--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/CTT/CTT.thy Thu Sep 16 12:20:38 1993 +0200
@@ -0,0 +1,253 @@
+(* Title: CTT/ctt.thy
+ ID: $Id$
+ Author: Lawrence C Paulson, Cambridge University Computer Laboratory
+ Copyright 1993 University of Cambridge
+
+Constructive Type Theory
+*)
+
+CTT = Pure +
+
+types i,t,o 0
+
+arities i,t,o :: logic
+
+consts
+ (*Types*)
+ F,T :: "t" (*F is empty, T contains one element*)
+ contr :: "i=>i"
+ tt :: "i"
+ (*Natural numbers*)
+ N :: "t"
+ succ :: "i=>i"
+ rec :: "[i, i, [i,i]=>i] => i"
+ (*Unions*)
+ inl,inr :: "i=>i"
+ when :: "[i, i=>i, i=>i]=>i"
+ (*General Sum and Binary Product*)
+ Sum :: "[t, i=>t]=>t"
+ fst,snd :: "i=>i"
+ split :: "[i, [i,i]=>i] =>i"
+ (*General Product and Function Space*)
+ Prod :: "[t, i=>t]=>t"
+ (*Equality type*)
+ Eq :: "[t,i,i]=>t"
+ eq :: "i"
+ (*Judgements*)
+ Type :: "t => prop" ("(_ type)" [10] 5)
+ Eqtype :: "[t,t]=>prop" ("(3_ =/ _)" [10,10] 5)
+ Elem :: "[i, t]=>prop" ("(_ /: _)" [10,10] 5)
+ Eqelem :: "[i,i,t]=>prop" ("(3_ =/ _ :/ _)" [10,10,10] 5)
+ Reduce :: "[i,i]=>prop" ("Reduce[_,_]")
+ (*Types*)
+ "@PROD" :: "[id,t,t]=>t" ("(3PROD _:_./ _)" 10)
+ "@SUM" :: "[id,t,t]=>t" ("(3SUM _:_./ _)" 10)
+ "+" :: "[t,t]=>t" (infixr 40)
+ (*Invisible infixes!*)
+ "@-->" :: "[t,t]=>t" ("(_ -->/ _)" [31,30] 30)
+ "@*" :: "[t,t]=>t" ("(_ */ _)" [51,50] 50)
+ (*Functions*)
+ lambda :: "(i => i) => i" (binder "lam " 10)
+ "`" :: "[i,i]=>i" (infixl 60)
+ (*Natural numbers*)
+ "0" :: "i" ("0")
+ (*Pairing*)
+ pair :: "[i,i]=>i" ("(1<_,/_>)")
+
+translations
+ "PROD x:A. B" => "Prod(A, %x. B)"
+ "SUM x:A. B" => "Sum(A, %x. B)"
+
+rules
+
+ (*Reduction: a weaker notion than equality; a hack for simplification.
