author | huffman |
Thu, 04 Sep 2008 17:19:57 +0200 | |
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parent 26462 | dac4e2bce00d |
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permissions | -rw-r--r-- |
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(* Title: HOL/Arith_Tools.thy |
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ID: $Id$ |
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Author: Lawrence C Paulson, Cambridge University Computer Laboratory |
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Author: Amine Chaieb, TU Muenchen |
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*) |
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header {* Setup of arithmetic tools *} |
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theory Arith_Tools |
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imports Groebner_Basis |
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uses |
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"~~/src/Provers/Arith/cancel_numeral_factor.ML" |
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"~~/src/Provers/Arith/extract_common_term.ML" |
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"int_factor_simprocs.ML" |
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"nat_simprocs.ML" |
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begin |
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subsection {* Simprocs for the Naturals *} |
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declaration {* K nat_simprocs_setup *} |
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subsubsection{*For simplifying @{term "Suc m - K"} and @{term "K - Suc m"}*} |
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text{*Where K above is a literal*} |
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lemma Suc_diff_eq_diff_pred: "Numeral0 < n ==> Suc m - n = m - (n - Numeral1)" |
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by (simp add: numeral_0_eq_0 numeral_1_eq_1 split add: nat_diff_split) |
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text {*Now just instantiating @{text n} to @{text "number_of v"} does |
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the right simplification, but with some redundant inequality |
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tests.*} |
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lemma neg_number_of_pred_iff_0: |
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"neg (number_of (Int.pred v)::int) = (number_of v = (0::nat))" |
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apply (subgoal_tac "neg (number_of (Int.pred v)) = (number_of v < Suc 0) ") |
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apply (simp only: less_Suc_eq_le le_0_eq) |
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apply (subst less_number_of_Suc, simp) |
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done |
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text{*No longer required as a simprule because of the @{text inverse_fold} |
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simproc*} |
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lemma Suc_diff_number_of: |
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"neg (number_of (uminus v)::int) ==> |
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Suc m - (number_of v) = m - (number_of (Int.pred v))" |
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apply (subst Suc_diff_eq_diff_pred) |
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apply simp |
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apply (simp del: nat_numeral_1_eq_1) |
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apply (auto simp only: diff_nat_number_of less_0_number_of [symmetric] |
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neg_number_of_pred_iff_0) |
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done |
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lemma diff_Suc_eq_diff_pred: "m - Suc n = (m - 1) - n" |
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by (simp add: numerals split add: nat_diff_split) |
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subsubsection{*For @{term nat_case} and @{term nat_rec}*} |
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lemma nat_case_number_of [simp]: |
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"nat_case a f (number_of v) = |
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(let pv = number_of (Int.pred v) in |
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if neg pv then a else f (nat pv))" |
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by (simp split add: nat.split add: Let_def neg_number_of_pred_iff_0) |
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lemma nat_case_add_eq_if [simp]: |
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"nat_case a f ((number_of v) + n) = |
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(let pv = number_of (Int.pred v) in |
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if neg pv then nat_case a f n else f (nat pv + n))" |
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apply (subst add_eq_if) |
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apply (simp split add: nat.split |
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del: nat_numeral_1_eq_1 |
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add: numeral_1_eq_Suc_0 [symmetric] Let_def |
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neg_imp_number_of_eq_0 neg_number_of_pred_iff_0) |
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done |
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lemma nat_rec_number_of [simp]: |
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"nat_rec a f (number_of v) = |
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(let pv = number_of (Int.pred v) in |
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if neg pv then a else f (nat pv) (nat_rec a f (nat pv)))" |
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apply (case_tac " (number_of v) ::nat") |
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apply (simp_all (no_asm_simp) add: Let_def neg_number_of_pred_iff_0) |
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apply (simp split add: split_if_asm) |
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done |
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lemma nat_rec_add_eq_if [simp]: |
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"nat_rec a f (number_of v + n) = |
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(let pv = number_of (Int.pred v) in |
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if neg pv then nat_rec a f n |
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else f (nat pv + n) (nat_rec a f (nat pv + n)))" |
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apply (subst add_eq_if) |
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apply (simp split add: nat.split |
