src/HOL/Library/Quotient_Sum.thy
 author kuncar Fri Apr 20 18:29:21 2012 +0200 (2012-04-20) changeset 47634 091bcd569441 parent 47624 16d433895d2e child 47635 ebb79474262c permissions -rw-r--r--
hide the invariant constant for relators: invariant_commute infrastracture
```     1 (*  Title:      HOL/Library/Quotient_Sum.thy
```
```     2     Author:     Cezary Kaliszyk, Christian Urban and Brian Huffman
```
```     3 *)
```
```     4
```
```     5 header {* Quotient infrastructure for the sum type *}
```
```     6
```
```     7 theory Quotient_Sum
```
```     8 imports Main Quotient_Syntax
```
```     9 begin
```
```    10
```
```    11 subsection {* Relator for sum type *}
```
```    12
```
```    13 fun
```
```    14   sum_rel :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> 'c + 'd \<Rightarrow> bool"
```
```    15 where
```
```    16   "sum_rel R1 R2 (Inl a1) (Inl b1) = R1 a1 b1"
```
```    17 | "sum_rel R1 R2 (Inl a1) (Inr b2) = False"
```
```    18 | "sum_rel R1 R2 (Inr a2) (Inl b1) = False"
```
```    19 | "sum_rel R1 R2 (Inr a2) (Inr b2) = R2 a2 b2"
```
```    20
```
```    21 lemma sum_rel_unfold:
```
```    22   "sum_rel R1 R2 x y = (case (x, y) of (Inl x, Inl y) \<Rightarrow> R1 x y
```
```    23     | (Inr x, Inr y) \<Rightarrow> R2 x y
```
```    24     | _ \<Rightarrow> False)"
```
```    25   by (cases x) (cases y, simp_all)+
```
```    26
```
```    27 lemma sum_rel_map1:
```
```    28   "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"
```
```    29   by (simp add: sum_rel_unfold split: sum.split)
```
```    30
```
```    31 lemma sum_rel_map2:
```
```    32   "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"
```
```    33   by (simp add: sum_rel_unfold split: sum.split)
```
```    34
```
```    35 lemma sum_map_id [id_simps]:
```
```    36   "sum_map id id = id"
```
```    37   by (simp add: id_def sum_map.identity fun_eq_iff)
```
```    38
```
```    39 lemma sum_rel_eq [id_simps, relator_eq]:
```
```    40   "sum_rel (op =) (op =) = (op =)"
```
```    41   by (simp add: sum_rel_unfold fun_eq_iff split: sum.split)
```
```    42
```
```    43 lemma split_sum_all: "(\<forall>x. P x) \<longleftrightarrow> (\<forall>x. P (Inl x)) \<and> (\<forall>x. P (Inr x))"
```
```    44   by (metis sum.exhaust) (* TODO: move to Sum_Type.thy *)
```
```    45
```
```    46 lemma split_sum_ex: "(\<exists>x. P x) \<longleftrightarrow> (\<exists>x. P (Inl x)) \<or> (\<exists>x. P (Inr x))"
```
```    47   by (metis sum.exhaust) (* TODO: move to Sum_Type.thy *)
```
```    48
```
```    49 lemma sum_reflp:
```
```    50   "reflp R1 \<Longrightarrow> reflp R2 \<Longrightarrow> reflp (sum_rel R1 R2)"
```
```    51   unfolding reflp_def split_sum_all sum_rel.simps by fast
```
```    52
```
```    53 lemma sum_symp:
```
```    54   "symp R1 \<Longrightarrow> symp R2 \<Longrightarrow> symp (sum_rel R1 R2)"
```
```    55   unfolding symp_def split_sum_all sum_rel.simps by fast
```
```    56
```
```    57 lemma sum_transp:
```
```    58   "transp R1 \<Longrightarrow> transp R2 \<Longrightarrow> transp (sum_rel R1 R2)"
```
```    59   unfolding transp_def split_sum_all sum_rel.simps by fast
```
```    60
```
```    61 lemma sum_equivp [quot_equiv]:
```
```    62   "equivp R1 \<Longrightarrow> equivp R2 \<Longrightarrow> equivp (sum_rel R1 R2)"
```
```    63   by (blast intro: equivpI sum_reflp sum_symp sum_transp elim: equivpE)
```
```    64
```
```    65 lemma right_total_sum_rel [transfer_rule]:
```
```    66   "right_total R1 \<Longrightarrow> right_total R2 \<Longrightarrow> right_total (sum_rel R1 R2)"
```
```    67   unfolding right_total_def split_sum_all split_sum_ex by simp
```
```    68
```
```    69 lemma right_unique_sum_rel [transfer_rule]:
```
```    70   "right_unique R1 \<Longrightarrow> right_unique R2 \<Longrightarrow> right_unique (sum_rel R1 R2)"
```
```    71   unfolding right_unique_def split_sum_all by simp
```
```    72
```
```    73 lemma bi_total_sum_rel [transfer_rule]:
```
```    74   "bi_total R1 \<Longrightarrow> bi_total R2 \<Longrightarrow> bi_total (sum_rel R1 R2)"
```
```    75   using assms unfolding bi_total_def split_sum_all split_sum_ex by simp
```
```    76
```
