author  wenzelm 
Tue, 11 Nov 2014 15:55:31 +0100  
changeset 58977  9576b510f6a2 
parent 58976  b38a54bbfbd6 
child 59164  ff40c53d1af9 
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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section {* 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\<Rightarrow>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\<Rightarrow>i" 
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rec :: "[i, i, [i,i]\<Rightarrow>i] \<Rightarrow> i" 

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(*Unions*) 
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inl :: "i\<Rightarrow>i" 
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inr :: "i\<Rightarrow>i" 

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when :: "[i, i\<Rightarrow>i, i\<Rightarrow>i]\<Rightarrow>i" 

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(*General Sum and Binary Product*) 
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Sum :: "[t, i\<Rightarrow>t]\<Rightarrow>t" 
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fst :: "i\<Rightarrow>i" 

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snd :: "i\<Rightarrow>i" 

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split :: "[i, [i,i]\<Rightarrow>i] \<Rightarrow>i" 

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(*General Product and Function Space*) 
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Prod :: "[t, i\<Rightarrow>t]\<Rightarrow>t" 
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(*Types*) 
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Plus :: "[t,t]\<Rightarrow>t" (infixr "+" 40) 
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(*Equality type*) 
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Eq :: "[t,i,i]\<Rightarrow>t" 
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eq :: "i" 
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(*Judgements*) 

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Type :: "t \<Rightarrow> prop" ("(_ type)" [10] 5) 
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Eqtype :: "[t,t]\<Rightarrow>prop" ("(_ =/ _)" [10,10] 5) 

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

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

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Reduce :: "[i,i]\<Rightarrow>prop" ("Reduce[_,_]") 

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(*Types*) 
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(*Functions*) 
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lambda :: "(i \<Rightarrow> i) \<Rightarrow> i" (binder "lam " 10) 
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app :: "[i,i]\<Rightarrow>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]\<Rightarrow>i" ("(1<_,/_>)") 
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syntax 
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"_PROD" :: "[idt,t,t]\<Rightarrow>t" ("(3PROD _:_./ _)" 10) 
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"_SUM" :: "[idt,t,t]\<Rightarrow>t" ("(3SUM _:_./ _)" 10) 

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translations 
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"PROD x:A. B" == "CONST Prod(A, \<lambda>x. B)" 
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"SUM x:A. B" == "CONST Sum(A, \<lambda>x. B)" 

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abbreviation 

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Arrow :: "[t,t]\<Rightarrow>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]\<Rightarrow>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] \<Rightarrow> t" ("(3\<Pi> _\<in>_./ _)" 10) 
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"_SUM" :: "[idt,t,t] \<Rightarrow> t" ("(3\<Sigma> _\<in>_./ _)" 10) 

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

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

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(*Reflexivity*) 

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

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(*Symmetry*) 

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

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(*Transitivity*) 

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trans_type: "\<And>A B C. \<lbrakk>A = B; B = C\<rbrakk> \<Longrightarrow> A = C" and 
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trans_elem: "\<And>a b c A. \<lbrakk>a = b : A; b = c : A\<rbrakk> \<Longrightarrow> a = c : A" and 

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equal_types: "\<And>a A B. \<lbrakk>a : A; A = B\<rbrakk> \<Longrightarrow> a : B" and 
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equal_typesL: "\<And>a b A B. \<lbrakk>a = b : A; A = B\<rbrakk> \<Longrightarrow> a = b : B" and 

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(*Substitution*) 

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subst_type: "\<And>a A B. \<lbrakk>a : A; \<And>z. z:A \<Longrightarrow> B(z) type\<rbrakk> \<Longrightarrow> B(a) type" and 
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subst_typeL: "\<And>a c A B D. \<lbrakk>a = c : A; \<And>z. z:A \<Longrightarrow> B(z) = D(z)\<rbrakk> \<Longrightarrow> B(a) = D(c)" and 

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subst_elem: "\<And>a b A B. \<lbrakk>a : A; \<And>z. z:A \<Longrightarrow> b(z):B(z)\<rbrakk> \<Longrightarrow> b(a):B(a)" and 
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subst_elemL: 
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"\<And>a b c d A B. \<lbrakk>a = c : A; \<And>z. z:A \<Longrightarrow> b(z)=d(z) : B(z)\<rbrakk> \<Longrightarrow> b(a)=d(c) : B(a)" and 
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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 \<Longrightarrow> succ(a) : N" and 
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NI_succL: "\<And>a b. a = b : N \<Longrightarrow> succ(a) = succ(b) : N" and 

