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(* Title: CTT/ctt.thy


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ID: $Id$


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Author: Lawrence C Paulson, Cambridge University Computer Laboratory


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Copyright 1993 University of Cambridge


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Constructive Type Theory


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*)


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CTT = Pure +


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283

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types


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i


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t


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o

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283

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arities


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i,t,o :: logic

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consts


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(*Types*)


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F,T :: "t" (*F is empty, T contains one element*)


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contr :: "i=>i"


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tt :: "i"


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(*Natural numbers*)


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N :: "t"


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succ :: "i=>i"


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rec :: "[i, i, [i,i]=>i] => i"


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(*Unions*)


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inl,inr :: "i=>i"


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when :: "[i, i=>i, i=>i]=>i"


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(*General Sum and Binary Product*)


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Sum :: "[t, i=>t]=>t"


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fst,snd :: "i=>i"


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split :: "[i, [i,i]=>i] =>i"


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(*General Product and Function Space*)


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Prod :: "[t, i=>t]=>t"


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(*Equality type*)


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Eq :: "[t,i,i]=>t"


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eq :: "i"


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(*Judgements*)


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Type :: "t => prop" ("(_ type)" [10] 5)


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Eqtype :: "[t,t]=>prop" ("(3_ =/ _)" [10,10] 5)


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Elem :: "[i, t]=>prop" ("(_ /: _)" [10,10] 5)


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Eqelem :: "[i,i,t]=>prop" ("(3_ =/ _ :/ _)" [10,10,10] 5)


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Reduce :: "[i,i]=>prop" ("Reduce[_,_]")


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(*Types*)

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"@PROD" :: "[idt,t,t]=>t" ("(3PROD _:_./ _)" 10)


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"@SUM" :: "[idt,t,t]=>t" ("(3SUM _:_./ _)" 10)

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"+" :: "[t,t]=>t" (infixr 40)


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(*Invisible infixes!*)


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"@>" :: "[t,t]=>t" ("(_ >/ _)" [31,30] 30)


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"@*" :: "[t,t]=>t" ("(_ */ _)" [51,50] 50)


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(*Functions*)


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lambda :: "(i => i) => i" (binder "lam " 10)


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"`" :: "[i,i]=>i" (infixl 60)


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(*Natural numbers*)


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"0" :: "i" ("0")


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(*Pairing*)


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pair :: "[i,i]=>i" ("(1<_,/_>)")


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translations


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"PROD x:A. B" => "Prod(A, %x. B)"

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"A > B" => "Prod(A, _K(B))"

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

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"A * B" => "Sum(A, _K(B))"

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rules


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(*Reduction: a weaker notion than equality; a hack for simplification.


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Reduce[a,b] means either that a=b:A for some A or else that "a" and "b"


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are textually identical.*)


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(*does not verify a:A! Sound because only trans_red uses a Reduce premise


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No new theorems can be proved about the standard judgements.*)


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refl_red "Reduce[a,a]"


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red_if_equal "a = b : A ==> Reduce[a,b]"


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trans_red "[ a = b : A; Reduce[b,c] ] ==> a = c : A"


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(*Reflexivity*)


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refl_type "A type ==> A = A"


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refl_elem "a : A ==> a = a : A"


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(*Symmetry*)


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sym_type "A = B ==> B = A"


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sym_elem "a = b : A ==> b = a : A"


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(*Transitivity*)


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trans_type "[ A = B; B = C ] ==> A = C"


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trans_elem "[ a = b : A; b = c : A ] ==> a = c : A"


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equal_types "[ a : A; A = B ] ==> a : B"


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equal_typesL "[ a = b : A; A = B ] ==> a = b : B"


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(*Substitution*)


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subst_type "[ a : A; !!z. z:A ==> B(z) type ] ==> B(a) type"


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subst_typeL "[ a = c : A; !!z. z:A ==> B(z) = D(z) ] ==> B(a) = D(c)"


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subst_elem "[ a : A; !!z. z:A ==> b(z):B(z) ] ==> b(a):B(a)"


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subst_elemL


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"[ a=c : A; !!z. z:A ==> b(z)=d(z) : B(z) ] ==> b(a)=d(c) : B(a)"


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(*The type N  natural numbers*)


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NF "N type"


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NI0 "0 : N"


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NI_succ "a : N ==> succ(a) : N"


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NI_succL "a = b : N ==> succ(a) = succ(b) : N"


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NE

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"[ p: N; a: C(0); !!u v. [ u: N; v: C(u) ] ==> b(u,v): C(succ(u)) ]

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==> rec(p, a, %u v. b(u,v)) : C(p)"

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NEL

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"[ p = q : N; a = c : C(0);


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!!u v. [ u: N; v: C(u) ] ==> b(u,v) = d(u,v): C(succ(u)) ]

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==> rec(p, a, %u v. b(u,v)) = rec(q,c,d) : C(p)"

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NC0

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"[ a: C(0); !!u v. [ u: N; v: C(u) ] ==> b(u,v): C(succ(u)) ]

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==> rec(0, a, %u v. b(u,v)) = a : C(0)"

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NC_succ

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"[ p: N; a: C(0);


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!!u v. [ u: N; v: C(u) ] ==> b(u,v): C(succ(u)) ] ==>

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rec(succ(p), a, %u v. b(u,v)) = b(p, rec(p, a, %u v. b(u,v))) : C(succ(p))"

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(*The fourth Peano axiom. See page 91 of MartinLof's book*)


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zero_ne_succ


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"[ a: N; 0 = succ(a) : N ] ==> 0: F"


