author  wenzelm 
Mon, 24 Oct 2016 11:10:17 +0200  
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parent 63505  42e1dece537a 
child 64980  7dc25cf5793e 
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(* Title: CTT/Arith.thy 
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory 
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Copyright 1991 University of Cambridge 
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*) 

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section \<open>Elementary arithmetic\<close> 
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theory Arith 

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imports Bool 
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begin 
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subsection \<open>Arithmetic operators and their definitions\<close> 
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definition add :: "[i,i]\<Rightarrow>i" (infixr "#+" 65) 
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where "a#+b \<equiv> rec(a, b, \<lambda>u v. succ(v))" 

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definition diff :: "[i,i]\<Rightarrow>i" (infixr "" 65) 
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where "ab \<equiv> rec(b, a, \<lambda>u v. rec(v, 0, \<lambda>x y. x))" 

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definition absdiff :: "[i,i]\<Rightarrow>i" (infixr "" 65) 
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where "ab \<equiv> (ab) #+ (ba)" 

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definition mult :: "[i,i]\<Rightarrow>i" (infixr "#*" 70) 
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where "a#*b \<equiv> rec(a, 0, \<lambda>u v. b #+ v)" 

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xsymbol support for Pi, Sigma, >, : (membership)
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definition mod :: "[i,i]\<Rightarrow>i" (infixr "mod" 70) 
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where "a mod b \<equiv> rec(a, 0, \<lambda>u v. rec(succ(v)  b, 0, \<lambda>x y. succ(v)))" 

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definition div :: "[i,i]\<Rightarrow>i" (infixr "div" 70) 
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where "a div b \<equiv> rec(a, 0, \<lambda>u v. rec(succ(u) mod b, succ(v), \<lambda>x y. v))" 

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lemmas arith_defs = add_def diff_def absdiff_def mult_def mod_def div_def 
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subsection \<open>Proofs about elementary arithmetic: addition, multiplication, etc.\<close> 
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subsubsection \<open>Addition\<close> 
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text \<open>Typing of \<open>add\<close>: short and long versions.\<close> 
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lemma add_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #+ b : N" 
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unfolding arith_defs by typechk 
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lemma add_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a #+ b = c #+ d : N" 
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unfolding arith_defs by equal 
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text \<open>Computation for \<open>add\<close>: 0 and successor cases.\<close> 
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lemma addC0: "b:N \<Longrightarrow> 0 #+ b = b : N" 
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unfolding arith_defs by rew 
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lemma addC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) #+ b = succ(a #+ b) : N" 
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unfolding arith_defs by rew 
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subsubsection \<open>Multiplication\<close> 
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text \<open>Typing of \<open>mult\<close>: short and long versions.\<close> 
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lemma mult_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #* b : N" 
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unfolding arith_defs by (typechk add_typing) 
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lemma mult_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a #* b = c #* d : N" 
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unfolding arith_defs by (equal add_typingL) 
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text \<open>Computation for \<open>mult\<close>: 0 and successor cases.\<close> 

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lemma multC0: "b:N \<Longrightarrow> 0 #* b = 0 : N" 
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unfolding arith_defs by rew 
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lemma multC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a) #* b = b #+ (a #* b) : N" 
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unfolding arith_defs by rew 
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subsubsection \<open>Difference\<close> 
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text \<open>Typing of difference.\<close> 
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lemma diff_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a  b : N" 
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unfolding arith_defs by typechk 
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lemma diff_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a  b = c  d : N" 
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unfolding arith_defs by equal 
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text \<open>Computation for difference: 0 and successor cases.\<close> 
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lemma diffC0: "a:N \<Longrightarrow> a  0 = a : N" 
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unfolding arith_defs by rew 
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text \<open>Note: \<open>rec(a, 0, \<lambda>z w.z)\<close> is \<open>pred(a).\<close>\<close> 
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lemma diff_0_eq_0: "b:N \<Longrightarrow> 0  b = 0 : N" 
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unfolding arith_defs 
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apply (NE b) 

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apply hyp_rew 

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done 

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text \<open> 
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Essential to simplify FIRST!! (Else we get a critical pair) 

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\<open>succ(a)  succ(b)\<close> rewrites to \<open>pred(succ(a)  b)\<close>. 

