(* Title: HOL/BNF_LFP.thy
Author: Dmitriy Traytel, TU Muenchen
Author: Lorenz Panny, TU Muenchen
Author: Jasmin Blanchette, TU Muenchen
Copyright 2012, 2013
Least fixed point operation on bounded natural functors.
*)
header {* Least Fixed Point Operation on Bounded Natural Functors *}
theory BNF_LFP
imports BNF_FP_Base
keywords
"datatype_new" :: thy_decl and
"datatype_compat" :: thy_decl
begin
lemma subset_emptyI: "(\<And>x. x \<in> A \<Longrightarrow> False) \<Longrightarrow> A \<subseteq> {}"
by blast
lemma image_Collect_subsetI:
"(\<And>x. P x \<Longrightarrow> f x \<in> B) \<Longrightarrow> f ` {x. P x} \<subseteq> B"
by blast
lemma Collect_restrict: "{x. x \<in> X \<and> P x} \<subseteq> X"
by auto
lemma prop_restrict: "\<lbrakk>x \<in> Z; Z \<subseteq> {x. x \<in> X \<and> P x}\<rbrakk> \<Longrightarrow> P x"
by auto
lemma underS_I: "\<lbrakk>i \<noteq> j; (i, j) \<in> R\<rbrakk> \<Longrightarrow> i \<in> underS R j"
unfolding underS_def by simp
lemma underS_E: "i \<in> underS R j \<Longrightarrow> i \<noteq> j \<and> (i, j) \<in> R"
unfolding underS_def by simp
lemma underS_Field: "i \<in> underS R j \<Longrightarrow> i \<in> Field R"
unfolding underS_def Field_def by auto
lemma FieldI2: "(i, j) \<in> R \<Longrightarrow> j \<in> Field R"
unfolding Field_def by auto
lemma fst_convol': "fst (<f, g> x) = f x"
using fst_convol unfolding convol_def by simp
lemma snd_convol': "snd (<f, g> x) = g x"
using snd_convol unfolding convol_def by simp
lemma convol_expand_snd: "fst o f = g \<Longrightarrow> <g, snd o f> = f"
unfolding convol_def by auto
lemma convol_expand_snd':
assumes "(fst o f = g)"
shows "h = snd o f \<longleftrightarrow> <g, h> = f"
proof -
from assms have *: "<g, snd o f> = f" by (rule convol_expand_snd)
then have "h = snd o f \<longleftrightarrow> h = snd o <g, snd o f>" by simp
moreover have "\<dots> \<longleftrightarrow> h = snd o f" by (simp add: snd_convol)
moreover have "\<dots> \<longleftrightarrow> <g, h> = f" by (subst (2) *[symmetric]) (auto simp: convol_def fun_eq_iff)
ultimately show ?thesis by simp
qed
definition inver where
"inver g f A = (ALL a : A. g (f a) = a)"
lemma bij_betw_iff_ex:
"bij_betw f A B = (EX g. g ` B = A \<and> inver g f A \<and> inver f g B)" (is "?L = ?R")
proof (rule iffI)
assume ?L
hence f: "f ` A = B" and inj_f: "inj_on f A" unfolding bij_betw_def by auto
let ?phi = "% b a. a : A \<and> f a = b"
have "ALL b : B. EX a. ?phi b a" using f by blast
then obtain g where g: "ALL b : B. g b : A \<and> f (g b) = b"
using bchoice[of B ?phi] by blast
hence gg: "ALL b : f ` A. g b : A \<and> f (g b) = b" using f by blast
have gf: "inver g f A" unfolding inver_def
proof
fix a assume "a \<in> A"
then show "g (f a) = a" using the_inv_into_f_f[OF inj_f, of "g (f a)"]
the_inv_into_f_f[OF inj_f, of a] gg imageI[of a A f] by auto
qed
moreover have "g ` B \<le> A \<and> inver f g B" using g unfolding inver_def by blast
moreover have "A \<le> g ` B"
proof safe
fix a assume a: "a : A"
hence "f a : B" using f by auto
moreover have "a = g (f a)" using a gf unfolding inver_def by auto
ultimately show "a : g ` B" by blast
qed
ultimately show ?R by blast
next
assume ?R
then obtain g where g: "g ` B = A \<and> inver g f A \<and> inver f g B" by blast
show ?L unfolding bij_betw_def
proof safe
show "inj_on f A" unfolding inj_on_def
proof safe
