author  wenzelm 
Mon, 31 Mar 2014 12:35:39 +0200  
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parent 56250  2c9f841f36b8 
child 58889  5b7a9633cfa8 
permissions  rwrr 
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(* Title: CTT/CTT.thy 
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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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ML_file "~~/src/Provers/typedsimp.ML" 
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setup Pure_Thy.old_appl_syntax_setup 
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setup PureThy.old_appl_syntax_setup  theory Pure provides regular application syntax by default;
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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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Plus :: "[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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app :: "[i,i]=>i" (infixl "`" 60) 
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(*Natural numbers*) 
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Zero :: "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" == "CONST Prod(A, %x. B)" 
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"SUM x:A. B" == "CONST 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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(*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.*) 

99 

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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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axiomatization where 
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refl_red: "\<And>a. Reduce[a,a]" and 

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red_if_equal: "\<And>a b A. a = b : A ==> Reduce[a,b]" and 

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trans_red: "\<And>a b c A. [ a = b : A; Reduce[b,c] ] ==> a = c : A" and 

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107 
(*Reflexivity*) 

108 

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refl_type: "\<And>A. A type ==> A = A" and 
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refl_elem: "\<And>a A. a : A ==> a = a : A" and 

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(*Symmetry*) 

113 

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sym_type: "\<And>A B. A = B ==> B = A" and 
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sym_elem: "\<And>a b A. a = b : A ==> b = a : A" and 

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(*Transitivity*) 

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trans_type: "\<And>A B C. [ A = B; B = C ] ==> A = C" and 
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trans_elem: "\<And>a b c A. [ a = b : A; b = c : A ] ==> a = c : A" and 

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equal_types: "\<And>a A B. [ a : A; A = B ] ==> a : B" and 
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equal_typesL: "\<And>a b A B. [ a = b : A; A = B ] ==> a = b : B" and 

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125 
(*Substitution*) 

126 

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subst_type: "\<And>a A B. [ a : A; !!z. z:A ==> B(z) type ] ==> B(a) type" and 
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subst_typeL: "\<And>a c A B D. [ a = c : A; !!z. z:A ==> B(z) = D(z) ] ==> B(a) = D(c)" and 

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subst_elem: "\<And>a b A B. [ a : A; !!z. z:A ==> b(z):B(z) ] ==> b(a):B(a)" and 
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subst_elemL: 
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"\<And>a b c d A B. [ a=c : A; !!z. z:A ==> b(z)=d(z) : B(z) ] ==> b(a)=d(c) : B(a)" and 
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134 

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(*The type N  natural numbers*) 

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NF: "N type" and 
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NI0: "0 : N" and 

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NI_succ: "\<And>a. a : N ==> succ(a) : N" and 

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NI_succL: "\<And>a b. a = b : N ==> succ(a) = succ(b) : N" and 

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NE: 
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"\<And>p a b C. [ 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)" and 

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NEL: 
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"\<And>p q a b c d C. [ 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)" and 
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NC0: 
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"\<And>a b C. [ 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)" and 

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NC_succ: 
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"\<And>p a b C. [ 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))" and 
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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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"\<And>a. [ a: N; 0 = succ(a) : N ] ==> 0: F" and 
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164 

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(*The Product of a family of types*) 

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ProdF: "\<And>A B. [ A type; !!x. x:A ==> B(x) type ] ==> PROD x:A. B(x) type" and 
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ProdFL: 
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"\<And>A B C D. [ A = C; !!x. x:A ==> B(x) = D(x) ] ==> 
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PROD x:A. B(x) = PROD x:C. D(x)" and 

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ProdI: 
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"\<And>b A B. [ A type; !!x. x:A ==> b(x):B(x)] ==> lam x. b(x) : PROD x:A. B(x)" and 
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ProdIL: 
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"\<And>b c A B. [ 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)" and 

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

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ProdC: 
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"\<And>a b A B. [ a : A; !!x. x:A ==> b(x) : B(x)] ==> 
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(lam x. b(x)) ` a = b(a) : B(a)" and 

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ProdC2: 
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"\<And>p A B. p : PROD x:A. B(x) ==> (lam x. p`x) = p : PROD x:A. B(x)" and 
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190 

191 
(*The Sum of a family of types*) 

