theory QuotList+
−
imports QuotMain List+
−
begin+
−
+
−
fun+
−
  list_rel+
−
where+
−
  "list_rel R [] [] = True"+
−
| "list_rel R (x#xs) [] = False"+
−
| "list_rel R [] (x#xs) = False"+
−
| "list_rel R (x#xs) (y#ys) = (R x y \<and> list_rel R xs ys)"+
−
+
−
declare [[map list = (map, list_rel)]]+
−
+
−
lemma list_equivp[quot_equiv]:+
−
  assumes a: "equivp R"+
−
  shows "equivp (list_rel R)"+
−
  unfolding equivp_def+
−
  apply(rule allI)++
−
  apply(induct_tac x y rule: list_induct2')+
−
  apply(simp_all add: expand_fun_eq)+
−
  apply(metis list_rel.simps(1) list_rel.simps(2))+
−
  apply(metis list_rel.simps(1) list_rel.simps(2))+
−
  apply(rule iffI)+
−
  apply(rule allI)+
−
  apply(case_tac x)+
−
  apply(simp_all)+
−
  using a+
−
  apply(unfold equivp_def)+
−
  apply(auto)[1]+
−
  apply(metis list_rel.simps(4))+
−
  done+
−
+
−
lemma list_rel_rel:+
−
  assumes q: "Quotient R Abs Rep"+
−
  shows "list_rel R r s = (list_rel R r r \<and> list_rel R s s \<and> (map Abs r = map Abs s))"+
−
  apply(induct r s rule: list_induct2')+
−
  apply(simp_all)+
−
  using Quotient_rel[OF q]+
−
  apply(metis)+
−
  done+
−
+
−
lemma list_quotient[quot_thm]:+
−
  assumes q: "Quotient R Abs Rep"+
−
  shows "Quotient (list_rel R) (map Abs) (map Rep)"+
−
  unfolding Quotient_def+
−
  apply(rule conjI)+
−
  apply(rule allI)+
−
  apply(induct_tac a)+
−
  apply(simp)+
−
  apply(simp add: Quotient_abs_rep[OF q])+
−
  apply(rule conjI)+
−
  apply(rule allI)+
−
  apply(induct_tac a)+
−
  apply(simp)+
−
  apply(simp)+
−
  apply(simp add: Quotient_rep_reflp[OF q])+
−
  apply(rule allI)++
−
  apply(rule list_rel_rel[OF q])+
−
  done+
−
+
−
lemma map_id[id_simps]: "map id \<equiv> id"+
−
  apply (rule eq_reflection)+
−
  apply (rule ext)+
−
  apply (rule_tac list="x" in list.induct)+
−
  apply (simp_all)+
−
  done+
−
+
−
lemma cons_prs_aux:+
−
  assumes q: "Quotient R Abs Rep"+
−
  shows "(map Abs) ((Rep h) # (map Rep t)) = h # t"+
−
by (induct t) (simp_all add: Quotient_abs_rep[OF q])+
−
+
−
lemma cons_prs[quot_preserve]:+
−
  assumes q: "Quotient R Abs Rep"+
−
  shows "(Rep ---> (map Rep) ---> (map Abs)) (op #) = (op #)"+
−
by (simp only: expand_fun_eq fun_map.simps cons_prs_aux[OF q])+
−
   (simp)+
−
+
−
lemma cons_rsp[quot_respect]:+
−
  assumes q: "Quotient R Abs Rep"+
−
  shows "(R ===> list_rel R ===> list_rel R) op # op #"+
−
by (auto)+
−
+
−
lemma nil_prs[quot_preserve]:+
−
  assumes q: "Quotient R Abs Rep"+
−
  shows "map Abs [] \<equiv> []"+
−
by (simp)+
−
+
−
lemma nil_rsp[quot_respect]:+
−
  assumes q: "Quotient R Abs Rep"+
−
  shows "list_rel R [] []"+
−
by simp+
−
+
−
lemma map_prs_aux:+
−
  assumes a: "Quotient R1 abs1 rep1"+
−
  and     b: "Quotient R2 abs2 rep2"+
−
  shows "(map abs2) (map ((abs1 ---> rep2) f) (map rep1 l)) = map f l"+
−
by (induct l) (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b])+
−
+
−
+
−
lemma map_prs[quot_preserve]:+
−
  assumes a: "Quotient R1 abs1 rep1"+
−
  and     b: "Quotient R2 abs2 rep2"+
−
  shows "((abs1 ---> rep2) ---> (map rep1) ---> (map abs2)) map = map"+
−
by (simp only: expand_fun_eq fun_map.simps map_prs_aux[OF a b])+
−
   (simp)+
−
+
−
+
−
lemma map_rsp[quot_respect]:+
−
  assumes q1: "Quotient R1 Abs1 Rep1"+
