author  wenzelm 
Wed, 22 Aug 2012 22:55:41 +0200  
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parent 47255  30a1692557b0 
child 54249  ce00f2fef556 
permissions  rwrr 
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(* Author: Various *) 
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header {* Combination and Cancellation Simprocs for Numeral Expressions *} 

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theory Numeral_Simprocs 

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imports Divides 

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begin 

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ML_file "~~/src/Provers/Arith/assoc_fold.ML" 
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ML_file "~~/src/Provers/Arith/cancel_numerals.ML" 

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ML_file "~~/src/Provers/Arith/combine_numerals.ML" 

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ML_file "~~/src/Provers/Arith/cancel_numeral_factor.ML" 

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ML_file "~~/src/Provers/Arith/extract_common_term.ML" 

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lemmas semiring_norm = 
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Let_def arith_simps nat_arith rel_simps 
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if_False if_True 
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add_0 add_Suc add_numeral_left 
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add_neg_numeral_left mult_numeral_left 
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numeral_1_eq_1 [symmetric] Suc_eq_plus1 
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eq_numeral_iff_iszero not_iszero_Numeral1 
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declare split_div [of _ _ "numeral k", arith_split] for k 
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declare split_mod [of _ _ "numeral k", arith_split] for k 
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text {* For @{text combine_numerals} *} 

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lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)" 

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by (simp add: add_mult_distrib) 

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text {* For @{text cancel_numerals} *} 

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lemma nat_diff_add_eq1: 

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"j <= (i::nat) ==> ((i*u + m)  (j*u + n)) = (((ij)*u + m)  n)" 

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by (simp split add: nat_diff_split add: add_mult_distrib) 

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lemma nat_diff_add_eq2: 

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"i <= (j::nat) ==> ((i*u + m)  (j*u + n)) = (m  ((ji)*u + n))" 

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by (simp split add: nat_diff_split add: add_mult_distrib) 

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lemma nat_eq_add_iff1: 

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"j <= (i::nat) ==> (i*u + m = j*u + n) = ((ij)*u + m = n)" 

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by (auto split add: nat_diff_split simp add: add_mult_distrib) 

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lemma nat_eq_add_iff2: 

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"i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (ji)*u + n)" 

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by (auto split add: nat_diff_split simp add: add_mult_distrib) 

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lemma nat_less_add_iff1: 

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"j <= (i::nat) ==> (i*u + m < j*u + n) = ((ij)*u + m < n)" 

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by (auto split add: nat_diff_split simp add: add_mult_distrib) 

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lemma nat_less_add_iff2: 

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"i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (ji)*u + n)" 

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by (auto split add: nat_diff_split simp add: add_mult_distrib) 

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lemma nat_le_add_iff1: 

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"j <= (i::nat) ==> (i*u + m <= j*u + n) = ((ij)*u + m <= n)" 

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by (auto split add: nat_diff_split simp add: add_mult_distrib) 

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lemma nat_le_add_iff2: 

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"i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (ji)*u + n)" 

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by (auto split add: nat_diff_split simp add: add_mult_distrib) 

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text {* For @{text cancel_numeral_factors} *} 

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lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)" 

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by auto 

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lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)" 

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by auto 

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lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)" 

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by auto 

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lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)" 

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by auto 

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lemma nat_mult_dvd_cancel_disj[simp]: 

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"(k*m) dvd (k*n) = (k=0  m dvd (n::nat))" 

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by (auto simp: dvd_eq_mod_eq_0 mod_mult_mult1) 
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lemma nat_mult_dvd_cancel1: "0 < k \<Longrightarrow> (k*m) dvd (k*n::nat) = (m dvd n)" 

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by(auto) 

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text {* For @{text cancel_factor} *} 

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lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k > m<=n)" 

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by auto 

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lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)" 

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by auto 

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lemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat)  m=n)" 

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by auto 

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lemma nat_mult_div_cancel_disj[simp]: 

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"(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)" 

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by (simp add: nat_mult_div_cancel1) 

