(* Title: LCF/LCF.thy
Author: Tobias Nipkow
Copyright 1992 University of Cambridge
*)
header {* LCF on top of First-Order Logic *}
theory LCF
imports FOL
begin
text {* This theory is based on Lawrence Paulson's book Logic and Computation. *}
subsection {* Natural Deduction Rules for LCF *}
classes cpo < "term"
default_sort cpo
typedecl tr
typedecl void
typedecl ('a,'b) "*" (infixl "*" 6)
typedecl ('a,'b) "+" (infixl "+" 5)
arities
"fun" :: (cpo, cpo) cpo
"*" :: (cpo, cpo) cpo
"+" :: (cpo, cpo) cpo
tr :: cpo
void :: cpo
consts
UU :: "'a"
TT :: "tr"
FF :: "tr"
FIX :: "('a => 'a) => 'a"
FST :: "'a*'b => 'a"
SND :: "'a*'b => 'b"
INL :: "'a => 'a+'b"
INR :: "'b => 'a+'b"
WHEN :: "['a=>'c, 'b=>'c, 'a+'b] => 'c"
adm :: "('a => o) => o"
VOID :: "void" ("'(')")
PAIR :: "['a,'b] => 'a*'b" ("(1<_,/_>)" [0,0] 100)
COND :: "[tr,'a,'a] => 'a" ("(_ =>/ (_ |/ _))" [60,60,60] 60)
less :: "['a,'a] => o" (infixl "<<" 50)
axioms
(** DOMAIN THEORY **)
eq_def: "x=y == x << y & y << x"
less_trans: "[| x << y; y << z |] ==> x << z"
less_ext: "(ALL x. f(x) << g(x)) ==> f << g"
mono: "[| f << g; x << y |] ==> f(x) << g(y)"
minimal: "UU << x"
FIX_eq: "f(FIX(f)) = FIX(f)"
(** TR **)
tr_cases: "p=UU | p=TT | p=FF"
not_TT_less_FF: "~ TT << FF"
not_FF_less_TT: "~ FF << TT"
not_TT_less_UU: "~ TT << UU"
not_FF_less_UU: "~ FF << UU"
COND_UU: "UU => x | y = UU"
COND_TT: "TT => x | y = x"
COND_FF: "FF => x | y = y"
(** PAIRS **)
surj_pairing: "<FST(z),SND(z)> = z"
FST: "FST(<x,y>) = x"
SND: "SND(<x,y>) = y"
(*** STRICT SUM ***)
INL_DEF: "~x=UU ==> ~INL(x)=UU"
INR_DEF: "~x=UU ==> ~INR(x)=UU"
INL_STRICT: "INL(UU) = UU"
INR_STRICT: "INR(UU) = UU"
WHEN_UU: "WHEN(f,g,UU) = UU"
WHEN_INL: "~x=UU ==> WHEN(f,g,INL(x)) = f(x)"
WHEN_INR: "~x=UU ==> WHEN(f,g,INR(x)) = g(x)"
SUM_EXHAUSTION:
"z = UU | (EX x. ~x=UU & z = INL(x)) | (EX y. ~y=UU & z = INR(y))"
(** VOID **)
void_cases: "(x::void) = UU"
(** INDUCTION **)
induct: "[| adm(P); P(UU); ALL x. P(x) --> P(f(x)) |] ==> P(FIX(f))"
(** 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: "adm(%x. t(x) << u(x))"
adm_not_less: "adm(%x.~ t(x) << u)"
adm_not_free: "adm(%x. A)"
adm_subst: "adm(P) ==> adm(%x. P(t(x)))"
adm_conj: "[| adm(P); adm(Q) |] ==> adm(%x. P(x)&Q(x))"
adm_disj: "[| adm(P); adm(Q) |] ==> adm(%x. P(x)|Q(x))"
adm_imp: "[| adm(%x.~P(x)); adm(Q) |] ==> adm(%x. P(x)-->Q(x))"
adm_all: "(!!y. adm(P(y))) ==> adm(%x. ALL y. P(y,x))"
lemma eq_imp_less1: "x = y ==> x << y"
by (simp add: eq_def)
lemma eq_imp_less2: "x = y ==> 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: "[| x << y; y << x |] ==> x=y"
by (simp add: eq_def)
lemma ext: "(!!x::'a::cpo. f(x)=(g(x)::'b::cpo)) ==> (%x. f(x))=(%x. g(x))"
apply (rule less_anti_sym)
apply (rule less_ext)
apply simp
apply simp
done
lemma cong: "[| f=g; x=y |] ==> f(x)=g(y)"
by simp
lemma less_ap_term: "x << y ==> f(x) << f(y)"
by (rule less_refl [THEN mono])
lemma less_ap_thm: "f << g ==> f(x) << g(x)"
