clasohm@1268: (*  Title:      FOL/IFOL.thy
lcp@35:     ID:         $Id$
wenzelm@79:     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
lcp@35:     Copyright   1993  University of Cambridge
lcp@35: 
lcp@35: Intuitionistic first-order logic
lcp@35: *)
lcp@35: 
wenzelm@79: IFOL = Pure +
clasohm@0: 
wenzelm@79: classes
wenzelm@79:   term < logic
wenzelm@79: 
wenzelm@79: default
wenzelm@79:   term
clasohm@0: 
wenzelm@79: types
clasohm@278:   o
clasohm@0: 
wenzelm@79: arities
wenzelm@79:   o :: logic
clasohm@0: 
wenzelm@79: 
wenzelm@79: consts
clasohm@0: 
wenzelm@79:   Trueprop      :: "o => prop"                  ("(_)" 5)
wenzelm@79:   True, False   :: "o"
wenzelm@79: 
wenzelm@79:   (* Connectives *)
wenzelm@79: 
wenzelm@79:   "="           :: "['a, 'a] => o"              (infixl 50)
lcp@35: 
wenzelm@79:   Not           :: "o => o"                     ("~ _" [40] 40)
wenzelm@79:   "&"           :: "[o, o] => o"                (infixr 35)
wenzelm@79:   "|"           :: "[o, o] => o"                (infixr 30)
wenzelm@79:   "-->"         :: "[o, o] => o"                (infixr 25)
wenzelm@79:   "<->"         :: "[o, o] => o"                (infixr 25)
wenzelm@79: 
wenzelm@79:   (* Quantifiers *)
wenzelm@79: 
wenzelm@79:   All           :: "('a => o) => o"             (binder "ALL " 10)
wenzelm@79:   Ex            :: "('a => o) => o"             (binder "EX " 10)
wenzelm@79:   Ex1           :: "('a => o) => o"             (binder "EX! " 10)
wenzelm@79: 
clasohm@0: 
lcp@928: syntax
lcp@928:   "~="          :: "['a, 'a] => o"              (infixl 50)
lcp@928: 
lcp@35: translations
wenzelm@79:   "x ~= y"      == "~ (x = y)"
wenzelm@79: 
lcp@35: 
clasohm@0: rules
clasohm@0: 
wenzelm@79:   (* Equality *)
clasohm@0: 
wenzelm@79:   refl          "a=a"
wenzelm@79:   subst         "[| a=b;  P(a) |] ==> P(b)"
clasohm@0: 
wenzelm@79:   (* Propositional logic *)
clasohm@0: 
wenzelm@79:   conjI         "[| P;  Q |] ==> P&Q"
wenzelm@79:   conjunct1     "P&Q ==> P"
wenzelm@79:   conjunct2     "P&Q ==> Q"
clasohm@0: 
wenzelm@79:   disjI1        "P ==> P|Q"
wenzelm@79:   disjI2        "Q ==> P|Q"
wenzelm@79:   disjE         "[| P|Q;  P ==> R;  Q ==> R |] ==> R"
clasohm@0: 
wenzelm@79:   impI          "(P ==> Q) ==> P-->Q"
wenzelm@79:   mp            "[| P-->Q;  P |] ==> Q"
clasohm@0: 
wenzelm@79:   FalseE        "False ==> P"
clasohm@0: 
wenzelm@79:   (* Definitions *)
clasohm@0: 
wenzelm@79:   True_def      "True  == False-->False"
wenzelm@79:   not_def       "~P    == P-->False"
wenzelm@79:   iff_def       "P<->Q == (P-->Q) & (Q-->P)"
wenzelm@79: 
wenzelm@79:   (* Unique existence *)
clasohm@0: 
wenzelm@79:   ex1_def       "EX! x. P(x) == EX x. P(x) & (ALL y. P(y) --> y=x)"
clasohm@0: 
wenzelm@79:   (* Quantifiers *)
clasohm@0: 
wenzelm@79:   allI          "(!!x. P(x)) ==> (ALL x.P(x))"
wenzelm@79:   spec          "(ALL x.P(x)) ==> P(x)"
clasohm@0: 
wenzelm@79:   exI           "P(x) ==> (EX x.P(x))"
wenzelm@79:   exE           "[| EX x.P(x);  !!x. P(x) ==> R |] ==> R"
clasohm@0: 
clasohm@0:   (* Reflection *)
clasohm@0: 
wenzelm@79:   eq_reflection   "(x=y)   ==> (x==y)"
wenzelm@79:   iff_reflection  "(P<->Q) ==> (P==Q)"
clasohm@0: 
clasohm@0: end
wenzelm@79: