2: "Quotient3 R2 Abs2 Rep2"
shows "(R1 ===> sum_rel R1 R2) Inl Inl"
by auto

lemma sum_Inr_rsp [quot_respect]:
assumes q1: "Quotient3 R1 Abs1 Rep1"
assumes q2: "Quotient3 R2 Abs2 Rep2"
shows "(R2 ===> sum_rel R1 R2) Inr Inr"
by auto

lemma sum_Inl_prs [quot_preserve]:
assumes q1: "Quotient3 R1 Abs1 Rep1"
assumes q2: "Quotient3 R2 Abs2 Rep2"
shows "(Rep1 ---> sum_map Abs1 Abs2) Inl = Inl"