src/HOL/Numeral_Simprocs.thy
author webertj
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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)) = (((i-j)*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 - ((j-i)*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) = ((i-j)*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 = (j-i)*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) = ((i-j)*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 < (j-i)*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) = ((i-j)*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 <= (j-i)*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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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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ML_file "Tools/nat_numeral_simprocs.ML"
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simproc_setup nat_combine_numerals
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  ("(i::nat) + j" | "Suc (i + j)") =
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  {* fn phi => Nat_Numeral_Simprocs.combine_numerals *}
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simproc_setup nateq_cancel_numerals
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  ("(l::nat) + m = n" | "(l::nat) = m + n" |
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   "(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.eq_cancel_numerals *}
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simproc_setup natless_cancel_numerals
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  ("(l::nat) + m < n" | "(l::nat) < m + n" |
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   "(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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simproc_setup natle_cancel_numerals
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  ("(l::nat) + m \<le> n" | "(l::nat) \<le> m + n" |
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   "(l::nat) * m \<le> n" | "(l::nat) \<le> m * n" |
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   "Suc m \<le> n" | "m \<le> Suc n") =
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  {* fn phi => Nat_Numeral_Simprocs.le_cancel_numerals *}
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simproc_setup natdiff_cancel_numerals
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  ("((l::nat) + m) - n" | "(l::nat) - (m + n)" |
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   "(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.diff_cancel_numerals *}
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simproc_setup nat_eq_cancel_numeral_factor
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  ("(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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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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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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simproc_setup nat_div_cancel_numeral_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_numeral_factor *}
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simproc_setup nat_dvd_cancel_numeral_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_numeral_factor *}
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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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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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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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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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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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(* FIXME: duplicate rule warnings for:
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  ring_distribs
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  numeral_plus_numeral numeral_times_numeral
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  numeral_eq_iff numeral_less_iff numeral_le_iff
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  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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     @{thm numeral_plus_numeral}, @{thm diff_nat_numeral}, @{thm numeral_times_numeral},
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     @{thm numeral_eq_iff}, @{thm numeral_less_iff}, @{thm numeral_le_iff},
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     @{thm le_Suc_numeral}, @{thm le_numeral_Suc},
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     @{thm less_Suc_numeral}, @{thm less_numeral_Suc},
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     @{thm Suc_eq_numeral}, @{thm eq_numeral_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 numeral_neq_zero}, @{thm zero_neq_numeral}, @{thm zero_less_numeral},
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     @{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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       @{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
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      [@{simproc nat_combine_numerals},
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       @{simproc nateq_cancel_numerals},
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       @{simproc natless_cancel_numerals},
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       @{simproc natle_cancel_numerals},
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       @{simproc natdiff_cancel_numerals}])
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*}
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end