--- a/src/CTT/ctt.thy Sat Apr 05 16:00:00 2003 +0200
+++ /dev/null Thu Jan 01 00:00:00 1970 +0000
@@ -1,256 +0,0 @@
-(* 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
-
-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" :: "[idt,t,t]=>t" ("(3PROD _:_./ _)" 10)
- "@SUM" :: "[idt,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)"
- "A --> B" => "Prod(A, _K(B))"
- "SUM x:A. B" => "Sum(A, %x. B)"
- "A * B" => "Sum(A, _K(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 print_translation =
- [("Prod", dependent_tr' ("@PROD", "@-->")),
- ("Sum", dependent_tr' ("@SUM", "@*"))];
-