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(* Author: Various *)
section \<open>Combination and Cancellation Simprocs for Numeral Expressions\<close>
theory Numeral_Simprocs
imports Divides
begin
ML_file \<open>~~/src/Provers/Arith/assoc_fold.ML\<close>
ML_file \<open>~~/src/Provers/Arith/cancel_numerals.ML\<close>
ML_file \<open>~~/src/Provers/Arith/combine_numerals.ML\<close>
ML_file \<open>~~/src/Provers/Arith/cancel_numeral_factor.ML\<close>
ML_file \<open>~~/src/Provers/Arith/extract_common_term.ML\<close>
lemmas semiring_norm =
Let_def arith_simps diff_nat_numeral rel_simps
if_False if_True
add_0 add_Suc add_numeral_left
add_neg_numeral_left mult_numeral_left
numeral_One [symmetric] uminus_numeral_One [symmetric] Suc_eq_plus1
eq_numeral_iff_iszero not_iszero_Numeral1
declare split_div [of _ _ "numeral k", arith_split] for k
declare split_mod [of _ _ "numeral k", arith_split] for k
text \<open>For \<open>combine_numerals\<close>\<close>
lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"
by (simp add: add_mult_distrib)
text \<open>For \<open>cancel_numerals\<close>\<close>
lemma nat_diff_add_eq1:
"j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"
by (simp split: 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: 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: 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: 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: 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: 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: 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: nat_diff_split simp add: add_mult_distrib)
text \<open>For \<open>cancel_numeral_factors\<close>\<close>
lemma nat_mult_le_cancel1: "(0::nat) < k ==> (k*m <= k*n) = (m<=n)"
by auto
lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m
by auto
lemma nat_mult_eq_cancel1: "(0::nat) < k ==> (k*m = k*n) = (m=n)"
by auto
lemma nat_mult_div_cancel1: "(0::nat) < k ==> (k*m) div (k*n) = (m div n)"
by auto
lemma 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 \<open>For \<open>cancel_factor\<close>\<close>
lemmas nat_mult_le_cancel_disj = mult_le_cancel1
lemmas nat_mult_less_cancel_disj = mult_less_cancel1
lemma nat_mult_eq_cancel_disj:
fixes k m n :: nat
shows "k * m = k * n \ k = 0 \ m = n"
by auto
lemma nat_mult_div_cancel_disj [simp]:
fixes k m n :: nat
shows "(k * m) div (k * n) = (if k = 0 then 0 else m div n)"
by (fact div_mult_mult1_if)
lemma numeral_times_minus_swap:
fixes x:: "'a::comm_ring_1" shows "numeral w * -x = x * - numeral w"
by (simp add: mult.commute)
ML_file \<open>Tools/numeral_simprocs.ML\<close>
simproc_setup semiring_assoc_fold
("(a::'a::comm_semiring_1_cancel) * b") =
\<open>fn phi => Numeral_Simprocs.assoc_fold\<close>
(* 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") =
\<open>fn phi => Numeral_Simprocs.combine_numerals\<close>
simproc_setup field_combine_numerals
("(i::'a::{field,ring_char_0}) + j"
|"(i::'a::{field,ring_char_0}) - j") =
\<open>fn phi => Numeral_Simprocs.field_combine_numerals\<close>
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") =
\<open>fn phi => Numeral_Simprocs.eq_cancel_numerals\<close>
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") =
