author  blanchet 
Fri, 05 Sep 2014 00:41:01 +0200  
changeset 58186  a6c3962ea907 
parent 58177  166131276380 
child 58821  11e226e8a095 
permissions  rwrr 
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(* Title: HOL/Lifting.thy 
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Author: Brian Huffman and Ondrej Kuncar 

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Author: Cezary Kaliszyk and Christian Urban 

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*) 

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header {* Lifting package *} 

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theory Lifting 

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imports Equiv_Relations Transfer 
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keywords 
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"parametric" and 
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"print_quot_maps" "print_quotients" :: diag and 
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"lift_definition" :: thy_goal and 
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"setup_lifting" "lifting_forget" "lifting_update" :: thy_decl 
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begin 
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subsection {* Function map *} 
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context 
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begin 
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interpretation lifting_syntax . 
47308  22 

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lemma map_fun_id: 

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"(id > id) = id" 

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by (simp add: fun_eq_iff) 

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subsection {* Quotient Predicate *} 

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29 
definition 

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"Quotient R Abs Rep T \<longleftrightarrow> 

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(\<forall>a. Abs (Rep a) = a) \<and> 
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(\<forall>a. R (Rep a) (Rep a)) \<and> 
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(\<forall>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s) \<and> 

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T = (\<lambda>x y. R x x \<and> Abs x = y)" 

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lemma QuotientI: 

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assumes "\<And>a. Abs (Rep a) = a" 

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and "\<And>a. R (Rep a) (Rep a)" 

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and "\<And>r s. R r s \<longleftrightarrow> R r r \<and> R s s \<and> Abs r = Abs s" 

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and "T = (\<lambda>x y. R x x \<and> Abs x = y)" 

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shows "Quotient R Abs Rep T" 

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using assms unfolding Quotient_def by blast 

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47536  44 
context 
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fixes R Abs Rep T 

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assumes a: "Quotient R Abs Rep T" 
47536  47 
begin 
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lemma Quotient_abs_rep: "Abs (Rep a) = a" 

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using a unfolding Quotient_def 

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by simp 
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47536  53 
lemma Quotient_rep_reflp: "R (Rep a) (Rep a)" 
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using a unfolding Quotient_def 

47308  55 
by blast 
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lemma Quotient_rel: 

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"R r r \<and> R s s \<and> Abs r = Abs s \<longleftrightarrow> R r s"  {* orientation does not loop on rewriting *} 
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using a unfolding Quotient_def 

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by blast 
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47536  62 
lemma Quotient_cr_rel: "T = (\<lambda>x y. R x x \<and> Abs x = y)" 
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using a unfolding Quotient_def 
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by blast 

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lemma Quotient_refl1: "R r s \<Longrightarrow> R r r" 
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using a unfolding Quotient_def 

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by fast 

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lemma Quotient_refl2: "R r s \<Longrightarrow> R s s" 

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using a unfolding Quotient_def 

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by fast 

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lemma Quotient_rel_rep: "R (Rep a) (Rep b) \<longleftrightarrow> a = b" 

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using a unfolding Quotient_def 

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by metis 

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lemma Quotient_rep_abs: "R r r \<Longrightarrow> R (Rep (Abs r)) r" 

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using a unfolding Quotient_def 
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by blast 

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lemma Quotient_rep_abs_eq: "R t t \<Longrightarrow> R \<le> op= \<Longrightarrow> Rep (Abs t) = t" 
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using a unfolding Quotient_def 
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by blast 
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lemma Quotient_rep_abs_fold_unmap: 
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assumes "x' \<equiv> Abs x" and "R x x" and "Rep x' \<equiv> Rep' x'" 

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shows "R (Rep' x') x" 
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proof  
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have "R (Rep x') x" using assms(12) Quotient_rep_abs by auto 
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then show ?thesis using assms(3) by simp 
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qed 
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lemma Quotient_Rep_eq: 
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assumes "x' \<equiv> Abs x" 
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shows "Rep x' \<equiv> Rep x'" 
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by simp 
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lemma Quotient_rel_abs: "R r s \<Longrightarrow> Abs r = Abs s" 
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using a unfolding Quotient_def 

