src/HOL/BNF_Fixpoint_Base.thy
author haftmann
Mon Jun 05 15:59:41 2017 +0200 (2017-06-05)
changeset 66010 2f7d39285a1a
parent 63046 8053ef5a0174
child 67091 1393c2340eec
permissions -rw-r--r--
executable domain membership checks
     1 (*  Title:      HOL/BNF_Fixpoint_Base.thy
     2     Author:     Lorenz Panny, TU Muenchen
     3     Author:     Dmitriy Traytel, TU Muenchen
     4     Author:     Jasmin Blanchette, TU Muenchen
     5     Author:     Martin Desharnais, TU Muenchen
     6     Copyright   2012, 2013, 2014
     7 
     8 Shared fixpoint operations on bounded natural functors.
     9 *)
    10 
    11 section \<open>Shared Fixpoint Operations on Bounded Natural Functors\<close>
    12 
    13 theory BNF_Fixpoint_Base
    14 imports BNF_Composition Basic_BNFs
    15 begin
    16 
    17 lemma conj_imp_eq_imp_imp: "(P \<and> Q \<Longrightarrow> PROP R) \<equiv> (P \<Longrightarrow> Q \<Longrightarrow> PROP R)"
    18   by standard simp_all
    19 
    20 lemma predicate2D_conj: "P \<le> Q \<and> R \<Longrightarrow> R \<and> (P x y \<longrightarrow> Q x y)"
    21   by blast
    22 
    23 lemma eq_sym_Unity_conv: "(x = (() = ())) = x"
    24   by blast
    25 
    26 lemma case_unit_Unity: "(case u of () \<Rightarrow> f) = f"
    27   by (cases u) (hypsubst, rule unit.case)
    28 
    29 lemma case_prod_Pair_iden: "(case p of (x, y) \<Rightarrow> (x, y)) = p"
    30   by simp
    31 
    32 lemma unit_all_impI: "(P () \<Longrightarrow> Q ()) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
    33   by simp
    34 
    35 lemma pointfree_idE: "f \<circ> g = id \<Longrightarrow> f (g x) = x"
    36   unfolding comp_def fun_eq_iff by simp
    37 
    38 lemma o_bij:
    39   assumes gf: "g \<circ> f = id" and fg: "f \<circ> g = id"
    40   shows "bij f"
    41 unfolding bij_def inj_on_def surj_def proof safe
    42   fix a1 a2 assume "f a1 = f a2"
    43   hence "g ( f a1) = g (f a2)" by simp
    44   thus "a1 = a2" using gf unfolding fun_eq_iff by simp
    45 next
    46   fix b
    47   have "b = f (g b)"
    48   using fg unfolding fun_eq_iff by simp
    49   thus "EX a. b = f a" by blast
    50 qed
    51 
    52 lemma case_sum_step:
    53   "case_sum (case_sum f' g') g (Inl p) = case_sum f' g' p"
    54   "case_sum f (case_sum f' g') (Inr p) = case_sum f' g' p"
    55   by auto
    56 
    57 lemma obj_one_pointE: "\<forall>x. s = x \<longrightarrow> P \<Longrightarrow> P"
    58   by blast
    59 
    60 lemma type_copy_obj_one_point_absE:
    61   assumes "type_definition Rep Abs UNIV" "\<forall>x. s = Abs x \<longrightarrow> P" shows P
    62   using type_definition.Rep_inverse[OF assms(1)]
    63   by (intro mp[OF spec[OF assms(2), of "Rep s"]]) simp
    64 
    65 lemma obj_sumE_f:
    66   assumes "\<forall>x. s = f (Inl x) \<longrightarrow> P" "\<forall>x. s = f (Inr x) \<longrightarrow> P"
    67   shows "\<forall>x. s = f x \<longrightarrow> P"
    68 proof
    69   fix x from assms show "s = f x \<longrightarrow> P" by (cases x) auto
    70 qed
    71 
    72 lemma case_sum_if:
    73   "case_sum f g (if p then Inl x else Inr y) = (if p then f x else g y)"
    74   by simp
    75 
    76 lemma prod_set_simps[simp]:
    77   "fsts (x, y) = {x}"
    78   "snds (x, y) = {y}"
    79   unfolding prod_set_defs by simp+
    80 
    81 lemma sum_set_simps[simp]:
    82   "setl (Inl x) = {x}"
    83   "setl (Inr x) = {}"
