# Theory Numeral_Simprocs

```(* Author: Various *)

section ‹Combination and Cancellation Simprocs for Numeral Expressions›

theory Numeral_Simprocs
imports Parity
begin

ML_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 diff_nat_numeral rel_simps
if_False if_True
numeral_One [symmetric] uminus_numeral_One [symmetric] Suc_eq_plus1
eq_numeral_iff_iszero not_iszero_Numeral1

text ‹For ‹combine_numerals››

lemma left_add_mult_distrib: "i*u + (j*u + k) = (i+j)*u + (k::nat)"

text ‹For ‹cancel_numerals››

"j <= (i::nat) ==> ((i*u + m) - (j*u + n)) = (((i-j)*u + m) - n)"

"i <= (j::nat) ==> ((i*u + m) - (j*u + n)) = (m - ((j-i)*u + n))"

"j <= (i::nat) ==> (i*u + m = j*u + n) = ((i-j)*u + m = n)"

"i <= (j::nat) ==> (i*u + m = j*u + n) = (m = (j-i)*u + n)"

"j <= (i::nat) ==> (i*u + m < j*u + n) = ((i-j)*u + m < n)"

"i <= (j::nat) ==> (i*u + m < j*u + n) = (m < (j-i)*u + n)"

"j <= (i::nat) ==> (i*u + m <= j*u + n) = ((i-j)*u + m <= n)"

"i <= (j::nat) ==> (i*u + m <= j*u + n) = (m <= (j-i)*u + n)"

text ‹For ‹cancel_numeral_factors››

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<n)"
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 ‹For ‹cancel_factor››

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 (fact mult_cancel_left)

lemma nat_mult_div_cancel_disj:
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"

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,ring_char_0}) + j"
|"(i::'a::{field,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::{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)") =
‹fn phi => Numeral_Simprocs.div_cancel_numeral_factor›

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)") =
‹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::euclidean_semiring_cancel) * m) div n"
|"(l::'a::euclidean_semiring_cancel) div (m * n)") =
‹fn phi => Numeral_Simprocs.div_cancel_factor›

simproc_setup int_mod_cancel_factor
("((l::'a::euclidean_semiring_cancel) * m) mod n"
|"(l::'a::euclidean_semiring_cancel) 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) * m) / n"
|"(l::'a::field) / (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›

declaration ‹
[\<^simproc>‹semiring_assoc_fold›,
\<^simproc>‹int_combine_numerals›,
\<^simproc>‹inteq_cancel_numerals›,
\<^simproc>‹intless_cancel_numerals›,
\<^simproc>‹intle_cancel_numerals›,
\<^simproc>‹field_combine_numerals›,
\<^simproc>‹nat_combine_numerals›,
\<^simproc>‹nateq_cancel_numerals›,
\<^simproc>‹natless_cancel_numerals›,
\<^simproc>‹natle_cancel_numerals›,
\<^simproc>‹natdiff_cancel_numerals›,
Numeral_Simprocs.field_divide_cancel_numeral_factor])
›

end
```