(* Title: FOL/IFOL.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
Intuitionistic first-order logic.
*)
IFOL = Pure +
classes
term < logic
default
term
types
o
arities
o :: logic
consts
Trueprop :: o => prop ("(_)" 5)
True, False :: o
(* Connectives *)
"=" :: ['a, 'a] => o (infixl 50)
Not :: o => o ("~ _" [40] 40)
"&" :: [o, o] => o (infixr 35)
"|" :: [o, o] => o (infixr 30)
"-->" :: [o, o] => o (infixr 25)
"<->" :: [o, o] => o (infixr 25)
(* Quantifiers *)
All :: ('a => o) => o (binder "ALL " 10)
Ex :: ('a => o) => o (binder "EX " 10)
Ex1 :: ('a => o) => o (binder "EX! " 10)
syntax
"~=" :: ['a, 'a] => o (infixl 50)
translations
"x ~= y" == "~ (x = y)"
syntax (symbols)
Not :: o => o ("\\<not> _" [40] 40)
"op &" :: [o, o] => o (infixr "\\<and>" 35)
"op |" :: [o, o] => o (infixr "\\<or>" 30)
"op -->" :: [o, o] => o (infixr "\\<midarrow>\\<rightarrow>" 25)
"op <->" :: [o, o] => o (infixr "\\<leftarrow>\\<rightarrow>" 25)
"ALL " :: [idts, o] => o ("(3\\<forall>_./ _)" [0, 10] 10)
"EX " :: [idts, o] => o ("(3\\<exists>_./ _)" [0, 10] 10)
"EX! " :: [idts, o] => o ("(3\\<exists>!_./ _)" [0, 10] 10)
"op ~=" :: ['a, 'a] => o (infixl "\\<noteq>" 50)
rules
(* Equality *)
refl "a=a"
subst "[| a=b; P(a) |] ==> P(b)"
(* Propositional logic *)
conjI "[| P; Q |] ==> P&Q"
conjunct1 "P&Q ==> P"
conjunct2 "P&Q ==> Q"
disjI1 "P ==> P|Q"
disjI2 "Q ==> P|Q"
disjE "[| P|Q; P ==> R; Q ==> R |] ==> R"
impI "(P ==> Q) ==> P-->Q"
mp "[| P-->Q; P |] ==> Q"
FalseE "False ==> P"
(* Definitions *)
True_def "True == False-->False"
not_def "~P == P-->False"
iff_def "P<->Q == (P-->Q) & (Q-->P)"
(* Unique existence *)
ex1_def "EX! x. P(x) == EX x. P(x) & (ALL y. P(y) --> y=x)"
(* Quantifiers *)
allI "(!!x. P(x)) ==> (ALL x. P(x))"
spec "(ALL x. P(x)) ==> P(x)"
exI "P(x) ==> (EX x. P(x))"
exE "[| EX x. P(x); !!x. P(x) ==> R |] ==> R"
(* Reflection *)
eq_reflection "(x=y) ==> (x==y)"
iff_reflection "(P<->Q) ==> (P==Q)"
end