1 (* Title: HOL/BNF_Def.thy
2 Author: Dmitriy Traytel, TU Muenchen
3 Author: Jasmin Blanchette, TU Muenchen
4 Copyright 2012, 2013, 2014
6 Definition of bounded natural functors.
9 section \<open>Definition of Bounded Natural Functors\<close>
12 imports BNF_Cardinal_Arithmetic Fun_Def_Base
14 "print_bnfs" :: diag and
18 lemma Collect_splitD: "x \<in> Collect (case_prod A) \<Longrightarrow> A (fst x) (snd x)"
22 rel_sum :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> 'c + 'd \<Rightarrow> bool" for R1 R2
24 "R1 a c \<Longrightarrow> rel_sum R1 R2 (Inl a) (Inl c)"
25 | "R2 b d \<Longrightarrow> rel_sum R1 R2 (Inr b) (Inr d)"
30 rel_fun :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool"
32 "rel_fun A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))"
34 lemma rel_funI [intro]:
35 assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)"
36 shows "rel_fun A B f g"
37 using assms by (simp add: rel_fun_def)
40 assumes "rel_fun A B f g" and "A x y"
42 using assms by (simp add: rel_fun_def)
45 "\<lbrakk> rel_fun X A f g; \<And>x y. Y x y \<longrightarrow> X x y; \<And>x y. A x y \<Longrightarrow> B x y \<rbrakk> \<Longrightarrow> rel_fun Y B f g"
46 by(simp add: rel_fun_def)
48 lemma rel_fun_mono' [mono]:
49 "\<lbrakk> \<And>x y. Y x y \<longrightarrow> X x y; \<And>x y. A x y \<longrightarrow> B x y \<rbrakk> \<Longrightarrow> rel_fun X A f g \<longrightarrow> rel_fun Y B f g"
50 by(simp add: rel_fun_def)
52 definition rel_set :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool"
53 where "rel_set R = (\<lambda>A B. (\<forall>x\<in>A. \<exists>y\<in>B. R x y) \<and> (\<forall>y\<in>B. \<exists>x\<in>A. R x y))"
56 assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y"
57 assumes "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y"
59 using assms unfolding rel_set_def by simp
61 lemma predicate2_transferD:
62 "\<lbrakk>rel_fun R1 (rel_fun R2 (op =)) P Q; a \<in> A; b \<in> B; A \<subseteq> {(x, y). R1 x y}; B \<subseteq> {(x, y). R2 x y}\<rbrakk> \<Longrightarrow>
63 P (fst a) (fst b) \<longleftrightarrow> Q (snd a) (snd b)"
64 unfolding rel_fun_def by (blast dest!: Collect_splitD)
66 definition collect where
67 "collect F x = (\<Union>f \<in> F. f x)"
69 lemma fstI: "x = (y, z) \<Longrightarrow> fst x = y"
72 lemma sndI: "x = (y, z) \<Longrightarrow> snd x = z"
75 lemma bijI': "\<lbrakk>\<And>x y. (f x = f y) = (x = y); \<And>y. \<exists>x. y = f x\<rbrakk> \<Longrightarrow> bij f"
76 unfolding bij_def inj_on_def by auto blast
79 definition "Gr A f = {(a, f a) | a. a \<in> A}"
81 definition "Grp A f = (\<lambda>a b. b = f a \<and> a \<in> A)"
83 definition vimage2p where
84 "vimage2p f g R = (\<lambda>x y. R (f x) (g y))"
86 lemma collect_comp: "collect F \<circ> g = collect ((\<lambda>f. f \<circ> g) ` F)"
87 by (rule ext) (auto simp only: comp_apply collect_def)
89 definition convol ("\<langle>(_,/ _)\<rangle>") where
90 "\<langle>f, g\<rangle> \<equiv> \<lambda>a. (f a, g a)"
92 lemma fst_convol: "fst \<circ> \<langle>f, g\<rangle> = f"
