# Theory Numeral_Simprocs

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theory Numeral_Simprocs
imports Divides
`(* Author: Various *)header {* Combination and Cancellation Simprocs for Numeral Expressions *}theory Numeral_Simprocsimports DividesbeginML_file "~~/src/Provers/Arith/assoc_fold.ML"ML_file "~~/src/Provers/Arith/cancel_numerals.ML"ML_file "~~/src/Provers/Arith/combine_numerals.ML"ML_file "~~/src/Provers/Arith/cancel_numeral_factor.ML"ML_file "~~/src/Provers/Arith/extract_common_term.ML"lemmas semiring_norm =  Let_def arith_simps nat_arith rel_simps  if_False if_True  add_0 add_Suc add_numeral_left  add_neg_numeral_left mult_numeral_left  numeral_1_eq_1 [symmetric] Suc_eq_plus1  eq_numeral_iff_iszero not_iszero_Numeral1declare split_div [of _ _ "numeral k", arith_split] for kdeclare split_mod [of _ _ "numeral k", arith_split] for ktext {* For @{text combine_numerals} *}lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"by (simp add: add_mult_distrib)text {* For @{text cancel_numerals} *}lemma nat_diff_add_eq1:     "j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"by (simp split add: nat_diff_split add: add_mult_distrib)lemma nat_diff_add_eq2:     "i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"by (simp split add: nat_diff_split add: add_mult_distrib)lemma nat_eq_add_iff1:     "j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"by (auto split add: nat_diff_split simp add: add_mult_distrib)lemma nat_eq_add_iff2:     "i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"by (auto split add: nat_diff_split simp add: add_mult_distrib)lemma nat_less_add_iff1:     "j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"by (auto split add: nat_diff_split simp add: add_mult_distrib)lemma nat_less_add_iff2:     "i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"by (auto split add: nat_diff_split simp add: add_mult_distrib)lemma nat_le_add_iff1:     "j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"by (auto split add: nat_diff_split simp add: add_mult_distrib)lemma nat_le_add_iff2:     "i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"by (auto split add: nat_diff_split simp add: add_mult_distrib)text {* For @{text cancel_numeral_factors} *}lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"by autolemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m<n)"by autolemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"by autolemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"by autolemma nat_mult_dvd_cancel_disj[simp]:  "(k*m) dvd (k*n) = (k=0 | m dvd (n::nat))"by (auto simp: dvd_eq_mod_eq_0 mod_mult_mult1)lemma nat_mult_dvd_cancel1: "0 < k ==> (k*m) dvd (k*n::nat) = (m dvd n)"by(auto)text {* For @{text cancel_factor} *}lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)"by autolemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m<n)"by autolemma nat_mult_eq_cancel_disj: "(k*m = k*n) = (k = (0::nat) | m=n)"by autolemma nat_mult_div_cancel_disj[simp]:     "(k*m) div (k*n) = (if k = (0::nat) then 0 else m div n)"by (simp add: nat_mult_div_cancel1)ML_file "Tools/numeral_simprocs.ML"simproc_setup semiring_assoc_fold  ("(a::'a::comm_semiring_1_cancel) * b") =  {* fn phi => Numeral_Simprocs.assoc_fold *}(* TODO: see whether the type class can be generalized further *)simproc_setup int_combine_numerals  ("(i::'a::comm_ring_1) + j" | "(i::'a::comm_ring_1) - j") =  {* fn phi => Numeral_Simprocs.combine_numerals *}simproc_setup field_combine_numerals  ("(i::'a::{field_inverse_zero,ring_char_0}) + j"  |"(i::'a::{field_inverse_zero,ring_char_0}) - j") =  {* fn phi => Numeral_Simprocs.field_combine_numerals *}simproc_setup inteq_cancel_numerals  ("(l::'a::comm_ring_1) + m = n"  |"(l::'a::comm_ring_1) = m + n"  |"(l::'a::comm_ring_1) - m = n"  |"(l::'a::comm_ring_1) = m - n"  |"(l::'a::comm_ring_1) * m = n"  |"(l::'a::comm_ring_1) = m * n"  |"- (l::'a::comm_ring_1) = m"  |"(l::'a::comm_ring_1) = - m") =  {* fn phi => Numeral_Simprocs.eq_cancel_numerals *}simproc_setup intless_cancel_numerals  ("(l::'a::linordered_idom) + m < n"  |"(l::'a::linordered_idom) < m + n"  |"(l::'a::linordered_idom) - m < n"  |"(l::'a::linordered_idom) < m - n"  |"(l::'a::linordered_idom) * m < n"  |"(l::'a::linordered_idom) < m * n"  |"- (l::'a::linordered_idom) < m"  |"(l::'a::linordered_idom) < - m") =  {* fn phi => Numeral_Simprocs.less_cancel_numerals *}simproc_setup intle_cancel_numerals  ("(l::'a::linordered_idom) + m ≤ n"  |"(l::'a::linordered_idom) ≤ m + n"  |"(l::'a::linordered_idom) - m ≤ n"  |"(l::'a::linordered_idom) ≤ m - n"  |"(l::'a::linordered_idom) * m ≤ n"  |"(l::'a::linordered_idom) ≤ m * n"  |"- (l::'a::linordered_idom) ≤ m"  |"(l::'a::linordered_idom) ≤ - m") =  {* fn phi => Numeral_Simprocs.le_cancel_numerals *}simproc_setup ring_eq_cancel_numeral_factor  ("(l::'a::{idom,ring_char_0}) * m = n"  |"(l::'a::{idom,ring_char_0}) = m * n") =  {* fn phi => Numeral_Simprocs.eq_cancel_numeral_factor *}simproc_setup ring_less_cancel_numeral_factor  ("(l::'a::linordered_idom) * m < n"  |"(l::'a::linordered_idom) < m * n") =  {* fn phi => Numeral_Simprocs.less_cancel_numeral_factor *}simproc_setup ring_le_cancel_numeral_factor  ("(l::'a::linordered_idom) * m <= n"  |"(l::'a::linordered_idom) <= m * n") =  {* fn phi => Numeral_Simprocs.le_cancel_numeral_factor *}(* TODO: remove comm_ring_1 constraint if possible *)simproc_setup int_div_cancel_numeral_factors  ("((l::'a::{semiring_div,comm_ring_1,ring_char_0}) * m) div n"  |"(l::'a::{semiring_div,comm_ring_1,ring_char_0}) div (m * n)") =  {* fn phi => Numeral_Simprocs.div_cancel_numeral_factor *}simproc_setup divide_cancel_numeral_factor  ("((l::'a::{field_inverse_zero,ring_char_0}) * m) / n"  |"(l::'a::{field_inverse_zero,ring_char_0}) / (m * n)"  |"((numeral v)::'a::{field_inverse_zero,ring_char_0}) / (numeral w)") =  {* fn phi => Numeral_Simprocs.divide_cancel_numeral_factor *}simproc_setup ring_eq_cancel_factor  ("(l::'a::idom) * m = n" | "(l::'a::idom) = m * n") =  {* fn phi => Numeral_Simprocs.eq_cancel_factor *}simproc_setup linordered_ring_le_cancel_factor  ("(l::'a::linordered_idom) * m <= n"  |"(l::'a::linordered_idom) <= m * n") =  {* fn phi => Numeral_Simprocs.le_cancel_factor *}simproc_setup linordered_ring_less_cancel_factor  ("(l::'a::linordered_idom) * m < n"  |"(l::'a::linordered_idom) < m * n") =  {* fn phi => Numeral_Simprocs.less_cancel_factor *}simproc_setup int_div_cancel_factor  ("((l::'a::semiring_div) * m) div n"  |"(l::'a::semiring_div) div (m * n)") =  {* fn phi => Numeral_Simprocs.div_cancel_factor *}simproc_setup int_mod_cancel_factor  ("((l::'a::semiring_div) * m) mod n"  |"(l::'a::semiring_div) mod (m * n)") =  {* fn phi => Numeral_Simprocs.mod_cancel_factor *}simproc_setup dvd_cancel_factor  ("((l::'a::idom) * m) dvd n"  |"(l::'a::idom) dvd (m * n)") =  {* fn phi => Numeral_Simprocs.dvd_cancel_factor *}simproc_setup divide_cancel_factor  ("((l::'a::field_inverse_zero) * m) / n"  |"(l::'a::field_inverse_zero) / (m * n)") =  {* fn phi => Numeral_Simprocs.divide_cancel_factor *}ML_file "Tools/nat_numeral_simprocs.ML"simproc_setup nat_combine_numerals  ("(i::nat) + j" | "Suc (i + j)") =  {* fn phi => Nat_Numeral_Simprocs.combine_numerals *}simproc_setup nateq_cancel_numerals  ("(l::nat) + m = n" | "(l::nat) = m + n" |   "(l::nat) * m = n" | "(l::nat) = m * n" |   "Suc m = n" | "m = Suc n") =  {* fn