author  paulson 
Thu, 14 Jun 2018 14:23:48 +0100  
changeset 68446  92ddca1edc43 
parent 67443  3abf6a722518 
child 69593  3dda49e08b9d 
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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theory CTT 
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imports Pure 

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begin 

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section \<open>Constructive Type Theory: axiomatic basis\<close> 
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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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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 B 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>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.\<close> 
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TF: "T type" and 
248 
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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19761  253 

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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> 
261 
Introduction rules. OMITTED: 

262 
\<^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> 
268 
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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by (rule sym_elem) (rule SumIL; rule sym_elem) 
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288 
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. 

293 
\<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)) 
308 
 is_rigid_elem (Const(@{const_name Eqelem},_) $ a $ _ $ _) = not(is_Var (head_of a)) 

309 
 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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312 
in 

313 

314 
(*Try solving a:A or a=b:A by assumption provided a is rigid!*) 

63505  315 
fun test_assume_tac ctxt = SUBGOAL (fn (prem, i) => 
316 
if is_rigid_elem (Logic.strip_assums_concl prem) 

317 
then assume_tac ctxt i else no_tac) 

19761  318 

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319 
fun ASSUME ctxt tf i = test_assume_tac ctxt i ORELSE tf i 
19761  320 

63505  321 
end 
60770  322 
\<close> 
19761  323 

63505  324 
text \<open> 
325 
For simplification: type formation and checking, 

326 
but no equalities between terms. 

327 
\<close> 

19761  328 
lemmas routine_rls = form_rls formL_rls refl_type element_rls 
329 

60770  330 
ML \<open> 
59164  331 
fun routine_tac rls ctxt prems = 
332 
ASSUME ctxt (filt_resolve_from_net_tac ctxt 4 (Tactic.build_net (prems @ rls))); 

19761  333 

334 
(*Solve all subgoals "A type" using formation rules. *) 

59164  335 
val form_net = Tactic.build_net @{thms form_rls}; 
336 
fun form_tac ctxt = 

337 
REPEAT_FIRST (ASSUME ctxt (filt_resolve_from_net_tac ctxt 1 form_net)); 

19761  338 

339 
(*Type checking: solve a:A (a rigid, A flexible) by intro and elim rules. *) 

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340 
fun typechk_tac ctxt thms = 
59164  341 
let val tac = 
342 
filt_resolve_from_net_tac ctxt 3 

343 
(Tactic.build_net (thms @ @{thms form_rls} @ @{thms element_rls})) 

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344 
in REPEAT_FIRST (ASSUME ctxt tac) end 
19761  345 

346 
(*Solve a:A (a flexible, A rigid) by introduction rules. 

347 
Cannot use stringtrees (filt_resolve_tac) since 

348 
goals like ?a:SUM(A,B) have a trivial headstring *) 

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349 
fun intr_tac ctxt thms = 
59164  350 
let val tac = 
351 
filt_resolve_from_net_tac ctxt 1 

352 
(Tactic.build_net (thms @ @{thms form_rls} @ @{thms intr_rls})) 

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353 
in REPEAT_FIRST (ASSUME ctxt tac) end 
19761  354 

355 
(*Equality proving: solve a=b:A (where a is rigid) by long rules. *) 

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356 
fun equal_tac ctxt thms = 
59164  357 
REPEAT_FIRST 
63505  358 
(ASSUME ctxt 
359 
(filt_resolve_from_net_tac ctxt 3 

360 
(Tactic.build_net (thms @ @{thms form_rls element_rls intrL_rls elimL_rls refl_elem})))) 

60770  361 
\<close> 
19761  362 

60770  363 
method_setup form = \<open>Scan.succeed (fn ctxt => SIMPLE_METHOD (form_tac ctxt))\<close> 
364 
method_setup typechk = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (typechk_tac ctxt ths))\<close> 

365 
method_setup intr = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (intr_tac ctxt ths))\<close> 

366 
method_setup equal = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (equal_tac ctxt ths))\<close> 

19761  367 

368 

60770  369 
subsection \<open>Simplification\<close> 
19761  370 

63505  371 
text \<open>To simplify the type in a goal.\<close> 
58977  372 
lemma replace_type: "\<lbrakk>B = A; a : A\<rbrakk> \<Longrightarrow> a : B" 
63505  373 
apply (rule equal_types) 
374 
apply (rule_tac [2] sym_type) 

