src/HOL/BNF/BNF_FP_Base.thy
author blanchet
Tue Oct 01 14:05:25 2013 +0200 (2013-10-01)
changeset 54006 9fe1bd54d437
parent 53904 446076262e92
child 54246 8fdb4dc08ed1
permissions -rw-r--r--
renamed theory file
     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 
     7 Shared fixed point operations on bounded natural functors, including
     8 *)
     9 
    10 header {* Shared Fixed Point Operations on Bounded Natural Functors *}
    11 
    12 theory BNF_FP_Base
    13 imports BNF_Comp Ctr_Sugar
    14 begin
    15 
    16 lemma not_TrueE: "\<not> True \<Longrightarrow> P"
    17 by (erule notE, rule TrueI)
    18 
    19 lemma neq_eq_eq_contradict: "\<lbrakk>t \<noteq> u; s = t; s = u\<rbrakk> \<Longrightarrow> P"
    20 by fast
    21 
    22 lemma mp_conj: "(P \<longrightarrow> Q) \<and> R \<Longrightarrow> P \<Longrightarrow> R \<and> Q"
    23 by auto
    24 
    25 lemma eq_sym_Unity_conv: "(x = (() = ())) = x"
    26 by blast
    27 
    28 lemma unit_case_Unity: "(case u of () \<Rightarrow> f) = f"
    29 by (cases u) (hypsubst, rule unit.cases)
    30 
    31 lemma prod_case_Pair_iden: "(case p of (x, y) \<Rightarrow> (x, y)) = p"
    32 by simp
    33 
    34 lemma unit_all_impI: "(P () \<Longrightarrow> Q ()) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
    35 by simp
    36 
    37 lemma prod_all_impI: "(\<And>x y. P (x, y) \<Longrightarrow> Q (x, y)) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
    38 by clarify
    39 
    40 lemma prod_all_impI_step: "(\<And>x. \<forall>y. P (x, y) \<longrightarrow> Q (x, y)) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
    41 by auto
    42 
    43 lemma pointfree_idE: "f \<circ> g = id \<Longrightarrow> f (g x) = x"
    44 unfolding o_def fun_eq_iff by simp
    45 
    46 lemma o_bij:
    47   assumes gf: "g \<circ> f = id" and fg: "f \<circ> g = id"
    48   shows "bij f"
    49 unfolding bij_def inj_on_def surj_def proof safe
    50   fix a1 a2 assume "f a1 = f a2"
    51   hence "g ( f a1) = g (f a2)" by simp
    52   thus "a1 = a2" using gf unfolding fun_eq_iff by simp
    53 next
    54   fix b
    55   have "b = f (g b)"
    56   using fg unfolding fun_eq_iff by simp
    57   thus "EX a. b = f a" by blast
    58 qed
    59 
    60 lemma ssubst_mem: "\<lbrakk>t = s; s \<in> X\<rbrakk> \<Longrightarrow> t \<in> X" by simp
    61 
    62 lemma sum_case_step:
    63 "sum_case (sum_case f' g') g (Inl p) = sum_case f' g' p"
    64 "sum_case f (sum_case f' g') (Inr p) = sum_case f' g' p"
    65 by auto
    66 
    67 lemma one_pointE: "\<lbrakk>\<And>x. s = x \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
    68 by simp
    69 
    70 lemma obj_one_pointE: "\<forall>x. s = x \<longrightarrow> P \<Longrightarrow> P"
    71 by blast
    72 
    73 lemma obj_sumE_f:
    74 "\<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"
    75 by (rule allI) (metis sumE)
    76 
    77 lemma obj_sumE: "\<lbrakk>\<forall>x. s = Inl x \<longrightarrow> P; \<forall>x. s = Inr x \<longrightarrow> P\<rbrakk> \<Longrightarrow> P"
    78 by (cases s) auto
    79 
    80 lemma sum_case_if:
    81 "sum_case f g (if p then Inl x else Inr y) = (if p then f x else g y)"
    82 by simp
    83 
    84 lemma mem_UN_compreh_eq: "(z : \<Union>{y. \<exists>x\<in>A. y = F x}) = (\<exists>x\<in>A. z : F x)"
    85 by blast
    86 
    87 lemma UN_compreh_eq_eq:
    88 "\<Union>{y. \<exists>x\<in>A. y = {}} = {}"
    89 "\<Union>{y. \<exists>x\<in>A. y = {x}} = A"
    90 by blast+
