src/HOL/Library/Quotient_List.thy
 author huffman Fri Dec 09 14:14:05 2011 +0100 (2011-12-09) changeset 45803 fe44c0b216ef parent 40820 fd9c98ead9a9 child 45806 0f1c049c147e permissions -rw-r--r--
remove some duplicate lemmas, simplify some proofs
```     1 (*  Title:      HOL/Library/Quotient_List.thy
```
```     2     Author:     Cezary Kaliszyk and Christian Urban
```
```     3 *)
```
```     4
```
```     5 header {* Quotient infrastructure for the list type *}
```
```     6
```
```     7 theory Quotient_List
```
```     8 imports Main Quotient_Syntax
```
```     9 begin
```
```    10
```
```    11 declare [[map list = (map, list_all2)]]
```
```    12
```
```    13 lemma map_id [id_simps]:
```
```    14   "map id = id"
```
```    15   by (fact map.id)
```
```    16
```
```    17 lemma list_all2_eq [id_simps]:
```
```    18   "list_all2 (op =) = (op =)"
```
```    19 proof (rule ext)+
```
```    20   fix xs ys
```
```    21   show "list_all2 (op =) xs ys \<longleftrightarrow> xs = ys"
```
```    22     by (induct xs ys rule: list_induct2') simp_all
```
```    23 qed
```
```    24
```
```    25 lemma list_reflp:
```
```    26   assumes "reflp R"
```
```    27   shows "reflp (list_all2 R)"
```
```    28 proof (rule reflpI)
```
```    29   from assms have *: "\<And>xs. R xs xs" by (rule reflpE)
```
```    30   fix xs
```
```    31   show "list_all2 R xs xs"
```
```    32     by (induct xs) (simp_all add: *)
```
```    33 qed
```
```    34
```
```    35 lemma list_symp:
```
```    36   assumes "symp R"
```
```    37   shows "symp (list_all2 R)"
```
```    38 proof (rule sympI)
```
```    39   from assms have *: "\<And>xs ys. R xs ys \<Longrightarrow> R ys xs" by (rule sympE)
```
```    40   fix xs ys
```
```    41   assume "list_all2 R xs ys"
```
```    42   then show "list_all2 R ys xs"
```
```    43     by (induct xs ys rule: list_induct2') (simp_all add: *)
```
```    44 qed
```
```    45
```
```    46 lemma list_transp:
```
```    47   assumes "transp R"
```
```    48   shows "transp (list_all2 R)"
```
```    49 proof (rule transpI)
```
```    50   from assms have *: "\<And>xs ys zs. R xs ys \<Longrightarrow> R ys zs \<Longrightarrow> R xs zs" by (rule transpE)
```
```    51   fix xs ys zs
```
```    52   assume "list_all2 R xs ys" and "list_all2 R ys zs"
```
```    53   then show "list_all2 R xs zs"
```
```    54     by (induct arbitrary: zs) (auto simp: list_all2_Cons1 intro: *)
```
```    55 qed
```
```    56
```
```    57 lemma list_equivp [quot_equiv]:
```
```    58   "equivp R \<Longrightarrow> equivp (list_all2 R)"
```
```    59   by (blast intro: equivpI list_reflp list_symp list_transp elim: equivpE)
```
```    60
```
```    61 lemma list_quotient [quot_thm]:
```
```    62   assumes "Quotient R Abs Rep"
```
```    63   shows "Quotient (list_all2 R) (map Abs) (map Rep)"
```
```    64 proof (rule QuotientI)
```
```    65   from assms have "\<And>x. Abs (Rep x) = x" by (rule Quotient_abs_rep)
```
```    66   then show "\<And>xs. map Abs (map Rep xs) = xs" by (simp add: comp_def)
```
```    67 next
```
```    68   from assms have "\<And>x y. R (Rep x) (Rep y) \<longleftrightarrow> x = y" by (rule Quotient_rel_rep)
```
```    69   then show "\<And>xs. list_all2 R (map Rep xs) (map Rep xs)"
```
```    70     by (simp add: list_all2_map1 list_all2_map2 list_all2_eq)
```
```    71 next
```
```    72   fix xs ys
```