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by blast 

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lemma Quotient_rel_abs2: 
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assumes "R (Rep x) y" 
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shows "x = Abs y" 
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proof  
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from assms have "Abs (Rep x) = Abs y" by (auto intro: Quotient_rel_abs) 
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then show ?thesis using assms(1) by (simp add: Quotient_abs_rep) 
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qed 
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lemma Quotient_symp: "symp R" 
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using a unfolding Quotient_def using sympI by (metis (full_types)) 
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lemma Quotient_transp: "transp R" 
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using a unfolding Quotient_def using transpI by (metis (full_types)) 
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lemma Quotient_part_equivp: "part_equivp R" 
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by (metis Quotient_rep_reflp Quotient_symp Quotient_transp part_equivpI) 

119 

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end 

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lemma identity_quotient: "Quotient (op =) id id (op =)" 

58186  123 
unfolding Quotient_def by simp 
47308  124 

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text {* TODO: Use one of these alternatives as the real definition. *} 
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lemma Quotient_alt_def: 
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"Quotient R Abs Rep T \<longleftrightarrow> 

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(\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> 

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(\<forall>b. T (Rep b) b) \<and> 

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(\<forall>x y. R x y \<longleftrightarrow> T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y)" 

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apply safe 

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apply (simp (no_asm_use) only: Quotient_def, fast) 

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apply (simp (no_asm_use) only: Quotient_def, fast) 

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apply (simp (no_asm_use) only: Quotient_def, fast) 

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apply (simp (no_asm_use) only: Quotient_def, fast) 

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apply (simp (no_asm_use) only: Quotient_def, fast) 

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apply (simp (no_asm_use) only: Quotient_def, fast) 

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apply (rule QuotientI) 

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apply simp 

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apply metis 

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apply simp 

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apply (rule ext, rule ext, metis) 

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done 

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lemma Quotient_alt_def2: 

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"Quotient R Abs Rep T \<longleftrightarrow> 

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(\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> 

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(\<forall>b. T (Rep b) b) \<and> 

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(\<forall>x y. R x y \<longleftrightarrow> T x (Abs y) \<and> T y (Abs x))" 

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unfolding Quotient_alt_def by (safe, metis+) 

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lemma Quotient_alt_def3: 
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"Quotient R Abs Rep T \<longleftrightarrow> 
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(\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> 
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(\<forall>x y. R x y \<longleftrightarrow> (\<exists>z. T x z \<and> T y z))" 
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unfolding Quotient_alt_def2 by (safe, metis+) 
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lemma Quotient_alt_def4: 
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"Quotient R Abs Rep T \<longleftrightarrow> 
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(\<forall>a b. T a b \<longrightarrow> Abs a = b) \<and> (\<forall>b. T (Rep b) b) \<and> R = T OO conversep T" 
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unfolding Quotient_alt_def3 fun_eq_iff by auto 
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lemma Quotient_alt_def5: 
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"Quotient R Abs Rep T \<longleftrightarrow> 
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T \<le> BNF_Def.Grp UNIV Abs \<and> BNF_Def.Grp UNIV Rep \<le> T\<inverse>\<inverse> \<and> R = T OO T\<inverse>\<inverse>" 
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unfolding Quotient_alt_def4 Grp_def by blast 
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47308  169 
lemma fun_quotient: 
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assumes 1: "Quotient R1 abs1 rep1 T1" 

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assumes 2: "Quotient R2 abs2 rep2 T2" 

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shows "Quotient (R1 ===> R2) (rep1 > abs2) (abs1 > rep2) (T1 ===> T2)" 

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using assms unfolding Quotient_alt_def2 

55945  174 
unfolding rel_fun_def fun_eq_iff map_fun_apply 
47308  175 
by (safe, metis+) 
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lemma apply_rsp: 

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fixes f g::"'a \<Rightarrow> 'c" 

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assumes q: "Quotient R1 Abs1 Rep1 T1" 

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and a: "(R1 ===> R2) f g" "R1 x y" 

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shows "R2 (f x) (g y)" 

55945  182 
using a by (auto elim: rel_funE) 
47308  183 

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lemma apply_rsp': 

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assumes a: "(R1 ===> R2) f g" "R1 x y" 

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shows "R2 (f x) (g y)" 

55945  187 
using a by (auto elim: rel_funE) 
47308  188 

189 
lemma apply_rsp'': 

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assumes "Quotient R Abs Rep T" 

191 
and "(R ===> S) f f" 