+ Reduce[a,b] means either that a=b:A for some A or else that "a" and "b"
+ are textually identical.*)
+
+ (*does not verify a:A! Sound because only trans_red uses a Reduce premise
+ No new theorems can be proved about the standard judgements.*)
+ refl_red "Reduce[a,a]"
+ red_if_equal "a = b : A ==> Reduce[a,b]"
+ trans_red "[| a = b : A; Reduce[b,c] |] ==> a = c : A"
+
+ (*Reflexivity*)
+
+ refl_type "A type ==> A = A"
+ refl_elem "a : A ==> a = a : A"
+
+ (*Symmetry*)
+
+ sym_type "A = B ==> B = A"
+ sym_elem "a = b : A ==> b = a : A"
+
+ (*Transitivity*)
+
+ trans_type "[| A = B; B = C |] ==> A = C"
+ trans_elem "[| a = b : A; b = c : A |] ==> a = c : A"
+
+ equal_types "[| a : A; A = B |] ==> a : B"
+ equal_typesL "[| a = b : A; A = B |] ==> a = b : B"
+
+ (*Substitution*)
+
+ subst_type "[| a : A; !!z. z:A ==> B(z) type |] ==> B(a) type"
+ subst_typeL "[| a = c : A; !!z. z:A ==> B(z) = D(z) |] ==> B(a) = D(c)"
+
+ subst_elem "[| a : A; !!z. z:A ==> b(z):B(z) |] ==> b(a):B(a)"
+ subst_elemL
+ "[| a=c : A; !!z. z:A ==> b(z)=d(z) : B(z) |] ==> b(a)=d(c) : B(a)"
+
+
+ (*The type N -- natural numbers*)
+
+ NF "N type"
+ NI0 "0 : N"
+ NI_succ "a : N ==> succ(a) : N"
+ NI_succL "a = b : N ==> succ(a) = succ(b) : N"
+
+ NE
+ "[| p: N; a: C(0); !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |] \
+\ ==> rec(p, a, %u v.b(u,v)) : C(p)"
+
+ NEL
+ "[| p = q : N; a = c : C(0); \
+\ !!u v. [| u: N; v: C(u) |] ==> b(u,v) = d(u,v): C(succ(u)) |] \
+\ ==> rec(p, a, %u v.b(u,v)) = rec(q,c,d) : C(p)"
+
+ NC0
+ "[| a: C(0); !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |] \
+\ ==> rec(0, a, %u v.b(u,v)) = a : C(0)"
+
+ NC_succ
+ "[| p: N; a: C(0); \
+\ !!u v. [| u: N; v: C(u) |] ==> b(u,v): C(succ(u)) |] ==> \
+\ rec(succ(p), a, %u v.b(u,v)) = b(p, rec(p, a, %u v.b(u,v))) : C(succ(p))"
+
+ (*The fourth Peano axiom. See page 91 of Martin-Lof's book*)
+ zero_ne_succ
+ "[| a: N; 0 = succ(a) : N |] ==> 0: F"
+
+
+ (*The Product of a family of types*)
+
+ ProdF "[| A type; !!x. x:A ==> B(x) type |] ==> PROD x:A.B(x) type"
+
+ ProdFL
+ "[| A = C; !!x. x:A ==> B(x) = D(x) |] ==> \
+\ PROD x:A.B(x) = PROD x:C.D(x)"
+
+ ProdI
+ "[| A type; !!x. x:A ==> b(x):B(x)|] ==> lam x.b(x) : PROD x:A.B(x)"
+
+ ProdIL
+ "[| A type; !!x. x:A ==> b(x) = c(x) : B(x)|] ==> \
+\ lam x.b(x) = lam x.c(x) : PROD x:A.B(x)"
+
+ ProdE "[| p : PROD x:A.B(x); a : A |] ==> p`a : B(a)"
+ ProdEL "[| p=q: PROD x:A.B(x); a=b : A |] ==> p`a = q`b : B(a)"
+
+ ProdC
+ "[| a : A; !!x. x:A ==> b(x) : B(x)|] ==> \
+\ (lam x.b(x)) ` a = b(a) : B(a)"
+
+ ProdC2
+ "p : PROD x:A.B(x) ==> (lam x. p`x) = p : PROD x:A.B(x)"
+
+
+ (*The Sum of a family of types*)
+
+ SumF "[| A type; !!x. x:A ==> B(x) type |] ==> SUM x:A.B(x) type"
+ SumFL