```    73   from assms have "\<And>x y. R x x \<and> R y y \<and> Abs x = Abs y \<longleftrightarrow> R x y" by (rule Quotient_rel)
```
```    74   then show "list_all2 R xs ys \<longleftrightarrow> list_all2 R xs xs \<and> list_all2 R ys ys \<and> map Abs xs = map Abs ys"
```
```    75     by (induct xs ys rule: list_induct2') auto
```
```    76 qed
```
```    77
```
```    78 lemma cons_prs [quot_preserve]:
```
```    79   assumes q: "Quotient R Abs Rep"
```
```    80   shows "(Rep ---> (map Rep) ---> (map Abs)) (op #) = (op #)"
```
```    81   by (auto simp add: fun_eq_iff comp_def Quotient_abs_rep [OF q])
```
```    82
```
```    83 lemma cons_rsp [quot_respect]:
```
```    84   assumes q: "Quotient R Abs Rep"
```
```    85   shows "(R ===> list_all2 R ===> list_all2 R) (op #) (op #)"
```
```    86   by auto
```
```    87
```
```    88 lemma nil_prs [quot_preserve]:
```
```    89   assumes q: "Quotient R Abs Rep"
```
```    90   shows "map Abs [] = []"
```
```    91   by simp
```
```    92
```
```    93 lemma nil_rsp [quot_respect]:
```
```    94   assumes q: "Quotient R Abs Rep"
```
```    95   shows "list_all2 R [] []"
```
```    96   by simp
```
```    97
```
```    98 lemma map_prs_aux:
```
```    99   assumes a: "Quotient R1 abs1 rep1"
```
```   100   and     b: "Quotient R2 abs2 rep2"
```
```   101   shows "(map abs2) (map ((abs1 ---> rep2) f) (map rep1 l)) = map f l"
```
```   102   by (induct l)
```
```   103      (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b])
```
```   104
```
```   105 lemma map_prs [quot_preserve]:
```
```   106   assumes a: "Quotient R1 abs1 rep1"
```
```   107   and     b: "Quotient R2 abs2 rep2"
```
```   108   shows "((abs1 ---> rep2) ---> (map rep1) ---> (map abs2)) map = map"
```
```   109   and   "((abs1 ---> id) ---> map rep1 ---> id) map = map"
```
```   110   by (simp_all only: fun_eq_iff map_prs_aux[OF a b] comp_def)
```
```   111     (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b])
```
```   112
```
```   113 lemma map_rsp [quot_respect]:
```
```   114   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   115   and     q2: "Quotient R2 Abs2 Rep2"
```
```   116   shows "((R1 ===> R2) ===> (list_all2 R1) ===> list_all2 R2) map map"
```
```   117   and   "((R1 ===> op =) ===> (list_all2 R1) ===> op =) map map"
```
```   118   apply (simp_all add: fun_rel_def)
```
```   119   apply(rule_tac [!] allI)+
```
```   120   apply(rule_tac [!] impI)
```
```   121   apply(rule_tac [!] allI)+
```
```   122   apply (induct_tac [!] xa ya rule: list_induct2')
```
```   123   apply simp_all
```
```   124   done
```
```   125
```
```   126 lemma foldr_prs_aux:
```
```   127   assumes a: "Quotient R1 abs1 rep1"
```
```   128   and     b: "Quotient R2 abs2 rep2"
```
```   129   shows "abs2 (foldr ((abs1 ---> abs2 ---> rep2) f) (map rep1 l) (rep2 e)) = foldr f l e"
```
```   130   by (induct l) (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b])
```
```   131
```
```   132 lemma foldr_prs [quot_preserve]:
```
```   133   assumes a: "Quotient R1 abs1 rep1"
```
```   134   and     b: "Quotient R2 abs2 rep2"
```
```   135   shows "((abs1 ---> abs2 ---> rep2) ---> (map rep1) ---> rep2 ---> abs2) foldr = foldr"
```
```   136   apply (simp add: fun_eq_iff)
```
```   137   by (simp only: fun_eq_iff foldr_prs_aux[OF a b])
```
```   138      (simp)
```
```   139
```
```   140 lemma foldl_prs_aux:
```
```   141   assumes a: "Quotient R1 abs1 rep1"
```
```   142   and     b: "Quotient R2 abs2 rep2"