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shows "S (f (Rep x)) (f (Rep x))" 

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proof  

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from assms(1) have "R (Rep x) (Rep x)" by (rule Quotient_rep_reflp) 

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then show ?thesis using assms(2) by (auto intro: apply_rsp') 

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qed 

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subsection {* Quotient composition *} 

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lemma Quotient_compose: 

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assumes 1: "Quotient R1 Abs1 Rep1 T1" 

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assumes 2: "Quotient R2 Abs2 Rep2 T2" 

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shows "Quotient (T1 OO R2 OO conversep T1) (Abs2 \<circ> Abs1) (Rep1 \<circ> Rep2) (T1 OO T2)" 

51994  204 
using assms unfolding Quotient_alt_def4 by fastforce 
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lemma equivp_reflp2: 
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"equivp R \<Longrightarrow> reflp R" 
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by (erule equivpE) 
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47544  210 
subsection {* Respects predicate *} 
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definition Respects :: "('a \<Rightarrow> 'a \<Rightarrow> bool) \<Rightarrow> 'a set" 

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where "Respects R = {x. R x x}" 

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lemma in_respects: "x \<in> Respects R \<longleftrightarrow> R x x" 

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unfolding Respects_def by simp 

217 

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lemma UNIV_typedef_to_Quotient: 
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assumes "type_definition Rep Abs UNIV" 
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and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)" 
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shows "Quotient (op =) Abs Rep T" 
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proof  

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interpret type_definition Rep Abs UNIV by fact 

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from Abs_inject Rep_inverse Abs_inverse T_def show ?thesis 
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by (fastforce intro!: QuotientI fun_eq_iff) 
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qed 
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lemma UNIV_typedef_to_equivp: 
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fixes Abs :: "'a \<Rightarrow> 'b" 
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and Rep :: "'b \<Rightarrow> 'a" 

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assumes "type_definition Rep Abs (UNIV::'a set)" 

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shows "equivp (op=::'a\<Rightarrow>'a\<Rightarrow>bool)" 

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by (rule identity_equivp) 

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lemma typedef_to_Quotient: 
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assumes "type_definition Rep Abs S" 
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and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)" 
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shows "Quotient (eq_onp (\<lambda>x. x \<in> S)) Abs Rep T" 
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proof  
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interpret type_definition Rep Abs S by fact 
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from Rep Abs_inject Rep_inverse Abs_inverse T_def show ?thesis 
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by (auto intro!: QuotientI simp: eq_onp_def fun_eq_iff) 
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qed 
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lemma typedef_to_part_equivp: 
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assumes "type_definition Rep Abs S" 
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shows "part_equivp (eq_onp (\<lambda>x. x \<in> S))" 
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proof (intro part_equivpI) 
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interpret type_definition Rep Abs S by fact 
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show "\<exists>x. eq_onp (\<lambda>x. x \<in> S) x x" using Rep by (auto simp: eq_onp_def) 
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next 
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show "symp (eq_onp (\<lambda>x. x \<in> S))" by (auto intro: sympI simp: eq_onp_def) 
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next 
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show "transp (eq_onp (\<lambda>x. x \<in> S))" by (auto intro: transpI simp: eq_onp_def) 
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qed 
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256 

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lemma open_typedef_to_Quotient: 
47308  258 
assumes "type_definition Rep Abs {x. P x}" 
47354  259 
and T_def: "T \<equiv> (\<lambda>x y. x = Rep y)" 
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shows "Quotient (eq_onp P) Abs Rep T" 
47651  261 
using typedef_to_Quotient [OF assms] by simp 
47308  262 

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lemma open_typedef_to_part_equivp: 
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assumes "type_definition Rep Abs {x. P x}" 
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shows "part_equivp (eq_onp P)" 
47651  266 
using typedef_to_part_equivp [OF assms] by simp 
47308  267 

47376  268 
text {* Generating transfer rules for quotients. *} 
269 

47537  270 
context 
271 
fixes R Abs Rep T 

272 
assumes 1: "Quotient R Abs Rep T" 

273 
begin 

47376  274 

47537  275 
lemma Quotient_right_unique: "right_unique T" 
276 
using 1 unfolding Quotient_alt_def right_unique_def by metis 

277 

278 
lemma Quotient_right_total: "right_total T" 

