src/HOL/Library/Quotient_List.thy
 author huffman Sun Apr 22 11:05:04 2012 +0200 (2012-04-22) changeset 47660 7a5c681c0265 parent 47650 33ecf14d5ee9 child 47777 f29e7dcd7c40 permissions -rw-r--r--
new example theory for quotient/transfer
```     1 (*  Title:      HOL/Library/Quotient_List.thy
```
```     2     Author:     Cezary Kaliszyk, Christian Urban and Brian Huffman
```
```     3 *)
```
```     4
```
```     5 header {* Quotient infrastructure for the list type *}
```
```     6
```
```     7 theory Quotient_List
```
```     8 imports Main Quotient_Set
```
```     9 begin
```
```    10
```
```    11 subsection {* Relator for list type *}
```
```    12
```
```    13 lemma map_id [id_simps]:
```
```    14   "map id = id"
```
```    15   by (fact List.map.id)
```
```    16
```
```    17 lemma list_all2_eq [id_simps, relator_eq]:
```
```    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_all2_OO: "list_all2 (A OO B) = list_all2 A OO list_all2 B"
```
```    26 proof (intro ext iffI)
```
```    27   fix xs ys
```
```    28   assume "list_all2 (A OO B) xs ys"
```
```    29   thus "(list_all2 A OO list_all2 B) xs ys"
```
```    30     unfolding OO_def
```
```    31     by (induct, simp, simp add: list_all2_Cons1 list_all2_Cons2, fast)
```
```    32 next
```
```    33   fix xs ys
```
```    34   assume "(list_all2 A OO list_all2 B) xs ys"
```
```    35   then obtain zs where "list_all2 A xs zs" and "list_all2 B zs ys" ..
```
```    36   thus "list_all2 (A OO B) xs ys"
```
```    37     by (induct arbitrary: ys, simp, clarsimp simp add: list_all2_Cons1, fast)
```
```    38 qed
```
```    39
```
```    40 lemma list_reflp:
```
```    41   assumes "reflp R"
```
```    42   shows "reflp (list_all2 R)"
```
```    43 proof (rule reflpI)
```
```    44   from assms have *: "\<And>xs. R xs xs" by (rule reflpE)
```
```    45   fix xs
```
```    46   show "list_all2 R xs xs"
```
```    47     by (induct xs) (simp_all add: *)
```
```    48 qed
```
```    49
```
```    50 lemma list_symp:
```
```    51   assumes "symp R"
```
```    52   shows "symp (list_all2 R)"
```
```    53 proof (rule sympI)
```
```    54   from assms have *: "\<And>xs ys. R xs ys \<Longrightarrow> R ys xs" by (rule sympE)
```
```    55   fix xs ys
```
```    56   assume "list_all2 R xs ys"
```
```    57   then show "list_all2 R ys xs"
```
```    58     by (induct xs ys rule: list_induct2') (simp_all add: *)
```
```    59 qed
```
```    60
```
```    61 lemma list_transp:
```
```    62   assumes "transp R"
```
```    63   shows "transp (list_all2 R)"
```
```    64 proof (rule transpI)
```
```    65   from assms have *: "\<And>xs ys zs. R xs ys \<Longrightarrow> R ys zs \<Longrightarrow> R xs zs" by (rule transpE)
```
```    66   fix xs ys zs
```
```    67   assume "list_all2 R xs ys" and "list_all2 R ys zs"
```
```    68   then show "list_all2 R xs zs"
```
```    69     by (induct arbitrary: zs) (auto simp: list_all2_Cons1 intro: *)
```
```    70 qed
```
```    71
```
```    72 lemma list_equivp [quot_equiv]:
```
```    73   "equivp R \<Longrightarrow> equivp (list_all2 R)"
```
```    74   by (blast intro: equivpI list_reflp list_symp list_transp elim: equivpE)
```
```    75
```
```    76 lemma right_total_list_all2 [transfer_rule]:
```
```    77   "right_total R \<Longrightarrow> right_total (list_all2 R)"
```
```    78   unfolding right_total_def
```
```    79   by (rule allI, induct_tac y, simp, simp add: list_all2_Cons2)
```
```    80
```