fix a1 a2 assume a: "a1 : A" "a2 : A" and "f a1 = f a2"
hence "g (f a1) = g (f a2)" by simp
thus "a1 = a2" using a g unfolding inver_def by simp
qed
next
fix a assume "a : A"
then obtain b where b: "b : B" and a: "a = g b" using g by blast
hence "b = f (g b)" using g unfolding inver_def by auto
thus "f a : B" unfolding a using b by simp
next
fix b assume "b : B"
hence "g b : A \<and> b = f (g b)" using g unfolding inver_def by auto
thus "b : f ` A" by auto
qed
qed
lemma bij_betw_ex_weakE:
"\<lbrakk>bij_betw f A B\<rbrakk> \<Longrightarrow> \<exists>g. g ` B \<subseteq> A \<and> inver g f A \<and> inver f g B"
by (auto simp only: bij_betw_iff_ex)
lemma inver_surj: "\<lbrakk>g ` B \<subseteq> A; f ` A \<subseteq> B; inver g f A\<rbrakk> \<Longrightarrow> g ` B = A"
unfolding inver_def by auto (rule rev_image_eqI, auto)
lemma inver_mono: "\<lbrakk>A \<subseteq> B; inver f g B\<rbrakk> \<Longrightarrow> inver f g A"
unfolding inver_def by auto
lemma inver_pointfree: "inver f g A = (\<forall>a \<in> A. (f o g) a = a)"
unfolding inver_def by simp
lemma bij_betwE: "bij_betw f A B \<Longrightarrow> \<forall>a\<in>A. f a \<in> B"
unfolding bij_betw_def by auto
lemma bij_betw_imageE: "bij_betw f A B \<Longrightarrow> f ` A = B"
unfolding bij_betw_def by auto
lemma inverE: "\<lbrakk>inver f f' A; x \<in> A\<rbrakk> \<Longrightarrow> f (f' x) = x"
unfolding inver_def by auto
lemma bij_betw_inver1: "bij_betw f A B \<Longrightarrow> inver (inv_into A f) f A"
unfolding bij_betw_def inver_def by auto
lemma bij_betw_inver2: "bij_betw f A B \<Longrightarrow> inver f (inv_into A f) B"
unfolding bij_betw_def inver_def by auto
lemma bij_betwI: "\<lbrakk>bij_betw g B A; inver g f A; inver f g B\<rbrakk> \<Longrightarrow> bij_betw f A B"
by (drule bij_betw_imageE, unfold bij_betw_iff_ex) blast
lemma bij_betwI':
"\<lbrakk>\<And>x y. \<lbrakk>x \<in> X; y \<in> X\<rbrakk> \<Longrightarrow> (f x = f y) = (x = y);
\<And>x. x \<in> X \<Longrightarrow> f x \<in> Y;
\<And>y. y \<in> Y \<Longrightarrow> \<exists>x \<in> X. y = f x\<rbrakk> \<Longrightarrow> bij_betw f X Y"
unfolding bij_betw_def inj_on_def by blast
lemma surj_fun_eq:
assumes surj_on: "f ` X = UNIV" and eq_on: "\<forall>x \<in> X. (g1 o f) x = (g2 o f) x"
shows "g1 = g2"
proof (rule ext)
fix y
from surj_on obtain x where "x \<in> X" and "y = f x" by blast
thus "g1 y = g2 y" using eq_on by simp
qed
lemma Card_order_wo_rel: "Card_order r \<Longrightarrow> wo_rel r"
unfolding wo_rel_def card_order_on_def by blast
lemma Cinfinite_limit: "\<lbrakk>x \<in> Field r; Cinfinite r\<rbrakk> \<Longrightarrow>
\<exists>y \<in> Field r. x \<noteq> y \<and> (x, y) \<in> r"
unfolding cinfinite_def by (auto simp add: infinite_Card_order_limit)
lemma Card_order_trans:
"\<lbrakk>Card_order r; x \<noteq> y; (x, y) \<in> r; y \<noteq> z; (y, z) \<in> r\<rbrakk> \<Longrightarrow> x \<noteq> z \<and> (x, z) \<in> r"
unfolding card_order_on_def well_order_on_def linear_order_on_def
partial_order_on_def preorder_on_def trans_def antisym_def by blast
lemma Cinfinite_limit2:
assumes x1: "x1 \<in> Field r" and x2: "x2 \<in> Field r" and r: "Cinfinite r"
shows "\<exists>y \<in> Field r. (x1 \<noteq> y \<and> (x1, y) \<in> r) \<and> (x2 \<noteq> y \<and> (x2, y) \<in> r)"
proof -
from r have trans: "trans r" and total: "Total r" and antisym: "antisym r"