192 

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SumF: "\<And>A B. [ A type; !!x. x:A ==> B(x) type ] ==> SUM x:A. B(x) type" and 
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SumFL: 
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"\<And>A B C D. [ A = C; !!x. x:A ==> B(x) = D(x) ] ==> SUM x:A. B(x) = SUM x:C. D(x)" and 
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SumI: "\<And>a b A B. [ a : A; b : B(a) ] ==> <a,b> : SUM x:A. B(x)" and 
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SumIL: "\<And>a b c d A B. [ a=c:A; b=d:B(a) ] ==> <a,b> = <c,d> : SUM x:A. B(x)" and 

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SumE: 
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"\<And>p c A B C. [ 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)" and 

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SumEL: 
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"\<And>p q c d A B C. [ 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)" and 
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SumC: 
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"\<And>a b c A B C. [ 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>)" and 

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fst_def: "\<And>a. fst(a) == split(a, %x y. x)" and 
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snd_def: "\<And>a. snd(a) == split(a, %x y. y)" and 

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216 

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(*The sum of two types*) 

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PlusF: "\<And>A B. [ A type; B type ] ==> A+B type" and 
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PlusFL: "\<And>A B C D. [ A = C; B = D ] ==> A+B = C+D" and 

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51308  222 
PlusI_inl: "\<And>a A B. [ a : A; B type ] ==> inl(a) : A+B" and 
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PlusI_inlL: "\<And>a c A B. [ a = c : A; B type ] ==> inl(a) = inl(c) : A+B" and 

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PlusI_inr: "\<And>b A B. [ A type; b : B ] ==> inr(b) : A+B" and 
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PlusI_inrL: "\<And>b d A B. [ A type; b = d : B ] ==> inr(b) = inr(d) : A+B" and 

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17441  228 
PlusE: 
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"\<And>p c d A B C. [ 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)" and 
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PlusEL: 
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"\<And>p q c d e f A B C. [ 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)" and 
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17441  238 
PlusC_inl: 
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"\<And>a c d A C. [ 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))" and 
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17441  243 
PlusC_inr: 
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"\<And>b c d A B C. [ b: B; !!x. x:A ==> c(x): C(inl(x)); 
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!!y. y:B ==> d(y): C(inr(y)) ] 
51308  246 
==> when(inr(b), %x. c(x), %y. d(y)) = d(b) : C(inr(b))" and 
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248 

249 
(*The type Eq*) 

250 

51308  251 
EqF: "\<And>a b A. [ A type; a : A; b : A ] ==> Eq(A,a,b) type" and 
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EqFL: "\<And>a b c d A B. [ A=B; a=c: A; b=d : A ] ==> Eq(A,a,b) = Eq(B,c,d)" and 

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EqI: "\<And>a b A. a = b : A ==> eq : Eq(A,a,b)" and 

254 
EqE: "\<And>p a b A. p : Eq(A,a,b) ==> a = b : A" and 

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256 
(*By equality of types, can prove C(p) from C(eq), an elimination rule*) 

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EqC: "\<And>p a b A. p : Eq(A,a,b) ==> p = eq : Eq(A,a,b)" and 
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(*The type F*) 

260 

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FF: "F type" and 
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FE: "\<And>p C. [ p: F; C type ] ==> contr(p) : C" and 

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FEL: "\<And>p q C. [ p = q : F; C type ] ==> contr(p) = contr(q) : C" and 

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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. 

269 
Also computation can be derived from elimination. *) 

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51308  271 
TF: "T type" and 
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TI: "tt : T" and 

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TE: "\<And>p c C. [ p : T; c : C(tt) ] ==> c : C(p)" and 

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TEL: "\<And>p q c d C. [ p = q : T; c = d : C(tt) ] ==> c = d : C(p)" and 

275 
TC: "\<And>p. p : T ==> p = tt : T" 

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subsection "Tactics and derived rules for Constructive Type Theory" 

279 

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(*Formation rules*) 

281 
lemmas form_rls = NF ProdF SumF PlusF EqF FF TF 

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and formL_rls = ProdFL SumFL PlusFL EqFL 