```    81 lemma right_unique_list_all2 [transfer_rule]:
```
```    82   "right_unique R \<Longrightarrow> right_unique (list_all2 R)"
```
```    83   unfolding right_unique_def
```
```    84   apply (rule allI, rename_tac xs, induct_tac xs)
```
```    85   apply (auto simp add: list_all2_Cons1)
```
```    86   done
```
```    87
```
```    88 lemma bi_total_list_all2 [transfer_rule]:
```
```    89   "bi_total A \<Longrightarrow> bi_total (list_all2 A)"
```
```    90   unfolding bi_total_def
```
```    91   apply safe
```
```    92   apply (rename_tac xs, induct_tac xs, simp, simp add: list_all2_Cons1)
```
```    93   apply (rename_tac ys, induct_tac ys, simp, simp add: list_all2_Cons2)
```
```    94   done
```
```    95
```
```    96 lemma bi_unique_list_all2 [transfer_rule]:
```
```    97   "bi_unique A \<Longrightarrow> bi_unique (list_all2 A)"
```
```    98   unfolding bi_unique_def
```
```    99   apply (rule conjI)
```
```   100   apply (rule allI, rename_tac xs, induct_tac xs)
```
```   101   apply (simp, force simp add: list_all2_Cons1)
```
```   102   apply (subst (2) all_comm, subst (1) all_comm)
```
```   103   apply (rule allI, rename_tac xs, induct_tac xs)
```
```   104   apply (simp, force simp add: list_all2_Cons2)
```
```   105   done
```
```   106
```
```   107 subsection {* Transfer rules for transfer package *}
```
```   108
```
```   109 lemma Nil_transfer [transfer_rule]: "(list_all2 A) [] []"
```
```   110   by simp
```
```   111
```
```   112 lemma Cons_transfer [transfer_rule]:
```
```   113   "(A ===> list_all2 A ===> list_all2 A) Cons Cons"
```
```   114   unfolding fun_rel_def by simp
```
```   115
```
```   116 lemma list_case_transfer [transfer_rule]:
```
```   117   "(B ===> (A ===> list_all2 A ===> B) ===> list_all2 A ===> B)
```
```   118     list_case list_case"
```
```   119   unfolding fun_rel_def by (simp split: list.split)
```
```   120
```
```   121 lemma list_rec_transfer [transfer_rule]:
```
```   122   "(B ===> (A ===> list_all2 A ===> B ===> B) ===> list_all2 A ===> B)
```
```   123     list_rec list_rec"
```
```   124   unfolding fun_rel_def by (clarify, erule list_all2_induct, simp_all)
```
```   125
```
```   126 lemma map_transfer [transfer_rule]:
```
```   127   "((A ===> B) ===> list_all2 A ===> list_all2 B) map map"
```
```   128   unfolding List.map_def by transfer_prover
```
```   129
```
```   130 lemma append_transfer [transfer_rule]:
```
```   131   "(list_all2 A ===> list_all2 A ===> list_all2 A) append append"
```
```   132   unfolding List.append_def by transfer_prover
```
```   133
```
```   134 lemma filter_transfer [transfer_rule]:
```
```   135   "((A ===> op =) ===> list_all2 A ===> list_all2 A) filter filter"
```
```   136   unfolding List.filter_def by transfer_prover
```
```   137
```
```   138 lemma foldr_transfer [transfer_rule]:
```
```   139   "((A ===> B ===> B) ===> list_all2 A ===> B ===> B) foldr foldr"
```
```   140   unfolding List.foldr_def by transfer_prover
```
```   141
```
```   142 lemma foldl_transfer [transfer_rule]:
```
```   143   "((B ===> A ===> B) ===> B ===> list_all2 A ===> B) foldl foldl"
```
```   144   unfolding List.foldl_def by transfer_prover
```
```   145
```
```   146 lemma concat_transfer [transfer_rule]:
```
```   147   "(list_all2 (list_all2 A) ===> list_all2 A) concat concat"
```
```   148   unfolding List.concat_def by transfer_prover
```
```   149
```
```   150 lemma drop_transfer [transfer_rule]:
```
