src/HOL/BNF_FP_Base.thy
author blanchet
Wed Feb 12 08:35:57 2014 +0100 (2014-02-12)
changeset 55415 05f5fdb8d093
parent 55414 eab03e9cee8a
child 55538 6a5986170c1d
permissions -rw-r--r--
renamed 'nat_{case,rec}' to '{case,rec}_nat'
     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.cases)
    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 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 case_sum_if:
    75 "case_sum f g (if p then Inl x else Inr y) = (if p then f x else g y)"
    76 by simp
    77 
    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:
    82 "\<Union>{y. \<exists>x\<in>A. y = {}} = {}"
    83 "\<Union>{y. \<exists>x\<in>A. y = {x}} = A"
    84 by blast+
    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 ML_file "Tools/BNF/bnf_fp_util.ML"
   152 ML_file "Tools/BNF/bnf_fp_def_sugar_tactics.ML"
   153 ML_file "Tools/BNF/bnf_fp_def_sugar.ML"
   154 ML_file "Tools/BNF/bnf_fp_n2m_tactics.ML"
   155 ML_file "Tools/BNF/bnf_fp_n2m.ML"
   156 ML_file "Tools/BNF/bnf_fp_n2m_sugar.ML"
   157 ML_file "Tools/BNF/bnf_fp_rec_sugar_util.ML"
   158 
   159 end