    84   "setr (Inl x) = {}"
    85   "setr (Inr x) = {x}"
    86   unfolding sum_set_defs by simp+
    87 
    88 lemma Inl_Inr_False: "(Inl x = Inr y) = False"
    89   by simp
    90 
    91 lemma Inr_Inl_False: "(Inr x = Inl y) = False"
    92   by simp
    93 
    94 lemma spec2: "\<forall>x y. P x y \<Longrightarrow> P x y"
    95   by blast
    96 
    97 lemma rewriteR_comp_comp: "\<lbrakk>g \<circ> h = r\<rbrakk> \<Longrightarrow> f \<circ> g \<circ> h = f \<circ> r"
    98   unfolding comp_def fun_eq_iff by auto
    99 
   100 lemma rewriteR_comp_comp2: "\<lbrakk>g \<circ> h = r1 \<circ> r2; f \<circ> r1 = l\<rbrakk> \<Longrightarrow> f \<circ> g \<circ> h = l \<circ> r2"
   101   unfolding comp_def fun_eq_iff by auto
   102 
   103 lemma rewriteL_comp_comp: "\<lbrakk>f \<circ> g = l\<rbrakk> \<Longrightarrow> f \<circ> (g \<circ> h) = l \<circ> h"
   104   unfolding comp_def fun_eq_iff by auto
   105 
   106 lemma rewriteL_comp_comp2: "\<lbrakk>f \<circ> g = l1 \<circ> l2; l2 \<circ> h = r\<rbrakk> \<Longrightarrow> f \<circ> (g \<circ> h) = l1 \<circ> r"
   107   unfolding comp_def fun_eq_iff by auto
   108 
   109 lemma convol_o: "\<langle>f, g\<rangle> \<circ> h = \<langle>f \<circ> h, g \<circ> h\<rangle>"
   110   unfolding convol_def by auto
   111 
   112 lemma map_prod_o_convol: "map_prod h1 h2 \<circ> \<langle>f, g\<rangle> = \<langle>h1 \<circ> f, h2 \<circ> g\<rangle>"
   113   unfolding convol_def by auto
   114 
   115 lemma map_prod_o_convol_id: "(map_prod f id \<circ> \<langle>id, g\<rangle>) x = \<langle>id \<circ> f, g\<rangle> x"
   116   unfolding map_prod_o_convol id_comp comp_id ..
   117 
   118 lemma o_case_sum: "h \<circ> case_sum f g = case_sum (h \<circ> f) (h \<circ> g)"
   119   unfolding comp_def by (auto split: sum.splits)
   120 
   121 lemma case_sum_o_map_sum: "case_sum f g \<circ> map_sum h1 h2 = case_sum (f \<circ> h1) (g \<circ> h2)"
   122   unfolding comp_def by (auto split: sum.splits)
   123 
   124 lemma case_sum_o_map_sum_id: "(case_sum id g \<circ> map_sum f id) x = case_sum (f \<circ> id) g x"
   125   unfolding case_sum_o_map_sum id_comp comp_id ..
   126 
   127 lemma rel_fun_def_butlast:
   128   "rel_fun R (rel_fun S T) f g = (\<forall>x y. R x y \<longrightarrow> (rel_fun S T) (f x) (g y))"
   129   unfolding rel_fun_def ..
   130 
   131 lemma subst_eq_imp: "(\<forall>a b. a = b \<longrightarrow> P a b) \<equiv> (\<forall>a. P a a)"
   132   by auto
   133 
   134 lemma eq_subset: "op = \<le> (\<lambda>a b. P a b \<or> a = b)"
   135   by auto
   136 
   137 lemma eq_le_Grp_id_iff: "(op = \<le> Grp (Collect R) id) = (All R)"
   138   unfolding Grp_def id_apply by blast
   139 
   140 lemma Grp_id_mono_subst: "(\<And>x y. Grp P id x y \<Longrightarrow> Grp Q id (f x) (f y)) \<equiv>
   141    (\<And>x. x \<in> P \<Longrightarrow> f x \<in> Q)"
   142   unfolding Grp_def by rule auto
   143 
   144 lemma vimage2p_mono: "vimage2p f g R x y \<Longrightarrow> R \<le> S \<Longrightarrow> vimage2p f g S x y"
   145   unfolding vimage2p_def by blast
   146 
   147 lemma vimage2p_refl: "(\<And>x. R x x) \<Longrightarrow> vimage2p f f R x x"
   148   unfolding vimage2p_def by auto
   149 
   150 lemma
   151   assumes "type_definition Rep Abs UNIV"
   152   shows type_copy_Rep_o_Abs: "Rep \<circ> Abs = id" and type_copy_Abs_o_Rep: "Abs \<circ> Rep = id"
   153   unfolding fun_eq_iff comp_apply id_apply
