# Theory LCF

```(*  Title:      LCF/LCF.thy
Author:     Tobias Nipkow
Copyright   1992  University of Cambridge
*)

section ‹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›

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 ⇒ '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)

axiomatization where
(** DOMAIN THEORY **)

eq_def:        "x=y == x << y ∧ y << x" and

less_trans:    "⟦x << y; y << z⟧ ⟹ x << z" and

less_ext:      "(∀x. f(x) << g(x)) ⟹ f << g" and

mono:          "⟦f << g; x << y⟧ ⟹ f(x) << g(y)" and

minimal:       "UU << x" and

FIX_eq:        "⋀f. f(FIX(f)) = FIX(f)"

axiomatization where
(** TR **)

tr_cases:      "p=UU ∨ p=TT ∨ p=FF" and

not_TT_less_FF: "¬ TT << FF" and
not_FF_less_TT: "¬ FF << TT" and
not_TT_less_UU: "¬ TT << UU" and
not_FF_less_UU: "¬ FF << UU" and

COND_UU:       "UU ⇒ x | y  =  UU" and
COND_TT:       "TT ⇒ x | y  =  x" and
COND_FF:       "FF ⇒ 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: "¬x=UU ⟹ ¬INL(x)=UU" and
INR_DEF: "¬x=UU ⟹ ¬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: "¬x=UU ⟹ WHEN(f,g,INL(x)) = f(x)" and
WHEN_INR: "¬x=UU ⟹ WHEN(f,g,INR(x)) = g(x)" and

SUM_EXHAUSTION:
"z = UU ∨ (∃x. ¬x=UU ∧ z = INL(x)) ∨ (∃y. ¬y=UU ∧ z = INR(y))"

axiomatization where
(** VOID **)

void_cases:    "(x::void) = UU"

(** INDUCTION **)

axiomatization where
induct: "⟦adm(P); P(UU); ∀x. P(x) ⟶ P(f(x))⟧ ⟹ 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:      "⋀t u. adm(λx. t(x) << u(x))" and
adm_not_less:  "⋀t u. adm(λx.¬ t(x) << u)" and

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)) ⟹ (λx. f(x))=(λx. 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: "(λx::'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)⟧ ⟹ ∀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 ⟹ x ≠ y"
apply (erule Contrapos)
apply simp
done

lemma not_less_imp_not_eq2: "¬ y << x ⟹ x ≠ y"
apply (erule Contrapos)
apply simp
done

lemma not_UU_eq_TT: "UU ≠ TT"
by (rule not_less_imp_not_eq2) (rule not_TT_less_UU)
lemma not_UU_eq_FF: "UU ≠ FF"
by (rule not_less_imp_not_eq2) (rule not_FF_less_UU)
lemma not_TT_eq_UU: "TT ≠ UU"
by (rule not_less_imp_not_eq1) (rule not_TT_less_UU)
lemma not_TT_eq_FF: "TT ≠ FF"
by (rule not_less_imp_not_eq1) (rule not_TT_less_FF)
lemma not_FF_eq_UU: "FF ≠ UU"
by (rule not_less_imp_not_eq1) (rule not_FF_less_UU)
lemma not_FF_eq_TT: "FF ≠ TT"
by (rule not_less_imp_not_eq1) (rule not_FF_less_TT)

lemma COND_cases_iff [rule_format]:
"∀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: "(∀p. P(p)) ⟷ (∀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›

apply (unfold eq_def)
done

by simp

lemma not_eq_TT: "∀p. ¬p=TT ⟷ (p=FF ∨ p=UU)"
and not_eq_FF: "∀p. ¬p=FF ⟷ (p=TT ∨ p=UU)"
and not_eq_UU: "∀p. ¬p=UU ⟷ (p=TT ∨ p=FF)"
by (rule tr_induct, simp_all)+

apply (rule tr_induct)
apply (simp_all add: not_eq_TT not_eq_FF not_eq_UU)
done

method_setup induct = ‹
Scan.lift Parse.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)))
›

lemma least_FIX: "f(p) = p ⟹ 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: "∀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(λp.<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(λp. <f(FST(p)),g(SND(p))>))"
by (rule FIX_pair [unfolded PROD_eq FST SND, THEN conjunct1])

lemma FIX2: "FIX(g) = SND(FIX(λp. <f(FST(p)),g(SND(p))>))"
by (rule FIX_pair [unfolded PROD_eq FST SND, THEN conjunct2])

lemma induct2:
assumes 1: "adm(λp. P(FST(p),SND(p)))"
and 2: "P(UU::'a,UU::'b)"
and 3: "∀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 = "λx. <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 =
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)
›

end
```