375 
apply assumption+ 

376 
done 

19761  377 

63505  378 
text \<open>Simplify the parameter of a unary type operator.\<close> 
19761  379 
lemma subst_eqtyparg: 
23467  380 
assumes 1: "a=c : A" 
58977  381 
and 2: "\<And>z. z:A \<Longrightarrow> B(z) type" 
63505  382 
shows "B(a) = B(c)" 
383 
apply (rule subst_typeL) 

384 
apply (rule_tac [2] refl_type) 

385 
apply (rule 1) 

386 
apply (erule 2) 

387 
done 

19761  388 

63505  389 
text \<open>Simplification rules for Constructive Type Theory.\<close> 
19761  390 
lemmas reduction_rls = comp_rls [THEN trans_elem] 
391 

60770  392 
ML \<open> 
19761  393 
(*Converts each goal "e : Eq(A,a,b)" into "a=b:A" for simplification. 
394 
Uses other intro rules to avoid changing flexible goals.*) 

59164  395 
val eqintr_net = Tactic.build_net @{thms EqI intr_rls} 
58963
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396 
fun eqintr_tac ctxt = 
59164  397 
REPEAT_FIRST (ASSUME ctxt (filt_resolve_from_net_tac ctxt 1 eqintr_net)) 
19761  398 

399 
(** Tactics that instantiate CTTrules. 

400 
Vars in the given terms will be incremented! 

401 
The (rtac EqE i) lets them apply to equality judgements. **) 

402 

27208
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403 
fun NE_tac ctxt sp i = 
60754  404 
TRY (resolve_tac ctxt @{thms EqE} i) THEN 
59780  405 
Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), sp)] [] @{thm NE} i 
19761  406 

27208
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407 
fun SumE_tac ctxt sp i = 
60754  408 
TRY (resolve_tac ctxt @{thms EqE} i) THEN 
59780  409 
Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), sp)] [] @{thm SumE} i 
19761  410 

27208
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411 
fun PlusE_tac ctxt sp i = 
60754  412 
TRY (resolve_tac ctxt @{thms EqE} i) THEN 
59780  413 
Rule_Insts.res_inst_tac ctxt [((("p", 0), Position.none), sp)] [] @{thm PlusE} i 
19761  414 

415 
(** Predicate logic reasoning, WITH THINNING!! Procedures adapted from NJ. **) 

416 

417 
(*Finds f:Prod(A,B) and a:A in the assumptions, concludes there is z:B(a) *) 

58963
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418 
fun add_mp_tac ctxt i = 
60754  419 
resolve_tac ctxt @{thms subst_prodE} i THEN assume_tac ctxt i THEN assume_tac ctxt i 
19761  420 

61391  421 
(*Finds P\<longrightarrow>Q and P in the assumptions, replaces implication by Q *) 
60754  422 
fun mp_tac ctxt i = eresolve_tac ctxt @{thms subst_prodE} i THEN assume_tac ctxt i 
19761  423 

424 
(*"safe" when regarded as predicate calculus rules*) 

425 
val safe_brls = sort (make_ord lessb) 

27208
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changeset

426 
[ (true, @{thm FE}), (true,asm_rl), 
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427 
(false, @{thm ProdI}), (true, @{thm SumE}), (true, @{thm PlusE}) ] 
19761  428 

429 
val unsafe_brls = 

27208
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430 
[ (false, @{thm PlusI_inl}), (false, @{thm PlusI_inr}), (false, @{thm SumI}), 
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changeset

431 
(true, @{thm subst_prodE}) ] 
19761  432 

433 
(*0 subgoals vs 1 or more*) 

434 
val (safe0_brls, safep_brls) = 

67405
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uniform use of Standard ML opinfix  eliminated warnings;
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changeset

435 
List.partition (curry (op =) 0 o subgoals_of_brl) safe_brls 
19761  436 

58963
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437 
fun safestep_tac ctxt thms i = 
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438 
form_tac ctxt ORELSE 
59498
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proper context for resolve_tac, eresolve_tac, dresolve_tac, forward_tac etc.;
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changeset

439 
resolve_tac ctxt thms i ORELSE 
50b60f501b05
proper context for resolve_tac, eresolve_tac, dresolve_tac, forward_tac etc.;
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diff
changeset

440 
biresolve_tac ctxt safe0_brls i ORELSE mp_tac ctxt i ORELSE 
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proper context for resolve_tac, eresolve_tac, dresolve_tac, forward_tac etc.;
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441 
DETERM (biresolve_tac ctxt safep_brls i) 
19761  442 

