author | nipkow |
Mon, 30 May 2022 20:58:45 +0200 | |
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parent 74886 | fa5476c54731 |
permissions | -rw-r--r-- |
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(* Title: HOL/BNF_Def.thy |
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Author: Dmitriy Traytel, TU Muenchen |
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Author: Jasmin Blanchette, TU Muenchen |
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Copyright 2012, 2013, 2014 |
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Definition of bounded natural functors. |
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*) |
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section \<open>Definition of Bounded Natural Functors\<close> |
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theory BNF_Def |
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imports BNF_Cardinal_Arithmetic Fun_Def_Base |
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keywords |
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"print_bnfs" :: diag and |
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"bnf" :: thy_goal_defn |
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begin |
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lemma Collect_case_prodD: "x \<in> Collect (case_prod A) \<Longrightarrow> A (fst x) (snd x)" |
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by auto |
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||
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inductive |
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rel_sum :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> 'a + 'b \<Rightarrow> 'c + 'd \<Rightarrow> bool" for R1 R2 |
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where |
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"R1 a c \<Longrightarrow> rel_sum R1 R2 (Inl a) (Inl c)" |
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| "R2 b d \<Longrightarrow> rel_sum R1 R2 (Inr b) (Inr d)" |
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||
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definition |
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rel_fun :: "('a \<Rightarrow> 'c \<Rightarrow> bool) \<Rightarrow> ('b \<Rightarrow> 'd \<Rightarrow> bool) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('c \<Rightarrow> 'd) \<Rightarrow> bool" |
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where |
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"rel_fun A B = (\<lambda>f g. \<forall>x y. A x y \<longrightarrow> B (f x) (g y))" |
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lemma rel_funI [intro]: |
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assumes "\<And>x y. A x y \<Longrightarrow> B (f x) (g y)" |
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shows "rel_fun A B f g" |
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using assms by (simp add: rel_fun_def) |
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lemma rel_funD: |
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assumes "rel_fun A B f g" and "A x y" |
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shows "B (f x) (g y)" |
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using assms by (simp add: rel_fun_def) |
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lemma rel_fun_mono: |
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"\<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" |
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by(simp add: rel_fun_def) |
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lemma rel_fun_mono' [mono]: |
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"\<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" |
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by(simp add: rel_fun_def) |
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definition rel_set :: "('a \<Rightarrow> 'b \<Rightarrow> bool) \<Rightarrow> 'a set \<Rightarrow> 'b set \<Rightarrow> bool" |
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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))" |
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lemma rel_setI: |
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assumes "\<And>x. x \<in> A \<Longrightarrow> \<exists>y\<in>B. R x y" |
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assumes "\<And>y. y \<in> B \<Longrightarrow> \<exists>x\<in>A. R x y" |
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shows "rel_set R A B" |
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using assms unfolding rel_set_def by simp |
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lemma predicate2_transferD: |
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"\<lbrakk>rel_fun R1 (rel_fun R2 (=)) P Q; a \<in> A; b \<in> B; A \<subseteq> {(x, y). R1 x y}; B \<subseteq> {(x, y). R2 x y}\<rbrakk> \<Longrightarrow> |
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P (fst a) (fst b) \<longleftrightarrow> Q (snd a) (snd b)" |
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unfolding rel_fun_def by (blast dest!: Collect_case_prodD) |
