| Quellcodebibliothek Statistik Leitseite products/sources/formale Sprachen/Isabelle/HOL/ (Beweissystem Isabelle Version 2025-1©) Datei vom 16.11.2025 mit Größe 10 kB |
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Quelle Numeral_Simprocs.thy Sprache: Isabelle |
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(* Author: Various *)
section \<open>Combination and Cancellation Simprocs for Numeral Expressions\<close> theory Numeral_Simprocs imports Parity 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_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 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 \ by (auto simp: dvd_eq_mod_eq_0 mod_mult_mult1) lemma nat_mult_dvd_cancel1: "0 < k \ 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 \ 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" by (simp add: ac_simps) ML_file \<open>Tools/numeral_simprocs.ML\<close> simproc_setup semiring_assoc_fold ("(a::'a::comm_semiring_1_cancel) * b") = \<open>K 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>K 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>K 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>K 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>K Numeral_Simprocs.less_cancel_numerals\<close> simproc_setup intle_cancel_numerals ("(l::'a::linordered_idom) + m \ |"(l::'a::linordered_idom) \ |"(l::'a::linordered_idom) - m \ |"(l::'a::linordered_idom) \ |"(l::'a::linordered_idom) * m \ |"(l::'a::linordered_idom) \ |"- (l::'a::linordered_idom) \ |"(l::'a::linordered_idom) \ \<open>K 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>K 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>K 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>K 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>K 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>K Numeral_Simprocs.divide_cancel_numeral_factor\<close> simproc_setup ring_eq_cancel_factor ("(l::'a::idom) * m = n" | "(l::'a::idom) = m * n") = \<open>K 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>K 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>K 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>K 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>K Numeral_Simprocs.mod_cancel_factor\<close> simproc_setup dvd_cancel_factor ("((l::'a::idom) * m) dvd n" |"(l::'a::idom) dvd (m * n)") = \<open>K Numeral_Simprocs.dvd_cancel_factor\<close> simproc_setup divide_cancel_factor ("((l::'a::field) * m) / n" |"(l::'a::field) / (m * n)") = \<open>K 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>K 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>K 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>K Nat_Numeral_Simprocs.less_cancel_numerals\<close> simproc_setup natle_cancel_numerals ("(l::nat) + m \ "(l::nat) * m \ "Suc m \ \<open>K 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>K Nat_Numeral_Simprocs.diff_cancel_numerals\<close> simproc_setup nat_eq_cancel_numeral_factor ("(l::nat) * m = n" | "(l::nat) = m * n") = \<open>K Nat_Numeral_Simprocs.eq_cancel_numeral_factor\<close> simproc_setup nat_less_cancel_numeral_factor ("(l::nat) * m < n" | "(l::nat) < m * n") = \<open>K Nat_Numeral_Simprocs.less_cancel_numeral_factor\<close> simproc_setup nat_le_cancel_numeral_factor ("(l::nat) * m <= n" | "(l::nat) <= m * n") = \<open>K 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>K 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>K Nat_Numeral_Simprocs.dvd_cancel_numeral_factor\<close> simproc_setup nat_eq_cancel_factor ("(l::nat) * m = n" | "(l::nat) = m * n") = \<open>K Nat_Numeral_Simprocs.eq_cancel_factor\<close> simproc_setup nat_less_cancel_factor ("(l::nat) * m < n" | "(l::nat) < m * n") = \<open>K Nat_Numeral_Simprocs.less_cancel_factor\<close> simproc_setup nat_le_cancel_factor ("(l::nat) * m <= n" | "(l::nat) <= m * n") = \<open>K Nat_Numeral_Simprocs.le_cancel_factor\<close> simproc_setup nat_div_cancel_factor ("((l::nat) * m) div n" | "(l::nat) div (m * n)") = \<open>K Nat_Numeral_Simprocs.div_cancel_factor\<close> simproc_setup nat_dvd_cancel_factor ("((l::nat) * m) dvd n" | "(l::nat) dvd (m * n)") = \<open>K 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> end ¤ Dauer der Verarbeitung: 0.10 Sekunden (vorverarbeitet) ¤*© Formatika GbR, Deutschland |
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2026-03-28