author  huffman 
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parent 45284  ae78a4ffa81d 
child 45308  2e84e5f0463b 
permissions  rwrr 
33366  1 
(* 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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uses 

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

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

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

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

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

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("Tools/numeral_simprocs.ML") 

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("Tools/nat_numeral_simprocs.ML") 

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begin 

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declare split_div [of _ _ "number_of k", standard, arith_split] 

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declare split_mod [of _ _ "number_of k", standard, arith_split] 

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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_distrib2[symmetric]) 

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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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use "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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simproc_setup int_combine_numerals 
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("(i::'a::number_ring) + j"  "(i::'a::number_ring)  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, number_ring}) + j" 
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"(i::'a::{field_inverse_zero, number_ring})  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::number_ring) + m = n" 
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"(l::'a::number_ring) = m + n" 
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"(l::'a::number_ring)  m = n" 
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"(l::'a::number_ring) = m  n" 
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"(l::'a::number_ring) * m = n" 
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"(l::'a::number_ring) = m * n") = 
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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,number_ring}) + m < n" 
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"(l::'a::{linordered_idom,number_ring}) < m + n" 
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"(l::'a::{linordered_idom,number_ring})  m < n" 
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"(l::'a::{linordered_idom,number_ring}) < m  n" 
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"(l::'a::{linordered_idom,number_ring}) * m < n" 
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"(l::'a::{linordered_idom,number_ring}) < m * n") = 
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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,number_ring}) + m \<le> n" 
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"(l::'a::{linordered_idom,number_ring}) \<le> m + n" 
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"(l::'a::{linordered_idom,number_ring})  m \<le> n" 
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"(l::'a::{linordered_idom,number_ring}) \<le> m  n" 
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"(l::'a::{linordered_idom,number_ring}) * m \<le> n" 
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"(l::'a::{linordered_idom,number_ring}) \<le> m * n") = 
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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,number_ring}) * m = n" 
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"(l::'a::{idom,number_ring}) = 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,number_ring}) * m < n" 
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"(l::'a::{linordered_idom,number_ring}) < 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,number_ring}) * m <= n" 
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"(l::'a::{linordered_idom,number_ring}) <= m * n") = 
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{* fn phi => Numeral_Simprocs.le_cancel_numeral_factor *} 
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simproc_setup int_div_cancel_numeral_factors 
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("((l::'a::{semiring_div,number_ring}) * m) div n" 
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"(l::'a::{semiring_div,number_ring}) 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,number_ring}) * m) / n" 
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"(l::'a::{field_inverse_zero,number_ring}) / (m * n)" 
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"((number_of v)::'a::{field_inverse_zero,number_ring}) / (number_of 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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simproc_setup int_div_cancel_factor 
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("((l::'a::semiring_div) * m) div n" 
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"(l::'a::semiring_div) div (m * n)") = 
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{* fn phi => Numeral_Simprocs.div_cancel_factor *} 
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simproc_setup int_mod_cancel_factor 
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("((l::'a::semiring_div) * m) mod n" 
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"(l::'a::semiring_div) mod (m * n)") = 
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{* fn phi => Numeral_Simprocs.mod_cancel_factor *} 
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simproc_setup dvd_cancel_factor 
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("((l::'a::idom) * m) dvd n" 
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"(l::'a::idom) dvd (m * n)") = 
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{* fn phi => Numeral_Simprocs.dvd_cancel_factor *} 
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simproc_setup divide_cancel_factor 
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("((l::'a::field_inverse_zero) * m) / n" 
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"(l::'a::field_inverse_zero) / (m * n)") = 
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{* fn phi => Numeral_Simprocs.divide_cancel_factor *} 
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use "Tools/nat_numeral_simprocs.ML" 
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declaration {* 

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K (Lin_Arith.add_simps (@{thms neg_simps} @ [@{thm Suc_nat_number_of}, @{thm int_nat_number_of}]) 

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#> Lin_Arith.add_simps (@{thms ring_distribs} @ [@{thm Let_number_of}, @{thm Let_0}, @{thm Let_1}, 

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@{thm nat_0}, @{thm nat_1}, 

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@{thm add_nat_number_of}, @{thm diff_nat_number_of}, @{thm mult_nat_number_of}, 

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@{thm eq_nat_number_of}, @{thm less_nat_number_of}, @{thm le_number_of_eq_not_less}, 

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@{thm le_Suc_number_of}, @{thm le_number_of_Suc}, 

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@{thm less_Suc_number_of}, @{thm less_number_of_Suc}, 

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@{thm Suc_eq_number_of}, @{thm eq_number_of_Suc}, 

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@{thm mult_Suc}, @{thm mult_Suc_right}, 

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@{thm add_Suc}, @{thm add_Suc_right}, 

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@{thm eq_number_of_0}, @{thm eq_0_number_of}, @{thm less_0_number_of}, 

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@{thm of_int_number_of_eq}, @{thm of_nat_number_of_eq}, @{thm nat_number_of}, 

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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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@{simproc int_combine_numerals}, 
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@{simproc inteq_cancel_numerals}, 
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@{simproc intless_cancel_numerals}, 
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@{simproc intle_cancel_numerals}] 
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#> Lin_Arith.add_simprocs (Nat_Numeral_Simprocs.combine_numerals :: Nat_Numeral_Simprocs.cancel_numerals)) 
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*} 

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end 