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\<close> 

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lemma diff_succ_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a)  succ(b) = a  b : N" 
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unfolding arith_defs 
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apply hyp_rew 

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apply (NE b) 

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apply hyp_rew 

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done 

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subsection \<open>Simplification\<close> 
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lemmas arith_typing_rls = add_typing mult_typing diff_typing 

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and arith_congr_rls = add_typingL mult_typingL diff_typingL 

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lemmas congr_rls = arith_congr_rls intrL2_rls elimL_rls 
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lemmas arithC_rls = 

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addC0 addC_succ 

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multC0 multC_succ 

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diffC0 diff_0_eq_0 diff_succ_succ 

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ML \<open> 
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structure Arith_simp = TSimpFun( 
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val refl = @{thm refl_elem} 

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val sym = @{thm sym_elem} 

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val trans = @{thm trans_elem} 

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val refl_red = @{thm refl_red} 

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val trans_red = @{thm trans_red} 

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val red_if_equal = @{thm red_if_equal} 

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val default_rls = @{thms arithC_rls comp_rls} 

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val routine_tac = routine_tac @{thms arith_typing_rls routine_rls} 

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) 

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fun arith_rew_tac ctxt prems = 
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make_rew_tac ctxt (Arith_simp.norm_tac ctxt (@{thms congr_rls}, prems)) 

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fun hyp_arith_rew_tac ctxt prems = 
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make_rew_tac ctxt 

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(Arith_simp.cond_norm_tac ctxt (prove_cond_tac ctxt, @{thms congr_rls}, prems)) 

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\<close> 
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method_setup arith_rew = \<open> 
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Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (arith_rew_tac ctxt ths)) 
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\<close> 
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method_setup hyp_arith_rew = \<open> 
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Attrib.thms >> (fn ths => fn ctxt => SIMPLE_METHOD (hyp_arith_rew_tac ctxt ths)) 
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\<close> 
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subsection \<open>Addition\<close> 
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text \<open>Associative law for addition.\<close> 
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lemma add_assoc: "\<lbrakk>a:N; b:N; c:N\<rbrakk> \<Longrightarrow> (a #+ b) #+ c = a #+ (b #+ c) : N" 
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apply (NE a) 
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apply hyp_arith_rew 

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done 

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text \<open>Commutative law for addition. Can be proved using three inductions. 
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Must simplify after first induction! Orientation of rewrites is delicate.\<close> 

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lemma add_commute: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #+ b = b #+ a : N" 
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apply (NE a) 
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apply hyp_arith_rew 

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apply (rule sym_elem) 

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prefer 2 

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apply (NE b) 

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prefer 4 

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apply (NE b) 

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apply hyp_arith_rew 

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done 

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subsection \<open>Multiplication\<close> 
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text \<open>Right annihilation in product.\<close> 
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lemma mult_0_right: "a:N \<Longrightarrow> a #* 0 = 0 : N" 
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apply (NE a) 
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apply hyp_arith_rew 

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done 

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text \<open>Right successor law for multiplication.\<close> 
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lemma mult_succ_right: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #* succ(b) = a #+ (a #* b) : N" 
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apply (NE a) 
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apply (hyp_arith_rew add_assoc [THEN sym_elem]) 

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apply (assumption  rule add_commute mult_typingL add_typingL intrL_rls refl_elem)+ 

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done 

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text \<open>Commutative law for multiplication.\<close> 
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lemma mult_commute: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a #* b = b #* a : N" 
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apply (NE a) 
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apply (hyp_arith_rew mult_0_right mult_succ_right) 

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done 

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text \<open>Addition distributes over multiplication.\<close> 
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lemma add_mult_distrib: "\<lbrakk>a:N; b:N; c:N\<rbrakk> \<Longrightarrow> (a #+ b) #* c = (a #* c) #+ (b #* c) : N" 
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apply (NE a) 
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apply (hyp_arith_rew add_assoc [THEN sym_elem]) 

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done 

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text \<open>Associative law for multiplication.\<close> 
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lemma mult_assoc: "\<lbrakk>a:N; b:N; c:N\<rbrakk> \<Longrightarrow> (a #* b) #* c = a #* (b #* c) : N" 
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apply (NE a) 
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apply (hyp_arith_rew add_mult_distrib) 

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done 

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subsection \<open>Difference\<close> 
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text \<open> 
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Difference on natural numbers, without negative numbers 
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\<^item> \<open>a  b = 0\<close> iff \<open>a \<le> b\<close> 

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\<^item> \<open>a  b = succ(c)\<close> iff \<open>a > b\<close> 

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\<close> 

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lemma diff_self_eq_0: "a:N \<Longrightarrow> a  a = 0 : N" 
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apply (NE a) 
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apply hyp_arith_rew 

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done 

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lemma add_0_right: "\<lbrakk>c : N; 0 : N; c : N\<rbrakk> \<Longrightarrow> c #+ 0 = c : N" 
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by (rule addC0 [THEN [3] add_commute [THEN trans_elem]]) 
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text \<open> 
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Addition is the inverse of subtraction: if \<open>b \<le> x\<close> then \<open>b #+ (x  b) = x\<close>. 