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NE: 
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"\<And>p a b C. \<lbrakk>p: N; a: C(0); \<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v): C(succ(u))\<rbrakk> 
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\<Longrightarrow> rec(p, a, \<lambda>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. \<lbrakk>p = q : N; a = c : C(0); 
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\<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v) = d(u,v): C(succ(u))\<rbrakk> 

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\<Longrightarrow> rec(p, a, \<lambda>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. \<lbrakk>a: C(0); \<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v): C(succ(u))\<rbrakk> 
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\<Longrightarrow> rec(0, a, \<lambda>u v. b(u,v)) = a : C(0)" and 

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NC_succ: 
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"\<And>p a b C. \<lbrakk>p: N; a: C(0); \<And>u v. \<lbrakk>u: N; v: C(u)\<rbrakk> \<Longrightarrow> b(u,v): C(succ(u))\<rbrakk> \<Longrightarrow> 
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rec(succ(p), a, \<lambda>u v. b(u,v)) = b(p, rec(p, a, \<lambda>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: "\<And>a. \<lbrakk>a: N; 0 = succ(a) : N\<rbrakk> \<Longrightarrow> 0: F" and 
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(*The Product of a family of types*) 

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ProdF: "\<And>A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> B(x) type\<rbrakk> \<Longrightarrow> PROD x:A. B(x) type" and 
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ProdFL: 
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"\<And>A B C D. \<lbrakk>A = C; \<And>x. x:A \<Longrightarrow> B(x) = D(x)\<rbrakk> \<Longrightarrow> PROD x:A. B(x) = PROD x:C. D(x)" and 
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ProdI: 
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"\<And>b A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> b(x):B(x)\<rbrakk> \<Longrightarrow> lam x. b(x) : PROD x:A. B(x)" and 
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ProdIL: "\<And>b c A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> b(x) = c(x) : B(x)\<rbrakk> \<Longrightarrow> 
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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. \<lbrakk>p : PROD x:A. B(x); a : A\<rbrakk> \<Longrightarrow> p`a : B(a)" and 
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ProdEL: "\<And>p q a b A B. \<lbrakk>p = q: PROD x:A. B(x); a = b : A\<rbrakk> \<Longrightarrow> p`a = q`b : B(a)" and 

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ProdC: "\<And>a b A B. \<lbrakk>a : A; \<And>x. x:A \<Longrightarrow> b(x) : B(x)\<rbrakk> \<Longrightarrow> (lam x. b(x)) ` a = b(a) : B(a)" and 
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ProdC2: "\<And>p A B. p : PROD x:A. B(x) \<Longrightarrow> (lam x. p`x) = p : PROD x:A. B(x)" and 
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(*The Sum of a family of types*) 

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SumF: "\<And>A B. \<lbrakk>A type; \<And>x. x:A \<Longrightarrow> B(x) type\<rbrakk> \<Longrightarrow> SUM x:A. B(x) type" and 
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SumFL: "\<And>A B C D. \<lbrakk>A = C; \<And>x. x:A \<Longrightarrow> B(x) = D(x)\<rbrakk> \<Longrightarrow> SUM x:A. B(x) = SUM x:C. D(x)" and 

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SumI: "\<And>a b A B. \<lbrakk>a : A; b : B(a)\<rbrakk> \<Longrightarrow> <a,b> : SUM x:A. B(x)" and 
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SumIL: "\<And>a b c d A B. \<lbrakk> a = c : A; b = d : B(a)\<rbrakk> \<Longrightarrow> <a,b> = <c,d> : SUM x:A. B(x)" and 

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SumE: "\<And>p c A B C. \<lbrakk>p: SUM x:A. B(x); \<And>x y. \<lbrakk>x:A; y:B(x)\<rbrakk> \<Longrightarrow> c(x,y): C(<x,y>)\<rbrakk> 
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\<Longrightarrow> split(p, \<lambda>x y. c(x,y)) : C(p)" and 