279 
using 1 unfolding Quotient_alt_def right_total_def by metis 

280 

281 
lemma Quotient_rel_eq_transfer: "(T ===> T ===> op =) R (op =)" 

55945  282 
using 1 unfolding Quotient_alt_def rel_fun_def by simp 
47376  283 

47538  284 
lemma Quotient_abs_induct: 
285 
assumes "\<And>y. R y y \<Longrightarrow> P (Abs y)" shows "P x" 

286 
using 1 assms unfolding Quotient_def by metis 

287 

47537  288 
end 
289 

290 
text {* Generating transfer rules for total quotients. *} 

47376  291 

47537  292 
context 
293 
fixes R Abs Rep T 

294 
assumes 1: "Quotient R Abs Rep T" and 2: "reflp R" 

295 
begin 

47376  296 

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lemma Quotient_left_total: "left_total T" 
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using 1 2 unfolding Quotient_alt_def left_total_def reflp_def by auto 
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47537  300 
lemma Quotient_bi_total: "bi_total T" 
301 
using 1 2 unfolding Quotient_alt_def bi_total_def reflp_def by auto 

302 

303 
lemma Quotient_id_abs_transfer: "(op = ===> T) (\<lambda>x. x) Abs" 

55945  304 
using 1 2 unfolding Quotient_alt_def reflp_def rel_fun_def by simp 
47537  305 

47575  306 
lemma Quotient_total_abs_induct: "(\<And>y. P (Abs y)) \<Longrightarrow> P x" 
307 
using 1 2 assms unfolding Quotient_alt_def reflp_def by metis 

308 

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lemma Quotient_total_abs_eq_iff: "Abs x = Abs y \<longleftrightarrow> R x y" 
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using Quotient_rel [OF 1] 2 unfolding reflp_def by simp 
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311 

47537  312 
end 
47376  313 

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text {* Generating transfer rules for a type defined with @{text "typedef"}. *} 
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315 

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context 
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fixes Rep Abs A T 
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assumes type: "type_definition Rep Abs A" 
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assumes T_def: "T \<equiv> (\<lambda>(x::'a) (y::'b). x = Rep y)" 
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begin 
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51994  322 
lemma typedef_left_unique: "left_unique T" 
323 
unfolding left_unique_def T_def 

324 
by (simp add: type_definition.Rep_inject [OF type]) 

325 

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lemma typedef_bi_unique: "bi_unique T" 
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unfolding bi_unique_def T_def 
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by (simp add: type_definition.Rep_inject [OF type]) 
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329 

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(* the following two theorems are here only for convinience *) 
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331 

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lemma typedef_right_unique: "right_unique T" 
58186  333 
using T_def type Quotient_right_unique typedef_to_Quotient 
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by blast 
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335 

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lemma typedef_right_total: "right_total T" 
58186  337 
using T_def type Quotient_right_total typedef_to_Quotient 
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by blast 
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339 

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lemma typedef_rep_transfer: "(T ===> op =) (\<lambda>x. x) Rep" 
55945  341 
unfolding rel_fun_def T_def by simp 
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342 

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end 
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344 

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text {* Generating the correspondence rule for a constant defined with 
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346 
@{text "lift_definition"}. *} 
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347 

47351  348 
lemma Quotient_to_transfer: 
349 
assumes "Quotient R Abs Rep T" and "R c c" and "c' \<equiv> Abs c" 

350 
shows "T c c'" 

351 
using assms by (auto dest: Quotient_cr_rel) 

352 

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text {* Proving reflexivity *} 
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354 

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lemma Quotient_to_left_total: 
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356 
assumes q: "Quotient R Abs Rep T" 
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357 
and r_R: "reflp R" 
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358 
shows "left_total T" 
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359 
using r_R Quotient_cr_rel[OF q] unfolding left_total_def by (auto elim: reflpE) 
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360 

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lemma Quotient_composition_ge_eq: 
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362 
assumes "left_total T" 
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assumes "R \<ge> op=" 
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364 
shows "(T OO R OO T\<inverse>\<inverse>) \<ge> op=" 
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365 
using assms unfolding left_total_def by fast 
51994  366 

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lemma Quotient_composition_le_eq: 
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assumes "left_unique T" 
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assumes "R \<le> op=" 
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370 
shows "(T OO R OO T\<inverse>\<inverse>) \<le> op=" 
55604  371 
using assms unfolding left_unique_def by blast 
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372 

