--- a/src/CTT/CTT.thy Fri Sep 16 21:02:15 2005 +0200
+++ b/src/CTT/CTT.thy Fri Sep 16 23:01:29 2005 +0200
@@ -1,21 +1,23 @@
-(* Title: CTT/ctt.thy
+(* Title: CTT/CTT.thy
ID: $Id$
Author: Lawrence C Paulson, Cambridge University Computer Laboratory
Copyright 1993 University of Cambridge
-
-Constructive Type Theory
*)
-CTT = Pure +
+header {* Constructive Type Theory *}
-types
- i
- t
- o
+theory CTT
+imports Pure
+begin
+
+typedecl i
+typedecl t
+typedecl o
consts
(*Types*)
- F,T :: "t" (*F is empty, T contains one element*)
+ F :: "t"
+ T :: "t" (*F is empty, T contains one element*)
contr :: "i=>i"
tt :: "i"
(*Natural numbers*)
@@ -23,11 +25,13 @@
succ :: "i=>i"
rec :: "[i, i, [i,i]=>i] => i"
(*Unions*)
- inl,inr :: "i=>i"
+ inl :: "i=>i"
+ inr :: "i=>i"
when :: "[i, i=>i, i=>i]=>i"
(*General Sum and Binary Product*)
Sum :: "[t, i=>t]=>t"
- fst,snd :: "i=>i"
+ fst :: "i=>i"
+ snd :: "i=>i"
split :: "[i, [i,i]=>i] =>i"
(*General Product and Function Space*)
Prod :: "[t, i=>t]=>t"
@@ -64,24 +68,30 @@
"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" ("(_ \\<longrightarrow>/ _)" [31,30] 30)
- "@*" :: "[t,t]=>t" ("(_ \\<times>/ _)" [51,50] 50)
- Elem :: "[i, t]=>prop" ("(_ /\\<in> _)" [10,10] 5)
- Eqelem :: "[i,i,t]=>prop" ("(2_ =/ _ \\<in>/ _)" [10,10,10] 5)
- "@SUM" :: "[idt,t,t] => t" ("(3\\<Sigma> _\\<in>_./ _)" 10)
- "@PROD" :: "[idt,t,t] => t" ("(3\\<Pi> _\\<in>_./ _)" 10)
- "lam " :: "[idts, i] => i" ("(3\\<lambda>\\<lambda>_./ _)" 10)
+ "@-->" :: "[t,t]=>t" ("(_ \<longrightarrow>/ _)" [31,30] 30)
+ "@*" :: "[t,t]=>t" ("(_ \<times>/ _)" [51,50] 50)
+ Elem :: "[i, t]=>prop" ("(_ /\<in> _)" [10,10] 5)
+ Eqelem :: "[i,i,t]=>prop" ("(2_ =/ _ \<in>/ _)" [10,10,10] 5)
+ "@SUM" :: "[idt,t,t] => t" ("(3\<Sigma> _\<in>_./ _)" 10)
+ "@PROD" :: "[idt,t,t] => t" ("(3\<Pi> _\<in>_./ _)" 10)
+ "lam " :: "[idts, i] => i" ("(3\<lambda>\<lambda>_./ _)" 10)
syntax (HTML output)
- "@*" :: "[t,t]=>t" ("(_ \\<times>/ _)" [51,50] 50)
- Elem :: "[i, t]=>prop" ("(_ /\\<in> _)" [10,10] 5)
- Eqelem :: "[i,i,t]=>prop" ("(2_ =/ _ \\<in>/ _)" [10,10,10] 5)
- "@SUM" :: "[idt,t,t] => t" ("(3\\<Sigma> _\\<in>_./ _)" 10)
- "@PROD" :: "[idt,t,t] => t" ("(3\\<Pi> _\\<in>_./ _)" 10)
- "lam " :: "[idts, i] => i" ("(3\\<lambda>\\<lambda>_./ _)" 10)
+ "@*" :: "[t,t]=>t" ("(_ \<times>/ _)" [51,50] 50)
+ Elem :: "[i, t]=>prop" ("(_ /\<in> _)" [10,10] 5)
+ Eqelem :: "[i,i,t]=>prop" ("(2_ =/ _ \<in>/ _)" [10,10,10] 5)
+ "@SUM" :: "[idt,t,t] => t" ("(3\<Sigma> _\<in>_./ _)" 10)
+ "@PROD" :: "[idt,t,t] => t" ("(3\<Pi> _\<in>_./ _)" 10)
+ "lam " :: "[idts, i] => i" ("(3\<lambda>\<lambda>_./ _)" 10)
-rules
+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"
@@ -89,167 +99,167 @@
(*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"
+ 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"
+ 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"
+ 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"
+ 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"
+ 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_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
+ 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"
+ 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)) |]
+ 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)) |]
+ 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)) |]
+ 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)) |] ==>
+ 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
+ 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"
+ 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) |] ==>
+ ProdFL:
+ "[| A = C; !!x. x:A ==> B(x) = D(x) |] ==>
PROD x:A. B(x) = PROD x:C. D(x)"
- ProdI
+ 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)|] ==>
+ 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)"
+ 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)|] ==>
+ ProdC:
+ "[| a : A; !!x. x:A ==> b(x) : B(x)|] ==>
(lam x. b(x)) ` a = b(a) : B(a)"
- ProdC2
+ 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
+ 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)"
+ 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>) |]
+ 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>)|]
+ 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>) |]
+ 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)"
+ 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"
+ 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_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"
+ 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)) |]
+ 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)) |]
+ 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)) |]
+ 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)) |]
+ 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"
+ 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)"
+ 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"
+ 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.
@@ -257,17 +267,12 @@
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
+ 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
+ML {* use_legacy_bindings (the_context ()) *}
-val print_translation =
- [("Prod", dependent_tr' ("@PROD", "@-->")),
- ("Sum", dependent_tr' ("@SUM", "@*"))];
-
+end