283 

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(*Introduction rules 

285 
OMITTED: EqI, because its premise is an eqelem, not an elem*) 

286 
lemmas intr_rls = NI0 NI_succ ProdI SumI PlusI_inl PlusI_inr TI 

287 
and intrL_rls = NI_succL ProdIL SumIL PlusI_inlL PlusI_inrL 

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(*Elimination rules 

290 
OMITTED: EqE, because its conclusion is an eqelem, not an elem 

291 
TE, because it does not involve a constructor *) 

292 
lemmas elim_rls = NE ProdE SumE PlusE FE 

293 
and elimL_rls = NEL ProdEL SumEL PlusEL FEL 

294 

295 
(*OMITTED: eqC are TC because they make rewriting loop: p = un = un = ... *) 

296 
lemmas comp_rls = NC0 NC_succ ProdC SumC PlusC_inl PlusC_inr 

297 

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

299 
lemmas element_rls = intr_rls elim_rls 

300 

301 
(*Definitions are (meta)equality axioms*) 

302 
lemmas basic_defs = fst_def snd_def 

303 

304 
(*Compare with standard version: B is applied to UNSIMPLIFIED expression! *) 

305 
lemma SumIL2: "[ c=a : A; d=b : B(a) ] ==> <c,d> = <a,b> : Sum(A,B)" 

306 
apply (rule sym_elem) 

307 
apply (rule SumIL) 

308 
apply (rule_tac [!] sym_elem) 

309 
apply assumption+ 

310 
done 

311 

312 
lemmas intrL2_rls = NI_succL ProdIL SumIL2 PlusI_inlL PlusI_inrL 

313 

314 
(*Exploit p:Prod(A,B) to create the assumption z:B(a). 

315 
A more natural form of product elimination. *) 

316 
lemma subst_prodE: 

317 
assumes "p: Prod(A,B)" 

318 
and "a: A" 

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

320 
shows "c(p`a): C(p`a)" 

41526  321 
apply (rule assms ProdE)+ 
19761  322 
done 
323 

324 

325 
subsection {* Tactics for type checking *} 

326 

327 
ML {* 

328 

329 
local 

330 

56250  331 
fun is_rigid_elem (Const(@{const_name Elem},_) $ a $ _) = not(is_Var (head_of a)) 
332 
 is_rigid_elem (Const(@{const_name Eqelem},_) $ a $ _ $ _) = not(is_Var (head_of a)) 

333 
 is_rigid_elem (Const(@{const_name Type},_) $ a) = not(is_Var (head_of a)) 

19761  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 

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val equal_rls = @{thms form_rls} @ @{thms element_rls} @ @{thms intrL_rls} @ 
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@{thms elimL_rls} @ @{thms refl_elem} 
19761  357 
in 
358 

359 
fun routine_tac rls prems = ASSUME (filt_resolve_tac (prems @ rls) 4); 

360 

361 
(*Solve all subgoals "A type" using formation rules. *) 

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val form_tac = REPEAT_FIRST (ASSUME (filt_resolve_tac @{thms form_rls} 1)); 
19761  363 

364 
(*Type checking: solve a:A (a rigid, A flexible) by intro and elim rules. *) 

365 
fun typechk_tac thms = 

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let val tac = filt_resolve_tac (thms @ @{thms form_rls} @ @{thms element_rls}) 3 
19761  367 
in REPEAT_FIRST (ASSUME tac) end 
368 

369 
(*Solve a:A (a flexible, A rigid) by introduction rules. 

370 
Cannot use stringtrees (filt_resolve_tac) since 

371 
goals like ?a:SUM(A,B) have a trivial headstring *) 

372 
fun intr_tac thms = 

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let val tac = filt_resolve_tac(thms @ @{thms form_rls} @ @{thms intr_rls}) 1 
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in REPEAT_FIRST (ASSUME tac) end 
375 

376 
(*Equality proving: solve a=b:A (where a is rigid) by long rules. *) 

377 
fun equal_tac thms = 

378 
REPEAT_FIRST (ASSUME (filt_resolve_tac (thms @ equal_rls) 3)) 

0  379 

17441  380 
end 
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382 
*} 

383 

384 

385 
subsection {* Simplification *} 

386 

387 
(*To simplify the type in a goal*) 

388 
lemma replace_type: "[ B = A; a : A ] ==> a : B" 

389 
apply (rule equal_types) 

390 
apply (rule_tac [2] sym_type) 

391 
apply assumption+ 

392 
done 

393 

394 
(*Simplify the parameter of a unary type operator.*) 