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ML_file "Tools/numeral_simprocs.ML" 
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simproc_setup semiring_assoc_fold 
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("(a::'a::comm_semiring_1_cancel) * b") = 
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{* fn phi => Numeral_Simprocs.assoc_fold *} 
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(* TODO: see whether the type class can be generalized further *) 
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simproc_setup int_combine_numerals 
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("(i::'a::comm_ring_1) + j"  "(i::'a::comm_ring_1)  j") = 
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{* fn phi => Numeral_Simprocs.combine_numerals *} 
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simproc_setup field_combine_numerals 
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("(i::'a::{field_inverse_zero,ring_char_0}) + j" 
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"(i::'a::{field_inverse_zero,ring_char_0})  j") = 
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{* fn phi => Numeral_Simprocs.field_combine_numerals *} 
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simproc_setup inteq_cancel_numerals 
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("(l::'a::comm_ring_1) + m = n" 
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"(l::'a::comm_ring_1) = m + n" 
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"(l::'a::comm_ring_1)  m = n" 
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"(l::'a::comm_ring_1) = m  n" 
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"(l::'a::comm_ring_1) * m = n" 
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"(l::'a::comm_ring_1) = m * n" 
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" (l::'a::comm_ring_1) = m" 
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"(l::'a::comm_ring_1) =  m") = 
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{* fn phi => Numeral_Simprocs.eq_cancel_numerals *} 
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simproc_setup intless_cancel_numerals 
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("(l::'a::linordered_idom) + m < n" 
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"(l::'a::linordered_idom) < m + n" 
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"(l::'a::linordered_idom)  m < n" 
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"(l::'a::linordered_idom) < m  n" 
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"(l::'a::linordered_idom) * m < n" 
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"(l::'a::linordered_idom) < m * n" 
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" (l::'a::linordered_idom) < m" 
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"(l::'a::linordered_idom) <  m") = 
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{* fn phi => Numeral_Simprocs.less_cancel_numerals *} 
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simproc_setup intle_cancel_numerals 
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("(l::'a::linordered_idom) + m \<le> n" 
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"(l::'a::linordered_idom) \<le> m + n" 
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"(l::'a::linordered_idom)  m \<le> n" 
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"(l::'a::linordered_idom) \<le> m  n" 
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"(l::'a::linordered_idom) * m \<le> n" 
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"(l::'a::linordered_idom) \<le> m * n" 
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" (l::'a::linordered_idom) \<le> m" 
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"(l::'a::linordered_idom) \<le>  m") = 
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{* fn phi => Numeral_Simprocs.le_cancel_numerals *} 
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simproc_setup ring_eq_cancel_numeral_factor 
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("(l::'a::{idom,ring_char_0}) * m = n" 
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"(l::'a::{idom,ring_char_0}) = m * n") = 
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{* fn phi => Numeral_Simprocs.eq_cancel_numeral_factor *} 
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simproc_setup ring_less_cancel_numeral_factor 
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("(l::'a::linordered_idom) * m < n" 
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"(l::'a::linordered_idom) < m * n") = 
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{* fn phi => Numeral_Simprocs.less_cancel_numeral_factor *} 
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simproc_setup ring_le_cancel_numeral_factor 
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("(l::'a::linordered_idom) * m <= n" 
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"(l::'a::linordered_idom) <= m * n") = 
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{* fn phi => Numeral_Simprocs.le_cancel_numeral_factor *} 
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(* TODO: remove comm_ring_1 constraint if possible *) 
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simproc_setup int_div_cancel_numeral_factors 
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("((l::'a::{semiring_div,comm_ring_1,ring_char_0}) * m) div n" 
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"(l::'a::{semiring_div,comm_ring_1,ring_char_0}) div (m * n)") = 
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{* fn phi => Numeral_Simprocs.div_cancel_numeral_factor *} 
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simproc_setup divide_cancel_numeral_factor 
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("((l::'a::{field_inverse_zero,ring_char_0}) * m) / n" 
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"(l::'a::{field_inverse_zero,ring_char_0}) / (m * n)" 
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"((numeral v)::'a::{field_inverse_zero,ring_char_0}) / (numeral w)") = 
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{* fn phi => Numeral_Simprocs.divide_cancel_numeral_factor *} 
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simproc_setup ring_eq_cancel_factor 
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("(l::'a::idom) * m = n"  "(l::'a::idom) = m * n") = 
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{* fn phi => Numeral_Simprocs.eq_cancel_factor *} 
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simproc_setup linordered_ring_le_cancel_factor 
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("(l::'a::linordered_idom) * m <= n" 
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"(l::'a::linordered_idom) <= m * n") = 
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{* fn phi => Numeral_Simprocs.le_cancel_factor *} 
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simproc_setup linordered_ring_less_cancel_factor 
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("(l::'a::linordered_idom) * m < n" 
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"(l::'a::linordered_idom) < m * n") = 
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{* fn phi => Numeral_Simprocs.less_cancel_factor *} 
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190 