58963
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proper context for assume_tac (atac remains as fallback without context);
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443 
fun safe_tac ctxt thms i = DEPTH_SOLVE_1 (safestep_tac ctxt thms i) 
19761  444 

59498
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proper context for resolve_tac, eresolve_tac, dresolve_tac, forward_tac etc.;
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445 
fun step_tac ctxt thms = safestep_tac ctxt thms ORELSE' biresolve_tac ctxt unsafe_brls 
19761  446 

447 
(*Fails unless it solves the goal!*) 

58963
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448 
fun pc_tac ctxt thms = DEPTH_SOLVE_1 o (step_tac ctxt thms) 
60770  449 
\<close> 
19761  450 

60770  451 
method_setup eqintr = \<open>Scan.succeed (SIMPLE_METHOD o eqintr_tac)\<close> 
452 
method_setup NE = \<open> 

63120
629a4c5e953e
embedded content may be delimited via cartouches;
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61391
diff
changeset

453 
Scan.lift Args.embedded_inner_syntax >> (fn s => fn ctxt => SIMPLE_METHOD' (NE_tac ctxt s)) 
60770  454 
\<close> 
455 
method_setup pc = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD' (pc_tac ctxt ths))\<close> 

456 
method_setup add_mp = \<open>Scan.succeed (SIMPLE_METHOD' o add_mp_tac)\<close> 

58972  457 

48891  458 
ML_file "rew.ML" 
60770  459 
method_setup rew = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (rew_tac ctxt ths))\<close> 
460 
method_setup hyp_rew = \<open>Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (hyp_rew_tac ctxt ths))\<close> 

58972  461 

19761  462 

60770  463 
subsection \<open>The elimination rules for fst/snd\<close> 
19761  464 

58977  465 
lemma SumE_fst: "p : Sum(A,B) \<Longrightarrow> fst(p) : A" 
63505  466 
apply (unfold basic_defs) 
467 
apply (erule SumE) 

468 
apply assumption 

469 
done 

19761  470 

63505  471 
text \<open>The first premise must be \<open>p:Sum(A,B)\<close>!!.\<close> 
19761  472 
lemma SumE_snd: 
473 
assumes major: "p: Sum(A,B)" 

474 
and "A type" 

58977  475 
and "\<And>x. x:A \<Longrightarrow> B(x) type" 
19761  476 
shows "snd(p) : B(fst(p))" 
477 
apply (unfold basic_defs) 

478 
apply (rule major [THEN SumE]) 

479 
apply (rule SumC [THEN subst_eqtyparg, THEN replace_type]) 

63505  480 
apply (typechk assms) 
19761  481 
done 
482 

65447
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
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diff
changeset

483 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
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diff
changeset

484 
section \<open>The twoelement type (booleans and conditionals)\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
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diff
changeset

485 

fae6051ec192
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diff
changeset

486 
definition Bool :: "t" 
fae6051ec192
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diff
changeset

487 
where "Bool \<equiv> T+T" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
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diff
changeset

488 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
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diff
changeset

489 
definition true :: "i" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
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diff
changeset

490 
where "true \<equiv> inl(tt)" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

491 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
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diff
changeset

492 
definition false :: "i" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

493 
where "false \<equiv> inr(tt)" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
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diff
changeset

494 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
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diff
changeset

495 
definition cond :: "[i,i,i]\<Rightarrow>i" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
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diff
changeset

496 
where "cond(a,b,c) \<equiv> when(a, \<lambda>_. b, \<lambda>_. c)" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

497 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

498 
lemmas bool_defs = Bool_def true_def false_def cond_def 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

499 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

500 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

501 
subsection \<open>Derivation of rules for the type \<open>Bool\<close>\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

502 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

503 
text \<open>Formation rule.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
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diff
changeset

504 
lemma boolF: "Bool type" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
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diff
changeset

505 
unfolding bool_defs by typechk 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

506 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
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diff
changeset

507 
text \<open>Introduction rules for \<open>true\<close>, \<open>false\<close>.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

508 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

509 
lemma boolI_true: "true : Bool" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

510 
unfolding bool_defs by typechk 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

511 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

512 
lemma boolI_false: "false : Bool" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

513 
unfolding bool_defs by typechk 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

514 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

515 
text \<open>Elimination rule: typing of \<open>cond\<close>.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
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diff
changeset