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(*The Product of a family of types*)


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ProdF "[ A type; !!x. x:A ==> B(x) type ] ==> PROD x:A. B(x) type"

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ProdFL

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"[ A = C; !!x. x:A ==> B(x) = D(x) ] ==>

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PROD x:A. B(x) = PROD x:C. D(x)"

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ProdI

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"[ A type; !!x. x:A ==> b(x):B(x)] ==> lam x. b(x) : PROD x:A. B(x)"

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ProdIL

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"[ A type; !!x. x:A ==> b(x) = c(x) : B(x)] ==>

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lam x. b(x) = lam x. c(x) : PROD x:A. B(x)"

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ProdE "[ p : PROD x:A. B(x); a : A ] ==> p`a : B(a)"


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ProdEL "[ p=q: PROD x:A. B(x); a=b : A ] ==> p`a = q`b : B(a)"

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ProdC

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"[ a : A; !!x. x:A ==> b(x) : B(x)] ==>

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(lam x. b(x)) ` a = b(a) : B(a)"

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ProdC2

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"p : PROD x:A. B(x) ==> (lam x. p`x) = p : PROD x:A. B(x)"

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(*The Sum of a family of types*)


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SumF "[ A type; !!x. x:A ==> B(x) type ] ==> SUM x:A. B(x) type"

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SumFL

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"[ A = C; !!x. x:A ==> B(x) = D(x) ] ==> SUM x:A. B(x) = SUM x:C. D(x)"

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SumI "[ a : A; b : B(a) ] ==> <a,b> : SUM x:A. B(x)"


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SumIL "[ a=c:A; b=d:B(a) ] ==> <a,b> = <c,d> : SUM x:A. B(x)"

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SumE

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"[ p: SUM x:A. B(x); !!x y. [ x:A; y:B(x) ] ==> c(x,y): C(<x,y>) ]


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==> split(p, %x y. c(x,y)) : C(p)"

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SumEL

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"[ p=q : SUM x:A. B(x);

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!!x y. [ x:A; y:B(x) ] ==> c(x,y)=d(x,y): C(<x,y>)]

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==> split(p, %x y. c(x,y)) = split(q, % x y. d(x,y)) : C(p)"

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SumC

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"[ a: A; b: B(a); !!x y. [ x:A; y:B(x) ] ==> c(x,y): C(<x,y>) ]

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==> split(<a,b>, %x y. c(x,y)) = c(a,b) : C(<a,b>)"

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fst_def "fst(a) == split(a, %x y. x)"


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snd_def "snd(a) == split(a, %x y. y)"

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(*The sum of two types*)


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PlusF "[ A type; B type ] ==> A+B type"


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PlusFL "[ A = C; B = D ] ==> A+B = C+D"


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PlusI_inl "[ a : A; B type ] ==> inl(a) : A+B"


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PlusI_inlL "[ a = c : A; B type ] ==> inl(a) = inl(c) : A+B"


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PlusI_inr "[ A type; b : B ] ==> inr(b) : A+B"


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PlusI_inrL "[ A type; b = d : B ] ==> inr(b) = inr(d) : A+B"


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PlusE

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"[ p: A+B; !!x. x:A ==> c(x): C(inl(x));


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!!y. y:B ==> d(y): C(inr(y)) ]

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==> when(p, %x. c(x), %y. d(y)) : C(p)"

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PlusEL

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"[ p = q : A+B; !!x. x: A ==> c(x) = e(x) : C(inl(x));


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!!y. y: B ==> d(y) = f(y) : C(inr(y)) ]

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==> when(p, %x. c(x), %y. d(y)) = when(q, %x. e(x), %y. f(y)) : C(p)"

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PlusC_inl

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"[ a: A; !!x. x:A ==> c(x): C(inl(x));


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!!y. y:B ==> d(y): C(inr(y)) ]

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==> when(inl(a), %x. c(x), %y. d(y)) = c(a) : C(inl(a))"

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PlusC_inr

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"[ b: B; !!x. x:A ==> c(x): C(inl(x));


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!!y. y:B ==> d(y): C(inr(y)) ]

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==> when(inr(b), %x. c(x), %y. d(y)) = d(b) : C(inr(b))"

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(*The type Eq*)


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EqF "[ A type; a : A; b : A ] ==> Eq(A,a,b) type"


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EqFL "[ A=B; a=c: A; b=d : A ] ==> Eq(A,a,b) = Eq(B,c,d)"


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EqI "a = b : A ==> eq : Eq(A,a,b)"


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EqE "p : Eq(A,a,b) ==> a = b : A"


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(*By equality of types, can prove C(p) from C(eq), an elimination rule*)


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EqC "p : Eq(A,a,b) ==> p = eq : Eq(A,a,b)"


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(*The type F*)


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FF "F type"


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FE "[ p: F; C type ] ==> contr(p) : C"


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FEL "[ p = q : F; C type ] ==> contr(p) = contr(q) : C"


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(*The type T


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MartinLof's book (page 68) discusses elimination and computation.


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Elimination can be derived by computation and equality of types,


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but with an extra premise C(x) type x:T.


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Also computation can be derived from elimination. *)


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TF "T type"


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TI "tt : T"


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TE "[ p : T; c : C(tt) ] ==> c : C(p)"


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TEL "[ p = q : T; c = d : C(tt) ] ==> c = d : C(p)"


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TC "p : T ==> p = tt : T"


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end


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ML


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val print_translation =


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[("Prod", dependent_tr' ("@PROD", "@>")),


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("Sum", dependent_tr' ("@SUM", "@*"))];


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