(* Author: Various *) header {* Combination and Cancellation Simprocs for Numeral Expressions *} theory Numeral_Simprocs imports Divides uses "~~/src/Provers/Arith/assoc_fold.ML" "~~/src/Provers/Arith/cancel_numerals.ML" "~~/src/Provers/Arith/combine_numerals.ML" "~~/src/Provers/Arith/cancel_numeral_factor.ML" "~~/src/Provers/Arith/extract_common_term.ML" ("Tools/numeral_simprocs.ML") ("Tools/nat_numeral_simprocs.ML") begin declare split_div [of _ _ "numeral k", arith_split] for k declare split_mod [of _ _ "numeral k", arith_split] for k text {* 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 auto lemma nat_mult_less_cancel1: "(0::nat) < k ==> (k*m < k*n) = (m (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 @{text cancel_factor} *} lemma nat_mult_le_cancel_disj: "(k*m <= k*n) = ((0::nat) < k --> m<=n)" by auto lemma nat_mult_less_cancel_disj: "(k*m < k*n) = ((0::nat) < k & m 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 *} use "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