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definition collect where |
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"collect F x = (\<Union>f \<in> F. f x)" |
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lemma fstI: "x = (y, z) \<Longrightarrow> fst x = y" |
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by simp |
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lemma sndI: "x = (y, z) \<Longrightarrow> snd x = z" |
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by simp |
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lemma bijI': "\<lbrakk>\<And>x y. (f x = f y) = (x = y); \<And>y. \<exists>x. y = f x\<rbrakk> \<Longrightarrow> bij f" |
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unfolding bij_def inj_on_def by auto blast |
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(* Operator: *) |
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definition "Gr A f = {(a, f a) | a. a \<in> A}" |
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definition "Grp A f = (\<lambda>a b. b = f a \<and> a \<in> A)" |
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definition vimage2p where |
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"vimage2p f g R = (\<lambda>x y. R (f x) (g y))" |
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lemma collect_comp: "collect F \<circ> g = collect ((\<lambda>f. f \<circ> g) ` F)" |
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by (rule ext) (simp add: collect_def) |
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definition convol ("\<langle>(_,/ _)\<rangle>") where |
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"\<langle>f, g\<rangle> \<equiv> \<lambda>a. (f a, g a)" |
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lemma fst_convol: "fst \<circ> \<langle>f, g\<rangle> = f" |
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apply(rule ext) |
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unfolding convol_def by simp |
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lemma snd_convol: "snd \<circ> \<langle>f, g\<rangle> = g" |
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apply(rule ext) |
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unfolding convol_def by simp |
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lemma convol_mem_GrpI: |
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"x \<in> A \<Longrightarrow> \<langle>id, g\<rangle> x \<in> (Collect (case_prod (Grp A g)))" |
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unfolding convol_def Grp_def by auto |
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definition csquare where |
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"csquare A f1 f2 p1 p2 \<longleftrightarrow> (\<forall> a \<in> A. f1 (p1 a) = f2 (p2 a))" |
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lemma eq_alt: "(=) = Grp UNIV id" |
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unfolding Grp_def by auto |
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lemma leq_conversepI: "R = (=) \<Longrightarrow> R \<le> R\<inverse>\<inverse>" |
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by auto |
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lemma leq_OOI: "R = (=) \<Longrightarrow> R \<le> R OO R" |
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by auto |
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lemma OO_Grp_alt: "(Grp A f)\<inverse>\<inverse> OO Grp A g = (\<lambda>x y. \<exists>z. z \<in> A \<and> f z = x \<and> g z = y)" |
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unfolding Grp_def by auto |
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lemma Grp_UNIV_id: "f = id \<Longrightarrow> (Grp UNIV f)\<inverse>\<inverse> OO Grp UNIV f = Grp UNIV f" |
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unfolding Grp_def by auto |
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lemma Grp_UNIV_idI: "x = y \<Longrightarrow> Grp UNIV id x y" |
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unfolding Grp_def by auto |
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lemma Grp_mono: "A \<le> B \<Longrightarrow> Grp A f \<le> Grp B f" |
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unfolding Grp_def by auto |
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lemma GrpI: "\<lbrakk>f x = y; x \<in> A\<rbrakk> \<Longrightarrow> Grp A f x y" |
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unfolding Grp_def by auto |
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lemma GrpE: "Grp A f x y \<Longrightarrow> (\<lbrakk>f x = y; x \<in> A\<rbrakk> \<Longrightarrow> R) \<Longrightarrow> R" |
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unfolding Grp_def by auto |
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lemma Collect_case_prod_Grp_eqD: "z \<in> Collect (case_prod (Grp A f)) \<Longrightarrow> (f \<circ> fst) z = snd z" |
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unfolding Grp_def comp_def by auto |
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lemma Collect_case_prod_Grp_in: "z \<in> Collect (case_prod (Grp A f)) \<Longrightarrow> fst z \<in> A" |