−
  and     q2: "Quotient R2 Abs2 Rep2"+
−
  shows "((R1 ===> R2) ===> (list_rel R1) ===> list_rel R2) map map"+
−
apply(simp)+
−
apply(rule allI)++
−
apply(rule impI)+
−
apply(rule allI)++
−
apply (induct_tac xa ya rule: list_induct2')+
−
apply simp_all+
−
done+
−
+
−
lemma foldr_prs_aux:+
−
  assumes a: "Quotient R1 abs1 rep1"+
−
  and     b: "Quotient R2 abs2 rep2"+
−
  shows "abs2 (foldr ((abs1 ---> abs2 ---> rep2) f) (map rep1 l) (rep2 e)) = foldr f l e"+
−
by (induct l) (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b])+
−
+
−
lemma foldr_prs[quot_respect]:+
−
  assumes a: "Quotient R1 abs1 rep1"+
−
  and     b: "Quotient R2 abs2 rep2"+
−
  shows "((abs1 ---> abs2 ---> rep2) ---> (map rep1) ---> rep2 ---> abs2) foldr = foldr"+
−
by (simp only: expand_fun_eq fun_map.simps foldr_prs_aux[OF a b])+
−
   (simp)+
−
+
−
lemma foldl_prs_aux:+
−
  assumes a: "Quotient R1 abs1 rep1"+
−
  and     b: "Quotient R2 abs2 rep2"+
−
  shows "abs1 (foldl ((abs1 ---> abs2 ---> rep1) f) (rep1 e) (map rep2 l)) = foldl f e l"+
−
by (induct l arbitrary:e) (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b])+
−
+
−
+
−
lemma foldl_prs[quot_preserve]:+
−
  assumes a: "Quotient R1 abs1 rep1"+
−
  and     b: "Quotient R2 abs2 rep2"+
−
  shows "((abs1 ---> abs2 ---> rep1) ---> rep1 ---> (map rep2) ---> abs1) foldl = foldl"+
−
by (simp only: expand_fun_eq fun_map.simps foldl_prs_aux[OF a b])+
−
   (simp)+
−
+
−
lemma list_rel_empty: "list_rel R [] b \<Longrightarrow> length b = 0"+
−
by (induct b) (simp_all)+
−
+
−
lemma list_rel_len: "list_rel R a b \<Longrightarrow> length a = length b"+
−
apply (induct a arbitrary: b)+
−
apply (simp add: list_rel_empty)+
−
apply (case_tac b)+
−
apply simp_all+
−
done+
−
+
−
(* induct_tac doesn't accept 'arbitrary', so we manually 'spec' *)+
−
lemma foldl_rsp[quot_respect]:+
−
  assumes q1: "Quotient R1 Abs1 Rep1"+
−
  and     q2: "Quotient R2 Abs2 Rep2"+
−
  shows "((R1 ===> R2 ===> R1) ===> R1 ===> list_rel R2 ===> R1) foldl foldl"+
−
apply auto+
−
apply (subgoal_tac "R1 xa ya \<longrightarrow> list_rel R2 xb yb \<longrightarrow> R1 (foldl x xa xb) (foldl y ya yb)")+
−
apply simp+
−
apply (rule_tac x="xa" in spec)+
−
apply (rule_tac x="ya" in spec)+
−
apply (rule_tac xs="xb" and ys="yb" in list_induct2)+
−
apply (rule list_rel_len)+
−
apply (simp_all)+
−
done+
−
+
−
lemma foldr_rsp[quot_respect]:+
−
  assumes q1: "Quotient R1 Abs1 Rep1"+
−
  and     q2: "Quotient R2 Abs2 Rep2"+
−
  shows "((R1 ===> R2 ===> R2) ===> list_rel R1 ===> R2 ===> R2) foldr foldr"+
−
apply auto+
−
apply(subgoal_tac "R2 xb yb \<longrightarrow> list_rel R1 xa ya \<longrightarrow> R2 (foldr x xa xb) (foldr y ya yb)")+
−
apply simp+
−
apply (rule_tac xs="xa" and ys="ya" in list_induct2)+
−
apply (rule list_rel_len)+
−
apply (simp_all)+
−
done+
−
+
−
(* Rest are unused *)+
−
+
−
lemma list_rel_eq:+
−
  shows "list_rel (op =) \<equiv> (op =)"+
−
apply(rule eq_reflection)+
−
unfolding expand_fun_eq+
−
apply(rule allI)++
−
apply(induct_tac x xa rule: list_induct2')+
−
apply(simp_all)+
−
done+
−
+
−
lemma list_rel_refl:+
−
  assumes a: "\<And>x y. R x y = (R x = R y)"+
−
  shows "list_rel R x x"+
−
by (induct x) (auto simp add: a)+
−
+
−
end+
−