author  wenzelm 
Fri, 15 Jul 2016 15:19:04 +0200  
changeset 63505  42e1dece537a 
parent 63120  629a4c5e953e 
child 64980  7dc25cf5793e 
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 \<open>Constructive Type Theory\<close> 
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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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\<comment> \<open>Types\<close> 
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F :: "t" 
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T :: "t" \<comment> \<open>\<open>F\<close> is empty, \<open>T\<close> contains one element\<close> 
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contr :: "i\<Rightarrow>i" 
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tt :: "i" 
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\<comment> \<open>Natural numbers\<close> 
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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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\<comment> \<open>Unions\<close> 
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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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\<comment> \<open>General Sum and Binary Product\<close> 
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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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\<comment> \<open>General Product and Function Space\<close> 
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Prod :: "[t, i\<Rightarrow>t]\<Rightarrow>t" 
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\<comment> \<open>Types\<close> 
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Plus :: "[t,t]\<Rightarrow>t" (infixr "+" 40) 
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\<comment> \<open>Equality type\<close> 
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Eq :: "[t,i,i]\<Rightarrow>t" 
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eq :: "i" 
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\<comment> \<open>Judgements\<close> 
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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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\<comment> \<open>Types\<close> 
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\<comment> \<open>Functions\<close> 

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lambda :: "(i \<Rightarrow> i) \<Rightarrow> i" (binder "\<^bold>\<lambda>" 10) 
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app :: "[i,i]\<Rightarrow>i" (infixl "`" 60) 
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\<comment> \<open>Natural numbers\<close> 
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Zero :: "i" ("0") 
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\<comment> \<open>Pairing\<close> 
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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" ("(3\<Prod>_:_./ _)" 10) 
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"_SUM" :: "[idt,t,t]\<Rightarrow>t" ("(3\<Sum>_:_./ _)" 10) 

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

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abbreviation Arrow :: "[t,t]\<Rightarrow>t" (infixr "\<longrightarrow>" 30) 
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where "A \<longrightarrow> B \<equiv> \<Prod>_:A. B" 

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abbreviation Times :: "[t,t]\<Rightarrow>t" (infixr "\<times>" 50) 

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where "A \<times> B \<equiv> \<Sum>_:A. B" 

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text \<open> 
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Reduction: a weaker notion than equality; a hack for simplification. 

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\<open>Reduce[a,b]\<close> means either that \<open>a = b : A\<close> for some \<open>A\<close> or else 

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that \<open>a\<close> and \<open>b\<close> are textually identical. 

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Does not verify \<open>a:A\<close>! Sound because only \<open>trans_red\<close> uses a \<open>Reduce\<close> 
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premise. No new theorems can be proved about the standard judgements. 

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\<close> 

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axiomatization 

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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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\<comment> \<open>Reflexivity\<close> 
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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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\<comment> \<open>Symmetry\<close> 
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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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\<comment> \<open>Transitivity\<close> 
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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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\<comment> \<open>Substitution\<close> 
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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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\<comment> \<open>The type \<open>N\<close>  natural numbers\<close> 
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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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\<comment> \<open>The fourth Peano axiom. See page 91 of MartinLÃ¶f's book.\<close> 
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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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\<comment> \<open>The Product of a family of types\<close> 
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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> \<^bold>\<lambda>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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\<^bold>\<lambda>x. b(x) = \<^bold>\<lambda>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> (\<^bold>\<lambda>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> (\<^bold>\<lambda>x. p`x) = p : \<Prod>x:A. B(x)" and 
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\<comment> \<open>The Sum of a family of types\<close> 
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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) \<equiv> split(a, \<lambda>x y. x)" and 
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snd_def: "\<And>a. snd(a) \<equiv> split(a, \<lambda>x y. y)" and 

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\<comment> \<open>The sum of two types\<close> 
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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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\<comment> \<open>The type \<open>Eq\<close>\<close> 
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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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\<comment> \<open>By equality of types, can prove \<open>C(p)\<close> from \<open>C(eq)\<close>, an elimination rule\<close> 
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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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\<comment> \<open>The type \<open>F\<close>\<close> 

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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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\<comment> \<open>The type T\<close> 

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\<comment> \<open> 

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MartinLÃ¶f'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 \<open>C(x)\<close> type \<open>x:T\<close>. 

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Also computation can be derived from elimination. 

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\<close> 

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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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text \<open>Formation rules.\<close> 
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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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text \<open> 
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Introduction rules. OMITTED: 

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\<^item> \<open>EqI\<close>, because its premise is an \<open>eqelem\<close>, not an \<open>elem\<close>. 

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\<close> 

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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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text \<open> 
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Elimination rules. OMITTED: 

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\<^item> \<open>EqE\<close>, because its conclusion is an \<open>eqelem\<close>, not an \<open>elem\<close> 

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\<^item> \<open>TE\<close>, because it does not involve a constructor. 