    91 
    92 lemma Inl_Inr_False: "(Inl x = Inr y) = False"
    93 by simp
    94 
    95 lemma prod_set_simps:
    96 "fsts (x, y) = {x}"
    97 "snds (x, y) = {y}"
    98 unfolding fsts_def snds_def by simp+
    99 
   100 lemma sum_set_simps:
   101 "setl (Inl x) = {x}"
   102 "setl (Inr x) = {}"
   103 "setr (Inl x) = {}"
   104 "setr (Inr x) = {x}"
   105 unfolding sum_set_defs by simp+
   106 
   107 lemma prod_rel_simp:
   108 "prod_rel P Q (x, y) (x', y') \<longleftrightarrow> P x x' \<and> Q y y'"
   109 unfolding prod_rel_def by simp
   110 
   111 lemma sum_rel_simps:
   112 "sum_rel P Q (Inl x) (Inl x') \<longleftrightarrow> P x x'"
   113 "sum_rel P Q (Inr y) (Inr y') \<longleftrightarrow> Q y y'"
   114 "sum_rel P Q (Inl x) (Inr y') \<longleftrightarrow> False"
   115 "sum_rel P Q (Inr y) (Inl x') \<longleftrightarrow> False"
   116 unfolding sum_rel_def by simp+
   117 
   118 lemma spec2: "\<forall>x y. P x y \<Longrightarrow> P x y"
   119 by blast
   120 
   121 lemma rewriteR_comp_comp: "\<lbrakk>g o h = r\<rbrakk> \<Longrightarrow> f o g o h = f o r"
   122   unfolding o_def fun_eq_iff by auto
   123 
   124 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"
   125   unfolding o_def fun_eq_iff by auto
   126 
   127 lemma rewriteL_comp_comp: "\<lbrakk>f o g = l\<rbrakk> \<Longrightarrow> f o (g o h) = l o h"
   128   unfolding o_def fun_eq_iff by auto
   129 
   130 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"
   131   unfolding o_def fun_eq_iff by auto
   132 
   133 lemma convol_o: "<f, g> o h = <f o h, g o h>"
   134   unfolding convol_def by auto
   135 
   136 lemma map_pair_o_convol: "map_pair h1 h2 o <f, g> = <h1 o f, h2 o g>"
   137   unfolding convol_def by auto
   138 
   139 lemma map_pair_o_convol_id: "(map_pair f id \<circ> <id , g>) x = <id \<circ> f , g> x"
   140   unfolding map_pair_o_convol id_o o_id ..
   141 
   142 lemma o_sum_case: "h o sum_case f g = sum_case (h o f) (h o g)"
   143   unfolding o_def by (auto split: sum.splits)
   144 
   145 lemma sum_case_o_sum_map: "sum_case f g o sum_map h1 h2 = sum_case (f o h1) (g o h2)"
   146   unfolding o_def by (auto split: sum.splits)
   147 
   148 lemma sum_case_o_sum_map_id: "(sum_case id g o sum_map f id) x = sum_case (f o id) g x"
   149   unfolding sum_case_o_sum_map id_o o_id ..
   150 
   151 lemma fun_rel_def_butlast:
   152   "(fun_rel R (fun_rel S T)) f g = (\<forall>x y. R x y \<longrightarrow> (fun_rel S T) (f x) (g y))"
   153   unfolding fun_rel_def ..
   154 
   155 lemma subst_eq_imp: "(\<forall>a b. a = b \<longrightarrow> P a b) \<equiv> (\<forall>a. P a a)"
   156   by auto
   157 
   158 lemma eq_subset: "op = \<le> (\<lambda>a b. P a b \<or> a = b)"
   159   by auto
   160 
   161 lemma eq_le_Grp_id_iff: "(op = \<le> Grp (Collect R) id) = (All R)"
   162   unfolding Grp_def id_apply by blast
   163 
   164 lemma Grp_id_mono_subst: "(\<And>x y. Grp P id x y \<Longrightarrow> Grp Q id (f x) (f y)) \<equiv>
   165    (\<And>x. x \<in> P \<Longrightarrow> f x \<in> Q)"
   166   unfolding Grp_def by rule auto
   167 
   168 ML_file "Tools/bnf_fp_util.ML"
   169 ML_file "Tools/bnf_fp_def_sugar_tactics.ML"
   170 ML_file "Tools/bnf_fp_def_sugar.ML"
   171 ML_file "Tools/bnf_fp_n2m_tactics.ML"
   172 ML_file "Tools/bnf_fp_n2m.ML"
   173 ML_file "Tools/bnf_fp_n2m_sugar.ML"
   174 ML_file "Tools/bnf_fp_rec_sugar_util.ML"
   175 ML_file "Tools/bnf_fp_rec_sugar_tactics.ML"
   176 ML_file "Tools/bnf_fp_rec_sugar.ML"
   177 
   178 end