   154     type_definition.Abs_inverse[OF assms UNIV_I] type_definition.Rep_inverse[OF assms] by simp_all
   155 
   156 lemma type_copy_map_comp0_undo:
   157   assumes "type_definition Rep Abs UNIV"
   158           "type_definition Rep' Abs' UNIV"
   159           "type_definition Rep'' Abs'' UNIV"
   160   shows "Abs' \<circ> M \<circ> Rep'' = (Abs' \<circ> M1 \<circ> Rep) \<circ> (Abs \<circ> M2 \<circ> Rep'') \<Longrightarrow> M1 \<circ> M2 = M"
   161   by (rule sym) (auto simp: fun_eq_iff type_definition.Abs_inject[OF assms(2) UNIV_I UNIV_I]
   162     type_definition.Abs_inverse[OF assms(1) UNIV_I]
   163     type_definition.Abs_inverse[OF assms(3) UNIV_I] dest: spec[of _ "Abs'' x" for x])
   164 
   165 lemma vimage2p_id: "vimage2p id id R = R"
   166   unfolding vimage2p_def by auto
   167 
   168 lemma vimage2p_comp: "vimage2p (f1 \<circ> f2) (g1 \<circ> g2) = vimage2p f2 g2 \<circ> vimage2p f1 g1"
   169   unfolding fun_eq_iff vimage2p_def o_apply by simp
   170 
   171 lemma vimage2p_rel_fun: "rel_fun (vimage2p f g R) R f g"
   172   unfolding rel_fun_def vimage2p_def by auto
   173 
   174 lemma fun_cong_unused_0: "f = (\<lambda>x. g) \<Longrightarrow> f (\<lambda>x. 0) = g"
   175   by (erule arg_cong)
   176 
   177 lemma inj_on_convol_ident: "inj_on (\<lambda>x. (x, f x)) X"
   178   unfolding inj_on_def by simp
   179 
   180 lemma map_sum_if_distrib_then:
   181   "\<And>f g e x y. map_sum f g (if e then Inl x else y) = (if e then Inl (f x) else map_sum f g y)"
   182   "\<And>f g e x y. map_sum f g (if e then Inr x else y) = (if e then Inr (g x) else map_sum f g y)"
   183   by simp_all
   184 
   185 lemma map_sum_if_distrib_else:
   186   "\<And>f g e x y. map_sum f g (if e then x else Inl y) = (if e then map_sum f g x else Inl (f y))"
   187   "\<And>f g e x y. map_sum f g (if e then x else Inr y) = (if e then map_sum f g x else Inr (g y))"
   188   by simp_all
   189 
   190 lemma case_prod_app: "case_prod f x y = case_prod (\<lambda>l r. f l r y) x"
   191   by (case_tac x) simp
   192 
   193 lemma case_sum_map_sum: "case_sum l r (map_sum f g x) = case_sum (l \<circ> f) (r \<circ> g) x"
   194   by (case_tac x) simp+
   195 
   196 lemma case_sum_transfer:
   197   "rel_fun (rel_fun R T) (rel_fun (rel_fun S T) (rel_fun (rel_sum R S) T)) case_sum case_sum"
   198   unfolding rel_fun_def by (auto split: sum.splits)
   199 
   200 lemma case_prod_map_prod: "case_prod h (map_prod f g x) = case_prod (\<lambda>l r. h (f l) (g r)) x"
   201   by (case_tac x) simp+
   202 
   203 lemma case_prod_o_map_prod: "case_prod f \<circ> map_prod g1 g2 = case_prod (\<lambda>l r. f (g1 l) (g2 r))"
   204   unfolding comp_def by auto
   205 
   206 lemma case_prod_transfer:
   207   "(rel_fun (rel_fun A (rel_fun B C)) (rel_fun (rel_prod A B) C)) case_prod case_prod"
   208   unfolding rel_fun_def by simp
   209 
   210 lemma eq_ifI: "(P \<longrightarrow> t = u1) \<Longrightarrow> (\<not> P \<longrightarrow> t = u2) \<Longrightarrow> t = (if P then u1 else u2)"
   211   by simp
   212 
   213 lemma comp_transfer:
   214   "rel_fun (rel_fun B C) (rel_fun (rel_fun A B) (rel_fun A C)) (op \<circ>) (op \<circ>)"
   215   unfolding rel_fun_def by simp
   216 
   217 lemma If_transfer: "rel_fun (op =) (rel_fun A (rel_fun A A)) If If"
   218   unfolding rel_fun_def by simp
   219 
   220 lemma Abs_transfer:
   221   assumes type_copy1: "type_definition Rep1 Abs1 UNIV"