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\<close> 

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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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text \<open>OMITTED: \<open>eqC\<close> are \<open>TC\<close> because they make rewriting loop: \<open>p = un = un = \<dots>\<close>\<close> 
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lemmas comp_rls = NC0 NC_succ ProdC SumC PlusC_inl PlusC_inr 
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text \<open>Rules with conclusion \<open>a:A\<close>, an elem judgement.\<close> 
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lemmas element_rls = intr_rls elim_rls 
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text \<open>Definitions are (meta)equality axioms.\<close> 
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lemmas basic_defs = fst_def snd_def 
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text \<open>Compare with standard version: \<open>B\<close> is applied to UNSIMPLIFIED expression!\<close> 
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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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text \<open> 
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Exploit \<open>p:Prod(A,B)\<close> to create the assumption \<open>z:B(a)\<close>. 

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A more natural form of product elimination. 

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\<close> 

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lemma subst_prodE: 
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assumes "p: Prod(A,B)" 

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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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by (rule assms ProdE)+ 
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subsection \<open>Tactics for type checking\<close> 
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ML \<open> 
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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 
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end 
60770  328 
\<close> 
19761  329 

63505  330 
text \<open> 
331 
For simplification: type formation and checking, 

332 
but no equalities between terms. 

333 
\<close> 

19761  334 
lemmas routine_rls = form_rls formL_rls refl_type element_rls 
335 

60770  336 
ML \<open> 
59164  337 
fun routine_tac rls ctxt prems = 
338 
ASSUME ctxt (filt_resolve_from_net_tac ctxt 4 (Tactic.build_net (prems @ rls))); 

19761  339 

340 
(*Solve all subgoals "A type" using formation rules. *) 

59164  341 
val form_net = Tactic.build_net @{thms form_rls}; 
342 
fun form_tac ctxt = 

343 
REPEAT_FIRST (ASSUME ctxt (filt_resolve_from_net_tac ctxt 1 form_net)); 

19761  344 

345 
(*Type checking: solve a:A (a rigid, A flexible) by intro and elim rules. *) 

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fun typechk_tac ctxt thms = 
59164  347 
let val tac = 
348 
filt_resolve_from_net_tac ctxt 3 

349 
(Tactic.build_net (thms @ @{thms form_rls} @ @{thms element_rls})) 

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in REPEAT_FIRST (ASSUME ctxt tac) end 
19761  351 

352 
(*Solve a:A (a flexible, A rigid) by introduction rules. 

353 
Cannot use stringtrees (filt_resolve_tac) since 

354 
goals like ?a:SUM(A,B) have a trivial headstring *) 

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fun intr_tac ctxt thms = 
59164  356 
let val tac = 
357 
filt_resolve_from_net_tac ctxt 1 

358 
(Tactic.build_net (thms @ @{thms form_rls} @ @{thms intr_rls})) 

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in REPEAT_FIRST (ASSUME ctxt tac) end 
19761  360 

361 
(*Equality proving: solve a=b:A (where a is rigid) by long rules. *) 

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fun equal_tac ctxt thms = 
59164  363 
REPEAT_FIRST 
63505  364 
(ASSUME ctxt 
365 
(filt_resolve_from_net_tac ctxt 3 

366 
(Tactic.build_net (thms @ @{thms form_rls element_rls intrL_rls elimL_rls refl_elem})))) 

60770  367 
\<close> 
19761  368 

60770  369 
method_setup form = \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (form_tac ctxt))\<close> 
370 
method_setup typechk = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (typechk_tac ctxt ths))\<close> 

371 
method_setup intr = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (intr_tac ctxt ths))\<close> 

372 
method_setup equal = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (equal_tac ctxt ths))\<close> 

19761  373 

374 

60770  375 
subsection \<open>Simplification\<close> 
19761  376 

63505  377 
text \<open>To simplify the type in a goal.\<close> 
58977  378 
lemma replace_type: "\<lbrakk>B = A; a : A\<rbrakk> \<Longrightarrow> a : B" 
63505  379 
apply (rule equal_types) 
380 
apply (rule_tac [2] sym_type) 

381 
apply assumption+ 

382 
done 

19761  383 

63505  384 
text \<open>Simplify the parameter of a unary type operator.\<close> 
19761  385 
lemma subst_eqtyparg: 
23467  386 
assumes 1: "a=c : A" 
58977  387 
and 2: "\<And>z. z:A \<Longrightarrow> B(z) type" 
63505  388 
shows "B(a) = B(c)" 
389 
apply (rule subst_typeL) 

390 
apply (rule_tac [2] refl_type) 

391 
apply (rule 1) 

392 
apply (erule 2) 

393 
done 

19761  394 

63505  395 
text \<open>Simplification rules for Constructive Type Theory.\<close> 
19761  396 
lemmas reduction_rls = comp_rls [THEN trans_elem] 
397 

60770  398 
ML \<open> 
19761  399 
(*Converts each goal "e : Eq(A,a,b)" into "a=b:A" for simplification. 
400 
Uses other intro rules to avoid changing flexible goals.*) 

59164  401 
val eqintr_net = Tactic.build_net @{thms EqI intr_rls} 
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fun eqintr_tac ctxt = 
59164  403 
REPEAT_FIRST (ASSUME ctxt (filt_resolve_from_net_tac ctxt 1 eqintr_net)) 
19761  404 

405 
(** Tactics that instantiate CTTrules. 

