src/CTT/CTT.thy
 author wenzelm Fri, 07 Oct 2005 22:59:19 +0200 changeset 17782 b3846df9d643 parent 17441 5b5feca0344a child 19761 5cd82054c2c6 permissions -rw-r--r--
replaced _K by dummy abstraction;
```
(*  Title:      CTT/CTT.thy
ID:         \$Id\$
Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
*)

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, %_. B)"
"SUM x:A. B"  => "Sum(A, %x. B)"
"A * B"       => "Sum(A, %_. 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)

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)

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
```