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(* Title: FOL/ifol.thy
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ID: $Id$
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory
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Copyright 1993 University of Cambridge
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Intuitionistic first-order logic
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*)
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IFOL = Pure +
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classes
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term < logic
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default
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term
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types
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278
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o
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arities
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o :: logic
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consts
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Trueprop :: "o => prop" ("(_)" 5)
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True, False :: "o"
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(* Connectives *)
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"=" :: "['a, 'a] => o" (infixl 50)
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"~=" :: "['a, 'a] => o" ("(_ ~=/ _)" [50, 51] 50)
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Not :: "o => o" ("~ _" [40] 40)
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"&" :: "[o, o] => o" (infixr 35)
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"|" :: "[o, o] => o" (infixr 30)
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"-->" :: "[o, o] => o" (infixr 25)
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"<->" :: "[o, o] => o" (infixr 25)
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(* Quantifiers *)
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All :: "('a => o) => o" (binder "ALL " 10)
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Ex :: "('a => o) => o" (binder "EX " 10)
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Ex1 :: "('a => o) => o" (binder "EX! " 10)
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translations
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"x ~= y" == "~ (x = y)"
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rules
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(* Equality *)
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refl "a=a"
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subst "[| a=b; P(a) |] ==> P(b)"
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(* Propositional logic *)
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conjI "[| P; Q |] ==> P&Q"
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conjunct1 "P&Q ==> P"
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conjunct2 "P&Q ==> Q"
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disjI1 "P ==> P|Q"
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disjI2 "Q ==> P|Q"
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disjE "[| P|Q; P ==> R; Q ==> R |] ==> R"
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impI "(P ==> Q) ==> P-->Q"
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mp "[| P-->Q; P |] ==> Q"
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FalseE "False ==> P"
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(* Definitions *)
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True_def "True == False-->False"
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not_def "~P == P-->False"
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iff_def "P<->Q == (P-->Q) & (Q-->P)"
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(* Unique existence *)
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ex1_def "EX! x. P(x) == EX x. P(x) & (ALL y. P(y) --> y=x)"
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(* Quantifiers *)
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allI "(!!x. P(x)) ==> (ALL x.P(x))"
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spec "(ALL x.P(x)) ==> P(x)"
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exI "P(x) ==> (EX x.P(x))"
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exE "[| EX x.P(x); !!x. P(x) ==> R |] ==> R"
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(* Reflection *)
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eq_reflection "(x=y) ==> (x==y)"
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iff_reflection "(P<->Q) ==> (P==Q)"
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end
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