```   151   "(op = ===> list_all2 A ===> list_all2 A) drop drop"
```
```   152   unfolding List.drop_def by transfer_prover
```
```   153
```
```   154 lemma take_transfer [transfer_rule]:
```
```   155   "(op = ===> list_all2 A ===> list_all2 A) take take"
```
```   156   unfolding List.take_def by transfer_prover
```
```   157
```
```   158 lemma length_transfer [transfer_rule]:
```
```   159   "(list_all2 A ===> op =) length length"
```
```   160   unfolding list_size_overloaded_def by transfer_prover
```
```   161
```
```   162 lemma list_all_transfer [transfer_rule]:
```
```   163   "((A ===> op =) ===> list_all2 A ===> op =) list_all list_all"
```
```   164   unfolding fun_rel_def by (clarify, erule list_all2_induct, simp_all)
```
```   165
```
```   166 lemma list_all2_transfer [transfer_rule]:
```
```   167   "((A ===> B ===> op =) ===> list_all2 A ===> list_all2 B ===> op =)
```
```   168     list_all2 list_all2"
```
```   169   apply (rule fun_relI, rule fun_relI, erule list_all2_induct)
```
```   170   apply (rule fun_relI, erule list_all2_induct, simp, simp)
```
```   171   apply (rule fun_relI, erule list_all2_induct [of B])
```
```   172   apply (simp, simp add: fun_rel_def)
```
```   173   done
```
```   174
```
```   175 lemma set_transfer [transfer_rule]:
```
```   176   "(list_all2 A ===> set_rel A) set set"
```
```   177   unfolding set_def by transfer_prover
```
```   178
```
```   179 subsection {* Setup for lifting package *}
```
```   180
```
```   181 lemma Quotient_list:
```
```   182   assumes "Quotient R Abs Rep T"
```
```   183   shows "Quotient (list_all2 R) (map Abs) (map Rep) (list_all2 T)"
```
```   184 proof (unfold Quotient_alt_def, intro conjI allI impI)
```
```   185   from assms have 1: "\<And>x y. T x y \<Longrightarrow> Abs x = y"
```
```   186     unfolding Quotient_alt_def by simp
```
```   187   fix xs ys assume "list_all2 T xs ys" thus "map Abs xs = ys"
```
```   188     by (induct, simp, simp add: 1)
```
```   189 next
```
```   190   from assms have 2: "\<And>x. T (Rep x) x"
```
```   191     unfolding Quotient_alt_def by simp
```
```   192   fix xs show "list_all2 T (map Rep xs) xs"
```
```   193     by (induct xs, simp, simp add: 2)
```
```   194 next
```
```   195   from assms have 3: "\<And>x y. R x y \<longleftrightarrow> T x (Abs x) \<and> T y (Abs y) \<and> Abs x = Abs y"
```
```   196     unfolding Quotient_alt_def by simp
```
```   197   fix xs ys show "list_all2 R xs ys \<longleftrightarrow> list_all2 T xs (map Abs xs) \<and>
```
```   198     list_all2 T ys (map Abs ys) \<and> map Abs xs = map Abs ys"
```
```   199     by (induct xs ys rule: list_induct2', simp_all, metis 3)
```
```   200 qed
```
```   201
```
```   202 declare [[map list = (list_all2, Quotient_list)]]
```
```   203
```
```   204 lemma list_invariant_commute [invariant_commute]:
```
```   205   "list_all2 (Lifting.invariant P) = Lifting.invariant (list_all P)"
```
```   206   apply (simp add: fun_eq_iff list_all2_def list_all_iff Lifting.invariant_def Ball_def)
```
```   207   apply (intro allI)
```
```   208   apply (induct_tac rule: list_induct2')
```
```   209   apply simp_all
```
```   210   apply metis
```
```   211 done
```
```   212
```
```   213 subsection {* Rules for quotient package *}
```
```   214
```
```   215 lemma list_quotient3 [quot_thm]:
```
```   216   assumes "Quotient3 R Abs Rep"
```
```   217   shows "Quotient3 (list_all2 R) (map Abs) (map Rep)"
```