516 
lemma boolE: "\<lbrakk>p:Bool; a : C(true); b : C(false)\<rbrakk> \<Longrightarrow> cond(p,a,b) : C(p)" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

517 
unfolding bool_defs 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
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diff
changeset

518 
apply (typechk; erule TE) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

519 
apply typechk 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

520 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

521 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

522 
lemma boolEL: "\<lbrakk>p = q : Bool; a = c : C(true); b = d : C(false)\<rbrakk> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

523 
\<Longrightarrow> cond(p,a,b) = cond(q,c,d) : C(p)" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

524 
unfolding bool_defs 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

525 
apply (rule PlusEL) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

526 
apply (erule asm_rl refl_elem [THEN TEL])+ 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

527 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

528 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

529 
text \<open>Computation rules for \<open>true\<close>, \<open>false\<close>.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

530 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

531 
lemma boolC_true: "\<lbrakk>a : C(true); b : C(false)\<rbrakk> \<Longrightarrow> cond(true,a,b) = a : C(true)" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

532 
unfolding bool_defs 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

533 
apply (rule comp_rls) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

534 
apply typechk 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

535 
apply (erule_tac [!] TE) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

536 
apply typechk 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

537 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

538 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

539 
lemma boolC_false: "\<lbrakk>a : C(true); b : C(false)\<rbrakk> \<Longrightarrow> cond(false,a,b) = b : C(false)" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

540 
unfolding bool_defs 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

541 
apply (rule comp_rls) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

542 
apply typechk 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

543 
apply (erule_tac [!] TE) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

544 
apply typechk 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

545 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

546 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

547 
section \<open>Elementary arithmetic\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

548 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

549 
subsection \<open>Arithmetic operators and their definitions\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

550 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

551 
definition add :: "[i,i]\<Rightarrow>i" (infixr "#+" 65) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

552 
where "a#+b \<equiv> rec(a, b, \<lambda>u v. succ(v))" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

553 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

554 
definition diff :: "[i,i]\<Rightarrow>i" (infixr "" 65) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

555 
where "ab \<equiv> rec(b, a, \<lambda>u v. rec(v, 0, \<lambda>x y. x))" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

556 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

557 
definition absdiff :: "[i,i]\<Rightarrow>i" (infixr "" 65) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

558 
where "ab \<equiv> (ab) #+ (ba)" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

559 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

560 
definition mult :: "[i,i]\<Rightarrow>i" (infixr "#*" 70) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

561 
where "a#*b \<equiv> rec(a, 0, \<lambda>u v. b #+ v)" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

562 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

563 
definition mod :: "[i,i]\<Rightarrow>i" (infixr "mod" 70) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

564 
where "a mod b \<equiv> rec(a, 0, \<lambda>u v. rec(succ(v)  b, 0, \<lambda>x y. succ(v)))" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

565 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

566 
definition div :: "[i,i]\<Rightarrow>i" (infixr "div" 70) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

567 
where "a div b \<equiv> rec(a, 0, \<lambda>u v. rec(succ(u) mod b, succ(v), \<lambda>x y. v))" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

568 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

569 
lemmas arith_defs = add_def diff_def absdiff_def mult_def mod_def div_def 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

570 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

571 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

572 
subsection \<open>Proofs about elementary arithmetic: addition, multiplication, etc.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

573 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

574 
subsubsection \<open>Addition\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

575 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

576 
text \<open>Typing of \<open>add\<close>: short and long versions.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

577 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

578 
lemma add_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #+ b : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

579 
unfolding arith_defs by typechk 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

580 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

581 
lemma add_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a #+ b = c #+ d : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

582 
unfolding arith_defs by equal 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

583 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

584 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

585 
text \<open>Computation for \<open>add\<close>: 0 and successor cases.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

586 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

587 
lemma addC0: "b:N \<Longrightarrow> 0 #+ b = b : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

588 
unfolding arith_defs by rew 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

589 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

590 
lemma addC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) #+ b = succ(a #+ b) : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

591 
unfolding arith_defs by rew 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

592 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

593 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

594 
subsubsection \<open>Multiplication\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

595 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

596 
text \<open>Typing of \<open>mult\<close>: short and long versions.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

597 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

598 
lemma mult_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #* b : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

599 
unfolding arith_defs by (typechk add_typing) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

600 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

601 
lemma mult_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a #* b = c #* d : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

602 
unfolding arith_defs by (equal add_typingL) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

603 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

604 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

605 
text \<open>Computation for \<open>mult\<close>: 0 and successor cases.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

