(* Title: LCF/LCF.thy
Author: Tobias Nipkow
Copyright 1992 University of Cambridge
*)
section \<open>LCF on top of First-Order Logic\<close>
theory LCF
imports FOL
begin
text \<open>This theory is based on Lawrence Paulson's book Logic and Computation.\<close>
subsection \<open>Natural Deduction Rules for LCF\<close>
class cpo = "term"
default_sort cpo
typedecl tr
typedecl void
typedecl ('a,'b) prod (infixl "*" 6)
typedecl ('a,'b) sum (infixl "+" 5)
instance "fun" :: (cpo, cpo) cpo ..
instance prod :: (cpo, cpo) cpo ..
instance sum :: (cpo, cpo) cpo ..
instance tr :: cpo ..
instance void :: cpo ..
consts
UU :: "'a"
TT :: "tr"
FF :: "tr"
FIX :: "('a \<Rightarrow> 'a) \<Rightarrow> 'a"
FST :: "'a*'b \<Rightarrow> 'a"
SND :: "'a*'b \<Rightarrow> 'b"
INL :: "'a \<Rightarrow> 'a+'b"
INR :: "'b \<Rightarrow> 'a+'b"
WHEN :: "['a\<Rightarrow>'c, 'b\<Rightarrow>'c, 'a+'b] \<Rightarrow> 'c"
adm :: "('a \<Rightarrow> o) \<Rightarrow> o"
VOID :: "void" ("'(')")
PAIR :: "['a,'b] \<Rightarrow> 'a*'b" ("(1<_,/_>)" [0,0] 100)
COND :: "[tr,'a,'a] \<Rightarrow> 'a" ("(_ \<Rightarrow>/ (_ |/ _))" [60,60,60] 60)
less :: "['a,'a] \<Rightarrow> o" (infixl "<<" 50)
axiomatization where
(** DOMAIN THEORY **)
eq_def: "x=y == x << y \<and> y << x" and
less_trans: "\<lbrakk>x << y; y << z\<rbrakk> \<Longrightarrow> x << z" and
less_ext: "(\<forall>x. f(x) << g(x)) \<Longrightarrow> f << g" and
mono: "\<lbrakk>f << g; x << y\<rbrakk> \<Longrightarrow> f(x) << g(y)" and
minimal: "UU << x" and
FIX_eq: "\<And>f. f(FIX(f)) = FIX(f)"
axiomatization where
(** TR **)
tr_cases: "p=UU \<or> p=TT \<or> p=FF" and
not_TT_less_FF: "\<not> TT << FF" and
not_FF_less_TT: "\<not> FF << TT" and
not_TT_less_UU: "\<not> TT << UU" and
not_FF_less_UU: "\<not> FF << UU" and
COND_UU: "UU \<Rightarrow> x | y = UU" and
COND_TT: "TT \<Rightarrow> x | y = x" and
COND_FF: "FF \<Rightarrow> x | y = y"
axiomatization where
(** PAIRS **)
surj_pairing: "<FST(z),SND(z)> = z" and
FST: "FST(<x,y>) = x" and
SND: "SND(<x,y>) = y"
axiomatization where
(*** STRICT SUM ***)
INL_DEF: "\<not>x=UU \<Longrightarrow> \<not>INL(x)=UU" and
INR_DEF: "\<not>x=UU \<Longrightarrow> \<not>INR(x)=UU" and
INL_STRICT: "INL(UU) = UU" and
INR_STRICT: "INR(UU) = UU" and
WHEN_UU: "WHEN(f,g,UU) = UU" and
WHEN_INL: "\<not>x=UU \<Longrightarrow> WHEN(f,g,INL(x)) = f(x)" and
WHEN_INR: "\<not>x=UU \<Longrightarrow> WHEN(f,g,INR(x)) = g(x)" and
SUM_EXHAUSTION:
"z = UU \<or> (\<exists>x. \<not>x=UU \<and> z = INL(x)) \<or> (\<exists>y. \<not>y=UU \<and> z = INR(y))"
axiomatization where
(** VOID **)
void_cases: "(x::void) = UU"
(** INDUCTION **)
axiomatization where
induct: "\<lbrakk>adm(P); P(UU); \<forall>x. P(x) \<longrightarrow> P(f(x))\<rbrakk> \<Longrightarrow> P(FIX(f))"
axiomatization where
(** Admissibility / Chain Completeness **)
(* All rules can be found on pages 199--200 of Larry's LCF book.
