src/HOL/Library/Quotient_Set.thy
 author wenzelm Fri Apr 13 14:00:26 2012 +0200 (2012-04-13) changeset 47455 26315a545e26 parent 47308 9caab698dbe4 child 47626 f7b1034cb9ce permissions -rw-r--r--
```     1 (*  Title:      HOL/Library/Quotient_Set.thy
```
```     2     Author:     Cezary Kaliszyk and Christian Urban
```
```     3 *)
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```     4
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```     5 header {* Quotient infrastructure for the set type *}
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```     6
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```     7 theory Quotient_Set
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```     8 imports Main Quotient_Syntax
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```     9 begin
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```    10
```
```    11 lemma set_quotient [quot_thm]:
```
```    12   assumes "Quotient3 R Abs Rep"
```
```    13   shows "Quotient3 (set_rel R) (vimage Rep) (vimage Abs)"
```
```    14 proof (rule Quotient3I)
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```    15   from assms have "\<And>x. Abs (Rep x) = x" by (rule Quotient3_abs_rep)
```
```    16   then show "\<And>xs. Rep -` (Abs -` xs) = xs"
```
```    17     unfolding vimage_def by auto
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```    18 next
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```    19   show "\<And>xs. set_rel R (Abs -` xs) (Abs -` xs)"
```
```    20     unfolding set_rel_def vimage_def
```
```    21     by auto (metis Quotient3_rel_abs[OF assms])+
```
```    22 next
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```    23   fix r s
```
```    24   show "set_rel R r s = (set_rel R r r \<and> set_rel R s s \<and> Rep -` r = Rep -` s)"
```
```    25     unfolding set_rel_def vimage_def set_eq_iff
```
```    26     by auto (metis rep_abs_rsp[OF assms] assms[simplified Quotient3_def])+
```
```    27 qed
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```    28
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```    29 declare [[mapQ3 set = (set_rel, set_quotient)]]
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```    30
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```    31 lemma empty_set_rsp[quot_respect]:
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```    32   "set_rel R {} {}"
```
```    33   unfolding set_rel_def by simp
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```    34
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```    35 lemma collect_rsp[quot_respect]:
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```    36   assumes "Quotient3 R Abs Rep"
```
```    37   shows "((R ===> op =) ===> set_rel R) Collect Collect"
```
```    38   by (intro fun_relI) (simp add: fun_rel_def set_rel_def)
```
```    39
```
```    40 lemma collect_prs[quot_preserve]:
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```    41   assumes "Quotient3 R Abs Rep"
```
```    42   shows "((Abs ---> id) ---> op -` Rep) Collect = Collect"
```
```    43   unfolding fun_eq_iff
```
```    44   by (simp add: Quotient3_abs_rep[OF assms])
```
```    45
```
```    46 lemma union_rsp[quot_respect]:
```
```    47   assumes "Quotient3 R Abs Rep"
```
```    48   shows "(set_rel R ===> set_rel R ===> set_rel R) op \<union> op \<union>"
```
```    49   by (intro fun_relI) (simp add: set_rel_def)
```
```    50
```
```    51 lemma union_prs[quot_preserve]:
```
```    52   assumes "Quotient3 R Abs Rep"
```
```    53   shows "(op -` Abs ---> op -` Abs ---> op -` Rep) op \<union> = op \<union>"
```
```    54   unfolding fun_eq_iff
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```    55   by (simp add: Quotient3_abs_rep[OF set_quotient[OF assms]])
```
```    56
```
```    57 lemma diff_rsp[quot_respect]:
```
```    58   assumes "Quotient3 R Abs Rep"
```
```    59   shows "(set_rel R ===> set_rel R ===> set_rel R) op - op -"
```
```    60   by (intro fun_relI) (simp add: set_rel_def)
```
```    61
```
```    62 lemma diff_prs[quot_preserve]:
```
```    63   assumes "Quotient3 R Abs Rep"
```
```    64   shows "(op -` Abs ---> op -` Abs ---> op -` Rep) op - = op -"
```
```    65   unfolding fun_eq_iff
```
```    66   by (simp add: Quotient3_abs_rep[OF set_quotient[OF assms]] vimage_Diff)
```
```    67
```
```    68 lemma inter_rsp[quot_respect]:
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```    69   assumes "Quotient3 R Abs Rep"
```
```    70   shows "(set_rel R ===> set_rel R ===> set_rel R) op \<inter> op \<inter>"
```
```    71   by (intro fun_relI) (auto simp add: set_rel_def)
```
```    72
```
```    73 lemma inter_prs[quot_preserve]:
```
```    74   assumes "Quotient3 R Abs Rep"
```
```    75   shows "(op -` Abs ---> op -` Abs ---> op -` Rep) op \<inter> = op \<inter>"
```
```    76   unfolding fun_eq_iff
```
```    77   by (simp add: Quotient3_abs_rep[OF set_quotient[OF assms]])
```
```    78
```
```    79 lemma mem_prs[quot_preserve]:
```
```    80   assumes "Quotient3 R Abs Rep"
```
```    81   shows "(Rep ---> op -` Abs ---> id) op \<in> = op \<in>"
```
```    82   by (simp add: fun_eq_iff Quotient3_abs_rep[OF assms])
```
```    83
```
```    84 lemma mem_rsp[quot_respect]:
```
```    85   shows "(R ===> set_rel R ===> op =) op \<in> op \<in>"
```
```    86   by (intro fun_relI) (simp add: set_rel_def)
```
```    87
```
```    88 end
```