src/HOL/BNF_FP_Base.thy
 author blanchet Mon Jan 20 18:24:56 2014 +0100 (2014-01-20) changeset 55058 4e700eb471d4 parent 54485 src/HOL/BNF/BNF_FP_Base.thy@b61b8c9e4cf7 child 55059 ef2e0fb783c6 permissions -rw-r--r--
moved BNF files to 'HOL'
```     1 (*  Title:      HOL/BNF/BNF_FP_Base.thy
```
```     2     Author:     Lorenz Panny, TU Muenchen
```
```     3     Author:     Dmitriy Traytel, TU Muenchen
```
```     4     Author:     Jasmin Blanchette, TU Muenchen
```
```     5     Copyright   2012, 2013
```
```     6
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```     7 Shared fixed point operations on bounded natural functors, including
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```     8 *)
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```     9
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```    10 header {* Shared Fixed Point Operations on Bounded Natural Functors *}
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```    11
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```    12 theory BNF_FP_Base
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```    13 imports BNF_Comp Ctr_Sugar
```
```    14 begin
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```    15
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```    16 lemma mp_conj: "(P \<longrightarrow> Q) \<and> R \<Longrightarrow> P \<Longrightarrow> R \<and> Q"
```
```    17 by auto
```
```    18
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```    19 lemma eq_sym_Unity_conv: "(x = (() = ())) = x"
```
```    20 by blast
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```    21
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```    22 lemma unit_case_Unity: "(case u of () \<Rightarrow> f) = f"
```
```    23 by (cases u) (hypsubst, rule unit.cases)
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```    24
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```    25 lemma prod_case_Pair_iden: "(case p of (x, y) \<Rightarrow> (x, y)) = p"
```
```    26 by simp
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```    27
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```    28 lemma unit_all_impI: "(P () \<Longrightarrow> Q ()) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
```
```    29 by simp
```
```    30
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```    31 lemma prod_all_impI: "(\<And>x y. P (x, y) \<Longrightarrow> Q (x, y)) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
```
```    32 by clarify
```
```    33
```
```    34 lemma prod_all_impI_step: "(\<And>x. \<forall>y. P (x, y) \<longrightarrow> Q (x, y)) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
```
```    35 by auto
```
```    36
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```    37 lemma pointfree_idE: "f \<circ> g = id \<Longrightarrow> f (g x) = x"
```
```    38 unfolding o_def fun_eq_iff by simp
```
```    39
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```    40 lemma o_bij:
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```    41   assumes gf: "g \<circ> f = id" and fg: "f \<circ> g = id"
```
```    42   shows "bij f"
```
```    43 unfolding bij_def inj_on_def surj_def proof safe
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```    44   fix a1 a2 assume "f a1 = f a2"
```
```    45   hence "g ( f a1) = g (f a2)" by simp
```
```    46   thus "a1 = a2" using gf unfolding fun_eq_iff by simp
```
```    47 next
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```    48   fix b
```
```    49   have "b = f (g b)"
```
```    50   using fg unfolding fun_eq_iff by simp
```
```    51   thus "EX a. b = f a" by blast
```
```    52 qed
```
```    53
```
```    54 lemma ssubst_mem: "\<lbrakk>t = s; s \<in> X\<rbrakk> \<Longrightarrow> t \<in> X" by simp
```
```    55
```
```    56 lemma sum_case_step:
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```    57 "sum_case (sum_case f' g') g (Inl p) = sum_case f' g' p"
```
```    58 "sum_case f (sum_case f' g') (Inr p) = sum_case f' g' p"
```
```    59 by auto
```
```    60
```
```    61 lemma one_pointE: "\<lbrakk>\<And>x. s = x \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
```
```    62 by simp
```
```    63
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```    64 lemma obj_one_pointE: "\<forall>x. s = x \<longrightarrow> P \<Longrightarrow> P"
```
```    65 by blast
```
```    66
```
```    67 lemma obj_sumE_f:
```
```    68 "\<lbrakk>\<forall>x. s = f (Inl x) \<longrightarrow> P; \<forall>x. s = f (Inr x) \<longrightarrow> P\<rbrakk> \<Longrightarrow> \<forall>x. s = f x \<longrightarrow> P"
```
```    69 by (rule allI) (metis sumE)
```
```    70
```
```    71 lemma obj_sumE: "\<lbrakk>\<forall>x. s = Inl x \<longrightarrow> P; \<forall>x. s = Inr x \<longrightarrow> P\<rbrakk> \<Longrightarrow> P"
```
```    72 by (cases s) auto
```
```    73
```
```    74 lemma sum_case_if:
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```    75 "sum_case f g (if p then Inl x else Inr y) = (if p then f x else g y)"