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191 
simproc_setup int_div_cancel_factor 
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192 
("((l::'a::semiring_div) * m) div n" 
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"(l::'a::semiring_div) div (m * n)") = 
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194 
{* fn phi => Numeral_Simprocs.div_cancel_factor *} 
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195 

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196 
simproc_setup int_mod_cancel_factor 
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197 
("((l::'a::semiring_div) * m) mod n" 
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198 
"(l::'a::semiring_div) mod (m * n)") = 
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199 
{* fn phi => Numeral_Simprocs.mod_cancel_factor *} 
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200 

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201 
simproc_setup dvd_cancel_factor 
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202 
("((l::'a::idom) * m) dvd n" 
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203 
"(l::'a::idom) dvd (m * n)") = 
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204 
{* fn phi => Numeral_Simprocs.dvd_cancel_factor *} 
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205 

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206 
simproc_setup divide_cancel_factor 
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207 
("((l::'a::field_inverse_zero) * m) / n" 
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208 
"(l::'a::field_inverse_zero) / (m * n)") = 
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209 
{* fn phi => Numeral_Simprocs.divide_cancel_factor *} 
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210 

48891  211 
ML_file "Tools/nat_numeral_simprocs.ML" 
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213 
simproc_setup nat_combine_numerals 
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214 
("(i::nat) + j"  "Suc (i + j)") = 
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215 
{* fn phi => Nat_Numeral_Simprocs.combine_numerals *} 
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216 

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217 
simproc_setup nateq_cancel_numerals 
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218 
("(l::nat) + m = n"  "(l::nat) = m + n"  
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219 
"(l::nat) * m = n"  "(l::nat) = m * n"  
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220 
"Suc m = n"  "m = Suc n") = 
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221 
{* fn phi => Nat_Numeral_Simprocs.eq_cancel_numerals *} 
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222 

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223 
simproc_setup natless_cancel_numerals 
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224 
("(l::nat) + m < n"  "(l::nat) < m + n"  
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225 
"(l::nat) * m < n"  "(l::nat) < m * n"  
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226 
"Suc m < n"  "m < Suc n") = 
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227 
{* fn phi => Nat_Numeral_Simprocs.less_cancel_numerals *} 
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228 

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229 
simproc_setup natle_cancel_numerals 
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230 
("(l::nat) + m \<le> n"  "(l::nat) \<le> m + n"  
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231 
"(l::nat) * m \<le> n"  "(l::nat) \<le> m * n"  
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232 
"Suc m \<le> n"  "m \<le> Suc n") = 
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233 
{* fn phi => Nat_Numeral_Simprocs.le_cancel_numerals *} 
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234 

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235 
simproc_setup natdiff_cancel_numerals 
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236 
("((l::nat) + m)  n"  "(l::nat)  (m + n)"  
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237 
"(l::nat) * m  n"  "(l::nat)  m * n"  
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238 
"Suc m  n"  "m  Suc n") = 
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239 
{* fn phi => Nat_Numeral_Simprocs.diff_cancel_numerals *} 
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240 

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241 
simproc_setup nat_eq_cancel_numeral_factor 
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242 
("(l::nat) * m = n"  "(l::nat) = m * n") = 
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243 
{* fn phi => Nat_Numeral_Simprocs.eq_cancel_numeral_factor *} 
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244 

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245 
simproc_setup nat_less_cancel_numeral_factor 
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246 
("(l::nat) * m < n"  "(l::nat) < m * n") = 
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{* fn phi => Nat_Numeral_Simprocs.less_cancel_numeral_factor *} 
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248 

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249 
simproc_setup nat_le_cancel_numeral_factor 
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250 
("(l::nat) * m <= n"  "(l::nat) <= m * n") = 
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{* fn phi => Nat_Numeral_Simprocs.le_cancel_numeral_factor *} 
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252 