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SumEL: "\<And>p q c d A B C. \<lbrakk>p = q : SUM x:A. B(x); 
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\<And>x y. \<lbrakk>x:A; y:B(x)\<rbrakk> \<Longrightarrow> c(x,y)=d(x,y): C(<x,y>)\<rbrakk> 

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\<Longrightarrow> split(p, \<lambda>x y. c(x,y)) = split(q, \<lambda>x y. d(x,y)) : C(p)" and 

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SumC: "\<And>a b c A B C. \<lbrakk>a: A; b: B(a); \<And>x y. \<lbrakk>x:A; y:B(x)\<rbrakk> \<Longrightarrow> c(x,y): C(<x,y>)\<rbrakk> 
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\<Longrightarrow> split(<a,b>, \<lambda>x y. c(x,y)) = c(a,b) : C(<a,b>)" and 

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

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(*The sum of two types*) 

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PlusF: "\<And>A B. \<lbrakk>A type; B type\<rbrakk> \<Longrightarrow> A+B type" and 
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PlusFL: "\<And>A B C D. \<lbrakk>A = C; B = D\<rbrakk> \<Longrightarrow> A+B = C+D" and 

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PlusI_inl: "\<And>a A B. \<lbrakk>a : A; B type\<rbrakk> \<Longrightarrow> inl(a) : A+B" and 
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PlusI_inlL: "\<And>a c A B. \<lbrakk>a = c : A; B type\<rbrakk> \<Longrightarrow> inl(a) = inl(c) : A+B" and 

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

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PlusE: 
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"\<And>p c d A B C. \<lbrakk>p: A+B; 
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\<And>x. x:A \<Longrightarrow> c(x): C(inl(x)); 

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\<And>y. y:B \<Longrightarrow> d(y): C(inr(y)) \<rbrakk> \<Longrightarrow> when(p, \<lambda>x. c(x), \<lambda>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. \<lbrakk>p = q : A+B; 
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\<And>x. x: A \<Longrightarrow> c(x) = e(x) : C(inl(x)); 

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\<And>y. y: B \<Longrightarrow> d(y) = f(y) : C(inr(y))\<rbrakk> 

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\<Longrightarrow> when(p, \<lambda>x. c(x), \<lambda>y. d(y)) = when(q, \<lambda>x. e(x), \<lambda>y. f(y)) : C(p)" and 

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PlusC_inl: 
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"\<And>a c d A C. \<lbrakk>a: A; 
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\<And>x. x:A \<Longrightarrow> c(x): C(inl(x)); 

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\<And>y. y:B \<Longrightarrow> d(y): C(inr(y)) \<rbrakk> 

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\<Longrightarrow> when(inl(a), \<lambda>x. c(x), \<lambda>y. d(y)) = c(a) : C(inl(a))" and 

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PlusC_inr: 
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"\<And>b c d A B C. \<lbrakk>b: B; 
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\<And>x. x:A \<Longrightarrow> c(x): C(inl(x)); 

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\<And>y. y:B \<Longrightarrow> d(y): C(inr(y))\<rbrakk> 

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\<Longrightarrow> when(inr(b), \<lambda>x. c(x), \<lambda>y. d(y)) = d(b) : C(inr(b))" and 

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(*The type Eq*) 

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EqF: "\<And>a b A. \<lbrakk>A type; a : A; b : A\<rbrakk> \<Longrightarrow> Eq(A,a,b) type" and 
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EqFL: "\<And>a b c d A B. \<lbrakk>A = B; a = c : A; b = d : A\<rbrakk> \<Longrightarrow> Eq(A,a,b) = Eq(B,c,d)" and 

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

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

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

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FF: "F type" and 
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FE: "\<And>p C. \<lbrakk>p: F; C type\<rbrakk> \<Longrightarrow> contr(p) : C" and 
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FEL: "\<And>p q C. \<lbrakk>p = q : F; C type\<rbrakk> \<Longrightarrow> 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. 

