author  nipkow 
Mon, 10 Feb 2014 21:50:50 +0100  
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parent 54489  03ff4d1e6784 
child 58889  5b7a9633cfa8 
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 diff_nat_numeral 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_One [symmetric] uminus_numeral_One [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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lemmas nat_mult_le_cancel_disj = mult_le_cancel1 
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lemmas nat_mult_less_cancel_disj = mult_less_cancel1 
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lemma nat_mult_eq_cancel_disj: 
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fixes k m n :: nat 
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shows "k * m = k * n \<longleftrightarrow> k = 0 \<or> m = n" 
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by auto 
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lemma nat_mult_div_cancel_disj [simp]: 
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fixes k m n :: nat 
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shows "(k * m) div (k * n) = (if k = 0 then 0 else m div n)" 
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by (fact div_mult_mult1_if) 
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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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183 
("(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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186 

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187 
simproc_setup linordered_ring_less_cancel_factor 
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188 
("(l::'a::linordered_idom) * m < n" 
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189 
"(l::'a::linordered_idom) < m * n") = 
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{* fn phi => Numeral_Simprocs.less_cancel_factor *} 
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191 

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192 
simproc_setup int_div_cancel_factor 
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193 
("((l::'a::semiring_div) * m) div n" 
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194 
"(l::'a::semiring_div) div (m * n)") = 
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195 
{* fn phi => Numeral_Simprocs.div_cancel_factor *} 
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196 

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197 
simproc_setup int_mod_cancel_factor 
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198 
("((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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201 

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202 
simproc_setup dvd_cancel_factor 
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203 
("((l::'a::idom) * m) dvd n" 
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"(l::'a::idom) dvd (m * n)") = 
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205 
{* fn phi => Numeral_Simprocs.dvd_cancel_factor *} 
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206 

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207 
simproc_setup divide_cancel_factor 
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208 
("((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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211 

48891  212 
ML_file "Tools/nat_numeral_simprocs.ML" 
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214 
simproc_setup nat_combine_numerals 
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215 
("(i::nat) + j"  "Suc (i + j)") = 
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{* fn phi => Nat_Numeral_Simprocs.combine_numerals *} 
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217 

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218 
simproc_setup nateq_cancel_numerals 
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219 
("(l::nat) + m = n"  "(l::nat) = m + n"  
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"(l::nat) * m = n"  "(l::nat) = m * n"  
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221 
"Suc m = n"  "m = Suc n") = 
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222 
{* fn phi => Nat_Numeral_Simprocs.eq_cancel_numerals *} 
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223 

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224 
simproc_setup natless_cancel_numerals 
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225 
("(l::nat) + m < n"  "(l::nat) < m + n"  
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226 
"(l::nat) * m < n"  "(l::nat) < m * n"  
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"Suc m < n"  "m < Suc n") = 
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{* fn phi => Nat_Numeral_Simprocs.less_cancel_numerals *} 
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229 

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230 
simproc_setup natle_cancel_numerals 
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231 
("(l::nat) + m \<le> n"  "(l::nat) \<le> m + n"  
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232 
"(l::nat) * m \<le> n"  "(l::nat) \<le> m * n"  
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233 
"Suc m \<le> n"  "m \<le> Suc n") = 
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{* fn phi => Nat_Numeral_Simprocs.le_cancel_numerals *} 
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235 

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236 
simproc_setup natdiff_cancel_numerals 
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237 
("((l::nat) + m)  n"  "(l::nat)  (m + n)"  
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238 
"(l::nat) * m  n"  "(l::nat)  m * n"  
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239 
"Suc m  n"  "m  Suc n") = 
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240 
{* fn phi => Nat_Numeral_Simprocs.diff_cancel_numerals *} 
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241 

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242 
simproc_setup nat_eq_cancel_numeral_factor 
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243 
("(l::nat) * m = n"  "(l::nat) = m * n") = 
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{* fn phi => Nat_Numeral_Simprocs.eq_cancel_numeral_factor *} 
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245 

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246 
simproc_setup nat_less_cancel_numeral_factor 
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("(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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249 

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250 
simproc_setup nat_le_cancel_numeral_factor 
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("(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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253 

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254 
simproc_setup nat_div_cancel_numeral_factor 
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255 
("((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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257 

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258 
simproc_setup nat_dvd_cancel_numeral_factor 
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259 
("((l::nat) * m) dvd n"  "(l::nat) dvd (m * n)") = 
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{* fn phi => Nat_Numeral_Simprocs.dvd_cancel_numeral_factor *} 
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261 

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262 
simproc_setup nat_eq_cancel_factor 
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("(l::nat) * m = n"  "(l::nat) = m * n") = 
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{* fn phi => Nat_Numeral_Simprocs.eq_cancel_factor *} 
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265 

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266 
simproc_setup nat_less_cancel_factor 
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("(l::nat) * m < n"  "(l::nat) < m * n") = 
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{* fn phi => Nat_Numeral_Simprocs.less_cancel_factor *} 
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269 

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270 
simproc_setup nat_le_cancel_factor 
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("(l::nat) * m <= n"  "(l::nat) <= m * n") = 
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{* fn phi => Nat_Numeral_Simprocs.le_cancel_factor *} 
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273 

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simproc_setup nat_div_cancel_factor 
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("((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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277 

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278 
simproc_setup nat_dvd_cancel_factor 
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("((l::nat) * m) dvd n"  "(l::nat) dvd (m * n)") = 
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{* fn phi => Nat_Numeral_Simprocs.dvd_cancel_factor *} 
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281 

33366  282 
declaration {* 
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K (Lin_Arith.add_simprocs 
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[@{simproc semiring_assoc_fold}, 
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285 
@{simproc int_combine_numerals}, 
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286 
@{simproc inteq_cancel_numerals}, 
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@{simproc intless_cancel_numerals}, 
55375  288 
@{simproc intle_cancel_numerals}, 
289 
@{simproc field_combine_numerals}] 

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#> Lin_Arith.add_simprocs 
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[@{simproc nat_combine_numerals}, 
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@{simproc nateq_cancel_numerals}, 
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293 
@{simproc natless_cancel_numerals}, 
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294 
@{simproc natle_cancel_numerals}, 
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295 
@{simproc natdiff_cancel_numerals}]) 
33366  296 
*} 
297 

37886  298 
end 