src/HOL/Library/Quotient_Option.thy
 author Cezary Kaliszyk Fri Feb 19 13:54:19 2010 +0100 (2010-02-19) changeset 35222 4f1fba00f66d child 35788 f1deaca15ca3 permissions -rw-r--r--
Initial version of HOL quotient package.
```     1 (*  Title:      Quotient_Option.thy
```
```     2     Author:     Cezary Kaliszyk and Christian Urban
```
```     3 *)
```
```     4 theory Quotient_Option
```
```     5 imports Main Quotient_Syntax
```
```     6 begin
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```     7
```
```     8 section {* Quotient infrastructure for the option type. *}
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```     9
```
```    10 fun
```
```    11   option_rel
```
```    12 where
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```    13   "option_rel R None None = True"
```
```    14 | "option_rel R (Some x) None = False"
```
```    15 | "option_rel R None (Some x) = False"
```
```    16 | "option_rel R (Some x) (Some y) = R x y"
```
```    17
```
```    18 declare [[map option = (Option.map, option_rel)]]
```
```    19
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```    20 text {* should probably be in Option.thy *}
```
```    21 lemma split_option_all:
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```    22   shows "(\<forall>x. P x) \<longleftrightarrow> P None \<and> (\<forall>a. P (Some a))"
```
```    23   apply(auto)
```
```    24   apply(case_tac x)
```
```    25   apply(simp_all)
```
```    26   done
```
```    27
```
```    28 lemma option_quotient[quot_thm]:
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```    29   assumes q: "Quotient R Abs Rep"
```
```    30   shows "Quotient (option_rel R) (Option.map Abs) (Option.map Rep)"
```
```    31   unfolding Quotient_def
```
```    32   apply(simp add: split_option_all)
```
```    33   apply(simp add: Quotient_abs_rep[OF q] Quotient_rel_rep[OF q])
```
```    34   using q
```
```    35   unfolding Quotient_def
```
```    36   apply(blast)
```
```    37   done
```
```    38
```
```    39 lemma option_equivp[quot_equiv]:
```
```    40   assumes a: "equivp R"
```
```    41   shows "equivp (option_rel R)"
```
```    42   apply(rule equivpI)
```
```    43   unfolding reflp_def symp_def transp_def
```
```    44   apply(simp_all add: split_option_all)
```
```    45   apply(blast intro: equivp_reflp[OF a])
```
```    46   apply(blast intro: equivp_symp[OF a])
```
```    47   apply(blast intro: equivp_transp[OF a])
```
```    48   done
```
```    49
```
```    50 lemma option_None_rsp[quot_respect]:
```
```    51   assumes q: "Quotient R Abs Rep"
```
```    52   shows "option_rel R None None"
```
```    53   by simp
```
```    54
```
```    55 lemma option_Some_rsp[quot_respect]:
```
```    56   assumes q: "Quotient R Abs Rep"
```
```    57   shows "(R ===> option_rel R) Some Some"
```
```    58   by simp
```
```    59
```
```    60 lemma option_None_prs[quot_preserve]:
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```    61   assumes q: "Quotient R Abs Rep"
```
```    62   shows "Option.map Abs None = None"
```
```    63   by simp
```
```    64
```
```    65 lemma option_Some_prs[quot_preserve]:
```
```    66   assumes q: "Quotient R Abs Rep"
```
```    67   shows "(Rep ---> Option.map Abs) Some = Some"
```
```    68   apply(simp add: expand_fun_eq)
```
```    69   apply(simp add: Quotient_abs_rep[OF q])
```
```    70   done
```
```    71
```
```    72 lemma option_map_id[id_simps]:
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```    73   shows "Option.map id = id"
```
```    74   by (simp add: expand_fun_eq split_option_all)
```
```    75
```
```    76 lemma option_rel_eq[id_simps]:
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```    77   shows "option_rel (op =) = (op =)"
```
```    78   by (simp add: expand_fun_eq split_option_all)
```
```    79
```
```    80 end
```