by (rule less_refl [THEN [2] mono])
lemma ap_term: "(x::'a::cpo) = y ==> (f(x)::'b::cpo) = f(y)"
apply (rule cong [OF refl])
apply simp
done
lemma ap_thm: "f = g ==> f(x) = g(x)"
apply (erule cong)
apply (rule refl)
done
lemma UU_abs: "(%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 ==> x=UU"
apply (rule less_anti_sym)
apply assumption
apply (rule minimal)
done
lemma tr_induct: "[| P(UU); P(TT); P(FF) |] ==> ALL b. P(b)"
apply (rule allI)
apply (rule mp)
apply (rule_tac [2] p = b in tr_cases)
apply blast
done
lemma Contrapos: "~ B ==> (A ==> B) ==> ~A"
by blast
lemma not_less_imp_not_eq1: "~ x << y \<Longrightarrow> x \<noteq> y"
apply (erule Contrapos)
apply simp
done
lemma not_less_imp_not_eq2: "~ 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]:
"ALL b. P(b=>x|y) <-> (b=UU-->P(UU)) & (b=TT-->P(x)) & (b=FF-->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:
"[| x = UU --> P(UU); x = TT --> P(xa); x = FF --> P(y) |] ==> P(x => 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 {* Ordered pairs and products *}
lemma expand_all_PROD: "(ALL p. P(p)) <-> (ALL 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 <-> FST(p) << FST(q) & 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 <-> FST(p)=FST(q) & 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> <-> a<<c & b<<d"
by (simp add: PROD_less)
lemma PAIR_eq [simp]: "<a,b> = <c,d> <-> a=c & 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 {* Fixedpoint theory *}
lemma adm_eq: "adm(%x. t(x)=(u(x)::'a::cpo))"
apply (unfold eq_def)
apply (rule adm_conj adm_less)+
done
lemma adm_not_not: "adm(P) ==> adm(%x.~~P(x))"
by simp
lemma not_eq_TT: "ALL p. ~p=TT <-> (p=FF | p=UU)"
and not_eq_FF: "ALL p. ~p=FF <-> (p=TT | p=UU)"
and not_eq_UU: "ALL p. ~p=UU <-> (p=TT | p=FF)"
by (rule tr_induct, simp_all)+
lemma adm_not_eq_tr: "ALL p::tr. adm(%x. ~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
ML {*
fun induct_tac ctxt v i =
res_inst_tac ctxt [(("f", 0), v)] @{thm induct} i THEN
REPEAT (resolve_tac @{thms adm_lemmas} i)
*}
lemma least_FIX: "f(p) = p ==> FIX(f) << p"
apply (tactic {* induct_tac @{context} "f" 1 *})
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: "ALL q. f(q)=q --> 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(%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(%p. <f(FST(p)),g(SND(p))>))"
by (rule FIX_pair [unfolded PROD_eq FST SND, THEN conjunct1])
lemma FIX2: "FIX(g) = SND(FIX(%p. <f(FST(p)),g(SND(p))>))"
by (rule FIX_pair [unfolded PROD_eq FST SND, THEN conjunct2])
lemma induct2:
assumes 1: "adm(%p. P(FST(p),SND(p)))"
and 2: "P(UU::'a,UU::'b)"
and 3: "ALL x y. P(x,y) --> 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 = "%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 {*
fun induct2_tac ctxt (f, g) i =
res_inst_tac ctxt [(("f", 0), f), (("g", 0), g)] @{thm induct2} i THEN
REPEAT(resolve_tac @{thms adm_lemmas} i)
*}
end