606 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

607 
lemma multC0: "b:N \<Longrightarrow> 0 #* b = 0 : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

608 
unfolding arith_defs by rew 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

609 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

610 
lemma multC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) #* b = b #+ (a #* b) : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

611 
unfolding arith_defs by rew 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

612 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

613 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

614 
subsubsection \<open>Difference\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

615 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

616 
text \<open>Typing of difference.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

617 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

618 
lemma diff_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a  b : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

619 
unfolding arith_defs by typechk 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

620 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

621 
lemma diff_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a  b = c  d : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

622 
unfolding arith_defs by equal 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

623 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

624 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

625 
text \<open>Computation for difference: 0 and successor cases.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

626 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

627 
lemma diffC0: "a:N \<Longrightarrow> a  0 = a : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

628 
unfolding arith_defs by rew 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

629 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

630 
text \<open>Note: \<open>rec(a, 0, \<lambda>z w.z)\<close> is \<open>pred(a).\<close>\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

631 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

632 
lemma diff_0_eq_0: "b:N \<Longrightarrow> 0  b = 0 : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

633 
unfolding arith_defs 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

634 
apply (NE b) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

635 
apply hyp_rew 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

636 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

637 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

638 
text \<open> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

639 
Essential to simplify FIRST!! (Else we get a critical pair) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

640 
\<open>succ(a)  succ(b)\<close> rewrites to \<open>pred(succ(a)  b)\<close>. 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

641 
\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

642 
lemma diff_succ_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a)  succ(b) = a  b : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

643 
unfolding arith_defs 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

644 
apply hyp_rew 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

645 
apply (NE b) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

646 
apply hyp_rew 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

647 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

648 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

649 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

650 
subsection \<open>Simplification\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

651 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

652 
lemmas arith_typing_rls = add_typing mult_typing diff_typing 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

653 
and arith_congr_rls = add_typingL mult_typingL diff_typingL 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

654 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

655 
lemmas congr_rls = arith_congr_rls intrL2_rls elimL_rls 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

656 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

657 
lemmas arithC_rls = 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

658 
addC0 addC_succ 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

659 
multC0 multC_succ 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

660 
diffC0 diff_0_eq_0 diff_succ_succ 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

661 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

662 
ML \<open> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

663 
structure Arith_simp = TSimpFun( 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

664 
val refl = @{thm refl_elem} 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

665 
val sym = @{thm sym_elem} 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

666 
val trans = @{thm trans_elem} 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

667 
val refl_red = @{thm refl_red} 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

668 
val trans_red = @{thm trans_red} 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

669 
val red_if_equal = @{thm red_if_equal} 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

670 
val default_rls = @{thms arithC_rls comp_rls} 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

671 
val routine_tac = routine_tac @{thms arith_typing_rls routine_rls} 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

672 
) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

673 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

674 
fun arith_rew_tac ctxt prems = 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

675 
make_rew_tac ctxt (Arith_simp.norm_tac ctxt (@{thms congr_rls}, prems)) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

676 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

677 
fun hyp_arith_rew_tac ctxt prems = 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

678 
make_rew_tac ctxt 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

679 
(Arith_simp.cond_norm_tac ctxt (prove_cond_tac ctxt, @{thms congr_rls}, prems)) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

680 
\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

681 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

682 
method_setup arith_rew = \<open> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

683 
Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (arith_rew_tac ctxt ths)) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

684 
\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

685 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

686 
method_setup hyp_arith_rew = \<open> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

687 
Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (hyp_arith_rew_tac ctxt ths)) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

688 
\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

689 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

690 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

691 
subsection \<open>Addition\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

692 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

693 
text \<open>Associative law for addition.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

694 
lemma add_assoc: "\<lbrakk>a:N; b:N; c:N\<rbrakk> \<Longrightarrow> (a #+ b) #+ c = a #+ (b #+ c) : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

695 
apply (NE a) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

696 
apply hyp_arith_rew 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

697 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

698 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

699 
text \<open>Commutative law for addition. Can be proved using three inductions. 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

700 
Must simplify after first induction! Orientation of rewrites is delicate.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

701 
lemma add_commute: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #+ b = b #+ a : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

702 
apply (NE a) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

703 
apply hyp_arith_rew 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

704 
apply (rule sym_elem) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

705 
prefer 2 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

706 
apply (NE b) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

707 
prefer 4 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

708 
apply (NE b) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

709 
apply hyp_arith_rew 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

710 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

711 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

712 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

713 
subsection \<open>Multiplication\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