+ "[| A = C; !!x. x:A ==> B(x) = D(x) |] ==> SUM x:A.B(x) = SUM x:C.D(x)"
+
+ SumI "[| a : A; b : B(a) |] ==> <a,b> : SUM x:A.B(x)"
+ SumIL "[| a=c:A; b=d:B(a) |] ==> <a,b> = <c,d> : SUM x:A.B(x)"
+
+ SumE
+ "[| p: SUM x:A.B(x); !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) |] \
+\ ==> split(p, %x y.c(x,y)) : C(p)"
+
+ SumEL
+ "[| p=q : SUM x:A.B(x); \
+\ !!x y. [| x:A; y:B(x) |] ==> c(x,y)=d(x,y): C(<x,y>)|] \
+\ ==> split(p, %x y.c(x,y)) = split(q, % x y.d(x,y)) : C(p)"
+
+ SumC
+ "[| a: A; b: B(a); !!x y. [| x:A; y:B(x) |] ==> c(x,y): C(<x,y>) |] \
+\ ==> split(<a,b>, %x y.c(x,y)) = c(a,b) : C(<a,b>)"
+
+ fst_def "fst(a) == split(a, %x y.x)"
+ snd_def "snd(a) == split(a, %x y.y)"
+
+
+ (*The sum of two types*)
+
+ PlusF "[| A type; B type |] ==> A+B type"
+ PlusFL "[| A = C; B = D |] ==> A+B = C+D"
+
+ PlusI_inl "[| a : A; B type |] ==> inl(a) : A+B"
+ PlusI_inlL "[| a = c : A; B type |] ==> inl(a) = inl(c) : A+B"
+
+ PlusI_inr "[| A type; b : B |] ==> inr(b) : A+B"
+ PlusI_inrL "[| A type; b = d : B |] ==> inr(b) = inr(d) : A+B"
+
+ PlusE
+ "[| p: A+B; !!x. x:A ==> c(x): C(inl(x)); \
+\ !!y. y:B ==> d(y): C(inr(y)) |] \
+\ ==> when(p, %x.c(x), %y.d(y)) : C(p)"
+
+ PlusEL
+ "[| p = q : A+B; !!x. x: A ==> c(x) = e(x) : C(inl(x)); \
+\ !!y. y: B ==> d(y) = f(y) : C(inr(y)) |] \
+\ ==> when(p, %x.c(x), %y.d(y)) = when(q, %x.e(x), %y.f(y)) : C(p)"
+
+ PlusC_inl
+ "[| a: A; !!x. x:A ==> c(x): C(inl(x)); \
+\ !!y. y:B ==> d(y): C(inr(y)) |] \
+\ ==> when(inl(a), %x.c(x), %y.d(y)) = c(a) : C(inl(a))"
+
+ PlusC_inr
+ "[| b: B; !!x. x:A ==> c(x): C(inl(x)); \
+\ !!y. y:B ==> d(y): C(inr(y)) |] \
+\ ==> when(inr(b), %x.c(x), %y.d(y)) = d(b) : C(inr(b))"
+
+
+ (*The type Eq*)
+
+ EqF "[| A type; a : A; b : A |] ==> Eq(A,a,b) type"
+ EqFL "[| A=B; a=c: A; b=d : A |] ==> Eq(A,a,b) = Eq(B,c,d)"
+ EqI "a = b : A ==> eq : Eq(A,a,b)"
+ EqE "p : Eq(A,a,b) ==> a = b : A"
+
+ (*By equality of types, can prove C(p) from C(eq), an elimination rule*)
+ EqC "p : Eq(A,a,b) ==> p = eq : Eq(A,a,b)"
+
+ (*The type F*)
+
+ FF "F type"
+ FE "[| p: F; C type |] ==> contr(p) : C"
+ FEL "[| p = q : F; C type |] ==> contr(p) = contr(q) : C"
+
+ (*The type T
+ Martin-Lof's book (page 68) discusses elimination and computation.
+ Elimination can be derived by computation and equality of types,
+ but with an extra premise C(x) type x:T.
+ Also computation can be derived from elimination. *)
+
+ TF "T type"
+ TI "tt : T"
+ TE "[| p : T; c : C(tt) |] ==> c : C(p)"
+ TEL "[| p = q : T; c = d : C(tt) |] ==> c = d : C(p)"
+ TC "p : T ==> p = tt : T"
+end
+
+
+ML
+
+val parse_translation =
+ [("@-->", ndependent_tr "Prod"), ("@*", ndependent_tr "Sum")];
+
+val print_translation =
+ [("Prod", dependent_tr' ("@PROD", "@-->")),
+ ("Sum", dependent_tr' ("@SUM", "@*"))];
+