395 
lemma subst_eqtyparg: 

23467  396 
assumes 1: "a=c : A" 
397 
and 2: "!!z. z:A ==> B(z) type" 

19761  398 
shows "B(a)=B(c)" 
399 
apply (rule subst_typeL) 

400 
apply (rule_tac [2] refl_type) 

23467  401 
apply (rule 1) 
402 
apply (erule 2) 

19761  403 
done 
404 

405 
(*Simplification rules for Constructive Type Theory*) 

406 
lemmas reduction_rls = comp_rls [THEN trans_elem] 

407 

408 
ML {* 

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

410 
Uses other intro rules to avoid changing flexible goals.*) 

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val eqintr_tac = REPEAT_FIRST (ASSUME (filt_resolve_tac (@{thm EqI} :: @{thms intr_rls}) 1)) 
19761  412 

413 
(** Tactics that instantiate CTTrules. 

414 
Vars in the given terms will be incremented! 

415 
The (rtac EqE i) lets them apply to equality judgements. **) 

416 

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fun NE_tac ctxt sp i = 
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TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm NE} i 
19761  419 

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fun SumE_tac ctxt sp i = 
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TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm SumE} i 
19761  422 

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fun PlusE_tac ctxt sp i = 
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TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm PlusE} i 
19761  425 

426 
(** Predicate logic reasoning, WITH THINNING!! Procedures adapted from NJ. **) 

427 

428 
(*Finds f:Prod(A,B) and a:A in the assumptions, concludes there is z:B(a) *) 

429 
fun add_mp_tac i = 

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rtac @{thm subst_prodE} i THEN assume_tac i THEN assume_tac i 
19761  431 

432 
(*Finds P>Q and P in the assumptions, replaces implication by Q *) 

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fun mp_tac i = etac @{thm subst_prodE} i THEN assume_tac i 
19761  434 

435 
(*"safe" when regarded as predicate calculus rules*) 

436 
val safe_brls = sort (make_ord lessb) 

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[ (true, @{thm FE}), (true,asm_rl), 
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(false, @{thm ProdI}), (true, @{thm SumE}), (true, @{thm PlusE}) ] 
19761  439 

440 
val unsafe_brls = 

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[ (false, @{thm PlusI_inl}), (false, @{thm PlusI_inr}), (false, @{thm SumI}), 
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(true, @{thm subst_prodE}) ] 
19761  443 

444 
(*0 subgoals vs 1 or more*) 

445 
val (safe0_brls, safep_brls) = 

446 
List.partition (curry (op =) 0 o subgoals_of_brl) safe_brls 

447 

448 
fun safestep_tac thms i = 

449 
form_tac ORELSE 

450 
resolve_tac thms i ORELSE 

451 
biresolve_tac safe0_brls i ORELSE mp_tac i ORELSE 

452 
DETERM (biresolve_tac safep_brls i) 

453 

454 
fun safe_tac thms i = DEPTH_SOLVE_1 (safestep_tac thms i) 

455 

456 
fun step_tac thms = safestep_tac thms ORELSE' biresolve_tac unsafe_brls 

457 

458 
(*Fails unless it solves the goal!*) 

459 
fun pc_tac thms = DEPTH_SOLVE_1 o (step_tac thms) 

460 
*} 

461 

48891  462 
ML_file "rew.ML" 
19761  463 

464 

465 
subsection {* The elimination rules for fst/snd *} 

466 

467 
lemma SumE_fst: "p : Sum(A,B) ==> fst(p) : A" 

468 
apply (unfold basic_defs) 

469 
apply (erule SumE) 

470 
apply assumption 

471 
done 

472 

473 
(*The first premise must be p:Sum(A,B) !!*) 

474 
lemma SumE_snd: 

475 
assumes major: "p: Sum(A,B)" 

476 
and "A type" 

477 
and "!!x. x:A ==> B(x) type" 

478 
shows "snd(p) : B(fst(p))" 

479 
apply (unfold basic_defs) 

480 
apply (rule major [THEN SumE]) 

481 
apply (rule SumC [THEN subst_eqtyparg, THEN replace_type]) 

26391  482 
apply (tactic {* typechk_tac @{thms assms} *}) 
19761  483 
done 
484 

485 
end 