```   218 proof (rule Quotient3I)
```
```   219   from assms have "\<And>x. Abs (Rep x) = x" by (rule Quotient3_abs_rep)
```
```   220   then show "\<And>xs. map Abs (map Rep xs) = xs" by (simp add: comp_def)
```
```   221 next
```
```   222   from assms have "\<And>x y. R (Rep x) (Rep y) \<longleftrightarrow> x = y" by (rule Quotient3_rel_rep)
```
```   223   then show "\<And>xs. list_all2 R (map Rep xs) (map Rep xs)"
```
```   224     by (simp add: list_all2_map1 list_all2_map2 list_all2_eq)
```
```   225 next
```
```   226   fix xs ys
```
```   227   from assms have "\<And>x y. R x x \<and> R y y \<and> Abs x = Abs y \<longleftrightarrow> R x y" by (rule Quotient3_rel)
```
```   228   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"
```
```   229     by (induct xs ys rule: list_induct2') auto
```
```   230 qed
```
```   231
```
```   232 declare [[mapQ3 list = (list_all2, list_quotient3)]]
```
```   233
```
```   234 lemma cons_prs [quot_preserve]:
```
```   235   assumes q: "Quotient3 R Abs Rep"
```
```   236   shows "(Rep ---> (map Rep) ---> (map Abs)) (op #) = (op #)"
```
```   237   by (auto simp add: fun_eq_iff comp_def Quotient3_abs_rep [OF q])
```
```   238
```
```   239 lemma cons_rsp [quot_respect]:
```
```   240   assumes q: "Quotient3 R Abs Rep"
```
```   241   shows "(R ===> list_all2 R ===> list_all2 R) (op #) (op #)"
```
```   242   by auto
```
```   243
```
```   244 lemma nil_prs [quot_preserve]:
```
```   245   assumes q: "Quotient3 R Abs Rep"
```
```   246   shows "map Abs [] = []"
```
```   247   by simp
```
```   248
```
```   249 lemma nil_rsp [quot_respect]:
```
```   250   assumes q: "Quotient3 R Abs Rep"
```
```   251   shows "list_all2 R [] []"
```
```   252   by simp
```
```   253
```
```   254 lemma map_prs_aux:
```
```   255   assumes a: "Quotient3 R1 abs1 rep1"
```
```   256   and     b: "Quotient3 R2 abs2 rep2"
```
```   257   shows "(map abs2) (map ((abs1 ---> rep2) f) (map rep1 l)) = map f l"
```
```   258   by (induct l)
```
```   259      (simp_all add: Quotient3_abs_rep[OF a] Quotient3_abs_rep[OF b])
```
```   260
```
```   261 lemma map_prs [quot_preserve]:
```
```   262   assumes a: "Quotient3 R1 abs1 rep1"
```
```   263   and     b: "Quotient3 R2 abs2 rep2"
```
```   264   shows "((abs1 ---> rep2) ---> (map rep1) ---> (map abs2)) map = map"
```
```   265   and   "((abs1 ---> id) ---> map rep1 ---> id) map = map"
```
```   266   by (simp_all only: fun_eq_iff map_prs_aux[OF a b] comp_def)
```
```   267     (simp_all add: Quotient3_abs_rep[OF a] Quotient3_abs_rep[OF b])
```
```   268
```
```   269 lemma map_rsp [quot_respect]:
```
```   270   assumes q1: "Quotient3 R1 Abs1 Rep1"
```
```   271   and     q2: "Quotient3 R2 Abs2 Rep2"
```
```   272   shows "((R1 ===> R2) ===> (list_all2 R1) ===> list_all2 R2) map map"
```
```   273   and   "((R1 ===> op =) ===> (list_all2 R1) ===> op =) map map"
```
```   274   unfolding list_all2_eq [symmetric] by (rule map_transfer)+
```
```   275
```
```   276 lemma foldr_prs_aux:
```
```   277   assumes a: "Quotient3 R1 abs1 rep1"
```
```   278   and     b: "Quotient3 R2 abs2 rep2"
```
```   279   shows "abs2 (foldr ((abs1 ---> abs2 ---> rep2) f) (map rep1 l) (rep2 e)) = foldr f l e"
```
```   280   by (induct l) (simp_all add: Quotient3_abs_rep[OF a] Quotient3_abs_rep[OF b])
```
```   281
```
```   282 lemma foldr_prs [quot_preserve]:
```
```   283   assumes a: "Quotient3 R1 abs1 rep1"
```
```   284   and     b: "Quotient3 R2 abs2 rep2"