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373 
lemma eq_onp_le_eq: 
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"eq_onp P \<le> op=" unfolding eq_onp_def by blast 
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375 

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lemma reflp_ge_eq: 
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"reflp R \<Longrightarrow> R \<ge> op=" unfolding reflp_def by blast 
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378 

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379 
lemma ge_eq_refl: 
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380 
"R \<ge> op= \<Longrightarrow> R x x" by blast 
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381 

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text {* Proving a parametrized correspondence relation *} 
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383 

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definition POS :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where 
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385 
"POS A B \<equiv> A \<le> B" 
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386 

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387 
definition NEG :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> bool" where 
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388 
"NEG A B \<equiv> B \<le> A" 
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389 

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390 
lemma pos_OO_eq: 
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391 
shows "POS (A OO op=) A" 
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392 
unfolding POS_def OO_def by blast 
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393 

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394 
lemma pos_eq_OO: 
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shows "POS (op= OO A) A" 
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396 
unfolding POS_def OO_def by blast 
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397 

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398 
lemma neg_OO_eq: 
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399 
shows "NEG (A OO op=) A" 
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400 
unfolding NEG_def OO_def by auto 
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401 

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402 
lemma neg_eq_OO: 
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403 
shows "NEG (op= OO A) A" 
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404 
unfolding NEG_def OO_def by blast 
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405 

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406 
lemma POS_trans: 
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407 
assumes "POS A B" 
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408 
assumes "POS B C" 
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409 
shows "POS A C" 
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410 
using assms unfolding POS_def by auto 
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411 

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412 
lemma NEG_trans: 
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413 
assumes "NEG A B" 
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414 
assumes "NEG B C" 
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415 
shows "NEG A C" 
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416 
using assms unfolding NEG_def by auto 
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417 

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418 
lemma POS_NEG: 
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419 
"POS A B \<equiv> NEG B A" 
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420 
unfolding POS_def NEG_def by auto 
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421 

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422 
lemma NEG_POS: 
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423 
"NEG A B \<equiv> POS B A" 
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424 
unfolding POS_def NEG_def by auto 
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425 

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426 
lemma POS_pcr_rule: 
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427 
assumes "POS (A OO B) C" 
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parents:
51112
diff
changeset

428 
shows "POS (A OO B OO X) (C OO X)" 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

429 
using assms unfolding POS_def OO_def by blast 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

430 

84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

431 
lemma NEG_pcr_rule: 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

432 
assumes "NEG (A OO B) C" 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

433 
shows "NEG (A OO B OO X) (C OO X)" 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

434 
using assms unfolding NEG_def OO_def by blast 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

435 

84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

436 
lemma POS_apply: 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

437 
assumes "POS R R'" 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

438 
assumes "R f g" 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

439 
shows "R' f g" 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

440 
using assms unfolding POS_def by auto 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

441 

84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

442 
text {* Proving a parametrized correspondence relation *} 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

443 

84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

444 
lemma fun_mono: 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

445 
assumes "A \<ge> C" 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

446 
assumes "B \<le> D" 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

447 
shows "(A ===> B) \<le> (C ===> D)" 
55945  448 
using assms unfolding rel_fun_def by blast 
51374
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

449 

84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

450 
lemma pos_fun_distr: "((R ===> S) OO (R' ===> S')) \<le> ((R OO R') ===> (S OO S'))" 
55945  451 
unfolding OO_def rel_fun_def by blast 
51374
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

452 

84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

453 
lemma functional_relation: "right_unique R \<Longrightarrow> left_total R \<Longrightarrow> \<forall>x. \<exists>!y. R x y" 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

454 
unfolding right_unique_def left_total_def by blast 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

455 

84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

456 
lemma functional_converse_relation: "left_unique R \<Longrightarrow> right_total R \<Longrightarrow> \<forall>y. \<exists>!x. R x y" 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

457 
unfolding left_unique_def right_total_def by blast 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

458 

84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

459 
lemma neg_fun_distr1: 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

460 
assumes 1: "left_unique R" "right_total R" 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

461 
assumes 2: "right_unique R'" "left_total R'" 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

462 
shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S')) " 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

463 
using functional_relation[OF 2] functional_converse_relation[OF 1] 
55945  464 
unfolding rel_fun_def OO_def 
51374
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

