author  wenzelm 
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parent 21404  eb85850d3eb7 
child 22808  a7daa74e2980 
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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uses "~~/src/Provers/typedsimp.ML" ("rew.ML") 
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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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translations 
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"PROD x:A. B" == "Prod(A, %x. B)" 
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"SUM x:A. B" == "Sum(A, %x. B)" 

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abbreviation 

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Arrow :: "[t,t]=>t" (infixr ">" 30) where 
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"A > B == PROD _:A. B" 
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abbreviation 
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Times :: "[t,t]=>t" (infixr "*" 50) where 
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"A * B == SUM _:A. B" 
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notation (xsymbols) 
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lambda (binder "\<lambda>\<lambda>" 10) and 
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Elem ("(_ /\<in> _)" [10,10] 5) and 
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Eqelem ("(2_ =/ _ \<in>/ _)" [10,10,10] 5) and 
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Arrow (infixr "\<longrightarrow>" 30) and 
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Times (infixr "\<times>" 50) 
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notation (HTML output) 
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lambda (binder "\<lambda>\<lambda>" 10) and 
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Elem ("(_ /\<in> _)" [10,10] 5) and 
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Eqelem ("(2_ =/ _ \<in>/ _)" [10,10,10] 5) and 
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Times (infixr "\<times>" 50) 
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syntax (xsymbols) 
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"_PROD" :: "[idt,t,t] => t" ("(3\<Pi> _\<in>_./ _)" 10) 
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"_SUM" :: "[idt,t,t] => t" ("(3\<Sigma> _\<in>_./ _)" 10) 

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syntax (HTML output) 
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"_PROD" :: "[idt,t,t] => t" ("(3\<Pi> _\<in>_./ _)" 10) 
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"_SUM" :: "[idt,t,t] => t" ("(3\<Sigma> _\<in>_./ _)" 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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134 

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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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subsection "Tactics and derived rules for Constructive Type Theory" 

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(*Formation rules*) 

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lemmas form_rls = NF ProdF SumF PlusF EqF FF TF 

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and formL_rls = ProdFL SumFL PlusFL EqFL 

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(*Introduction rules 

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OMITTED: EqI, because its premise is an eqelem, not an elem*) 

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lemmas intr_rls = NI0 NI_succ ProdI SumI PlusI_inl PlusI_inr TI 

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and intrL_rls = NI_succL ProdIL SumIL PlusI_inlL PlusI_inrL 

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(*Elimination rules 

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OMITTED: EqE, because its conclusion is an eqelem, not an elem 

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TE, because it does not involve a constructor *) 

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lemmas elim_rls = NE ProdE SumE PlusE FE 

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and elimL_rls = NEL ProdEL SumEL PlusEL FEL 

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(*OMITTED: eqC are TC because they make rewriting loop: p = un = un = ... *) 

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lemmas comp_rls = NC0 NC_succ ProdC SumC PlusC_inl PlusC_inr 

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298 
(*rules with conclusion a:A, an elem judgement*) 

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lemmas element_rls = intr_rls elim_rls 

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(*Definitions are (meta)equality axioms*) 

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lemmas basic_defs = fst_def snd_def 

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(*Compare with standard version: B is applied to UNSIMPLIFIED expression! *) 

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lemma SumIL2: "[ c=a : A; d=b : B(a) ] ==> <c,d> = <a,b> : Sum(A,B)" 

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apply (rule sym_elem) 

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apply (rule SumIL) 

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apply (rule_tac [!] sym_elem) 

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apply assumption+ 

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done 

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312 
lemmas intrL2_rls = NI_succL ProdIL SumIL2 PlusI_inlL PlusI_inrL 

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(*Exploit p:Prod(A,B) to create the assumption z:B(a). 