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Also computation can be derived from elimination. *) 

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

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TE: "\<And>p c C. \<lbrakk>p : T; c : C(tt)\<rbrakk> \<Longrightarrow> c : C(p)" and 
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TEL: "\<And>p q c d C. \<lbrakk>p = q : T; c = d : C(tt)\<rbrakk> \<Longrightarrow> c = d : C(p)" and 

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TC: "\<And>p. p : T \<Longrightarrow> 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 

282 
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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(*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: "\<lbrakk>c = a : A; d = b : B(a)\<rbrakk> \<Longrightarrow> <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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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. *) 

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lemma subst_prodE: 

309 
assumes "p: Prod(A,B)" 

310 
and "a: A" 

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and "\<And>z. z: B(a) \<Longrightarrow> c(z): C(z)" 
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shows "c(p`a): C(p`a)" 
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apply (rule assms ProdE)+ 
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done 
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subsection {* Tactics for type checking *} 

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ML {* 

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local 

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

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

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 is_rigid_elem _ = false 
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in 

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

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fun test_assume_tac ctxt = SUBGOAL(fn (prem,i) => 
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if is_rigid_elem (Logic.strip_assums_concl prem) 
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then assume_tac ctxt i else no_tac) 
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fun ASSUME ctxt tf i = test_assume_tac ctxt i ORELSE tf i 
19761  336 

337 
end; 

338 

339 
*} 

340 

341 
(*For simplification: type formation and checking, 

342 
but no equalities between terms*) 

343 
lemmas routine_rls = form_rls formL_rls refl_type element_rls 

344 

345 
ML {* 

346 
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  349 
in 
350 

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fun routine_tac rls ctxt prems = ASSUME ctxt (filt_resolve_tac (prems @ rls) 4); 
19761  352 

353 
(*Solve all subgoals "A type" using formation rules. *) 

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fun form_tac ctxt = REPEAT_FIRST (ASSUME ctxt (filt_resolve_tac @{thms form_rls} 1)); 
19761  355 

356 
(*Type checking: solve a:A (a rigid, A flexible) by intro and elim rules. *) 

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fun typechk_tac ctxt thms = 
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let val tac = filt_resolve_tac (thms @ @{thms form_rls} @ @{thms element_rls}) 3 
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in REPEAT_FIRST (ASSUME ctxt tac) end 
19761  360 

361 
(*Solve a:A (a flexible, A rigid) by introduction rules. 

362 
Cannot use stringtrees (filt_resolve_tac) since 

363 
goals like ?a:SUM(A,B) have a trivial headstring *) 

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fun intr_tac ctxt 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 ctxt tac) end 
19761  367 

368 
(*Equality proving: solve a=b:A (where a is rigid) by long rules. *) 

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fun equal_tac ctxt thms = 
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REPEAT_FIRST (ASSUME ctxt (filt_resolve_tac (thms @ equal_rls) 3)) 
0  371 

17441  372 
end 
58972  373 
*} 
19761  374 

58976  375 
method_setup form = {* Scan.succeed (fn ctxt => SIMPLE_METHOD (form_tac ctxt)) *} 
376 
method_setup typechk = {* Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (typechk_tac ctxt ths)) *} 

377 
method_setup intr = {* Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (intr_tac ctxt ths)) *} 

378 
method_setup equal = {* Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (equal_tac ctxt ths)) *} 

19761  379 

380 

381 
subsection {* Simplification *} 

382 

383 
(*To simplify the type in a goal*) 

58977  384 
lemma replace_type: "\<lbrakk>B = A; a : A\<rbrakk> \<Longrightarrow> a : B" 
19761  385 
apply (rule equal_types) 
386 
apply (rule_tac [2] sym_type) 

387 
apply assumption+ 

388 
done 

389 

390 
(*Simplify the parameter of a unary type operator.*) 

391 
lemma subst_eqtyparg: 

23467  392 
assumes 1: "a=c : A" 
58977  393 
and 2: "\<And>z. z:A \<Longrightarrow> B(z) type" 
19761  394 
shows "B(a)=B(c)" 
395 
apply (rule subst_typeL) 

396 
apply (rule_tac [2] refl_type) 

23467  397 
apply (rule 1) 
398 
apply (erule 2) 

19761  399 
done 
400 

401 
(*Simplification rules for Constructive Type Theory*) 

402 
lemmas reduction_rls = comp_rls [THEN trans_elem] 

403 

404 
ML {* 

405 
(*Converts each goal "e : Eq(A,a,b)" into "a=b:A" for simplification. 