465 
apply clarify 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

466 
apply (subst all_comm) 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

467 
apply (subst all_conj_distrib[symmetric]) 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

468 
apply (intro choice) 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

469 
by metis 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

470 

84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

471 
lemma neg_fun_distr2: 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

472 
assumes 1: "right_unique R'" "left_total R'" 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

473 
assumes 2: "left_unique S'" "right_total S'" 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

474 
shows "(R OO R' ===> S OO S') \<le> ((R ===> S) OO (R' ===> S'))" 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

475 
using functional_converse_relation[OF 2] functional_relation[OF 1] 
55945  476 
unfolding rel_fun_def OO_def 
51374
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

477 
apply clarify 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

478 
apply (subst all_comm) 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

479 
apply (subst all_conj_distrib[symmetric]) 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

480 
apply (intro choice) 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

481 
by metis 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

482 

51956
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

483 
subsection {* Domains *} 
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

484 

56519
c1048f5bbb45
more appropriate name (Lifting.invariant > eq_onp)
kuncar
parents:
56518
diff
changeset

485 
lemma composed_equiv_rel_eq_onp: 
55731
66df76dd2640
rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents:
55610
diff
changeset

486 
assumes "left_unique R" 
66df76dd2640
rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents:
55610
diff
changeset

487 
assumes "(R ===> op=) P P'" 
66df76dd2640
rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents:
55610
diff
changeset

488 
assumes "Domainp R = P''" 
56519
c1048f5bbb45
more appropriate name (Lifting.invariant > eq_onp)
kuncar
parents:
56518
diff
changeset

489 
shows "(R OO eq_onp P' OO R\<inverse>\<inverse>) = eq_onp (inf P'' P)" 
c1048f5bbb45
more appropriate name (Lifting.invariant > eq_onp)
kuncar
parents:
56518
diff
changeset

490 
using assms unfolding OO_def conversep_iff Domainp_iff[abs_def] left_unique_def rel_fun_def eq_onp_def 
55731
66df76dd2640
rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents:
55610
diff
changeset

491 
fun_eq_iff by blast 
66df76dd2640
rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents:
55610
diff
changeset

492 

56519
c1048f5bbb45
more appropriate name (Lifting.invariant > eq_onp)
kuncar
parents:
56518
diff
changeset

493 
lemma composed_equiv_rel_eq_eq_onp: 
55731
66df76dd2640
rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents:
55610
diff
changeset

494 
assumes "left_unique R" 
66df76dd2640
rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents:
55610
diff
changeset

495 
assumes "Domainp R = P" 
56519
c1048f5bbb45
more appropriate name (Lifting.invariant > eq_onp)
kuncar
parents:
56518
diff
changeset

496 
shows "(R OO op= OO R\<inverse>\<inverse>) = eq_onp P" 
c1048f5bbb45
more appropriate name (Lifting.invariant > eq_onp)
kuncar
parents:
56518
diff
changeset

497 
using assms unfolding OO_def conversep_iff Domainp_iff[abs_def] left_unique_def eq_onp_def 
55731
66df76dd2640
rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents:
55610
diff
changeset

498 
fun_eq_iff is_equality_def by metis 
66df76dd2640
rewrite composition of quotients to a more readable form in a respectfulness goal that is presented to a user
kuncar
parents:
55610
diff
changeset

499 

51956
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

500 
lemma pcr_Domainp_par_left_total: 
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

501 
assumes "Domainp B = P" 
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

502 
assumes "left_total A" 
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

503 
assumes "(A ===> op=) P' P" 
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

504 
shows "Domainp (A OO B) = P'" 
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

505 
using assms 
58186  506 
unfolding Domainp_iff[abs_def] OO_def bi_unique_def left_total_def rel_fun_def 
51956
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

507 
by (fast intro: fun_eq_iff) 
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

508 

a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

509 
lemma pcr_Domainp_par: 
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

510 
assumes "Domainp B = P2" 
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

511 
assumes "Domainp A = P1" 
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

512 
assumes "(A ===> op=) P2' P2" 
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

513 
shows "Domainp (A OO B) = (inf P1 P2')" 
55945  514 
using assms unfolding rel_fun_def Domainp_iff[abs_def] OO_def 
51956
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