```
```    76 by simp
```
```    77
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```    78 lemma mem_UN_compreh_eq: "(z : \<Union>{y. \<exists>x\<in>A. y = F x}) = (\<exists>x\<in>A. z : F x)"
```
```    79 by blast
```
```    80
```
```    81 lemma UN_compreh_eq_eq:
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```    82 "\<Union>{y. \<exists>x\<in>A. y = {}} = {}"
```
```    83 "\<Union>{y. \<exists>x\<in>A. y = {x}} = A"
```
```    84 by blast+
```
```    85
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```    86 lemma Inl_Inr_False: "(Inl x = Inr y) = False"
```
```    87 by simp
```
```    88
```
```    89 lemma prod_set_simps:
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```    90 "fsts (x, y) = {x}"
```
```    91 "snds (x, y) = {y}"
```
```    92 unfolding fsts_def snds_def by simp+
```
```    93
```
```    94 lemma sum_set_simps:
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```    95 "setl (Inl x) = {x}"
```
```    96 "setl (Inr x) = {}"
```
```    97 "setr (Inl x) = {}"
```
```    98 "setr (Inr x) = {x}"
```
```    99 unfolding sum_set_defs by simp+
```
```   100
```
```   101 lemma prod_rel_simp:
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```   102 "prod_rel P Q (x, y) (x', y') \<longleftrightarrow> P x x' \<and> Q y y'"
```
```   103 unfolding prod_rel_def by simp
```
```   104
```
```   105 lemma sum_rel_simps:
```
```   106 "sum_rel P Q (Inl x) (Inl x') \<longleftrightarrow> P x x'"
```
```   107 "sum_rel P Q (Inr y) (Inr y') \<longleftrightarrow> Q y y'"
```
```   108 "sum_rel P Q (Inl x) (Inr y') \<longleftrightarrow> False"
```
```   109 "sum_rel P Q (Inr y) (Inl x') \<longleftrightarrow> False"
```
```   110 unfolding sum_rel_def by simp+
```
```   111
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```   112 lemma spec2: "\<forall>x y. P x y \<Longrightarrow> P x y"
```
```   113 by blast
```
```   114
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```   115 lemma rewriteR_comp_comp: "\<lbrakk>g o h = r\<rbrakk> \<Longrightarrow> f o g o h = f o r"
```
```   116   unfolding o_def fun_eq_iff by auto
```
```   117
```
```   118 lemma rewriteR_comp_comp2: "\<lbrakk>g o h = r1 o r2; f o r1 = l\<rbrakk> \<Longrightarrow> f o g o h = l o r2"
```
```   119   unfolding o_def fun_eq_iff by auto
```
```   120
```
```   121 lemma rewriteL_comp_comp: "\<lbrakk>f o g = l\<rbrakk> \<Longrightarrow> f o (g o h) = l o h"
```
```   122   unfolding o_def fun_eq_iff by auto
```
```   123
```
```   124 lemma rewriteL_comp_comp2: "\<lbrakk>f o g = l1 o l2; l2 o h = r\<rbrakk> \<Longrightarrow> f o (g o h) = l1 o r"
```
```   125   unfolding o_def fun_eq_iff by auto
```
```   126
```
```   127 lemma convol_o: "<f, g> o h = <f o h, g o h>"
```
```   128   unfolding convol_def by auto
```
```   129
```
```   130 lemma map_pair_o_convol: "map_pair h1 h2 o <f, g> = <h1 o f, h2 o g>"
```
```   131   unfolding convol_def by auto
```
```   132
```
```   133 lemma map_pair_o_convol_id: "(map_pair f id \<circ> <id , g>) x = <id \<circ> f , g> x"
```
```   134   unfolding map_pair_o_convol id_o o_id ..
```
```   135
```
```   136 lemma o_sum_case: "h o sum_case f g = sum_case (h o f) (h o g)"
```
```   137   unfolding o_def by (auto split: sum.splits)
```
```   138
```
```   139 lemma sum_case_o_sum_map: "sum_case f g o sum_map h1 h2 = sum_case (f o h1) (g o h2)"
```
```   140   unfolding o_def by (auto split: sum.splits)
```
```   141
```
```   142 lemma sum_case_o_sum_map_id: "(sum_case id g o sum_map f id) x = sum_case (f o id) g x"
```
```   143   unfolding sum_case_o_sum_map id_o o_id ..
```
```   144
```
```   145 lemma fun_rel_def_butlast:
```
```   146   "(fun_rel R (fun_rel S T)) f g = (\<forall>x y. R x y \<longrightarrow> (fun_rel S T) (f x) (g y))"
```
```   147   unfolding fun_rel_def ..
```
```   148
```
```   149 lemma subst_eq_imp: "(\<forall>a b. a = b \<longrightarrow> P a b) \<equiv> (\<forall>a. P a a)"
```
```   150   by auto
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```   151
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```   152 lemma eq_subset: "op = \<le> (\<lambda>a b. P a b \<or> a = b)"
```
```   153   by auto
```
```   154
```
```   155 lemma eq_le_Grp_id_iff: "(op = \<le> Grp (Collect R) id) = (All R)"
```
```   156   unfolding Grp_def id_apply by blast
```
```   157
```
```   158 lemma Grp_id_mono_subst: "(\<And>x y. Grp P id x y \<Longrightarrow> Grp Q id (f x) (f y)) \<equiv>
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```   159    (\<And>x. x \<in> P \<Longrightarrow> f x \<in> Q)"
```
```   160   unfolding Grp_def by rule auto
```
```   161
```
```   162 ML_file "Tools/bnf_fp_util.ML"
```
```   163 ML_file "Tools/bnf_fp_def_sugar_tactics.ML"
```
```   164 ML_file "Tools/bnf_fp_def_sugar.ML"
```
```   165 ML_file "Tools/bnf_fp_n2m_tactics.ML"
```
```   166 ML_file "Tools/bnf_fp_n2m.ML"
```
```   167 ML_file "Tools/bnf_fp_n2m_sugar.ML"
```
```   168 ML_file "Tools/bnf_fp_rec_sugar_util.ML"
```
```   169
```
```   170 end
```