src/HOL/BNF_FP_Base.thy
 author traytel Thu Mar 06 12:17:26 2014 +0100 (2014-03-06) changeset 55930 25a90cebbbe5 parent 55906 abf91ebd0820 child 55931 62156e694f3d permissions -rw-r--r--
more careful simplification of sets (cf. abf91ebd0820)---yields smaller terms
```     1 (*  Title:      HOL/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
```
```     7 Shared fixed point operations on bounded natural functors.
```
```     8 *)
```
```     9
```
```    10 header {* Shared Fixed Point Operations on Bounded Natural Functors *}
```
```    11
```
```    12 theory BNF_FP_Base
```
```    13 imports BNF_Comp
```
```    14 begin
```
```    15
```
```    16 lemma mp_conj: "(P \<longrightarrow> Q) \<and> R \<Longrightarrow> P \<Longrightarrow> R \<and> Q"
```
```    17 by auto
```
```    18
```
```    19 lemma eq_sym_Unity_conv: "(x = (() = ())) = x"
```
```    20 by blast
```
```    21
```
```    22 lemma case_unit_Unity: "(case u of () \<Rightarrow> f) = f"
```
```    23 by (cases u) (hypsubst, rule unit.case)
```
```    24
```
```    25 lemma case_prod_Pair_iden: "(case p of (x, y) \<Rightarrow> (x, y)) = p"
```
```    26 by simp
```
```    27
```
```    28 lemma unit_all_impI: "(P () \<Longrightarrow> Q ()) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
```
```    29 by simp
```
```    30
```
```    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
```
```    37 lemma pointfree_idE: "f \<circ> g = id \<Longrightarrow> f (g x) = x"
```
```    38 unfolding comp_def fun_eq_iff by simp
```
```    39
```
```    40 lemma o_bij:
```
```    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
```
```    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
```
```    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 case_sum_step:
```
```    57 "case_sum (case_sum f' g') g (Inl p) = case_sum f' g' p"
```
```    58 "case_sum f (case_sum f' g') (Inr p) = case_sum 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
```
```    64 lemma obj_one_pointE: "\<forall>x. s = x \<longrightarrow> P \<Longrightarrow> P"
```
```    65 by blast
```
```    66
```
```    67 lemma type_copy_obj_one_point_absE:
```
```    68   assumes "type_definition Rep Abs UNIV" "\<forall>x. s = Abs x \<longrightarrow> P" shows P
```
```    69   using type_definition.Rep_inverse[OF assms(1)]
```
```    70   by (intro mp[OF spec[OF assms(2), of "Rep s"]]) simp
```
```    71
```
```    72 lemma obj_sumE_f:
```
```    73   assumes "\<forall>x. s = f (Inl x) \<longrightarrow> P" "\<forall>x. s = f (Inr x) \<longrightarrow> P"
```
```    74   shows "\<forall>x. s = f x \<longrightarrow> P"
```
```    75 proof
```
```    76   fix x from assms show "s = f x \<longrightarrow> P" by (cases x) auto
```
```    77 qed
```
```    78
```
```    79 lemma obj_sumE: "\<lbrakk>\<forall>x. s = Inl x \<longrightarrow> P; \<forall>x. s = Inr x \<longrightarrow> P\<rbrakk> \<Longrightarrow> P"
```
```    80 by (cases s) auto
```
```    81
```
```    82 lemma case_sum_if:
```
```    83 "case_sum f g (if p then Inl x else Inr y) = (if p then f x else g y)"
```
```    84 by simp
```
```    85
```
```    86 lemma Inl_Inr_False: "(Inl x = Inr y) = False"
```
```    87 by simp
```
```    88
```
```    89 lemma prod_set_simps:
```
```    90 "fsts (x, y) = {x}"
```
```    91 "snds (x, y) = {y}"
```
```    92 unfolding fsts_def snds_def by simp+
```
```    93
```
```    94 lemma sum_set_simps:
```
```    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 spec2: "\<forall>x y. P x y \<Longrightarrow> P x y"
```
```   102 by blast
```
```   103
```
```   104 lemma rewriteR_comp_comp: "\<lbrakk>g o h = r\<rbrakk> \<Longrightarrow> f o g o h = f o r"
```
```   105   unfolding comp_def fun_eq_iff by auto
```
```   106
```
```   107 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"
```
```   108   unfolding comp_def fun_eq_iff by auto
```
```   109
```
```   110 lemma rewriteL_comp_comp: "\<lbrakk>f o g = l\<rbrakk> \<Longrightarrow> f o (g o h) = l o h"
```
```   111   unfolding comp_def fun_eq_iff by auto
```
```   112
```
```   113 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"
```
```   114   unfolding comp_def fun_eq_iff by auto
```
```   115
```
```   116 lemma convol_o: "<f, g> o h = <f o h, g o h>"
```
```   117   unfolding convol_def by auto
```
```   118
```
```   119 lemma map_pair_o_convol: "map_pair h1 h2 o <f, g> = <h1 o f, h2 o g>"
```
```   120   unfolding convol_def by auto
```
```   121
```
```   122 lemma map_pair_o_convol_id: "(map_pair f id \<circ> <id , g>) x = <id \<circ> f , g> x"
```
```   123   unfolding map_pair_o_convol id_comp comp_id ..