94 unfolding convol_def by simp
96 lemma snd_convol: "snd \<circ> \<langle>f, g\<rangle> = g"
98 unfolding convol_def by simp
100 lemma convol_mem_GrpI:
101 "x \<in> A \<Longrightarrow> \<langle>id, g\<rangle> x \<in> (Collect (case_prod (Grp A g)))"
102 unfolding convol_def Grp_def by auto
104 definition csquare where
105 "csquare A f1 f2 p1 p2 \<longleftrightarrow> (\<forall> a \<in> A. f1 (p1 a) = f2 (p2 a))"
107 lemma eq_alt: "op = = Grp UNIV id"
108 unfolding Grp_def by auto
110 lemma leq_conversepI: "R = op = \<Longrightarrow> R \<le> R^--1"
113 lemma leq_OOI: "R = op = \<Longrightarrow> R \<le> R OO R"
116 lemma OO_Grp_alt: "(Grp A f)^--1 OO Grp A g = (\<lambda>x y. \<exists>z. z \<in> A \<and> f z = x \<and> g z = y)"
117 unfolding Grp_def by auto
119 lemma Grp_UNIV_id: "f = id \<Longrightarrow> (Grp UNIV f)^--1 OO Grp UNIV f = Grp UNIV f"
120 unfolding Grp_def by auto
122 lemma Grp_UNIV_idI: "x = y \<Longrightarrow> Grp UNIV id x y"
123 unfolding Grp_def by auto
125 lemma Grp_mono: "A \<le> B \<Longrightarrow> Grp A f \<le> Grp B f"
126 unfolding Grp_def by auto
128 lemma GrpI: "\<lbrakk>f x = y; x \<in> A\<rbrakk> \<Longrightarrow> Grp A f x y"
129 unfolding Grp_def by auto
131 lemma GrpE: "Grp A f x y \<Longrightarrow> (\<lbrakk>f x = y; x \<in> A\<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R"
132 unfolding Grp_def by auto
134 lemma Collect_split_Grp_eqD: "z \<in> Collect (case_prod (Grp A f)) \<Longrightarrow> (f \<circ> fst) z = snd z"
135 unfolding Grp_def comp_def by auto
137 lemma Collect_split_Grp_inD: "z \<in> Collect (case_prod (Grp A f)) \<Longrightarrow> fst z \<in> A"
138 unfolding Grp_def comp_def by auto
140 definition "pick_middlep P Q a c = (SOME b. P a b \<and> Q b c)"
143 "(P OO Q) a c \<Longrightarrow> P a (pick_middlep P Q a c) \<and> Q (pick_middlep P Q a c) c"
144 unfolding pick_middlep_def apply(rule someI_ex) by auto
146 definition fstOp where
147 "fstOp P Q ac = (fst ac, pick_middlep P Q (fst ac) (snd ac))"
149 definition sndOp where
150 "sndOp P Q ac = (pick_middlep P Q (fst ac) (snd ac), (snd ac))"
152 lemma fstOp_in: "ac \<in> Collect (case_prod (P OO Q)) \<Longrightarrow> fstOp P Q ac \<in> Collect (case_prod P)"
153 unfolding fstOp_def mem_Collect_eq
154 by (subst (asm) surjective_pairing, unfold prod.case) (erule pick_middlep[THEN conjunct1])
156 lemma fst_fstOp: "fst bc = (fst \<circ> fstOp P Q) bc"
157 unfolding comp_def fstOp_def by simp
159 lemma snd_sndOp: "snd bc = (snd \<circ> sndOp P Q) bc"
160 unfolding comp_def sndOp_def by simp
162 lemma sndOp_in: "ac \<in> Collect (case_prod (P OO Q)) \<Longrightarrow> sndOp P Q ac \<in> Collect (case_prod Q)"
163 unfolding sndOp_def mem_Collect_eq
164 by (subst (asm) surjective_pairing, unfold prod.case) (erule pick_middlep[THEN conjunct2])
166 lemma csquare_fstOp_sndOp:
167 "csquare (Collect (f (P OO Q))) snd fst (fstOp P Q) (sndOp P Q)"
168 unfolding csquare_def fstOp_def sndOp_def using pick_middlep by simp
170 lemma snd_fst_flip: "snd xy = (fst \<circ> (%(x, y). (y, x))) xy"