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A more natural form of product elimination. *) 

316 
lemma subst_prodE: 

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assumes "p: Prod(A,B)" 

318 
and "a: A" 

319 
and "!!z. z: B(a) ==> c(z): C(z)" 

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shows "c(p`a): C(p`a)" 

321 
apply (rule prems ProdE)+ 

322 
done 

323 

324 

325 
subsection {* Tactics for type checking *} 

326 

327 
ML {* 

328 

329 
local 

330 

331 
fun is_rigid_elem (Const("CTT.Elem",_) $ a $ _) = not(is_Var (head_of a)) 

332 
 is_rigid_elem (Const("CTT.Eqelem",_) $ a $ _ $ _) = not(is_Var (head_of a)) 

333 
 is_rigid_elem (Const("CTT.Type",_) $ a) = not(is_Var (head_of a)) 

334 
 is_rigid_elem _ = false 

335 

336 
in 

337 

338 
(*Try solving a:A or a=b:A by assumption provided a is rigid!*) 

339 
val test_assume_tac = SUBGOAL(fn (prem,i) => 

340 
if is_rigid_elem (Logic.strip_assums_concl prem) 

341 
then assume_tac i else no_tac) 

342 

343 
fun ASSUME tf i = test_assume_tac i ORELSE tf i 

344 

345 
end; 

346 

347 
*} 

348 

349 
(*For simplification: type formation and checking, 

350 
but no equalities between terms*) 

351 
lemmas routine_rls = form_rls formL_rls refl_type element_rls 

352 

353 
ML {* 

354 
local 

355 
val routine_rls = thms "routine_rls"; 

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val form_rls = thms "form_rls"; 

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val intr_rls = thms "intr_rls"; 

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val element_rls = thms "element_rls"; 

359 
val equal_rls = form_rls @ element_rls @ thms "intrL_rls" @ 

360 
thms "elimL_rls" @ thms "refl_elem" 

361 
in 

362 

363 
fun routine_tac rls prems = ASSUME (filt_resolve_tac (prems @ rls) 4); 

364 

365 
(*Solve all subgoals "A type" using formation rules. *) 

366 
val form_tac = REPEAT_FIRST (ASSUME (filt_resolve_tac(form_rls) 1)); 

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368 
(*Type checking: solve a:A (a rigid, A flexible) by intro and elim rules. *) 

369 
fun typechk_tac thms = 

370 
let val tac = filt_resolve_tac (thms @ form_rls @ element_rls) 3 

371 
in REPEAT_FIRST (ASSUME tac) end 

372 

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(*Solve a:A (a flexible, A rigid) by introduction rules. 

374 
Cannot use stringtrees (filt_resolve_tac) since 

375 
goals like ?a:SUM(A,B) have a trivial headstring *) 

376 
fun intr_tac thms = 

377 
let val tac = filt_resolve_tac(thms@form_rls@intr_rls) 1 

378 
in REPEAT_FIRST (ASSUME tac) end 

379 

380 
(*Equality proving: solve a=b:A (where a is rigid) by long rules. *) 

381 
fun equal_tac thms = 

382 
REPEAT_FIRST (ASSUME (filt_resolve_tac (thms @ equal_rls) 3)) 

0  383 

17441  384 
end 
19761  385 

386 
*} 

387 

388 

389 
subsection {* Simplification *} 

390 

391 
(*To simplify the type in a goal*) 

392 
lemma replace_type: "[ B = A; a : A ] ==> a : B" 

393 
apply (rule equal_types) 

394 
apply (rule_tac [2] sym_type) 

395 
apply assumption+ 

396 
done 

397 

398 
(*Simplify the parameter of a unary type operator.*) 

399 
lemma subst_eqtyparg: 

400 
assumes "a=c : A" 

401 
and "!!z. z:A ==> B(z) type" 

402 
shows "B(a)=B(c)" 

403 
apply (rule subst_typeL) 

404 
apply (rule_tac [2] refl_type) 

405 
apply (rule prems) 

406 
apply assumption 

407 
done 

408 

409 
(*Simplification rules for Constructive Type Theory*) 

410 
lemmas reduction_rls = comp_rls [THEN trans_elem] 

411 

412 
ML {* 

413 
local 

414 
val EqI = thm "EqI"; 