714 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

715 
text \<open>Right annihilation in product.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

716 
lemma mult_0_right: "a:N \<Longrightarrow> a #* 0 = 0 : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

717 
apply (NE a) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

718 
apply hyp_arith_rew 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

719 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

720 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

721 
text \<open>Right successor law for multiplication.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

722 
lemma mult_succ_right: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #* succ(b) = a #+ (a #* b) : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

723 
apply (NE a) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

724 
apply (hyp_arith_rew add_assoc [THEN sym_elem]) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

725 
apply (assumption  rule add_commute mult_typingL add_typingL intrL_rls refl_elem)+ 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

726 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

727 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

728 
text \<open>Commutative law for multiplication.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

729 
lemma mult_commute: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #* b = b #* a : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

730 
apply (NE a) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

731 
apply (hyp_arith_rew mult_0_right mult_succ_right) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

732 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

733 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

734 
text \<open>Addition distributes over multiplication.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

735 
lemma add_mult_distrib: "\<lbrakk>a:N; b:N; c:N\<rbrakk> \<Longrightarrow> (a #+ b) #* c = (a #* c) #+ (b #* c) : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

736 
apply (NE a) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

737 
apply (hyp_arith_rew add_assoc [THEN sym_elem]) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

738 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

739 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

740 
text \<open>Associative law for multiplication.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

741 
lemma mult_assoc: "\<lbrakk>a:N; b:N; c:N\<rbrakk> \<Longrightarrow> (a #* b) #* c = a #* (b #* c) : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

742 
apply (NE a) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

743 
apply (hyp_arith_rew add_mult_distrib) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

744 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

745 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

746 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

747 
subsection \<open>Difference\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

748 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

749 
text \<open> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

750 
Difference on natural numbers, without negative numbers 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

751 
\<^item> \<open>a  b = 0\<close> iff \<open>a \<le> b\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

752 
\<^item> \<open>a  b = succ(c)\<close> iff \<open>a > b\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

753 
\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

754 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

755 
lemma diff_self_eq_0: "a:N \<Longrightarrow> a  a = 0 : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

756 
apply (NE a) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

757 
apply hyp_arith_rew 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

758 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

759 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

760 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

761 
lemma add_0_right: "\<lbrakk>c : N; 0 : N; c : N\<rbrakk> \<Longrightarrow> c #+ 0 = c : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

762 
by (rule addC0 [THEN [3] add_commute [THEN trans_elem]]) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

763 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

764 
text \<open> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

765 
Addition is the inverse of subtraction: if \<open>b \<le> x\<close> then \<open>b #+ (x  b) = x\<close>. 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

766 
An example of induction over a quantified formula (a product). 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

767 
Uses rewriting with a quantified, implicative inductive hypothesis. 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

768 
\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

769 
schematic_goal add_diff_inverse_lemma: 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

770 
"b:N \<Longrightarrow> ?a : \<Prod>x:N. Eq(N, bx, 0) \<longrightarrow> Eq(N, b #+ (xb), x)" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

771 
apply (NE b) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

772 
\<comment> \<open>strip one "universal quantifier" but not the "implication"\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

773 
apply (rule_tac [3] intr_rls) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

774 
\<comment> \<open>case analysis on \<open>x\<close> in \<open>succ(u) \<le> x \<longrightarrow> succ(u) #+ (x  succ(u)) = x\<close>\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

775 
prefer 4 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

776 
apply (NE x) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

777 
apply assumption 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

778 
\<comment> \<open>Prepare for simplification of types  the antecedent \<open>succ(u) \<le> x\<close>\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

779 
apply (rule_tac [2] replace_type) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

780 
apply (rule_tac [1] replace_type) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

781 
apply arith_rew 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

782 
\<comment> \<open>Solves first 0 goal, simplifies others. Two sugbgoals remain. 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

783 
Both follow by rewriting, (2) using quantified induction hyp.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

784 
apply intr \<comment> \<open>strips remaining \<open>\<Prod>\<close>s\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

785 
apply (hyp_arith_rew add_0_right) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

786 
apply assumption 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

787 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

788 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

789 
text \<open> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

790 
Version of above with premise \<open>b  a = 0\<close> i.e. \<open>a \<ge> b\<close>. 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

791 
Using @{thm ProdE} does not work  for \<open>?B(?a)\<close> is ambiguous. 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

