src/FOLP/ifolp.thy

+IFOLP = Pure +
+
+classes term < logic
+
+default term
+
+types p,o 0
+
+arities p,o :: logic
+
+consts	
+      (*** Judgements ***)
+ "@Proof"   	::   "[p,o]=>prop"	("(_ /: _)" [10,10] 5)
+ Proof  	::   "[o,p]=>prop"
+ EqProof        ::   "[p,p,o]=>prop"    ("(3_ /= _ :/ _)" [10,10,10] 5)
+        
+      (*** Logical Connectives -- Type Formers ***)
+ "="		::	"['a,'a] => o"	(infixl 50)
+ True,False	::	"o"
+ "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)
+      (*Rewriting gadgets*)
+ NORM		::	"o => o"
+ norm		::	"'a => 'a"
+
+      (*** Proof Term Formers ***)
+ tt		:: "p"
+ contr		:: "p=>p"
+ fst,snd	:: "p=>p"
+ pair		:: "[p,p]=>p"		("(1<_,/_>)")
+ split		:: "[p, [p,p]=>p] =>p"
+ inl,inr	:: "p=>p"
+ when		:: "[p, p=>p, p=>p]=>p"
+ lambda		:: "(p => p) => p"	(binder "lam " 20)
+ "`"		:: "[p,p]=>p"		(infixl 60)
+ alll           :: "['a=>p]=>p"         (binder "all " 15)
+ "^"            :: "[p,'a]=>p"          (infixl 50)
+ exists		:: "['a,p]=>p"		("(1[_,/_])")
+ xsplit         :: "[p,['a,p]=>p]=>p"
+ ideq           :: "'a=>p"
+ idpeel         :: "[p,'a=>p]=>p"
+ nrm, NRM       :: "p"
+
+rules
+
+(**** Propositional logic ****)
+
+(*Equality*)
+(* Like Intensional Equality in MLTT - but proofs distinct from terms *)
+
+ieqI	  "ideq(a) : a=a"
+ieqE      "[| p : a=b;  !!x.f(x) : P(x,x) |] ==> idpeel(p,f) : P(a,b)"
+
+(* Truth and Falsity *)
+
+TrueI     "tt : True"
+FalseE    "a:False ==> contr(a):P"
+
+(* Conjunction *)
+
+conjI     "[| a:P;  b:Q |] ==> <a,b> : P&Q"
+conjunct1 "p:P&Q ==> fst(p):P"
+conjunct2 "p:P&Q ==> snd(p):Q"
+
+(* Disjunction *)
+
+disjI1    "a:P ==> inl(a):P|Q"
+disjI2    "b:Q ==> inr(b):P|Q"
+disjE     "[| a:P|Q;  !!x.x:P ==> f(x):R;  !!x.x:Q ==> g(x):R \
+\          |] ==> when(a,f,g):R"
+
+(* Implication *)
+
+impI      "(!!x.x:P ==> f(x):Q) ==> lam x.f(x):P-->Q"
+mp        "[| f:P-->Q;  a:P |] ==> f`a:Q"
+
+(*Quantifiers*)
+
+allI	  "(!!x. f(x) : P(x)) ==> all x.f(x) : ALL x.P(x)"
+spec	  "(f:ALL x.P(x)) ==> f^x : P(x)"
+
+exI	  "p : P(x) ==> [x,p] : EX x.P(x)"
+exE	  "[| p: EX x.P(x);  !!x u. u:P(x) ==> f(x,u) : R |] ==> xsplit(p,f):R"
+
+(**** Equality between proofs ****)
+
+prefl     "a : P ==> a = a : P"
+psym      "a = b : P ==> b = a : P"
+ptrans    "[| a = b : P;  b = c : P |] ==> a = c : P"
+
+idpeelB   "[| !!x.f(x) : P(x,x) |] ==> idpeel(ideq(a),f) = f(a) : P(a,a)"
+
+fstB      "a:P ==> fst(<a,b>) = a : P"
+sndB      "b:Q ==> snd(<a,b>) = b : Q"
+pairEC    "p:P&Q ==> p = <fst(p),snd(p)> : P&Q"
+
+whenBinl  "[| a:P;  !!x.x:P ==> f(x) : Q |] ==> when(inl(a),f,g) = f(a) : Q"
+whenBinr  "[| b:P;  !!x.x:P ==> g(x) : Q |] ==> when(inr(b),f,g) = g(b) : Q"
+plusEC    "a:P|Q ==> when(a,%x.inl(x),%y.inr(y)) = p : P|Q"
+
+applyB     "[| a:P;  !!x.x:P ==> b(x) : Q |] ==> (lam x.b(x)) ` a = b(a) : Q"
+funEC      "f:P ==> f = lam x.f`x : P"
+
+specB      "[| !!x.f(x) : P(x) |] ==> (all x.f(x)) ^ a = f(a) : P(a)"
+
+
+(**** Definitions ****)
+
+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)"
+
+(*Rewriting -- special constants to flag normalized terms and formulae*)
+norm_eq	"nrm : norm(x) = x"
+NORM_iff	"NRM : NORM(P) <-> P"
+
+end
+
+ML
+
+(*show_proofs:=true displays the proof terms -- they are ENORMOUS*)
+val show_proofs = ref false;
+
+fun proof_tr [p,P] = Const("Proof",dummyT) $ P $ p;
+
+fun proof_tr' [P,p] = 
+    if !show_proofs then Const("@Proof",dummyT) $ p $ P 
+    else P  (*this case discards the proof term*);
+
+val  parse_translation = [("@Proof", proof_tr)];
+val print_translation  = [("Proof", proof_tr')];
+