171 by (simp split: prod.split)
173 lemma fst_snd_flip: "fst xy = (snd \<circ> (%(x, y). (y, x))) xy"
174 by (simp split: prod.split)
176 lemma flip_pred: "A \<subseteq> Collect (case_prod (R ^--1)) \<Longrightarrow> (%(x, y). (y, x)) ` A \<subseteq> Collect (case_prod R)"
179 lemma predicate2_eqD: "A = B \<Longrightarrow> A a b \<longleftrightarrow> B a b"
182 lemma case_sum_o_inj: "case_sum f g \<circ> Inl = f" "case_sum f g \<circ> Inr = g"
185 lemma map_sum_o_inj: "map_sum f g o Inl = Inl o f" "map_sum f g o Inr = Inr o g"
188 lemma card_order_csum_cone_cexp_def:
189 "card_order r \<Longrightarrow> ( |A1| +c cone) ^c r = |Func UNIV (Inl ` A1 \<union> {Inr ()})|"
190 unfolding cexp_def cone_def Field_csum Field_card_of by (auto dest: Field_card_order)
192 lemma If_the_inv_into_in_Func:
193 "\<lbrakk>inj_on g C; C \<subseteq> B \<union> {x}\<rbrakk> \<Longrightarrow>
194 (\<lambda>i. if i \<in> g ` C then the_inv_into C g i else x) \<in> Func UNIV (B \<union> {x})"
195 unfolding Func_def by (auto dest: the_inv_into_into)
197 lemma If_the_inv_into_f_f:
198 "\<lbrakk>i \<in> C; inj_on g C\<rbrakk> \<Longrightarrow> ((\<lambda>i. if i \<in> g ` C then the_inv_into C g i else x) \<circ> g) i = id i"
199 unfolding Func_def by (auto elim: the_inv_into_f_f)
201 lemma the_inv_f_o_f_id: "inj f \<Longrightarrow> (the_inv f \<circ> f) z = id z"
202 by (simp add: the_inv_f_f)
204 lemma vimage2pI: "R (f x) (g y) \<Longrightarrow> vimage2p f g R x y"
205 unfolding vimage2p_def by -
207 lemma rel_fun_iff_leq_vimage2p: "(rel_fun R S) f g = (R \<le> vimage2p f g S)"
208 unfolding rel_fun_def vimage2p_def by auto
210 lemma convol_image_vimage2p: "\<langle>f \<circ> fst, g \<circ> snd\<rangle> ` Collect (case_prod (vimage2p f g R)) \<subseteq> Collect (case_prod R)"
211 unfolding vimage2p_def convol_def by auto
213 lemma vimage2p_Grp: "vimage2p f g P = Grp UNIV f OO P OO (Grp UNIV g)\<inverse>\<inverse>"
214 unfolding vimage2p_def Grp_def by auto
216 lemma subst_Pair: "P x y \<Longrightarrow> a = (x, y) \<Longrightarrow> P (fst a) (snd a)"
219 lemma comp_apply_eq: "f (g x) = h (k x) \<Longrightarrow> (f \<circ> g) x = (h \<circ> k) x"
220 unfolding comp_apply by assumption
222 lemma refl_ge_eq: "(\<And>x. R x x) \<Longrightarrow> op = \<le> R"
225 lemma ge_eq_refl: "op = \<le> R \<Longrightarrow> R x x"
228 lemma reflp_eq: "reflp R = (op = \<le> R)"
229 by (auto simp: reflp_def fun_eq_iff)
231 lemma transp_relcompp: "transp r \<longleftrightarrow> r OO r \<le> r"
232 by (auto simp: transp_def)
234 lemma symp_conversep: "symp R = (R\<inverse>\<inverse> \<le> R)"
235 by (auto simp: symp_def fun_eq_iff)
237 lemma diag_imp_eq_le: "(\<And>x. x \<in> A \<Longrightarrow> R x x) \<Longrightarrow> \<forall>x y. x \<in> A \<longrightarrow> y \<in> A \<longrightarrow> x = y \<longrightarrow> R x y"
240 ML_file "Tools/BNF/bnf_util.ML"
241 ML_file "Tools/BNF/bnf_tactics.ML"
242 ML_file "Tools/BNF/bnf_def_tactics.ML"
243 ML_file "Tools/BNF/bnf_def.ML"