Note that "easiness" of types is not taken into account
because it cannot be expressed schematically; flatness could be. *)
adm_less: "\<And>t u. adm(\<lambda>x. t(x) << u(x))" and
adm_not_less: "\<And>t u. adm(\<lambda>x.\<not> t(x) << u)" and
adm_not_free: "\<And>A. adm(\<lambda>x. A)" and
adm_subst: "\<And>P t. adm(P) \<Longrightarrow> adm(\<lambda>x. P(t(x)))" and
adm_conj: "\<And>P Q. \<lbrakk>adm(P); adm(Q)\<rbrakk> \<Longrightarrow> adm(\<lambda>x. P(x)\<and>Q(x))" and
adm_disj: "\<And>P Q. \<lbrakk>adm(P); adm(Q)\<rbrakk> \<Longrightarrow> adm(\<lambda>x. P(x)\<or>Q(x))" and
adm_imp: "\<And>P Q. \<lbrakk>adm(\<lambda>x.\<not>P(x)); adm(Q)\<rbrakk> \<Longrightarrow> adm(\<lambda>x. P(x)\<longrightarrow>Q(x))" and
adm_all: "\<And>P. (\<And>y. adm(P(y))) \<Longrightarrow> adm(\<lambda>x. \<forall>y. P(y,x))"
lemma eq_imp_less1: "x = y \<Longrightarrow> x << y"
by (simp add: eq_def)
lemma eq_imp_less2: "x = y \<Longrightarrow> y << x"
by (simp add: eq_def)
lemma less_refl [simp]: "x << x"
apply (rule eq_imp_less1)
apply (rule refl)
done
lemma less_anti_sym: "\<lbrakk>x << y; y << x\<rbrakk> \<Longrightarrow> x=y"
by (simp add: eq_def)
lemma ext: "(\<And>x::'a::cpo. f(x)=(g(x)::'b::cpo)) \<Longrightarrow> (\<lambda>x. f(x))=(\<lambda>x. g(x))"
apply (rule less_anti_sym)
apply (rule less_ext)
apply simp
apply simp
done
lemma cong: "\<lbrakk>f = g; x = y\<rbrakk> \<Longrightarrow> f(x)=g(y)"
by simp
lemma less_ap_term: "x << y \<Longrightarrow> f(x) << f(y)"
by (rule less_refl [THEN mono])
lemma less_ap_thm: "f << g \<Longrightarrow> f(x) << g(x)"
by (rule less_refl [THEN [2] mono])
lemma ap_term: "(x::'a::cpo) = y \<Longrightarrow> (f(x)::'b::cpo) = f(y)"
apply (rule cong [OF refl])
apply simp
done
lemma ap_thm: "f = g \<Longrightarrow> f(x) = g(x)"
apply (erule cong)
apply (rule refl)
done
lemma UU_abs: "(\<lambda>x::'a::cpo. UU) = UU"
apply (rule less_anti_sym)
prefer 2
apply (rule minimal)
apply (rule less_ext)
apply (rule allI)
apply (rule minimal)
done
lemma UU_app: "UU(x) = UU"
by (rule UU_abs [symmetric, THEN ap_thm])
lemma less_UU: "x << UU \<Longrightarrow> x=UU"
apply (rule less_anti_sym)
apply assumption
apply (rule minimal)
done
lemma tr_induct: "\<lbrakk>P(UU); P(TT); P(FF)\<rbrakk> \<Longrightarrow> \<forall>b. P(b)"
apply (rule allI)
apply (rule mp)
apply (rule_tac [2] p = b in tr_cases)
apply blast
done
lemma Contrapos: "\<not> B \<Longrightarrow> (A \<Longrightarrow> B) \<Longrightarrow> \<not>A"
by blast
lemma not_less_imp_not_eq1: "\<not> x << y \<Longrightarrow> x \<noteq> y"
apply (erule Contrapos)
apply simp
done
lemma not_less_imp_not_eq2: "\<not> y << x \<Longrightarrow> x \<noteq> y"
apply (erule Contrapos)
apply simp
done
lemma not_UU_eq_TT: "UU \<noteq> TT"