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del: nat_numeral_1_eq_1 |
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add: numeral_1_eq_Suc_0 [symmetric] Let_def neg_imp_number_of_eq_0 |
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neg_number_of_pred_iff_0) |
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done |
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subsubsection{*Various Other Lemmas*} |
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text {*Evens and Odds, for Mutilated Chess Board*} |
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text{*Lemmas for specialist use, NOT as default simprules*} |
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lemma nat_mult_2: "2 * z = (z+z::nat)" |
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proof - |
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have "2*z = (1 + 1)*z" by simp |
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also have "... = z+z" by (simp add: left_distrib) |
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finally show ?thesis . |
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qed |
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lemma nat_mult_2_right: "z * 2 = (z+z::nat)" |
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by (subst mult_commute, rule nat_mult_2) |
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text{*Case analysis on @{term "n<2"}*} |
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lemma less_2_cases: "(n::nat) < 2 ==> n = 0 | n = Suc 0" |
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by arith |
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lemma div2_Suc_Suc [simp]: "Suc(Suc m) div 2 = Suc (m div 2)" |
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by arith |
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lemma add_self_div_2 [simp]: "(m + m) div 2 = (m::nat)" |
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by (simp add: nat_mult_2 [symmetric]) |
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lemma mod2_Suc_Suc [simp]: "Suc(Suc(m)) mod 2 = m mod 2" |
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apply (subgoal_tac "m mod 2 < 2") |
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apply (erule less_2_cases [THEN disjE]) |
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apply (simp_all (no_asm_simp) add: Let_def mod_Suc nat_1) |
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done |
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lemma mod2_gr_0 [simp]: "!!m::nat. (0 < m mod 2) = (m mod 2 = 1)" |
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apply (subgoal_tac "m mod 2 < 2") |
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apply (force simp del: mod_less_divisor, simp) |
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done |
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text{*Removal of Small Numerals: 0, 1 and (in additive positions) 2*} |
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lemma add_2_eq_Suc [simp]: "2 + n = Suc (Suc n)" |
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by simp |
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lemma add_2_eq_Suc' [simp]: "n + 2 = Suc (Suc n)" |
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by simp |
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text{*Can be used to eliminate long strings of Sucs, but not by default*} |
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lemma Suc3_eq_add_3: "Suc (Suc (Suc n)) = 3 + n" |
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by simp |
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text{*These lemmas collapse some needless occurrences of Suc: |
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at least three Sucs, since two and fewer are rewritten back to Suc again! |
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We already have some rules to simplify operands smaller than 3.*} |
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lemma div_Suc_eq_div_add3 [simp]: "m div (Suc (Suc (Suc n))) = m div (3+n)" |
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by (simp add: Suc3_eq_add_3) |
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lemma mod_Suc_eq_mod_add3 [simp]: "m mod (Suc (Suc (Suc n))) = m mod (3+n)" |
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by (simp add: Suc3_eq_add_3) |
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lemma Suc_div_eq_add3_div: "(Suc (Suc (Suc m))) div n = (3+m) div n" |
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by (simp add: Suc3_eq_add_3) |
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lemma Suc_mod_eq_add3_mod: "(Suc (Suc (Suc m))) mod n = (3+m) mod n" |
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by (simp add: Suc3_eq_add_3) |
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lemmas Suc_div_eq_add3_div_number_of = |
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Suc_div_eq_add3_div [of _ "number_of v", standard] |
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declare Suc_div_eq_add3_div_number_of [simp] |
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lemmas Suc_mod_eq_add3_mod_number_of = |
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Suc_mod_eq_add3_mod [of _ "number_of v", standard] |
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declare Suc_mod_eq_add3_mod_number_of [simp] |
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subsubsection{*Special Simplification for Constants*} |
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text{*These belong here, late in the development of HOL, to prevent their |
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interfering with proofs of abstract properties of instances of the function |
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@{term number_of}*} |
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text{*These distributive laws move literals inside sums and differences.*} |
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lemmas left_distrib_number_of = left_distrib [of _ _ "number_of v", standard] |
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declare left_distrib_number_of [simp] |
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lemmas right_distrib_number_of = right_distrib [of "number_of v", standard] |
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declare right_distrib_number_of [simp] |
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lemmas left_diff_distrib_number_of = |
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left_diff_distrib [of _ _ "number_of v", standard] |
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declare left_diff_distrib_number_of [simp] |
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lemmas right_diff_distrib_number_of = |
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right_diff_distrib [of "number_of v", standard] |
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declare right_diff_distrib_number_of [simp] |