406 
Vars in the given terms will be incremented! 

407 
The (rtac EqE i) lets them apply to equality judgements. **) 

408 

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fun NE_tac ctxt sp i = 
60754  410 
TRY (resolve_tac ctxt @{thms EqE} i) THEN 
59780  411 
Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), sp)] [] @{thm NE} i 
19761  412 

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fun SumE_tac ctxt sp i = 
60754  414 
TRY (resolve_tac ctxt @{thms EqE} i) THEN 
59780  415 
Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), sp)] [] @{thm SumE} i 
19761  416 

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fun PlusE_tac ctxt sp i = 
60754  418 
TRY (resolve_tac ctxt @{thms EqE} i) THEN 
59780  419 
Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), sp)] [] @{thm PlusE} i 
19761  420 

421 
(** Predicate logic reasoning, WITH THINNING!! Procedures adapted from NJ. **) 

422 

423 
(*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 = 
60754  425 
resolve_tac ctxt @{thms subst_prodE} i THEN assume_tac ctxt i THEN assume_tac ctxt i 
19761  426 

61391  427 
(*Finds P\<longrightarrow>Q and P in the assumptions, replaces implication by Q *) 
60754  428 
fun mp_tac ctxt i = eresolve_tac ctxt @{thms subst_prodE} i THEN assume_tac ctxt i 
19761  429 

430 
(*"safe" when regarded as predicate calculus rules*) 

431 
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  434 

435 
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  438 

439 
(*0 subgoals vs 1 or more*) 

440 
val (safe0_brls, safep_brls) = 

441 
List.partition (curry (op =) 0 o subgoals_of_brl) safe_brls 

442 

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fun safestep_tac ctxt thms i = 
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form_tac ctxt ORELSE 
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resolve_tac ctxt thms i ORELSE 
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biresolve_tac ctxt safe0_brls i ORELSE mp_tac ctxt i ORELSE 
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DETERM (biresolve_tac ctxt safep_brls i) 
19761  448 

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fun safe_tac ctxt thms i = DEPTH_SOLVE_1 (safestep_tac ctxt thms i) 
19761  450 

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fun step_tac ctxt thms = safestep_tac ctxt thms ORELSE' biresolve_tac ctxt unsafe_brls 
19761  452 

453 
(*Fails unless it solves the goal!*) 

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fun pc_tac ctxt thms = DEPTH_SOLVE_1 o (step_tac ctxt thms) 
60770  455 
\<close> 
19761  456 

60770  457 
method_setup eqintr = \<open>Scan.succeed (SIMPLE_METHOD o eqintr_tac)\<close> 
458 
method_setup NE = \<open> 

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Scan.lift Args.embedded_inner_syntax >> (fn s => fn ctxt => SIMPLE_METHOD' (NE_tac ctxt s)) 
60770  460 
\<close> 
461 
method_setup pc = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (pc_tac ctxt ths))\<close> 

462 
method_setup add_mp = \<open>Scan.succeed (SIMPLE_METHOD' o add_mp_tac)\<close> 

58972  463 

48891  464 
ML_file "rew.ML" 
60770  465 
method_setup rew = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (rew_tac ctxt ths))\<close> 
466 
method_setup hyp_rew = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (hyp_rew_tac ctxt ths))\<close> 

58972  467 

19761  468 

60770  469 
subsection \<open>The elimination rules for fst/snd\<close> 
19761  470 

58977  471 
lemma SumE_fst: "p : Sum(A,B) \<Longrightarrow> fst(p) : A" 
63505  472 
apply (unfold basic_defs) 
473 
apply (erule SumE) 

474 
apply assumption 

475 
done 

19761  476 

63505  477 
text \<open>The first premise must be \<open>p:Sum(A,B)\<close>!!.\<close> 
19761  478 
lemma SumE_snd: 
479 
assumes major: "p: Sum(A,B)" 

480 
and "A type" 

58977  481 
and "\<And>x. x:A \<Longrightarrow> B(x) type" 
19761  482 
shows "snd(p) : B(fst(p))" 
483 
apply (unfold basic_defs) 

484 
apply (rule major [THEN SumE]) 

485 
apply (rule SumC [THEN subst_eqtyparg, THEN replace_type]) 

63505  486 
apply (typechk assms) 
19761  487 
done 
488 

489 
end 