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An example of induction over a quantified formula (a product). 
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Uses rewriting with a quantified, implicative inductive hypothesis. 
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\<close> 

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schematic_goal add_diff_inverse_lemma: 
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"b:N \<Longrightarrow> ?a : \<Prod>x:N. Eq(N, bx, 0) \<longrightarrow> Eq(N, b #+ (xb), x)" 
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apply (NE b) 
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\<comment> \<open>strip one "universal quantifier" but not the "implication"\<close> 

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apply (rule_tac [3] intr_rls) 

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\<comment> \<open>case analysis on \<open>x\<close> in \<open>succ(u) \<le> x \<longrightarrow> succ(u) #+ (x  succ(u)) = x\<close>\<close> 

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prefer 4 

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apply (NE x) 

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apply assumption 

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\<comment> \<open>Prepare for simplification of types  the antecedent \<open>succ(u) \<le> x\<close>\<close> 

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apply (rule_tac [2] replace_type) 

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apply (rule_tac [1] replace_type) 

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apply arith_rew 

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\<comment> \<open>Solves first 0 goal, simplifies others. Two sugbgoals remain. 

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Both follow by rewriting, (2) using quantified induction hyp.\<close> 

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apply intr \<comment> \<open>strips remaining \<open>\<Prod>\<close>s\<close> 

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apply (hyp_arith_rew add_0_right) 

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apply assumption 

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done 

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text \<open> 
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Version of above with premise \<open>b  a = 0\<close> i.e. \<open>a \<ge> b\<close>. 

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Using @{thm ProdE} does not work  for \<open>?B(?a)\<close> is ambiguous. 

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Instead, @{thm add_diff_inverse_lemma} states the desired induction scheme; 

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the use of \<open>THEN\<close> below instantiates Vars in @{thm ProdE} automatically. 

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\<close> 

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lemma add_diff_inverse: "\<lbrakk>a:N; b:N; b  a = 0 : N\<rbrakk> \<Longrightarrow> b #+ (ab) = a : N" 
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apply (rule EqE) 
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apply (rule add_diff_inverse_lemma [THEN ProdE, THEN ProdE]) 

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apply (assumption  rule EqI)+ 

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done 

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subsection \<open>Absolute difference\<close> 
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text \<open>Typing of absolute difference: short and long versions.\<close> 
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lemma absdiff_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a  b : N" 
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unfolding arith_defs by typechk 
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lemma absdiff_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a  b = c  d : N" 
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unfolding arith_defs by equal 
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lemma absdiff_self_eq_0: "a:N \<Longrightarrow> a  a = 0 : N" 
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unfolding absdiff_def by (arith_rew diff_self_eq_0) 
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lemma absdiffC0: "a:N \<Longrightarrow> 0  a = a : N" 
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unfolding absdiff_def by hyp_arith_rew 
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lemma absdiff_succ_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> succ(a)  succ(b) = a  b : N" 
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unfolding absdiff_def by hyp_arith_rew 
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text \<open>Note how easy using commutative laws can be? ...not always...\<close> 
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lemma absdiff_commute: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a  b = b  a : N" 
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unfolding absdiff_def 
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apply (rule add_commute) 

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apply (typechk diff_typing) 

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done 

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text \<open>If \<open>a + b = 0\<close> then \<open>a = 0\<close>. Surprisingly tedious.\<close> 
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schematic_goal add_eq0_lemma: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> ?c : \<Prod>u: Eq(N,a#+b,0) . Eq(N,a,0)" 
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apply (NE a) 
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apply (rule_tac [3] replace_type) 

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apply arith_rew 

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apply intr \<comment> \<open>strips remaining \<open>\<Prod>\<close>s\<close> 

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apply (rule_tac [2] zero_ne_succ [THEN FE]) 

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apply (erule_tac [3] EqE [THEN sym_elem]) 

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apply (typechk add_typing) 

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done 

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text \<open> 
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Version of above with the premise \<open>a + b = 0\<close>. 

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Again, resolution instantiates variables in @{thm ProdE}. 

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\<close> 

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lemma add_eq0: "\<lbrakk>a:N; b:N; a #+ b = 0 : N\<rbrakk> \<Longrightarrow> a = 0 : N" 
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apply (rule EqE) 
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apply (rule add_eq0_lemma [THEN ProdE]) 

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apply (rule_tac [3] EqI) 

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apply typechk 

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done 

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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> 
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schematic_goal absdiff_eq0_lem: 
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"\<lbrakk>a:N; b:N; a  b = 0 : N\<rbrakk> \<Longrightarrow> ?a : \<Sum>v: Eq(N, ab, 0) . Eq(N, ba, 0)" 
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apply (unfold absdiff_def) 
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apply intr 

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apply eqintr 

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apply (rule_tac [2] add_eq0) 

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apply (rule add_eq0) 

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apply (rule_tac [6] add_commute [THEN trans_elem]) 

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apply (typechk diff_typing) 

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done 

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text \<open>If \<open>a  b = 0\<close> then \<open>a = b\<close> 
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proof: \<open>a  b = 0\<close> and \<open>b  a = 0\<close>, so \<open>b = a + (b  a) = a + 0 = a\<close>. 

