(* Title: CTT/CTT.thy ID: $Id: CTT.thy,v 1.11 2005/09/16 21:01:30 wenzelm Exp $ Author: Lawrence C Paulson, Cambridge University Computer Laboratory Copyright 1993 University of Cambridge *) header {* Constructive Type Theory *} theory CTT imports Pure begin typedecl i typedecl t typedecl o consts (*Types*) F :: "t" 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 :: "i=>i" inr :: "i=>i" when :: "[i, i=>i, i=>i]=>i" (*General Sum and Binary Product*) Sum :: "[t, i=>t]=>t" fst :: "i=>i" snd :: "i=>i" split :: "[i, [i,i]=>i] =>i" (*General Product and Function Space*) Prod :: "[t, i=>t]=>t" (*Types*) "+" :: "[t,t]=>t" (infixr 40) (*Equality type*) Eq :: "[t,i,i]=>t" eq :: "i" (*Judgements*) Type :: "t => prop" ("(_ type)" [10] 5) Eqtype :: "[t,t]=>prop" ("(_ =/ _)" [10,10] 5) Elem :: "[i, t]=>prop" ("(_ /: _)" [10,10] 5) Eqelem :: "[i,i,t]=>prop" ("(_ =/ _ :/ _)" [10,10,10] 5) Reduce :: "[i,i]=>prop" ("Reduce[_,_]") (*Types*) (*Functions*) lambda :: "(i => i) => i" (binder "lam " 10) "`" :: "[i,i]=>i" (infixl 60) (*Natural numbers*) "0" :: "i" ("0") (*Pairing*) pair :: "[i,i]=>i" ("(1<_,/_>)") syntax "@PROD" :: "[idt,t,t]=>t" ("(3PROD _:_./ _)" 10) "@SUM" :: "[idt,t,t]=>t" ("(3SUM _:_./ _)" 10) "@-->" :: "[t,t]=>t" ("(_ -->/ _)" [31,30] 30) "@*" :: "[t,t]=>t" ("(_ */ _)" [51,50] 50) 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))" print_translation {* [("Prod", dependent_tr' ("@PROD", "@-->")), ("Sum", dependent_tr' ("@SUM", "@*"))] *} syntax (xsymbols) "@-->" :: "[t,t]=>t" ("(_ -->/ _)" [31,30] 30) "@*" :: "[t,t]=>t" ("(_ ×/ _)" [51,50] 50) Elem :: "[i, t]=>prop" ("(_ /∈ _)" [10,10] 5) Eqelem :: "[i,i,t]=>prop" ("(2_ =/ _ ∈/ _)" [10,10,10] 5) "@SUM" :: "[idt,t,t] => t" ("(3Σ _∈_./ _)" 10) "@PROD" :: "[idt,t,t] => t" ("(3Π _∈_./ _)" 10) "lam " :: "[idts, i] => i" ("(3λλ_./ _)" 10) syntax (HTML output) "@*" :: "[t,t]=>t" ("(_ ×/ _)" [51,50] 50) Elem :: "[i, t]=>prop" ("(_ /∈ _)" [10,10] 5) Eqelem :: "[i,i,t]=>prop" ("(2_ =/ _ ∈/ _)" [10,10,10] 5) "@SUM" :: "[idt,t,t] => t" ("(3Σ _∈_./ _)" 10) "@PROD" :: "[idt,t,t] => t" ("(3Π _∈_./ _)" 10) "lam " :: "[idts, i] => i" ("(3λλ_./ _)" 10) axioms (*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" ML {* use_legacy_bindings (the_context ()) *} end
theorem SumIL2:
[| c = a ∈ A; d = b ∈ B(a) |] ==> <c,d> = <a,b> ∈ Sum(A, B)
theorem subst_prodE:
[| p ∈ Prod(A, B); a ∈ A; !!z. z ∈ B(a) ==> c(z) ∈ C(z) |] ==> c(p ` a) ∈ C(p ` a)
theorem replace_type:
[| B = A; a ∈ A |] ==> a ∈ B
theorem subst_eqtyparg:
[| a = c ∈ A; !!z. z ∈ A ==> B(z) type |] ==> B(a) = B(c)
theorem SumE_fst:
p ∈ Sum(A, B) ==> fst(p) ∈ A
theorem SumE_snd:
[| p ∈ Sum(A, B); A type; !!x. x ∈ A ==> B(x) type |] ==> snd(p) ∈ B(fst(p))