IFOLP = Pure +
classes term < logic
default term
types
p
o
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')];