src/HOL/BNF_Fixpoint_Base.thy
 author traytel Thu Apr 07 17:56:22 2016 +0200 (2016-04-07) changeset 62905 52c5a25e0c96 parent 62335 e85c42f4f30a child 63046 8053ef5a0174 permissions -rw-r--r--
derive (co)rec uniformly from (un)fold
```     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 ML_file "Tools/BNF/bnf_fp_util_tactics.ML"
```
```   288 ML_file "Tools/BNF/bnf_fp_util.ML"
```
```   289 ML_file "Tools/BNF/bnf_fp_def_sugar_tactics.ML"
```
```   290 ML_file "Tools/BNF/bnf_fp_def_sugar.ML"
```
```   291 ML_file "Tools/BNF/bnf_fp_n2m_tactics.ML"
```
```   292 ML_file "Tools/BNF/bnf_fp_n2m.ML"
```
```   293 ML_file "Tools/BNF/bnf_fp_n2m_sugar.ML"
```
```   294
```
```   295 end
```