```
```   285   shows "((abs1 ---> abs2 ---> rep2) ---> (map rep1) ---> rep2 ---> abs2) foldr = foldr"
```
```   286   apply (simp add: fun_eq_iff)
```
```   287   by (simp only: fun_eq_iff foldr_prs_aux[OF a b])
```
```   288      (simp)
```
```   289
```
```   290 lemma foldl_prs_aux:
```
```   291   assumes a: "Quotient3 R1 abs1 rep1"
```
```   292   and     b: "Quotient3 R2 abs2 rep2"
```
```   293   shows "abs1 (foldl ((abs1 ---> abs2 ---> rep1) f) (rep1 e) (map rep2 l)) = foldl f e l"
```
```   294   by (induct l arbitrary:e) (simp_all add: Quotient3_abs_rep[OF a] Quotient3_abs_rep[OF b])
```
```   295
```
```   296 lemma foldl_prs [quot_preserve]:
```
```   297   assumes a: "Quotient3 R1 abs1 rep1"
```
```   298   and     b: "Quotient3 R2 abs2 rep2"
```
```   299   shows "((abs1 ---> abs2 ---> rep1) ---> rep1 ---> (map rep2) ---> abs1) foldl = foldl"
```
```   300   by (simp add: fun_eq_iff foldl_prs_aux [OF a b])
```
```   301
```
```   302 (* induct_tac doesn't accept 'arbitrary', so we manually 'spec' *)
```
```   303 lemma foldl_rsp[quot_respect]:
```
```   304   assumes q1: "Quotient3 R1 Abs1 Rep1"
```
```   305   and     q2: "Quotient3 R2 Abs2 Rep2"
```
```   306   shows "((R1 ===> R2 ===> R1) ===> R1 ===> list_all2 R2 ===> R1) foldl foldl"
```
```   307   by (rule foldl_transfer)
```
```   308
```
```   309 lemma foldr_rsp[quot_respect]:
```
```   310   assumes q1: "Quotient3 R1 Abs1 Rep1"
```
```   311   and     q2: "Quotient3 R2 Abs2 Rep2"
```
```   312   shows "((R1 ===> R2 ===> R2) ===> list_all2 R1 ===> R2 ===> R2) foldr foldr"
```
```   313   by (rule foldr_transfer)
```
```   314
```
```   315 lemma list_all2_rsp:
```
```   316   assumes r: "\<forall>x y. R x y \<longrightarrow> (\<forall>a b. R a b \<longrightarrow> S x a = T y b)"
```
```   317   and l1: "list_all2 R x y"
```
```   318   and l2: "list_all2 R a b"
```
```   319   shows "list_all2 S x a = list_all2 T y b"
```
```   320   using l1 l2
```
```   321   by (induct arbitrary: a b rule: list_all2_induct,
```
```   322     auto simp: list_all2_Cons1 list_all2_Cons2 r)
```
```   323
```
```   324 lemma [quot_respect]:
```
```   325   "((R ===> R ===> op =) ===> list_all2 R ===> list_all2 R ===> op =) list_all2 list_all2"
```
```   326   by (rule list_all2_transfer)
```
```   327
```
```   328 lemma [quot_preserve]:
```
```   329   assumes a: "Quotient3 R abs1 rep1"
```
```   330   shows "((abs1 ---> abs1 ---> id) ---> map rep1 ---> map rep1 ---> id) list_all2 = list_all2"
```
```   331   apply (simp add: fun_eq_iff)
```
```   332   apply clarify
```
```   333   apply (induct_tac xa xb rule: list_induct2')
```
```   334   apply (simp_all add: Quotient3_abs_rep[OF a])
```
```   335   done
```
```   336
```
```   337 lemma [quot_preserve]:
```
```   338   assumes a: "Quotient3 R abs1 rep1"
```
```   339   shows "(list_all2 ((rep1 ---> rep1 ---> id) R) l m) = (l = m)"
```
```   340   by (induct l m rule: list_induct2') (simp_all add: Quotient3_rel_rep[OF a])
```
```   341
```
```   342 lemma list_all2_find_element:
```
```   343   assumes a: "x \<in> set a"
```
```   344   and b: "list_all2 R a b"
```
```   345   shows "\<exists>y. (y \<in> set b \<and> R x y)"
```
```   346   using b a by induct auto
```
```   347
```
```   348 lemma list_all2_refl:
```
```   349   assumes a: "\<And>x y. R x y = (R x = R y)"
```
```   350   shows "list_all2 R x x"
```
```   351   by (induct x) (auto simp add: a)
```
```   352
```
```   353 end
```