415 
val EqE = thm "EqE"; 

416 
val NE = thm "NE"; 

417 
val FE = thm "FE"; 

418 
val ProdI = thm "ProdI"; 

419 
val SumI = thm "SumI"; 

420 
val SumE = thm "SumE"; 

421 
val PlusE = thm "PlusE"; 

422 
val PlusI_inl = thm "PlusI_inl"; 

423 
val PlusI_inr = thm "PlusI_inr"; 

424 
val subst_prodE = thm "subst_prodE"; 

425 
val intr_rls = thms "intr_rls"; 

426 
in 

427 

428 
(*Converts each goal "e : Eq(A,a,b)" into "a=b:A" for simplification. 

429 
Uses other intro rules to avoid changing flexible goals.*) 

430 
val eqintr_tac = REPEAT_FIRST (ASSUME (filt_resolve_tac(EqI::intr_rls) 1)) 

431 

432 
(** Tactics that instantiate CTTrules. 

433 
Vars in the given terms will be incremented! 

434 
The (rtac EqE i) lets them apply to equality judgements. **) 

435 

436 
fun NE_tac (sp: string) i = 

437 
TRY (rtac EqE i) THEN res_inst_tac [ ("p",sp) ] NE i 

438 

439 
fun SumE_tac (sp: string) i = 

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TRY (rtac EqE i) THEN res_inst_tac [ ("p",sp) ] SumE i 

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fun PlusE_tac (sp: string) i = 

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TRY (rtac EqE i) THEN res_inst_tac [ ("p",sp) ] PlusE i 

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(** Predicate logic reasoning, WITH THINNING!! Procedures adapted from NJ. **) 

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(*Finds f:Prod(A,B) and a:A in the assumptions, concludes there is z:B(a) *) 

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fun add_mp_tac i = 

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rtac subst_prodE i THEN assume_tac i THEN assume_tac i 

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(*Finds P>Q and P in the assumptions, replaces implication by Q *) 

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fun mp_tac i = etac subst_prodE i THEN assume_tac i 

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(*"safe" when regarded as predicate calculus rules*) 

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val safe_brls = sort (make_ord lessb) 

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[ (true,FE), (true,asm_rl), 

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(false,ProdI), (true,SumE), (true,PlusE) ] 

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val unsafe_brls = 

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[ (false,PlusI_inl), (false,PlusI_inr), (false,SumI), 

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(true,subst_prodE) ] 

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(*0 subgoals vs 1 or more*) 

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val (safe0_brls, safep_brls) = 

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List.partition (curry (op =) 0 o subgoals_of_brl) safe_brls 

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fun safestep_tac thms i = 

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form_tac ORELSE 

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resolve_tac thms i ORELSE 

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biresolve_tac safe0_brls i ORELSE mp_tac i ORELSE 

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DETERM (biresolve_tac safep_brls i) 

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fun safe_tac thms i = DEPTH_SOLVE_1 (safestep_tac thms i) 

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fun step_tac thms = safestep_tac thms ORELSE' biresolve_tac unsafe_brls 

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(*Fails unless it solves the goal!*) 

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fun pc_tac thms = DEPTH_SOLVE_1 o (step_tac thms) 

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end 

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*} 

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use "rew.ML" 

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subsection {* The elimination rules for fst/snd *} 

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lemma SumE_fst: "p : Sum(A,B) ==> fst(p) : A" 

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apply (unfold basic_defs) 

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apply (erule SumE) 

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apply assumption 

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done 

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(*The first premise must be p:Sum(A,B) !!*) 

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lemma SumE_snd: 

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assumes major: "p: Sum(A,B)" 

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and "A type" 

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and "!!x. x:A ==> B(x) type" 

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shows "snd(p) : B(fst(p))" 

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apply (unfold basic_defs) 

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apply (rule major [THEN SumE]) 

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apply (rule SumC [THEN subst_eqtyparg, THEN replace_type]) 

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apply (tactic {* typechk_tac (thms "prems") *}) 

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done 

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end 