by (rule not_less_imp_not_eq2) (rule not_TT_less_UU)
lemma not_UU_eq_FF: "UU \<noteq> FF"
by (rule not_less_imp_not_eq2) (rule not_FF_less_UU)
lemma not_TT_eq_UU: "TT \<noteq> UU"
by (rule not_less_imp_not_eq1) (rule not_TT_less_UU)
lemma not_TT_eq_FF: "TT \<noteq> FF"
by (rule not_less_imp_not_eq1) (rule not_TT_less_FF)
lemma not_FF_eq_UU: "FF \<noteq> UU"
by (rule not_less_imp_not_eq1) (rule not_FF_less_UU)
lemma not_FF_eq_TT: "FF \<noteq> TT"
by (rule not_less_imp_not_eq1) (rule not_FF_less_TT)
lemma COND_cases_iff [rule_format]:
"\<forall>b. P(b\<Rightarrow>x|y) \<longleftrightarrow> (b=UU\<longrightarrow>P(UU)) \<and> (b=TT\<longrightarrow>P(x)) \<and> (b=FF\<longrightarrow>P(y))"
apply (insert not_UU_eq_TT not_UU_eq_FF not_TT_eq_UU
not_TT_eq_FF not_FF_eq_UU not_FF_eq_TT)
apply (rule tr_induct)
apply (simplesubst COND_UU)
apply blast
apply (simplesubst COND_TT)
apply blast
apply (simplesubst COND_FF)
apply blast
done
lemma COND_cases:
"\<lbrakk>x = UU \<longrightarrow> P(UU); x = TT \<longrightarrow> P(xa); x = FF \<longrightarrow> P(y)\<rbrakk> \<Longrightarrow> P(x \<Rightarrow> xa | y)"
apply (rule COND_cases_iff [THEN iffD2])
apply blast
done
lemmas [simp] =
minimal
UU_app
UU_app [THEN ap_thm]
UU_app [THEN ap_thm, THEN ap_thm]
not_TT_less_FF not_FF_less_TT not_TT_less_UU not_FF_less_UU not_UU_eq_TT
not_UU_eq_FF not_TT_eq_UU not_TT_eq_FF not_FF_eq_UU not_FF_eq_TT
COND_UU COND_TT COND_FF
surj_pairing FST SND
subsection \<open>Ordered pairs and products\<close>
lemma expand_all_PROD: "(\<forall>p. P(p)) \<longleftrightarrow> (\<forall>x y. P(<x,y>))"
apply (rule iffI)
apply blast
apply (rule allI)
apply (rule surj_pairing [THEN subst])
apply blast
done
lemma PROD_less: "(p::'a*'b) << q \<longleftrightarrow> FST(p) << FST(q) \<and> SND(p) << SND(q)"
apply (rule iffI)
apply (rule conjI)
apply (erule less_ap_term)
apply (erule less_ap_term)
apply (erule conjE)
apply (rule surj_pairing [of p, THEN subst])
apply (rule surj_pairing [of q, THEN subst])
apply (rule mono, erule less_ap_term, assumption)
done
lemma PROD_eq: "p=q \<longleftrightarrow> FST(p)=FST(q) \<and> SND(p)=SND(q)"
apply (rule iffI)
apply simp
apply (unfold eq_def)
apply (simp add: PROD_less)
done
lemma PAIR_less [simp]: "<a,b> << <c,d> \<longleftrightarrow> a<<c \<and> b<<d"
by (simp add: PROD_less)
lemma PAIR_eq [simp]: "<a,b> = <c,d> \<longleftrightarrow> a=c \<and> b=d"
by (simp add: PROD_eq)
lemma UU_is_UU_UU [simp]: "<UU,UU> = UU"
by (rule less_UU) (simp add: PROD_less)
lemma FST_STRICT [simp]: "FST(UU) = UU"
apply (rule subst [OF UU_is_UU_UU])
apply (simp del: UU_is_UU_UU)
done
lemma SND_STRICT [simp]: "SND(UU) = UU"
apply (rule subst [OF UU_is_UU_UU])
apply (simp del: UU_is_UU_UU)
done