   222   assumes type_copy2: "type_definition Rep2 Abs2 UNIV"
   223   shows "rel_fun R (vimage2p Rep1 Rep2 R) Abs1 Abs2"
   224   unfolding vimage2p_def rel_fun_def
   225     type_definition.Abs_inverse[OF type_copy1 UNIV_I]
   226     type_definition.Abs_inverse[OF type_copy2 UNIV_I] by simp
   227 
   228 lemma Inl_transfer:
   229   "rel_fun S (rel_sum S T) Inl Inl"
   230   by auto
   231 
   232 lemma Inr_transfer:
   233   "rel_fun T (rel_sum S T) Inr Inr"
   234   by auto
   235 
   236 lemma Pair_transfer: "rel_fun A (rel_fun B (rel_prod A B)) Pair Pair"
   237   unfolding rel_fun_def by simp
   238 
   239 lemma eq_onp_live_step: "x = y \<Longrightarrow> eq_onp P a a \<and> x \<longleftrightarrow> P a \<and> y"
   240   by (simp only: eq_onp_same_args)
   241 
   242 lemma top_conj: "top x \<and> P \<longleftrightarrow> P" "P \<and> top x \<longleftrightarrow> P"
   243   by blast+
   244 
   245 lemma fst_convol': "fst (\<langle>f, g\<rangle> x) = f x"
   246   using fst_convol unfolding convol_def by simp
   247 
   248 lemma snd_convol': "snd (\<langle>f, g\<rangle> x) = g x"
   249   using snd_convol unfolding convol_def by simp
   250 
   251 lemma convol_expand_snd: "fst o f = g \<Longrightarrow> \<langle>g, snd o f\<rangle> = f"
   252   unfolding convol_def by auto
   253 
   254 lemma convol_expand_snd':
   255   assumes "(fst o f = g)"
   256   shows "h = snd o f \<longleftrightarrow> \<langle>g, h\<rangle> = f"
   257 proof -
   258   from assms have *: "\<langle>g, snd o f\<rangle> = f" by (rule convol_expand_snd)
   259   then have "h = snd o f \<longleftrightarrow> h = snd o \<langle>g, snd o f\<rangle>" by simp
   260   moreover have "\<dots> \<longleftrightarrow> h = snd o f" by (simp add: snd_convol)
   261   moreover have "\<dots> \<longleftrightarrow> \<langle>g, h\<rangle> = f" by (subst (2) *[symmetric]) (auto simp: convol_def fun_eq_iff)
   262   ultimately show ?thesis by simp
   263 qed
   264 
   265 lemma case_sum_expand_Inr_pointfree: "f o Inl = g \<Longrightarrow> case_sum g (f o Inr) = f"
   266   by (auto split: sum.splits)
   267 
   268 lemma case_sum_expand_Inr': "f o Inl = g \<Longrightarrow> h = f o Inr \<longleftrightarrow> case_sum g h = f"
   269   by (rule iffI) (auto simp add: fun_eq_iff split: sum.splits)
   270 
   271 lemma case_sum_expand_Inr: "f o Inl = g \<Longrightarrow> f x = case_sum g (f o Inr) x"
   272   by (auto split: sum.splits)
   273 
   274 lemma id_transfer: "rel_fun A A id id"
   275   unfolding rel_fun_def by simp
   276 
   277 lemma fst_transfer: "rel_fun (rel_prod A B) A fst fst"
   278   unfolding rel_fun_def by simp
   279 
   280 lemma snd_transfer: "rel_fun (rel_prod A B) B snd snd"
   281   unfolding rel_fun_def by simp
   282 
   283 lemma convol_transfer:
   284   "rel_fun (rel_fun R S) (rel_fun (rel_fun R T) (rel_fun R (rel_prod S T))) BNF_Def.convol BNF_Def.convol"
   285   unfolding rel_fun_def convol_def by auto
   286 
   287 lemma Let_const: "Let x (\<lambda>_. c) = c"
   288   unfolding Let_def ..
   289 
   290 ML_file "Tools/BNF/bnf_fp_util_tactics.ML"
   291 ML_file "Tools/BNF/bnf_fp_util.ML"
   292 ML_file "Tools/BNF/bnf_fp_def_sugar_tactics.ML"
   293 ML_file "Tools/BNF/bnf_fp_def_sugar.ML"
   294 ML_file "Tools/BNF/bnf_fp_n2m_tactics.ML"
   295 ML_file "Tools/BNF/bnf_fp_n2m.ML"
   296 ML_file "Tools/BNF/bnf_fp_n2m_sugar.ML"
   297 
   298 end