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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 firstorder logic. 
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*) 
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theory IFOL = Pure 
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files ("IFOL_lemmas.ML") ("fologic.ML") ("hypsubstdata.ML") ("intprover.ML"): 
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global 
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classes "term" < logic 
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defaultsort "term" 
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typedecl o 
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consts 

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Trueprop :: "o => prop" ("(_)" 5) 
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True :: o 
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False :: o 
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(* Connectives *) 

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"=" :: "['a, 'a] => o" (infixl 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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syntax 
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"~=" :: "['a, 'a] => o" (infixl 50) 
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translations 
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"x ~= y" == "~ (x = y)" 
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syntax (symbols) 
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Not :: "o => o" ("\\<not> _" [40] 40) 
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"op &" :: "[o, o] => o" (infixr "\\<and>" 35) 
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"op " :: "[o, o] => o" (infixr "\\<or>" 30) 
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"op >" :: "[o, o] => o" (infixr "\\<midarrow>\\<rightarrow>" 25) 
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"op <>" :: "[o, o] => o" (infixr "\\<leftarrow>\\<rightarrow>" 25) 
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"ALL " :: "[idts, o] => o" ("(3\\<forall>_./ _)" [0, 10] 10) 
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"EX " :: "[idts, o] => o" ("(3\\<exists>_./ _)" [0, 10] 10) 
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"EX! " :: "[idts, o] => o" ("(3\\<exists>!_./ _)" [0, 10] 10) 
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"op ~=" :: "['a, 'a] => o" (infixl "\\<noteq>" 50) 
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syntax (xsymbols) 
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"op >" :: "[o, o] => o" (infixr "\\<longrightarrow>" 25) 
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"op <>" :: "[o, o] => o" (infixr "\\<longleftrightarrow>" 25) 
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syntax (HTML output) 
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Not :: "o => o" ("\\<not> _" [40] 40) 
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local 
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axioms 
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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 ==> PQ" 
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disjI2: "Q ==> PQ" 
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disjE: "[ PQ; 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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setup Simplifier.setup 
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use "IFOL_lemmas.ML" setup attrib_setup 
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use "fologic.ML" 
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use "hypsubstdata.ML" setup hypsubst_setup 
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use "intprover.ML" 
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end 