406 
Uses other intro rules to avoid changing flexible goals.*) 

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fun eqintr_tac ctxt = 
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REPEAT_FIRST (ASSUME ctxt (filt_resolve_tac (@{thm EqI} :: @{thms intr_rls}) 1)) 
19761  409 

410 
(** Tactics that instantiate CTTrules. 

411 
Vars in the given terms will be incremented! 

412 
The (rtac EqE i) lets them apply to equality judgements. **) 

413 

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fun NE_tac ctxt sp i = 
27239  415 
TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm NE} i 
19761  416 

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fun SumE_tac ctxt sp i = 
27239  418 
TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm SumE} i 
19761  419 

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fun PlusE_tac ctxt sp i = 
27239  421 
TRY (rtac @{thm EqE} i) THEN res_inst_tac ctxt [(("p", 0), sp)] @{thm PlusE} i 
19761  422 

423 
(** Predicate logic reasoning, WITH THINNING!! Procedures adapted from NJ. **) 

424 

425 
(*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 ctxt i = 
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rtac @{thm subst_prodE} i THEN assume_tac ctxt i THEN assume_tac ctxt i 
19761  428 

429 
(*Finds P>Q and P in the assumptions, replaces implication by Q *) 

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fun mp_tac ctxt i = etac @{thm subst_prodE} i THEN assume_tac ctxt i 
19761  431 

432 
(*"safe" when regarded as predicate calculus rules*) 

433 
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  436 

437 
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  440 

441 
(*0 subgoals vs 1 or more*) 

442 
val (safe0_brls, safep_brls) = 

443 
List.partition (curry (op =) 0 o subgoals_of_brl) safe_brls 

444 

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fun safestep_tac ctxt thms i = 
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form_tac ctxt ORELSE 
19761  447 
resolve_tac thms i ORELSE 
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biresolve_tac safe0_brls i ORELSE mp_tac ctxt i ORELSE 
19761  449 
DETERM (biresolve_tac safep_brls i) 
450 

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fun safe_tac ctxt thms i = DEPTH_SOLVE_1 (safestep_tac ctxt thms i) 
19761  452 

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fun step_tac ctxt thms = safestep_tac ctxt thms ORELSE' biresolve_tac unsafe_brls 
19761  454 

455 
(*Fails unless it solves the goal!*) 

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fun pc_tac ctxt thms = DEPTH_SOLVE_1 o (step_tac ctxt thms) 
19761  457 
*} 
458 

58976  459 
method_setup eqintr = {* Scan.succeed (SIMPLE_METHOD o eqintr_tac) *} 
58972  460 
method_setup NE = {* 
58975  461 
Scan.lift Args.name_inner_syntax >> (fn s => fn ctxt => SIMPLE_METHOD' (NE_tac ctxt s)) 
58972  462 
*} 
58976  463 
method_setup pc = {* Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (pc_tac ctxt ths)) *} 
464 
method_setup add_mp = {* Scan.succeed (SIMPLE_METHOD' o add_mp_tac) *} 

58972  465 

48891  466 
ML_file "rew.ML" 
58976  467 
method_setup rew = {* Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (rew_tac ctxt ths)) *} 
468 
method_setup hyp_rew = {* Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (hyp_rew_tac ctxt ths)) *} 

58972  469 

19761  470 

471 
subsection {* The elimination rules for fst/snd *} 

472 

58977  473 
lemma SumE_fst: "p : Sum(A,B) \<Longrightarrow> fst(p) : A" 
19761  474 
apply (unfold basic_defs) 
475 
apply (erule SumE) 

476 
apply assumption 

477 
done 

478 

479 
(*The first premise must be p:Sum(A,B) !!*) 

480 
lemma SumE_snd: 

481 
assumes major: "p: Sum(A,B)" 

482 
and "A type" 

58977  483 
and "\<And>x. x:A \<Longrightarrow> B(x) type" 
19761  484 
shows "snd(p) : B(fst(p))" 
485 
apply (unfold basic_defs) 

486 
apply (rule major [THEN SumE]) 

487 
apply (rule SumC [THEN subst_eqtyparg, THEN replace_type]) 

58972  488 
apply (typechk assms) 
19761  489 
done 
490 

491 
end 