phi => Nat_Numeral_Simprocs.eq_cancel_numerals *}simproc_setup natless_cancel_numerals  ("(l::nat) + m < n" | "(l::nat) < m + n" |   "(l::nat) * m < n" | "(l::nat) < m * n" |   "Suc m < n" | "m < Suc n") =  {* fn phi => Nat_Numeral_Simprocs.less_cancel_numerals *}simproc_setup natle_cancel_numerals  ("(l::nat) + m ≤ n" | "(l::nat) ≤ m + n" |   "(l::nat) * m ≤ n" | "(l::nat) ≤ m * n" |   "Suc m ≤ n" | "m ≤ Suc n") =  {* fn phi => Nat_Numeral_Simprocs.le_cancel_numerals *}simproc_setup natdiff_cancel_numerals  ("((l::nat) + m) - n" | "(l::nat) - (m + n)" |   "(l::nat) * m - n" | "(l::nat) - m * n" |   "Suc m - n" | "m - Suc n") =  {* fn phi => Nat_Numeral_Simprocs.diff_cancel_numerals *}simproc_setup nat_eq_cancel_numeral_factor  ("(l::nat) * m = n" | "(l::nat) = m * n") =  {* fn phi => Nat_Numeral_Simprocs.eq_cancel_numeral_factor *}simproc_setup nat_less_cancel_numeral_factor  ("(l::nat) * m < n" | "(l::nat) < m * n") =  {* fn phi => Nat_Numeral_Simprocs.less_cancel_numeral_factor *}simproc_setup nat_le_cancel_numeral_factor  ("(l::nat) * m <= n" | "(l::nat) <= m * n") =  {* fn phi => Nat_Numeral_Simprocs.le_cancel_numeral_factor *}simproc_setup nat_div_cancel_numeral_factor  ("((l::nat) * m) div n" | "(l::nat) div (m * n)") =  {* fn phi => Nat_Numeral_Simprocs.div_cancel_numeral_factor *}simproc_setup nat_dvd_cancel_numeral_factor  ("((l::nat) * m) dvd n" | "(l::nat) dvd (m * n)") =  {* fn phi => Nat_Numeral_Simprocs.dvd_cancel_numeral_factor *}simproc_setup nat_eq_cancel_factor  ("(l::nat) * m = n" | "(l::nat) = m * n") =  {* fn phi => Nat_Numeral_Simprocs.eq_cancel_factor *}simproc_setup nat_less_cancel_factor  ("(l::nat) * m < n" | "(l::nat) < m * n") =  {* fn phi => Nat_Numeral_Simprocs.less_cancel_factor *}simproc_setup nat_le_cancel_factor  ("(l::nat) * m <= n" | "(l::nat) <= m * n") =  {* fn phi => Nat_Numeral_Simprocs.le_cancel_factor *}simproc_setup nat_div_cancel_factor  ("((l::nat) * m) div n" | "(l::nat) div (m * n)") =  {* fn phi => Nat_Numeral_Simprocs.div_cancel_factor *}simproc_setup nat_dvd_cancel_factor  ("((l::nat) * m) dvd n" | "(l::nat) dvd (m * n)") =  {* fn phi => Nat_Numeral_Simprocs.dvd_cancel_factor *}(* FIXME: duplicate rule warnings for:  ring_distribs  numeral_plus_numeral numeral_times_numeral  numeral_eq_iff numeral_less_iff numeral_le_iff  numeral_neq_zero zero_neq_numeral zero_less_numeral  if_True if_False *)declaration {*   K (Lin_Arith.add_simps ([@{thm Suc_numeral}, @{thm int_numeral}])  #> Lin_Arith.add_simps (@{thms ring_distribs} @ [@{thm Let_numeral}, @{thm Let_neg_numeral}, @{thm Let_0}, @{thm Let_1},     @{thm nat_0}, @{thm nat_1},     @{thm numeral_plus_numeral}, @{thm diff_nat_numeral}, @{thm numeral_times_numeral},     @{thm numeral_eq_iff}, @{thm numeral_less_iff}, @{thm numeral_le_iff},     @{thm le_Suc_numeral}, @{thm le_numeral_Suc},     @{thm less_Suc_numeral}, @{thm less_numeral_Suc},     @{thm Suc_eq_numeral}, @{thm eq_numeral_Suc},     @{thm mult_Suc}, @{thm mult_Suc_right},     @{thm add_Suc}, @{thm add_Suc_right},     @{thm numeral_neq_zero}, @{thm zero_neq_numeral}, @{thm zero_less_numeral},     @{thm of_int_numeral}, @{thm of_nat_numeral}, @{thm nat_numeral},     @{thm if_True}, @{thm if_False}])  #> Lin_Arith.add_simprocs      [@{simproc semiring_assoc_fold},       @{simproc int_combine_numerals},       @{simproc inteq_cancel_numerals},       @{simproc intless_cancel_numerals},       @{simproc intle_cancel_numerals}]  #> Lin_Arith.add_simprocs      [@{simproc nat_combine_numerals},       @{simproc nateq_cancel_numerals},       @{simproc natless_cancel_numerals},       @{simproc natle_cancel_numerals},       @{simproc natdiff_cancel_numerals}])*}end`