subsection \<open>Fixedpoint theory\<close>
lemma adm_eq: "adm(\<lambda>x. t(x)=(u(x)::'a::cpo))"
apply (unfold eq_def)
apply (rule adm_conj adm_less)+
done
lemma adm_not_not: "adm(P) \<Longrightarrow> adm(\<lambda>x. \<not> \<not> P(x))"
by simp
lemma not_eq_TT: "\<forall>p. \<not>p=TT \<longleftrightarrow> (p=FF \<or> p=UU)"
and not_eq_FF: "\<forall>p. \<not>p=FF \<longleftrightarrow> (p=TT \<or> p=UU)"
and not_eq_UU: "\<forall>p. \<not>p=UU \<longleftrightarrow> (p=TT \<or> p=FF)"
by (rule tr_induct, simp_all)+
lemma adm_not_eq_tr: "\<forall>p::tr. adm(\<lambda>x. \<not>t(x)=p)"
apply (rule tr_induct)
apply (simp_all add: not_eq_TT not_eq_FF not_eq_UU)
apply (rule adm_disj adm_eq)+
done
lemmas adm_lemmas =
adm_not_free adm_eq adm_less adm_not_less
adm_not_eq_tr adm_conj adm_disj adm_imp adm_all
method_setup induct = \<open>
Scan.lift Args.embedded_inner_syntax >> (fn v => fn ctxt =>
SIMPLE_METHOD' (fn i =>
Rule_Insts.res_inst_tac ctxt [((("f", 0), Position.none), v)] [] @{thm induct} i THEN
REPEAT (resolve_tac ctxt @{thms adm_lemmas} i)))
\<close>
lemma least_FIX: "f(p) = p \<Longrightarrow> FIX(f) << p"
apply (induct f)
apply (rule minimal)
apply (intro strip)
apply (erule subst)
apply (erule less_ap_term)
done
lemma lfp_is_FIX:
assumes 1: "f(p) = p"
and 2: "\<forall>q. f(q)=q \<longrightarrow> p << q"
shows "p = FIX(f)"
apply (rule less_anti_sym)
apply (rule 2 [THEN spec, THEN mp])
apply (rule FIX_eq)
apply (rule least_FIX)
apply (rule 1)
done
lemma FIX_pair: "<FIX(f),FIX(g)> = FIX(\<lambda>p.<f(FST(p)),g(SND(p))>)"
apply (rule lfp_is_FIX)
apply (simp add: FIX_eq [of f] FIX_eq [of g])
apply (intro strip)
apply (simp add: PROD_less)
apply (rule conjI)
apply (rule least_FIX)
apply (erule subst, rule FST [symmetric])
apply (rule least_FIX)
apply (erule subst, rule SND [symmetric])
done
lemma FIX1: "FIX(f) = FST(FIX(\<lambda>p. <f(FST(p)),g(SND(p))>))"
by (rule FIX_pair [unfolded PROD_eq FST SND, THEN conjunct1])
lemma FIX2: "FIX(g) = SND(FIX(\<lambda>p. <f(FST(p)),g(SND(p))>))"
by (rule FIX_pair [unfolded PROD_eq FST SND, THEN conjunct2])
lemma induct2:
assumes 1: "adm(\<lambda>p. P(FST(p),SND(p)))"
and 2: "P(UU::'a,UU::'b)"
and 3: "\<forall>x y. P(x,y) \<longrightarrow> P(f(x),g(y))"
shows "P(FIX(f),FIX(g))"
apply (rule FIX1 [THEN ssubst, of _ f g])
apply (rule FIX2 [THEN ssubst, of _ f g])
apply (rule induct [where ?f = "\<lambda>x. <f(FST(x)),g(SND(x))>"])
apply (rule 1)
apply simp
apply (rule 2)
apply (simp add: expand_all_PROD)
apply (rule 3)
done
ML \<open>
fun induct2_tac ctxt (f, g) i =
Rule_Insts.res_inst_tac ctxt
[((("f", 0), Position.none), f), ((("g", 0), Position.none), g)] [] @{thm induct2} i THEN
REPEAT(resolve_tac ctxt @{thms adm_lemmas} i)
\<close>
end