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253 
simproc_setup nat_div_cancel_numeral_factor 
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254 
("((l::nat) * m) div n"  "(l::nat) div (m * n)") = 
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{* fn phi => Nat_Numeral_Simprocs.div_cancel_numeral_factor *} 
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256 

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257 
simproc_setup nat_dvd_cancel_numeral_factor 
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258 
("((l::nat) * m) dvd n"  "(l::nat) dvd (m * n)") = 
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259 
{* fn phi => Nat_Numeral_Simprocs.dvd_cancel_numeral_factor *} 
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260 

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261 
simproc_setup nat_eq_cancel_factor 
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262 
("(l::nat) * m = n"  "(l::nat) = m * n") = 
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{* fn phi => Nat_Numeral_Simprocs.eq_cancel_factor *} 
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264 

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265 
simproc_setup nat_less_cancel_factor 
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266 
("(l::nat) * m < n"  "(l::nat) < m * n") = 
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{* fn phi => Nat_Numeral_Simprocs.less_cancel_factor *} 
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268 

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269 
simproc_setup nat_le_cancel_factor 
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270 
("(l::nat) * m <= n"  "(l::nat) <= m * n") = 
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271 
{* fn phi => Nat_Numeral_Simprocs.le_cancel_factor *} 
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272 

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273 
simproc_setup nat_div_cancel_factor 
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274 
("((l::nat) * m) div n"  "(l::nat) div (m * n)") = 
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{* fn phi => Nat_Numeral_Simprocs.div_cancel_factor *} 
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276 

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277 
simproc_setup nat_dvd_cancel_factor 
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278 
("((l::nat) * m) dvd n"  "(l::nat) dvd (m * n)") = 
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279 
{* fn phi => Nat_Numeral_Simprocs.dvd_cancel_factor *} 
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280 

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281 
(* FIXME: duplicate rule warnings for: 
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282 
ring_distribs 
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283 
numeral_plus_numeral numeral_times_numeral 
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284 
numeral_eq_iff numeral_less_iff numeral_le_iff 
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285 
numeral_neq_zero zero_neq_numeral zero_less_numeral 
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if_True if_False *) 
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declaration {* 
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K (Lin_Arith.add_simps ([@{thm Suc_numeral}, @{thm int_numeral}]) 
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#> Lin_Arith.add_simps (@{thms ring_distribs} @ [@{thm Let_numeral}, @{thm Let_neg_numeral}, @{thm Let_0}, @{thm Let_1}, 
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@{thm nat_0}, @{thm nat_1}, 
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291 
@{thm numeral_plus_numeral}, @{thm diff_nat_numeral}, @{thm numeral_times_numeral}, 
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292 
@{thm numeral_eq_iff}, @{thm numeral_less_iff}, @{thm numeral_le_iff}, 
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293 
@{thm le_Suc_numeral}, @{thm le_numeral_Suc}, 
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294 
@{thm less_Suc_numeral}, @{thm less_numeral_Suc}, 
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295 
@{thm Suc_eq_numeral}, @{thm eq_numeral_Suc}, 
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@{thm mult_Suc}, @{thm mult_Suc_right}, 
297 
@{thm add_Suc}, @{thm add_Suc_right}, 

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@{thm numeral_neq_zero}, @{thm zero_neq_numeral}, @{thm zero_less_numeral}, 
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299 
@{thm of_int_numeral}, @{thm of_nat_numeral}, @{thm nat_numeral}, 
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@{thm if_True}, @{thm if_False}]) 
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#> Lin_Arith.add_simprocs 
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[@{simproc semiring_assoc_fold}, 
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303 
@{simproc int_combine_numerals}, 
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304 
@{simproc inteq_cancel_numerals}, 
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305 
@{simproc intless_cancel_numerals}, 
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306 
@{simproc intle_cancel_numerals}] 
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307 
#> Lin_Arith.add_simprocs 
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308 
[@{simproc nat_combine_numerals}, 
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309 
@{simproc nateq_cancel_numerals}, 
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@{simproc natless_cancel_numerals}, 
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311 
@{simproc natle_cancel_numerals}, 
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312 
@{simproc natdiff_cancel_numerals}]) 
33366  313 
*} 
314 

37886  315 
end 