```
```   143   shows "abs1 (foldl ((abs1 ---> abs2 ---> rep1) f) (rep1 e) (map rep2 l)) = foldl f e l"
```
```   144   by (induct l arbitrary:e) (simp_all add: Quotient_abs_rep[OF a] Quotient_abs_rep[OF b])
```
```   145
```
```   146 lemma foldl_prs [quot_preserve]:
```
```   147   assumes a: "Quotient R1 abs1 rep1"
```
```   148   and     b: "Quotient R2 abs2 rep2"
```
```   149   shows "((abs1 ---> abs2 ---> rep1) ---> rep1 ---> (map rep2) ---> abs1) foldl = foldl"
```
```   150   by (simp add: fun_eq_iff foldl_prs_aux [OF a b])
```
```   151
```
```   152 (* induct_tac doesn't accept 'arbitrary', so we manually 'spec' *)
```
```   153 lemma foldl_rsp[quot_respect]:
```
```   154   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   155   and     q2: "Quotient R2 Abs2 Rep2"
```
```   156   shows "((R1 ===> R2 ===> R1) ===> R1 ===> list_all2 R2 ===> R1) foldl foldl"
```
```   157   apply(auto simp add: fun_rel_def)
```
```   158   apply (erule_tac P="R1 xa ya" in rev_mp)
```
```   159   apply (rule_tac x="xa" in spec)
```
```   160   apply (rule_tac x="ya" in spec)
```
```   161   apply (erule list_all2_induct, simp_all)
```
```   162   done
```
```   163
```
```   164 lemma foldr_rsp[quot_respect]:
```
```   165   assumes q1: "Quotient R1 Abs1 Rep1"
```
```   166   and     q2: "Quotient R2 Abs2 Rep2"
```
```   167   shows "((R1 ===> R2 ===> R2) ===> list_all2 R1 ===> R2 ===> R2) foldr foldr"
```
```   168   apply (auto simp add: fun_rel_def)
```
```   169   apply (erule list_all2_induct, simp_all)
```
```   170   done
```
```   171
```
```   172 lemma list_all2_rsp:
```
```   173   assumes r: "\<forall>x y. R x y \<longrightarrow> (\<forall>a b. R a b \<longrightarrow> S x a = T y b)"
```
```   174   and l1: "list_all2 R x y"
```
```   175   and l2: "list_all2 R a b"
```
```   176   shows "list_all2 S x a = list_all2 T y b"
```
```   177   using l1 l2
```
```   178   by (induct arbitrary: a b rule: list_all2_induct,
```
```   179     auto simp: list_all2_Cons1 list_all2_Cons2 r)
```
```   180
```
```   181 lemma [quot_respect]:
```
```   182   "((R ===> R ===> op =) ===> list_all2 R ===> list_all2 R ===> op =) list_all2 list_all2"
```
```   183   by (simp add: list_all2_rsp fun_rel_def)
```
```   184
```
```   185 lemma [quot_preserve]:
```
```   186   assumes a: "Quotient R abs1 rep1"
```
```   187   shows "((abs1 ---> abs1 ---> id) ---> map rep1 ---> map rep1 ---> id) list_all2 = list_all2"
```
```   188   apply (simp add: fun_eq_iff)
```
```   189   apply clarify
```
```   190   apply (induct_tac xa xb rule: list_induct2')
```
```   191   apply (simp_all add: Quotient_abs_rep[OF a])
```
```   192   done
```
```   193
```
```   194 lemma [quot_preserve]:
```
```   195   assumes a: "Quotient R abs1 rep1"
```
```   196   shows "(list_all2 ((rep1 ---> rep1 ---> id) R) l m) = (l = m)"
```
```   197   by (induct l m rule: list_induct2') (simp_all add: Quotient_rel_rep[OF a])
```
```   198
```
```   199 lemma list_all2_find_element:
```
```   200   assumes a: "x \<in> set a"
```
```   201   and b: "list_all2 R a b"
```
```   202   shows "\<exists>y. (y \<in> set b \<and> R x y)"
```
```   203   using b a by induct auto
```
```   204
```
```   205 lemma list_all2_refl:
```
```   206   assumes a: "\<And>x y. R x y = (R x = R y)"
```
```   207   shows "list_all2 R x x"
```
```   208   by (induct x) (auto simp add: a)
```
```   209
```
```   210 end
```