unfolding card_order_on_def well_order_on_def linear_order_on_def
partial_order_on_def preorder_on_def by auto
obtain y1 where y1: "y1 \<in> Field r" "x1 \<noteq> y1" "(x1, y1) \<in> r"
using Cinfinite_limit[OF x1 r] by blast
obtain y2 where y2: "y2 \<in> Field r" "x2 \<noteq> y2" "(x2, y2) \<in> r"
using Cinfinite_limit[OF x2 r] by blast
show ?thesis
proof (cases "y1 = y2")
case True with y1 y2 show ?thesis by blast
next
case False
with y1(1) y2(1) total have "(y1, y2) \<in> r \<or> (y2, y1) \<in> r"
unfolding total_on_def by auto
thus ?thesis
proof
assume *: "(y1, y2) \<in> r"
with trans y1(3) have "(x1, y2) \<in> r" unfolding trans_def by blast
with False y1 y2 * antisym show ?thesis by (cases "x1 = y2") (auto simp: antisym_def)
next
assume *: "(y2, y1) \<in> r"
with trans y2(3) have "(x2, y1) \<in> r" unfolding trans_def by blast
with False y1 y2 * antisym show ?thesis by (cases "x2 = y1") (auto simp: antisym_def)
qed
qed
qed
lemma Cinfinite_limit_finite: "\<lbrakk>finite X; X \<subseteq> Field r; Cinfinite r\<rbrakk>
\<Longrightarrow> \<exists>y \<in> Field r. \<forall>x \<in> X. (x \<noteq> y \<and> (x, y) \<in> r)"
proof (induct X rule: finite_induct)
case empty thus ?case unfolding cinfinite_def using ex_in_conv[of "Field r"] finite.emptyI by auto
next
case (insert x X)
then obtain y where y: "y \<in> Field r" "\<forall>x \<in> X. (x \<noteq> y \<and> (x, y) \<in> r)" by blast
then obtain z where z: "z \<in> Field r" "x \<noteq> z \<and> (x, z) \<in> r" "y \<noteq> z \<and> (y, z) \<in> r"
using Cinfinite_limit2[OF _ y(1) insert(5), of x] insert(4) by blast
show ?case
apply (intro bexI ballI)
apply (erule insertE)
apply hypsubst
apply (rule z(2))
using Card_order_trans[OF insert(5)[THEN conjunct2]] y(2) z(3)
apply blast
apply (rule z(1))
done
qed
lemma insert_subsetI: "\<lbrakk>x \<in> A; X \<subseteq> A\<rbrakk> \<Longrightarrow> insert x X \<subseteq> A"
by auto
(*helps resolution*)
lemma well_order_induct_imp:
"wo_rel r \<Longrightarrow> (\<And>x. \<forall>y. y \<noteq> x \<and> (y, x) \<in> r \<longrightarrow> y \<in> Field r \<longrightarrow> P y \<Longrightarrow> x \<in> Field r \<longrightarrow> P x) \<Longrightarrow>
x \<in> Field r \<longrightarrow> P x"
by (erule wo_rel.well_order_induct)
lemma meta_spec2:
assumes "(\<And>x y. PROP P x y)"
shows "PROP P x y"
by (rule assms)
lemma nchotomy_relcomppE:
assumes "\<And>y. \<exists>x. y = f x" "(r OO s) a c" "\<And>b. r a (f b) \<Longrightarrow> s (f b) c \<Longrightarrow> P"
shows P
proof (rule relcompp.cases[OF assms(2)], hypsubst)
fix b assume "r a b" "s b c"
moreover from assms(1) obtain b' where "b = f b'" by blast
ultimately show P by (blast intro: assms(3))
qed
lemma vimage2p_rel_fun: "rel_fun (vimage2p f g R) R f g"
unfolding rel_fun_def vimage2p_def by auto
lemma predicate2D_vimage2p: "\<lbrakk>R \<le> vimage2p f g S; R x y\<rbrakk> \<Longrightarrow> S (f x) (g y)"
unfolding vimage2p_def by auto
lemma id_transfer: "rel_fun A A id id"
unfolding rel_fun_def by simp
lemma ssubst_Pair_rhs: "\<lbrakk>(r, s) \<in> R; s' = s\<rbrakk> \<Longrightarrow> (r, s') \<in> R"
by (rule ssubst)
ML_file "Tools/BNF/bnf_lfp_util.ML"
ML_file "Tools/BNF/bnf_lfp_tactics.ML"
ML_file "Tools/BNF/bnf_lfp.ML"
ML_file "Tools/BNF/bnf_lfp_compat.ML"
ML_file "Tools/BNF/bnf_lfp_rec_sugar_more.ML"
hide_fact (open) id_transfer
end