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IFOLP = Pure +
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classes term < logic
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default term
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283
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types
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p
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o
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0
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283
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arities
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p,o :: logic
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0
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consts
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(*** Judgements ***)
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"@Proof" :: "[p,o]=>prop" ("(_ /: _)" [10,10] 5)
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Proof :: "[o,p]=>prop"
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EqProof :: "[p,p,o]=>prop" ("(3_ /= _ :/ _)" [10,10,10] 5)
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(*** Logical Connectives -- Type Formers ***)
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"=" :: "['a,'a] => o" (infixl 50)
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True,False :: "o"
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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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(*Rewriting gadgets*)
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NORM :: "o => o"
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norm :: "'a => 'a"
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(*** Proof Term Formers ***)
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tt :: "p"
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contr :: "p=>p"
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fst,snd :: "p=>p"
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pair :: "[p,p]=>p" ("(1<_,/_>)")
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split :: "[p, [p,p]=>p] =>p"
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inl,inr :: "p=>p"
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when :: "[p, p=>p, p=>p]=>p"
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lambda :: "(p => p) => p" (binder "lam " 20)
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"`" :: "[p,p]=>p" (infixl 60)
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alll :: "['a=>p]=>p" (binder "all " 15)
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"^" :: "[p,'a]=>p" (infixl 50)
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exists :: "['a,p]=>p" ("(1[_,/_])")
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xsplit :: "[p,['a,p]=>p]=>p"
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ideq :: "'a=>p"
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idpeel :: "[p,'a=>p]=>p"
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nrm, NRM :: "p"
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rules
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(**** Propositional logic ****)
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(*Equality*)
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(* Like Intensional Equality in MLTT - but proofs distinct from terms *)
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ieqI "ideq(a) : a=a"
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ieqE "[| p : a=b; !!x.f(x) : P(x,x) |] ==> idpeel(p,f) : P(a,b)"
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(* Truth and Falsity *)
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TrueI "tt : True"
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FalseE "a:False ==> contr(a):P"
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(* Conjunction *)
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conjI "[| a:P; b:Q |] ==> <a,b> : P&Q"
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conjunct1 "p:P&Q ==> fst(p):P"
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conjunct2 "p:P&Q ==> snd(p):Q"
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(* Disjunction *)
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disjI1 "a:P ==> inl(a):P|Q"
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disjI2 "b:Q ==> inr(b):P|Q"
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disjE "[| a:P|Q; !!x.x:P ==> f(x):R; !!x.x:Q ==> g(x):R \
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\ |] ==> when(a,f,g):R"
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(* Implication *)
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impI "(!!x.x:P ==> f(x):Q) ==> lam x.f(x):P-->Q"
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mp "[| f:P-->Q; a:P |] ==> f`a:Q"
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(*Quantifiers*)
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allI "(!!x. f(x) : P(x)) ==> all x.f(x) : ALL x.P(x)"
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spec "(f:ALL x.P(x)) ==> f^x : P(x)"
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exI "p : P(x) ==> [x,p] : EX x.P(x)"
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exE "[| p: EX x.P(x); !!x u. u:P(x) ==> f(x,u) : R |] ==> xsplit(p,f):R"
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(**** Equality between proofs ****)
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prefl "a : P ==> a = a : P"
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psym "a = b : P ==> b = a : P"
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ptrans "[| a = b : P; b = c : P |] ==> a = c : P"
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idpeelB "[| !!x.f(x) : P(x,x) |] ==> idpeel(ideq(a),f) = f(a) : P(a,a)"
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fstB "a:P ==> fst(<a,b>) = a : P"
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sndB "b:Q ==> snd(<a,b>) = b : Q"
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pairEC "p:P&Q ==> p = <fst(p),snd(p)> : P&Q"
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whenBinl "[| a:P; !!x.x:P ==> f(x) : Q |] ==> when(inl(a),f,g) = f(a) : Q"
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whenBinr "[| b:P; !!x.x:P ==> g(x) : Q |] ==> when(inr(b),f,g) = g(b) : Q"
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plusEC "a:P|Q ==> when(a,%x.inl(x),%y.inr(y)) = p : P|Q"
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applyB "[| a:P; !!x.x:P ==> b(x) : Q |] ==> (lam x.b(x)) ` a = b(a) : Q"
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funEC "f:P ==> f = lam x.f`x : P"
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specB "[| !!x.f(x) : P(x) |] ==> (all x.f(x)) ^ a = f(a) : P(a)"
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(**** Definitions ****)
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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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(*Rewriting -- special constants to flag normalized terms and formulae*)
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norm_eq "nrm : norm(x) = x"
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NORM_iff "NRM : NORM(P) <-> P"
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end
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ML
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(*show_proofs:=true displays the proof terms -- they are ENORMOUS*)
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val show_proofs = ref false;
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fun proof_tr [p,P] = Const("Proof",dummyT) $ P $ p;
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fun proof_tr' [P,p] =
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if !show_proofs then Const("@Proof",dummyT) $ p $ P
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else P (*this case discards the proof term*);
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val parse_translation = [("@Proof", proof_tr)];
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val print_translation = [("Proof", proof_tr')];
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