src/LCF/LCF.thy
 author bulwahn Thu, 07 Jul 2011 23:33:14 +0200 changeset 43704 47b0be18ccbe parent 41310 65631ca437c9 child 47025 b2b8ae61d6ad permissions -rw-r--r--
floor and ceiling definitions are not code equations -- this enables trivial evaluation of floor and ceiling
```
(*  Title:      LCF/LCF.thy
Author:     Tobias Nipkow
*)

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) prod  (infixl "*" 6)
typedecl ('a,'b) sum  (infixl "+" 5)

arities
"fun" :: (cpo, cpo) cpo
prod :: (cpo, cpo) cpo
sum :: (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. *)

lemma eq_imp_less1: "x = y ==> x << y"

lemma eq_imp_less2: "x = y ==> y << x"

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"

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)
done

lemma PAIR_less [simp]: "<a,b> << <c,d> <-> a<<c & b<<d"

lemma PAIR_eq [simp]: "<a,b> = <c,d> <-> a=c & b=d"

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: "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)+

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

ML {*
fun induct_tac ctxt v i =
res_inst_tac ctxt [(("f", 0), v)] @{thm induct} i THEN
*}

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 (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:
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)