515 
by (fast intro: fun_eq_iff) 
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

516 

53151  517 
definition rel_pred_comp :: "('a => 'b => bool) => ('b => bool) => 'a => bool" 
51956
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

518 
where "rel_pred_comp R P \<equiv> \<lambda>x. \<exists>y. R x y \<and> P y" 
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

519 

a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

520 
lemma pcr_Domainp: 
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

521 
assumes "Domainp B = P" 
53151  522 
shows "Domainp (A OO B) = (\<lambda>x. \<exists>y. A x y \<and> P y)" 
523 
using assms by blast 

51956
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

524 

a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

525 
lemma pcr_Domainp_total: 
56518
beb3b6851665
left_total and left_unique rules are now transfer rules (cleaner solution, reflexvity_rule attribute not needed anymore)
kuncar
parents:
56517
diff
changeset

526 
assumes "left_total B" 
51956
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

527 
assumes "Domainp A = P" 
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

528 
shows "Domainp (A OO B) = P" 
58186  529 
using assms unfolding left_total_def 
51956
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

530 
by fast 
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

531 

a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

532 
lemma Quotient_to_Domainp: 
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

533 
assumes "Quotient R Abs Rep T" 
58186  534 
shows "Domainp T = (\<lambda>x. R x x)" 
51956
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

535 
by (simp add: Domainp_iff[abs_def] Quotient_cr_rel[OF assms]) 
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

536 

56519
c1048f5bbb45
more appropriate name (Lifting.invariant > eq_onp)
kuncar
parents:
56518
diff
changeset

537 
lemma eq_onp_to_Domainp: 
c1048f5bbb45
more appropriate name (Lifting.invariant > eq_onp)
kuncar
parents:
56518
diff
changeset

538 
assumes "Quotient (eq_onp P) Abs Rep T" 
51956
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

539 
shows "Domainp T = P" 
56519
c1048f5bbb45
more appropriate name (Lifting.invariant > eq_onp)
kuncar
parents:
56518
diff
changeset

540 
by (simp add: eq_onp_def Domainp_iff[abs_def] Quotient_cr_rel[OF assms]) 
51956
a4d81cdebf8b
better support for domains in Lifting/Transfer = replace Domainp T by the actual invariant in a transferred goal
kuncar
parents:
51374
diff
changeset

541 

53011
aeee0a4be6cf
introduce locale with syntax for fun_rel and map_fun and make thus ===> and > local
kuncar
parents:
52307
diff
changeset

542 
end 
aeee0a4be6cf
introduce locale with syntax for fun_rel and map_fun and make thus ===> and > local
kuncar
parents:
52307
diff
changeset

543 

47308  544 
subsection {* ML setup *} 
545 

48891  546 
ML_file "Tools/Lifting/lifting_util.ML" 
47308  547 

57961  548 
named_theorems relator_eq_onp 
549 
"theorems that a relator of an eq_onp is an eq_onp of the corresponding predicate" 

48891  550 
ML_file "Tools/Lifting/lifting_info.ML" 
47308  551 
setup Lifting_Info.setup 
552 

51374
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

553 
(* setup for the function type *) 
47777
f29e7dcd7c40
use a quot_map theorem attribute instead of the complicated map attribute
kuncar
parents:
47698
diff
changeset

554 
declare fun_quotient[quot_map] 
51374
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

555 
declare fun_mono[relator_mono] 
84d01fd733cf
lift_definition and setup_lifting generate parametric transfer rules if parametricity theorems are provided
kuncar
parents:
51112
diff
changeset

556 
lemmas [relator_distr] = pos_fun_distr neg_fun_distr1 neg_fun_distr2 
47308  557 

56524
f4ba736040fa
setup for Transfer and Lifting from BNF; tuned thm names
kuncar
parents:
56519
diff
changeset

558 
ML_file "Tools/Lifting/lifting_bnf.ML" 
48891  559 
ML_file "Tools/Lifting/lifting_term.ML" 
560 
ML_file "Tools/Lifting/lifting_def.ML" 

561 
ML_file "Tools/Lifting/lifting_setup.ML" 

58186  562 

56519
c1048f5bbb45
more appropriate name (Lifting.invariant > eq_onp)
kuncar
parents:
56518
diff
changeset

563 
hide_const (open) POS NEG 
47308  564 

565 
end 