\<open>fn phi => Numeral_Simprocs.less_cancel_numerals\<close>
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") =
\<open>fn phi => Numeral_Simprocs.le_cancel_numerals\<close>
simproc_setup ring_eq_cancel_numeral_factor
("(l::'a::{idom,ring_char_0}) * m = n"
|"(l::'a::{idom,ring_char_0}) = m * n") =
\<open>fn phi => Numeral_Simprocs.eq_cancel_numeral_factor\<close>
simproc_setup ring_less_cancel_numeral_factor
("(l::'a::linordered_idom) * m < n"
|"(l::'a::linordered_idom) < m * n") =
\<open>fn phi => Numeral_Simprocs.less_cancel_numeral_factor\<close>
simproc_setup ring_le_cancel_numeral_factor
("(l::'a::linordered_idom) * m <= n"
|"(l::'a::linordered_idom) <= m * n") =
\<open>fn phi => Numeral_Simprocs.le_cancel_numeral_factor\<close>
(* TODO: remove comm_ring_1 constraint if possible *)
simproc_setup int_div_cancel_numeral_factors
("((l::'a::{euclidean_semiring_cancel,comm_ring_1,ring_char_0}) * m) div n"
|"(l::'a::{euclidean_semiring_cancel,comm_ring_1,ring_char_0}) div (m * n)") =
\<open>fn phi => Numeral_Simprocs.div_cancel_numeral_factor\<close>
simproc_setup divide_cancel_numeral_factor
("((l::'a::{field,ring_char_0}) * m) / n"
|"(l::'a::{field,ring_char_0}) / (m * n)"
|"((numeral v)::'a::{field,ring_char_0}) / (numeral w)") =
\<open>fn phi => Numeral_Simprocs.divide_cancel_numeral_factor\<close>
simproc_setup ring_eq_cancel_factor
("(l::'a::idom) * m = n" | "(l::'a::idom) = m * n") =
\<open>fn phi => Numeral_Simprocs.eq_cancel_factor\<close>
simproc_setup linordered_ring_le_cancel_factor
("(l::'a::linordered_idom) * m <= n"
|"(l::'a::linordered_idom) <= m * n") =
\<open>fn phi => Numeral_Simprocs.le_cancel_factor\<close>
simproc_setup linordered_ring_less_cancel_factor
("(l::'a::linordered_idom) * m < n"
|"(l::'a::linordered_idom) < m * n") =
\<open>fn phi => Numeral_Simprocs.less_cancel_factor\<close>
simproc_setup int_div_cancel_factor
("((l::'a::euclidean_semiring_cancel) * m) div n"
|"(l::'a::euclidean_semiring_cancel) div (m * n)") =
\<open>fn phi => Numeral_Simprocs.div_cancel_factor\<close>
simproc_setup int_mod_cancel_factor
("((l::'a::euclidean_semiring_cancel) * m) mod n"
|"(l::'a::euclidean_semiring_cancel) mod (m * n)") =
\<open>fn phi => Numeral_Simprocs.mod_cancel_factor\<close>
simproc_setup dvd_cancel_factor
("((l::'a::idom) * m) dvd n"
|"(l::'a::idom) dvd (m * n)") =
\<open>fn phi => Numeral_Simprocs.dvd_cancel_factor\<close>
simproc_setup divide_cancel_factor
("((l::'a::field) * m) / n"
|"(l::'a::field) / (m * n)") =
\<open>fn phi => Numeral_Simprocs.divide_cancel_factor\<close>
ML_file \<open>Tools/nat_numeral_simprocs.ML\<close>
simproc_setup nat_combine_numerals
("(i::nat) + j" | "Suc (i + j)") =
\<open>fn phi => Nat_Numeral_Simprocs.combine_numerals\<close>
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") =
\<open>fn phi => Nat_Numeral_Simprocs.eq_cancel_numerals\<close>
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") =
\<open>fn phi => Nat_Numeral_Simprocs.less_cancel_numerals\<close>
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") =
\<open>fn phi => Nat_Numeral_Simprocs.le_cancel_numerals\<close>
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") =
\<open>fn phi => Nat_Numeral_Simprocs.diff_cancel_numerals\<close>