```    77 lemma bi_unique_sum_rel [transfer_rule]:
```
```    78   "bi_unique R1 \<Longrightarrow> bi_unique R2 \<Longrightarrow> bi_unique (sum_rel R1 R2)"
```
```    79   using assms unfolding bi_unique_def split_sum_all by simp
```
```    80
```
```    81 subsection {* Correspondence rules for transfer package *}
```
```    82
```
```    83 lemma Inl_transfer [transfer_rule]: "(A ===> sum_rel A B) Inl Inl"
```
```    84   unfolding fun_rel_def by simp
```
```    85
```
```    86 lemma Inr_transfer [transfer_rule]: "(B ===> sum_rel A B) Inr Inr"
```
```    87   unfolding fun_rel_def by simp
```
```    88
```
```    89 lemma sum_case_transfer [transfer_rule]:
```
```    90   "((A ===> C) ===> (B ===> C) ===> sum_rel A B ===> C) sum_case sum_case"
```
```    91   unfolding fun_rel_def sum_rel_unfold by (simp split: sum.split)
```
```    92
```
```    93 subsection {* Setup for lifting package *}
```
```    94
```
```    95 lemma Quotient_sum:
```
```    96   assumes "Quotient R1 Abs1 Rep1 T1"
```
```    97   assumes "Quotient R2 Abs2 Rep2 T2"
```
```    98   shows "Quotient (sum_rel R1 R2) (sum_map Abs1 Abs2)
```
```    99     (sum_map Rep1 Rep2) (sum_rel T1 T2)"
```
```   100   using assms unfolding Quotient_alt_def
```
```   101   by (simp add: split_sum_all)
```
```   102
```
```   103 declare [[map sum = (sum_rel, Quotient_sum)]]
```
```   104
```
```   105 fun sum_pred :: "('a \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> bool"
```
```   106 where
```
```   107   "sum_pred R1 R2 (Inl a) = R1 a"
```
```   108 | "sum_pred R1 R2 (Inr a) = R2 a"
```
```   109
```
```   110 lemma sum_invariant_commute [invariant_commute]:
```
```   111   "sum_rel (Lifting.invariant P1) (Lifting.invariant P2) = Lifting.invariant (sum_pred P1 P2)"
```
```   112   apply (simp add: fun_eq_iff Lifting.invariant_def)
```
```   113   apply (intro allI)
```
```   114   apply (case_tac x rule: sum.exhaust)
```
```   115   apply (case_tac xa rule: sum.exhaust)
```
```   116   apply auto[2]
```
```   117   apply (case_tac xa rule: sum.exhaust)
```
```   118   apply auto
```
```   119 done
```
```   120
```
```   121 subsection {* Rules for quotient package *}
```
```   122
```
```   123 lemma sum_quotient [quot_thm]:
```
```   124   assumes q1: "Quotient3 R1 Abs1 Rep1"
```
```   125   assumes q2: "Quotient3 R2 Abs2 Rep2"
```
```   126   shows "Quotient3 (sum_rel R1 R2) (sum_map Abs1 Abs2) (sum_map Rep1 Rep2)"
```
```   127   apply (rule Quotient3I)
```
```   128   apply (simp_all add: sum_map.compositionality comp_def sum_map.identity sum_rel_eq sum_rel_map1 sum_rel_map2
```
```   129     Quotient3_abs_rep [OF q1] Quotient3_rel_rep [OF q1] Quotient3_abs_rep [OF q2] Quotient3_rel_rep [OF q2])
```
```   130   using Quotient3_rel [OF q1] Quotient3_rel [OF q2]
```
```   131   apply (simp add: sum_rel_unfold comp_def split: sum.split)
```
```   132   done
```
```   133
```
```   134 declare [[mapQ3 sum = (sum_rel, sum_quotient)]]
```
```   135
```
```   136 lemma sum_Inl_rsp [quot_respect]:
```
```   137   assumes q1: "Quotient3 R1 Abs1 Rep1"
```
```   138   assumes q2: "Quotient3 R2 Abs2 Rep2"
```
```   139   shows "(R1 ===> sum_rel R1 R2) Inl Inl"
```
```   140   by auto
```
```   141
```
```   142 lemma sum_Inr_rsp [quot_respect]:
```
```   143   assumes q1: "Quotient3 R1 Abs1 Rep1"
```
```   144   assumes q2: "Quotient3 R2 Abs2 Rep2"
```
```   145   shows "(R2 ===> sum_rel R1 R2) Inr Inr"
```
```   146   by auto
```
```   147
```
```   148 lemma sum_Inl_prs [quot_preserve]:
```
```   149   assumes q1: "Quotient3 R1 Abs1 Rep1"
```
```   150   assumes q2: "Quotient3 R2 Abs2 Rep2"
```
```   151   shows "(Rep1 ---> sum_map Abs1 Abs2) Inl = Inl"
```
```   152   apply(simp add: fun_eq_iff)
```
```   153   apply(simp add: Quotient3_abs_rep[OF q1])
```
```   154   done
```
```   155
```
```   156 lemma sum_Inr_prs [quot_preserve]:
```
```   157   assumes q1: "Quotient3 R1 Abs1 Rep1"
```
```   158   assumes q2: "Quotient3 R2 Abs2 Rep2"
```
```   159   shows "(Rep2 ---> sum_map Abs1 Abs2) Inr = Inr"
```
```   160   apply(simp add: fun_eq_iff)
```
```   161   apply(simp add: Quotient3_abs_rep[OF q2])
```
```   162   done
```
```   163
```
```   164 end
```