author  wenzelm 
Fri, 16 Sep 2005 23:01:29 +0200  
changeset 17441  5b5feca0344a 
parent 14854  61bdf2ae4dc5 
child 17782  b3846df9d643 
permissions  rwrr 
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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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*) 

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header {* Constructive Type Theory *} 
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theory CTT 
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imports Pure 

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begin 

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typedecl i 

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typedecl t 

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typedecl o 

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consts 

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(*Types*) 

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F :: "t" 
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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 :: "i=>i" 
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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 :: "i=>i" 
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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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(*Types*) 
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"+" :: "[t,t]=>t" (infixr 40) 

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

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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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syntax 
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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" ("(_ >/ _)" [31,30] 30) 

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

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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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print_translation {* 
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[("Prod", dependent_tr' ("@PROD", "@>")), 

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

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*} 

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syntax (xsymbols) 
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"@>" :: "[t,t]=>t" ("(_ \<longrightarrow>/ _)" [31,30] 30) 
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"@*" :: "[t,t]=>t" ("(_ \<times>/ _)" [51,50] 50) 

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

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Eqelem :: "[i,i,t]=>prop" ("(2_ =/ _ \<in>/ _)" [10,10,10] 5) 

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

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

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"lam " :: "[idts, i] => i" ("(3\<lambda>\<lambda>_./ _)" 10) 

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syntax (HTML output) 
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"@*" :: "[t,t]=>t" ("(_ \<times>/ _)" [51,50] 50) 
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Elem :: "[i, t]=>prop" ("(_ /\<in> _)" [10,10] 5) 

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Eqelem :: "[i,i,t]=>prop" ("(2_ =/ _ \<in>/ _)" [10,10,10] 5) 

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

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

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"lam " :: "[idts, i] => i" ("(3\<lambda>\<lambda>_./ _)" 10) 

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axioms 
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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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ML {* use_legacy_bindings (the_context ()) *} 
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end 