simproc_setup nat_eq_cancel_numeral_factor
("(l::nat) * m = n" | "(l::nat) = m * n") =
\<open>fn phi => Nat_Numeral_Simprocs.eq_cancel_numeral_factor\<close>
simproc_setup nat_less_cancel_numeral_factor
("(l::nat) * m < n" | "(l::nat) < m * n") =
\<open>fn phi => Nat_Numeral_Simprocs.less_cancel_numeral_factor\<close>
simproc_setup nat_le_cancel_numeral_factor
("(l::nat) * m <= n" | "(l::nat) <= m * n") =
\<open>fn phi => Nat_Numeral_Simprocs.le_cancel_numeral_factor\<close>
simproc_setup nat_div_cancel_numeral_factor
("((l::nat) * m) div n" | "(l::nat) div (m * n)") =
\<open>fn phi => Nat_Numeral_Simprocs.div_cancel_numeral_factor\<close>
simproc_setup nat_dvd_cancel_numeral_factor
("((l::nat) * m) dvd n" | "(l::nat) dvd (m * n)") =
\<open>fn phi => Nat_Numeral_Simprocs.dvd_cancel_numeral_factor\<close>
simproc_setup nat_eq_cancel_factor
("(l::nat) * m = n" | "(l::nat) = m * n") =
\<open>fn phi => Nat_Numeral_Simprocs.eq_cancel_factor\<close>
simproc_setup nat_less_cancel_factor
("(l::nat) * m < n" | "(l::nat) < m * n") =
\<open>fn phi => Nat_Numeral_Simprocs.less_cancel_factor\<close>
simproc_setup nat_le_cancel_factor
("(l::nat) * m <= n" | "(l::nat) <= m * n") =
\<open>fn phi => Nat_Numeral_Simprocs.le_cancel_factor\<close>
simproc_setup nat_div_cancel_factor
("((l::nat) * m) div n" | "(l::nat) div (m * n)") =
\<open>fn phi => Nat_Numeral_Simprocs.div_cancel_factor\<close>
simproc_setup nat_dvd_cancel_factor
("((l::nat) * m) dvd n" | "(l::nat) dvd (m * n)") =
\<open>fn phi => Nat_Numeral_Simprocs.dvd_cancel_factor\<close>
declaration \<open>
K (Lin_Arith.add_simprocs
[\<^simproc>\<open>semiring_assoc_fold\<close>,
\<^simproc>\<open>int_combine_numerals\<close>,
\<^simproc>\<open>inteq_cancel_numerals\<close>,
\<^simproc>\<open>intless_cancel_numerals\<close>,
\<^simproc>\<open>intle_cancel_numerals\<close>,
\<^simproc>\<open>field_combine_numerals\<close>,
\<^simproc>\<open>nat_combine_numerals\<close>,
\<^simproc>\<open>nateq_cancel_numerals\<close>,
\<^simproc>\<open>natless_cancel_numerals\<close>,
\<^simproc>\<open>natle_cancel_numerals\<close>,
\<^simproc>\<open>natdiff_cancel_numerals\<close>,
Numeral_Simprocs.field_divide_cancel_numeral_factor])
\<close>
lemma bit_numeral_int_simps [simp]:
\<open>bit (1 :: int) (numeral n) \<longleftrightarrow> bit (0 :: int) (pred_numeral n)\<close>
\<open>bit (numeral (num.Bit0 w) :: int) (numeral n) \<longleftrightarrow> bit (numeral w :: int) (pred_numeral n)\<close>
\<open>bit (numeral (num.Bit1 w) :: int) (numeral n) \<longleftrightarrow> bit (numeral w :: int) (pred_numeral n)\<close>
\<open>bit (numeral (Num.BitM w) :: int) (numeral n) \<longleftrightarrow> \<not> bit (- numeral w :: int) (pred_numeral n)\<close>
\<open>bit (- numeral (num.Bit0 w) :: int) (numeral n) \<longleftrightarrow> bit (- numeral w :: int) (pred_numeral n)\<close>
\<open>bit (- numeral (num.Bit1 w) :: int) (numeral n) \<longleftrightarrow> \<not> bit (numeral w :: int) (pred_numeral n)\<close>
\<open>bit (- numeral (Num.BitM w) :: int) (numeral n) \<longleftrightarrow> bit (- (numeral w) :: int) (pred_numeral n)\<close>
by (simp_all add: bit_1_iff numeral_eq_Suc bit_Suc add_One sub_inc_One_eq bit_minus_int_iff)
end
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