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unfolding Grp_def comp_def by auto |
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definition "pick_middlep P Q a c = (SOME b. P a b \<and> Q b c)" |
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lemma pick_middlep: |
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"(P OO Q) a c \<Longrightarrow> P a (pick_middlep P Q a c) \<and> Q (pick_middlep P Q a c) c" |
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unfolding pick_middlep_def by (rule someI_ex) auto |
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definition fstOp where |
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"fstOp P Q ac = (fst ac, pick_middlep P Q (fst ac) (snd ac))" |
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definition sndOp where |
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"sndOp P Q ac = (pick_middlep P Q (fst ac) (snd ac), (snd ac))" |
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lemma fstOp_in: "ac \<in> Collect (case_prod (P OO Q)) \<Longrightarrow> fstOp P Q ac \<in> Collect (case_prod P)" |
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unfolding fstOp_def mem_Collect_eq |
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by (subst (asm) surjective_pairing, unfold prod.case) (erule pick_middlep[THEN conjunct1]) |
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lemma fst_fstOp: "fst bc = (fst \<circ> fstOp P Q) bc" |
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unfolding comp_def fstOp_def by simp |
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lemma snd_sndOp: "snd bc = (snd \<circ> sndOp P Q) bc" |
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unfolding comp_def sndOp_def by simp |
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lemma sndOp_in: "ac \<in> Collect (case_prod (P OO Q)) \<Longrightarrow> sndOp P Q ac \<in> Collect (case_prod Q)" |
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unfolding sndOp_def mem_Collect_eq |
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by (subst (asm) surjective_pairing, unfold prod.case) (erule pick_middlep[THEN conjunct2]) |
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lemma csquare_fstOp_sndOp: |
61423 | 165 |
"csquare (Collect (f (P OO Q))) snd fst (fstOp P Q) (sndOp P Q)" |
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unfolding csquare_def fstOp_def sndOp_def using pick_middlep by simp |
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lemma snd_fst_flip: "snd xy = (fst \<circ> (%(x, y). (y, x))) xy" |
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by (simp split: prod.split) |
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|
56635 | 171 |
lemma fst_snd_flip: "fst xy = (snd \<circ> (%(x, y). (y, x))) xy" |
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by (simp split: prod.split) |
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|
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lemma flip_pred: "A \<subseteq> Collect (case_prod (R \<inverse>\<inverse>)) \<Longrightarrow> (%(x, y). (y, x)) ` A \<subseteq> Collect (case_prod R)" |
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by auto |
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176 |
|
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store proper theorems even for fixed points that have no passive live variables
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lemma predicate2_eqD: "A = B \<Longrightarrow> A a b \<longleftrightarrow> B a b" |
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by simp |
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generate "rel_as_srel" and "rel_flip" properties
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|
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lemma case_sum_o_inj: "case_sum f g \<circ> Inl = f" "case_sum f g \<circ> Inr = g" |
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181 |
by auto |
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|
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lemma map_sum_o_inj: "map_sum f g \<circ> Inl = Inl \<circ> f" "map_sum f g \<circ> Inr = Inr \<circ> g" |
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by auto |
57802 | 185 |
|
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lemma card_order_csum_cone_cexp_def: |
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"card_order r \<Longrightarrow> ( |A1| +c cone) ^c r = |Func UNIV (Inl ` A1 \<union> {Inr ()})|" |
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unfolding cexp_def cone_def Field_csum Field_card_of by (auto dest: Field_card_order) |
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changeset
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189 |
|
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lemma If_the_inv_into_in_Func: |
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"\<lbrakk>inj_on g C; C \<subseteq> B \<union> {x}\<rbrakk> \<Longrightarrow> |
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(\<lambda>i. if i \<in> g ` C then the_inv_into C g i else x) \<in> Func UNIV (B \<union> {x})" |
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unfolding Func_def by (auto dest: the_inv_into_into) |
