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(* Title: LCF/lcf.thy

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ID: $Id$

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Author: Tobias Nipkow

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Copyright 1992 University of Cambridge


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Natural Deduction Rules for LCF


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*)


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LCF = FOL +


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classes cpo < term


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default cpo


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types


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tr


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void

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('a,'b) "*" (infixl 6)


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('a,'b) "+" (infixl 5)

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arities


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fun, "*", "+" :: (cpo,cpo)cpo


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tr,void :: cpo

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consts

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UU :: "'a"


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TT,FF :: "tr"


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FIX :: "('a => 'a) => 'a"


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FST :: "'a*'b => 'a"


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SND :: "'a*'b => 'b"

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INL :: "'a => 'a+'b"


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INR :: "'b => 'a+'b"


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WHEN :: "['a=>'c, 'b=>'c, 'a+'b] => 'c"

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adm :: "('a => o) => o"


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VOID :: "void" ("'(')")


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PAIR :: "['a,'b] => 'a*'b" ("(1<_,/_>)" [0,0] 100)


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COND :: "[tr,'a,'a] => 'a" ("(_ =>/ (_ / _))" [60,60,60] 60)


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"<<" :: "['a,'a] => o" (infixl 50)

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rules


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(** DOMAIN THEORY **)


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eq_def "x=y == x << y & y << x"

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less_trans "[ x << y; y << z ] ==> x << z"

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less_ext "(ALL x. f(x) << g(x)) ==> f << g"

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mono "[ f << g; x << y ] ==> f(x) << g(y)"

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minimal "UU << x"

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FIX_eq "f(FIX(f)) = FIX(f)"

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(** TR **)


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tr_cases "p=UU  p=TT  p=FF"

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not_TT_less_FF "~ TT << FF"


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not_FF_less_TT "~ FF << TT"


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not_TT_less_UU "~ TT << UU"


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not_FF_less_UU "~ FF << UU"


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COND_UU "UU => x  y = UU"


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COND_TT "TT => x  y = x"


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COND_FF "FF => x  y = y"

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(** PAIRS **)


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surj_pairing "<FST(z),SND(z)> = z"

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FST "FST(<x,y>) = x"


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SND "SND(<x,y>) = y"

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(*** STRICT SUM ***)


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INL_DEF "~x=UU ==> ~INL(x)=UU"


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INR_DEF "~x=UU ==> ~INR(x)=UU"


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INL_STRICT "INL(UU) = UU"


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INR_STRICT "INR(UU) = UU"


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WHEN_UU "WHEN(f,g,UU) = UU"


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WHEN_INL "~x=UU ==> WHEN(f,g,INL(x)) = f(x)"


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WHEN_INR "~x=UU ==> WHEN(f,g,INR(x)) = g(x)"


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SUM_EXHAUSTION


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"z = UU  (EX x. ~x=UU & z = INL(x))  (EX y. ~y=UU & z = INR(y))"


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(** VOID **)


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void_cases "(x::void) = UU"

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(** INDUCTION **)


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induct "[ adm(P); P(UU); ALL x. P(x) > P(f(x)) ] ==> P(FIX(f))"

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(** Admissibility / Chain Completeness **)


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(* All rules can be found on pages 199200 of Larry's LCF book.


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Note that "easiness" of types is not taken into account


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because it cannot be expressed schematically; flatness could be. *)


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adm_less "adm(%x.t(x) << u(x))"


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adm_not_less "adm(%x.~ t(x) << u)"

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adm_not_free "adm(%x.A)"

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adm_subst "adm(P) ==> adm(%x.P(t(x)))"


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adm_conj "[ adm(P); adm(Q) ] ==> adm(%x.P(x)&Q(x))"


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adm_disj "[ adm(P); adm(Q) ] ==> adm(%x.P(x)Q(x))"


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adm_imp "[ adm(%x.~P(x)); adm(Q) ] ==> adm(%x.P(x)>Q(x))"


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adm_all "(!!y.adm(P(y))) ==> adm(%x.ALL y.P(y,x))"

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end