```
```   124
```
```   125 lemma o_case_sum: "h o case_sum f g = case_sum (h o f) (h o g)"
```
```   126   unfolding comp_def by (auto split: sum.splits)
```
```   127
```
```   128 lemma case_sum_o_sum_map: "case_sum f g o sum_map h1 h2 = case_sum (f o h1) (g o h2)"
```
```   129   unfolding comp_def by (auto split: sum.splits)
```
```   130
```
```   131 lemma case_sum_o_sum_map_id: "(case_sum id g o sum_map f id) x = case_sum (f o id) g x"
```
```   132   unfolding case_sum_o_sum_map id_comp comp_id ..
```
```   133
```
```   134 lemma fun_rel_def_butlast:
```
```   135   "fun_rel R (fun_rel S T) f g = (\<forall>x y. R x y \<longrightarrow> (fun_rel S T) (f x) (g y))"
```
```   136   unfolding fun_rel_def ..
```
```   137
```
```   138 lemma subst_eq_imp: "(\<forall>a b. a = b \<longrightarrow> P a b) \<equiv> (\<forall>a. P a a)"
```
```   139   by auto
```
```   140
```
```   141 lemma eq_subset: "op = \<le> (\<lambda>a b. P a b \<or> a = b)"
```
```   142   by auto
```
```   143
```
```   144 lemma eq_le_Grp_id_iff: "(op = \<le> Grp (Collect R) id) = (All R)"
```
```   145   unfolding Grp_def id_apply by blast
```
```   146
```
```   147 lemma Grp_id_mono_subst: "(\<And>x y. Grp P id x y \<Longrightarrow> Grp Q id (f x) (f y)) \<equiv>
```
```   148    (\<And>x. x \<in> P \<Longrightarrow> f x \<in> Q)"
```
```   149   unfolding Grp_def by rule auto
```
```   150
```
```   151 lemma vimage2p_mono: "vimage2p f g R x y \<Longrightarrow> R \<le> S \<Longrightarrow> vimage2p f g S x y"
```
```   152   unfolding vimage2p_def by blast
```
```   153
```
```   154 lemma vimage2p_refl: "(\<And>x. R x x) \<Longrightarrow> vimage2p f f R x x"
```
```   155   unfolding vimage2p_def by auto
```
```   156
```
```   157 lemma
```
```   158   assumes "type_definition Rep Abs UNIV"
```
```   159   shows type_copy_Rep_o_Abs: "Rep \<circ> Abs = id" and type_copy_Abs_o_Rep: "Abs o Rep = id"
```
```   160   unfolding fun_eq_iff comp_apply id_apply
```
```   161     type_definition.Abs_inverse[OF assms UNIV_I] type_definition.Rep_inverse[OF assms] by simp_all
```
```   162
```
```   163 lemma type_copy_map_comp0_undo:
```
```   164   assumes "type_definition Rep Abs UNIV"
```
```   165           "type_definition Rep' Abs' UNIV"
```
```   166           "type_definition Rep'' Abs'' UNIV"
```
```   167   shows "Abs' o M o Rep'' = (Abs' o M1 o Rep) o (Abs o M2 o Rep'') \<Longrightarrow> M1 o M2 = M"
```
```   168   by (rule sym) (auto simp: fun_eq_iff type_definition.Abs_inject[OF assms(2) UNIV_I UNIV_I]
```
```   169     type_definition.Abs_inverse[OF assms(1) UNIV_I]
```
```   170     type_definition.Abs_inverse[OF assms(3) UNIV_I] dest: spec[of _ "Abs'' x" for x])
```
```   171
```
```   172 lemma vimage2p_id: "vimage2p id id R = R"
```
```   173   unfolding vimage2p_def by auto
```
```   174
```
```   175 lemma vimage2p_comp: "vimage2p (f1 o f2) (g1 o g2) = vimage2p f2 g2 o vimage2p f1 g1"
```
```   176   unfolding fun_eq_iff vimage2p_def o_apply by simp
```
```   177
```
```   178 ML_file "Tools/BNF/bnf_fp_util.ML"
```
```   179 ML_file "Tools/BNF/bnf_fp_def_sugar_tactics.ML"
```
```   180 ML_file "Tools/BNF/bnf_fp_def_sugar.ML"
```
```   181 ML_file "Tools/BNF/bnf_fp_n2m_tactics.ML"
```
```   182 ML_file "Tools/BNF/bnf_fp_n2m.ML"
```
```   183 ML_file "Tools/BNF/bnf_fp_n2m_sugar.ML"
```
```   184
```
```   185 end
```