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194 |
|
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lemma If_the_inv_into_f_f: |
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"\<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" |
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unfolding Func_def by (auto elim: the_inv_into_f_f) |
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198 |
|
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lemma the_inv_f_o_f_id: "inj f \<Longrightarrow> (the_inv f \<circ> f) z = id z" |
200 |
by (simp add: the_inv_f_f) |
|
201 |
||
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lemma vimage2pI: "R (f x) (g y) \<Longrightarrow> vimage2p f g R x y" |
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unfolding vimage2p_def . |
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|
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lemma rel_fun_iff_leq_vimage2p: "(rel_fun R S) f g = (R \<le> vimage2p f g S)" |
206 |
unfolding rel_fun_def vimage2p_def by auto |
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207 |
|
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lemma convol_image_vimage2p: "\<langle>f \<circ> fst, g \<circ> snd\<rangle> ` Collect (case_prod (vimage2p f g R)) \<subseteq> Collect (case_prod R)" |
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unfolding vimage2p_def convol_def by auto |
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|
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lemma vimage2p_Grp: "vimage2p f g P = Grp UNIV f OO P OO (Grp UNIV g)\<inverse>\<inverse>" |
212 |
unfolding vimage2p_def Grp_def by auto |
|
213 |
||
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lemma subst_Pair: "P x y \<Longrightarrow> a = (x, y) \<Longrightarrow> P (fst a) (snd a)" |
215 |
by simp |
|
216 |
||
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lemma comp_apply_eq: "f (g x) = h (k x) \<Longrightarrow> (f \<circ> g) x = (h \<circ> k) x" |
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unfolding comp_apply by assumption |
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219 |
|
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lemma refl_ge_eq: "(\<And>x. R x x) \<Longrightarrow> (=) \<le> R" |
59726 | 221 |
by auto |
222 |
||
67399 | 223 |
lemma ge_eq_refl: "(=) \<le> R \<Longrightarrow> R x x" |
59726 | 224 |
by auto |
225 |
||
67399 | 226 |
lemma reflp_eq: "reflp R = ((=) \<le> R)" |
61240 | 227 |
by (auto simp: reflp_def fun_eq_iff) |
228 |
||
229 |
lemma transp_relcompp: "transp r \<longleftrightarrow> r OO r \<le> r" |
|
230 |
by (auto simp: transp_def) |
|
231 |
||
232 |
lemma symp_conversep: "symp R = (R\<inverse>\<inverse> \<le> R)" |
|
233 |
by (auto simp: symp_def fun_eq_iff) |
|
234 |
||
235 |
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" |
|
236 |
by blast |
|
237 |
||
62324 | 238 |
definition eq_onp :: "('a \<Rightarrow> bool) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> bool" |
239 |
where "eq_onp R = (\<lambda>x y. R x \<and> x = y)" |
|
240 |
||
241 |
lemma eq_onp_Grp: "eq_onp P = BNF_Def.Grp (Collect P) id" |
|
242 |
unfolding eq_onp_def Grp_def by auto |
|
243 |
||
244 |
lemma eq_onp_to_eq: "eq_onp P x y \<Longrightarrow> x = y" |
|
245 |
by (simp add: eq_onp_def) |
|
246 |
||
67399 | 247 |
lemma eq_onp_top_eq_eq: "eq_onp top = (=)" |
62324 | 248 |
by (simp add: eq_onp_def) |
249 |
||
250 |
lemma eq_onp_same_args: "eq_onp P x x = P x" |
|
63092 | 251 |
by (auto simp add: eq_onp_def) |
62324 | 252 |
|
253 |
lemma eq_onp_eqD: "eq_onp P = Q \<Longrightarrow> P x = Q x x" |
|
254 |
unfolding eq_onp_def by blast |
|
255 |
||
256 |
lemma Ball_Collect: "Ball A P = (A \<subseteq> (Collect P))" |
|
257 |
by auto |
|
258 |
||
259 |
lemma eq_onp_mono0: "\<forall>x\<in>A. P x \<longrightarrow> Q x \<Longrightarrow> \<forall>x\<in>A. \<forall>y\<in>A. eq_onp P x y \<longrightarrow> eq_onp Q x y" |
|
260 |
unfolding eq_onp_def by auto |
|
261 |
||
67399 | 262 |
lemma eq_onp_True: "eq_onp (\<lambda>_. True) = (=)" |
62324 | 263 |
unfolding eq_onp_def by simp |
264 |
||
67091 | 265 |
lemma Ball_image_comp: "Ball (f ` A) g = Ball A (g \<circ> f)" |
62324 | 266 |
by auto |
267 |
||
62329 | 268 |
lemma rel_fun_Collect_case_prodD: |
67091 | 269 |
"rel_fun A B f g \<Longrightarrow> X \<subseteq> Collect (case_prod A) \<Longrightarrow> x \<in> X \<Longrightarrow> B ((f \<circ> fst) x) ((g \<circ> snd) x)" |
62329 | 270 |
unfolding rel_fun_def by auto |
271 |
||
63714 | 272 |
lemma eq_onp_mono_iff: "eq_onp P \<le> eq_onp Q \<longleftrightarrow> P \<le> Q" |
273 |
unfolding eq_onp_def by auto |
|
274 |
||
69605 | 275 |
ML_file \<open>Tools/BNF/bnf_util.ML\<close> |
276 |
ML_file \<open>Tools/BNF/bnf_tactics.ML\<close> |
|
277 |
ML_file \<open>Tools/BNF/bnf_def_tactics.ML\<close> |
|
278 |
ML_file \<open>Tools/BNF/bnf_def.ML\<close> |
|
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279 |
|
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added new (co)datatype package + theories of ordinals and cardinals (with Dmitriy and Andrei)
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280 |
end |