792 
Instead, @{thm add_diff_inverse_lemma} states the desired induction scheme; 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

793 
the use of \<open>THEN\<close> below instantiates Vars in @{thm ProdE} automatically. 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

794 
\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

795 
lemma add_diff_inverse: "\<lbrakk>a:N; b:N; b  a = 0 : N\<rbrakk> \<Longrightarrow> b #+ (ab) = a : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

796 
apply (rule EqE) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

797 
apply (rule add_diff_inverse_lemma [THEN ProdE, THEN ProdE]) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

798 
apply (assumption  rule EqI)+ 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

799 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

800 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

801 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

802 
subsection \<open>Absolute difference\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

803 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

804 
text \<open>Typing of absolute difference: short and long versions.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

805 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

806 
lemma absdiff_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a  b : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

807 
unfolding arith_defs by typechk 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

808 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

809 
lemma absdiff_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a  b = c  d : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

810 
unfolding arith_defs by equal 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

811 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

812 
lemma absdiff_self_eq_0: "a:N \<Longrightarrow> a  a = 0 : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

813 
unfolding absdiff_def by (arith_rew diff_self_eq_0) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

814 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

815 
lemma absdiffC0: "a:N \<Longrightarrow> 0  a = a : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

816 
unfolding absdiff_def by hyp_arith_rew 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

817 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

818 
lemma absdiff_succ_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a)  succ(b) = a  b : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

819 
unfolding absdiff_def by hyp_arith_rew 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

820 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

821 
text \<open>Note how easy using commutative laws can be? ...not always...\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

822 
lemma absdiff_commute: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a  b = b  a : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

823 
unfolding absdiff_def 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

824 
apply (rule add_commute) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

825 
apply (typechk diff_typing) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

826 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

827 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

828 
text \<open>If \<open>a + b = 0\<close> then \<open>a = 0\<close>. Surprisingly tedious.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

829 
schematic_goal add_eq0_lemma: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> ?c : Eq(N,a#+b,0) \<longrightarrow> Eq(N,a,0)" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

830 
apply (NE a) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

831 
apply (rule_tac [3] replace_type) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

832 
apply arith_rew 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

833 
apply intr \<comment> \<open>strips remaining \<open>\<Prod>\<close>s\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

834 
apply (rule_tac [2] zero_ne_succ [THEN FE]) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

835 
apply (erule_tac [3] EqE [THEN sym_elem]) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

836 
apply (typechk add_typing) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

837 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

838 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

839 
text \<open> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

840 
Version of above with the premise \<open>a + b = 0\<close>. 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

841 
Again, resolution instantiates variables in @{thm ProdE}. 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

842 
\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

843 
lemma add_eq0: "\<lbrakk>a:N; b:N; a #+ b = 0 : N\<rbrakk> \<Longrightarrow> a = 0 : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

844 
apply (rule EqE) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

845 
apply (rule add_eq0_lemma [THEN ProdE]) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

846 
apply (rule_tac [3] EqI) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

847 
apply typechk 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

848 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

849 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

850 
text \<open>Here is a lemma to infer \<open>a  b = 0\<close> and \<open>b  a = 0\<close> from \<open>a  b = 0\<close>, below.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

851 
schematic_goal absdiff_eq0_lem: 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

852 
"\<lbrakk>a:N; b:N; a  b = 0 : N\<rbrakk> \<Longrightarrow> ?a : Eq(N, ab, 0) \<times> Eq(N, ba, 0)" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

853 
apply (unfold absdiff_def) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

854 
apply intr 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

855 
apply eqintr 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

856 
apply (rule_tac [2] add_eq0) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

857 
apply (rule add_eq0) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

858 
apply (rule_tac [6] add_commute [THEN trans_elem]) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

859 
apply (typechk diff_typing) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

860 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

861 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

862 
text \<open>If \<open>a  b = 0\<close> then \<open>a = b\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

863 
proof: \<open>a  b = 0\<close> and \<open>b  a = 0\<close>, so \<open>b = a + (b  a) = a + 0 = a\<close>. 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

864 
\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

865 
lemma absdiff_eq0: "\<lbrakk>a  b = 0 : N; a:N; b:N\<rbrakk> \<Longrightarrow> a = b : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

866 
apply (rule EqE) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

867 
apply (rule absdiff_eq0_lem [THEN SumE]) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

868 
apply eqintr 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

869 
apply (rule add_diff_inverse [THEN sym_elem, THEN trans_elem]) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