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\<close> 

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lemma absdiff_eq0: "\<lbrakk>a  b = 0 : N; a:N; b:N\<rbrakk> \<Longrightarrow> a = b : N" 
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apply (rule EqE) 
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apply (rule absdiff_eq0_lem [THEN SumE]) 

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apply eqintr 

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apply (rule add_diff_inverse [THEN sym_elem, THEN trans_elem]) 

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apply (erule_tac [3] EqE) 

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apply (hyp_arith_rew add_0_right) 

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done 

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subsection \<open>Remainder and Quotient\<close> 
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text \<open>Typing of remainder: short and long versions.\<close> 
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lemma mod_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a mod b : N" 
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unfolding mod_def by (typechk absdiff_typing) 
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lemma mod_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a mod b = c mod d : N" 
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unfolding mod_def by (equal absdiff_typingL) 
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text \<open>Computation for \<open>mod\<close>: 0 and successor cases.\<close> 
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lemma modC0: "b:N \<Longrightarrow> 0 mod b = 0 : N" 
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unfolding mod_def by (rew absdiff_typing) 
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lemma modC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> 
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succ(a) mod b = rec(succ(a mod b)  b, 0, \<lambda>x y. succ(a mod b)) : N" 

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unfolding mod_def by (rew absdiff_typing) 
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text \<open>Typing of quotient: short and long versions.\<close> 
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lemma div_typing: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a div b : N" 
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unfolding div_def by (typechk absdiff_typing mod_typing) 
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lemma div_typingL: "\<lbrakk>a = c:N; b = d:N\<rbrakk> \<Longrightarrow> a div b = c div d : N" 
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unfolding div_def by (equal absdiff_typingL mod_typingL) 
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lemmas div_typing_rls = mod_typing div_typing absdiff_typing 

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text \<open>Computation for quotient: 0 and successor cases.\<close> 
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lemma divC0: "b:N \<Longrightarrow> 0 div b = 0 : N" 
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unfolding div_def by (rew mod_typing absdiff_typing) 
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lemma divC_succ: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> 
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succ(a) div b = rec(succ(a) mod b, succ(a div b), \<lambda>x y. a div b) : N" 

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unfolding div_def by (rew mod_typing) 
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text \<open>Version of above with same condition as the \<open>mod\<close> one.\<close> 
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lemma divC_succ2: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> 
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succ(a) div b =rec(succ(a mod b)  b, succ(a div b), \<lambda>x y. a div b) : N" 

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apply (rule divC_succ [THEN trans_elem]) 
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apply (rew div_typing_rls modC_succ) 

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apply (NE "succ (a mod b) b") 

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apply (rew mod_typing div_typing absdiff_typing) 

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done 

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text \<open>For case analysis on whether a number is 0 or a successor.\<close> 
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lemma iszero_decidable: "a:N \<Longrightarrow> rec(a, inl(eq), \<lambda>ka kb. inr(<ka, eq>)) : 
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Eq(N,a,0) + (\<Sum>x:N. Eq(N,a, succ(x)))" 
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apply (NE a) 

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apply (rule_tac [3] PlusI_inr) 

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apply (rule_tac [2] PlusI_inl) 

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apply eqintr 

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apply equal 

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done 

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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> 
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lemma mod_div_equality: "\<lbrakk>a:N; b:N\<rbrakk> \<Longrightarrow> a mod b #+ (a div b) #* b = a : N" 
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apply (NE a) 
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apply (arith_rew div_typing_rls modC0 modC_succ divC0 divC_succ2) 

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apply (rule EqE) 

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\<comment> \<open>case analysis on \<open>succ(u mod b)  b\<close>\<close> 

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apply (rule_tac a1 = "succ (u mod b)  b" in iszero_decidable [THEN PlusE]) 

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apply (erule_tac [3] SumE) 

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apply (hyp_arith_rew div_typing_rls modC0 modC_succ divC0 divC_succ2) 

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\<comment> \<open>Replace one occurrence of \<open>b\<close> by \<open>succ(u mod b)\<close>. Clumsy!\<close> 

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apply (rule add_typingL [THEN trans_elem]) 

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apply (erule EqE [THEN absdiff_eq0, THEN sym_elem]) 

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apply (rule_tac [3] refl_elem) 

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apply (hyp_arith_rew div_typing_rls) 

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done 

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end 