870 
apply (erule_tac [3] EqE) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

871 
apply (hyp_arith_rew add_0_right) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

872 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

873 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

874 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

875 
subsection \<open>Remainder and Quotient\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

876 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

877 
text \<open>Typing of remainder: short and long versions.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

878 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

879 
lemma mod_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a mod b : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

880 
unfolding mod_def by (typechk absdiff_typing) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

881 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

882 
lemma mod_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a mod b = c mod d : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

883 
unfolding mod_def by (equal absdiff_typingL) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

884 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

885 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

886 
text \<open>Computation for \<open>mod\<close>: 0 and successor cases.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

887 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

888 
lemma modC0: "b:N \<Longrightarrow> 0 mod b = 0 : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

889 
unfolding mod_def by (rew absdiff_typing) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

890 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

891 
lemma modC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

892 
succ(a) mod b = rec(succ(a mod b)  b, 0, \<lambda>x y. succ(a mod b)) : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

893 
unfolding mod_def by (rew absdiff_typing) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

894 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

895 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

896 
text \<open>Typing of quotient: short and long versions.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

897 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

898 
lemma div_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a div b : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

899 
unfolding div_def by (typechk absdiff_typing mod_typing) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

900 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

901 
lemma div_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a div b = c div d : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

902 
unfolding div_def by (equal absdiff_typingL mod_typingL) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

903 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

904 
lemmas div_typing_rls = mod_typing div_typing absdiff_typing 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

905 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

906 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

907 
text \<open>Computation for quotient: 0 and successor cases.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

908 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

909 
lemma divC0: "b:N \<Longrightarrow> 0 div b = 0 : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

910 
unfolding div_def by (rew mod_typing absdiff_typing) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

911 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

912 
lemma divC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

913 
succ(a) div b = rec(succ(a) mod b, succ(a div b), \<lambda>x y. a div b) : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

914 
unfolding div_def by (rew mod_typing) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

915 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

916 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

917 
text \<open>Version of above with same condition as the \<open>mod\<close> one.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

918 
lemma divC_succ2: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

919 
succ(a) div b =rec(succ(a mod b)  b, succ(a div b), \<lambda>x y. a div b) : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

920 
apply (rule divC_succ [THEN trans_elem]) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

921 
apply (rew div_typing_rls modC_succ) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

922 
apply (NE "succ (a mod b) b") 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

923 
apply (rew mod_typing div_typing absdiff_typing) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

924 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

925 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

926 
text \<open>For case analysis on whether a number is 0 or a successor.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

927 
lemma iszero_decidable: "a:N \<Longrightarrow> rec(a, inl(eq), \<lambda>ka kb. inr(<ka, eq>)) : 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

928 
Eq(N,a,0) + (\<Sum>x:N. Eq(N,a, succ(x)))" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

929 
apply (NE a) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

930 
apply (rule_tac [3] PlusI_inr) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

931 
apply (rule_tac [2] PlusI_inl) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

932 
apply eqintr 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

933 
apply equal 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

934 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

935 

fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

936 
text \<open>Main Result. Holds when \<open>b\<close> is 0 since \<open>a mod 0 = a\<close> and \<open>a div 0 = 0\<close>.\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

937 
lemma mod_div_equality: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a mod b #+ (a div b) #* b = a : N" 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

938 
apply (NE a) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

939 
apply (arith_rew div_typing_rls modC0 modC_succ divC0 divC_succ2) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

940 
apply (rule EqE) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

941 
\<comment> \<open>case analysis on \<open>succ(u mod b)  b\<close>\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

942 
apply (rule_tac a1 = "succ (u mod b)  b" in iszero_decidable [THEN PlusE]) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

943 
apply (erule_tac [3] SumE) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

944 
apply (hyp_arith_rew div_typing_rls modC0 modC_succ divC0 divC_succ2) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

945 
\<comment> \<open>Replace one occurrence of \<open>b\<close> by \<open>succ(u mod b)\<close>. Clumsy!\<close> 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

946 
apply (rule add_typingL [THEN trans_elem]) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

947 
apply (erule EqE [THEN absdiff_eq0, THEN sym_elem]) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

948 
apply (rule_tac [3] refl_elem) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

949 
apply (hyp_arith_rew div_typing_rls) 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

950 
done 
fae6051ec192
clarified main CTT.thy, and avoid name clash with global HOL/Main.thy;
wenzelm
parents:
65338
diff
changeset

951 

19761  952 
end 