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text{*These are actually for fields, like real: but where else to put them?*} |
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lemmas zero_less_divide_iff_number_of = |
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zero_less_divide_iff [of "number_of w", standard] |
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declare zero_less_divide_iff_number_of [simp,noatp] |
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lemmas divide_less_0_iff_number_of = |
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divide_less_0_iff [of "number_of w", standard] |
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declare divide_less_0_iff_number_of [simp,noatp] |
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lemmas zero_le_divide_iff_number_of = |
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zero_le_divide_iff [of "number_of w", standard] |
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declare zero_le_divide_iff_number_of [simp,noatp] |
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lemmas divide_le_0_iff_number_of = |
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divide_le_0_iff [of "number_of w", standard] |
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declare divide_le_0_iff_number_of [simp,noatp] |
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(**** |
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IF times_divide_eq_right and times_divide_eq_left are removed as simprules, |
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then these special-case declarations may be useful. |
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text{*These simprules move numerals into numerators and denominators.*} |
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lemma times_recip_eq_right [simp]: "a * (1/c) = a / (c::'a::field)" |
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by (simp add: times_divide_eq) |
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lemma times_recip_eq_left [simp]: "(1/c) * a = a / (c::'a::field)" |
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by (simp add: times_divide_eq) |
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lemmas times_divide_eq_right_number_of = |
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times_divide_eq_right [of "number_of w", standard] |
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declare times_divide_eq_right_number_of [simp] |
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lemmas times_divide_eq_right_number_of = |
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times_divide_eq_right [of _ _ "number_of w", standard] |
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declare times_divide_eq_right_number_of [simp] |
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lemmas times_divide_eq_left_number_of = |
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times_divide_eq_left [of _ "number_of w", standard] |
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declare times_divide_eq_left_number_of [simp] |
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lemmas times_divide_eq_left_number_of = |
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times_divide_eq_left [of _ _ "number_of w", standard] |
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declare times_divide_eq_left_number_of [simp] |
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****) |
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text {*Replaces @{text "inverse #nn"} by @{text "1/#nn"}. It looks |
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strange, but then other simprocs simplify the quotient.*} |
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lemmas inverse_eq_divide_number_of = |
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inverse_eq_divide [of "number_of w", standard] |
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declare inverse_eq_divide_number_of [simp] |
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text {*These laws simplify inequalities, moving unary minus from a term |
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into the literal.*} |
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lemmas less_minus_iff_number_of = |
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less_minus_iff [of "number_of v", standard] |
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declare less_minus_iff_number_of [simp,noatp] |
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lemmas le_minus_iff_number_of = |
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le_minus_iff [of "number_of v", standard] |
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declare le_minus_iff_number_of [simp,noatp] |
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lemmas equation_minus_iff_number_of = |
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equation_minus_iff [of "number_of v", standard] |
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declare equation_minus_iff_number_of [simp,noatp] |
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lemmas minus_less_iff_number_of = |
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minus_less_iff [of _ "number_of v", standard] |
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declare minus_less_iff_number_of [simp,noatp] |
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lemmas minus_le_iff_number_of = |
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minus_le_iff [of _ "number_of v", standard] |
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declare minus_le_iff_number_of [simp,noatp] |
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lemmas minus_equation_iff_number_of = |
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minus_equation_iff [of _ "number_of v", standard] |
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declare minus_equation_iff_number_of [simp,noatp] |
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text{*To Simplify Inequalities Where One Side is the Constant 1*} |
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lemma less_minus_iff_1 [simp,noatp]: |
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fixes b::"'b::{ordered_idom,number_ring}" |
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shows "(1 < - b) = (b < -1)" |
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by auto |
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lemma le_minus_iff_1 [simp,noatp]: |
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fixes b::"'b::{ordered_idom,number_ring}" |
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shows "(1 \<le> - b) = (b \<le> -1)" |
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by auto |
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lemma equation_minus_iff_1 [simp,noatp]: |
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fixes b::"'b::number_ring" |
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shows "(1 = - b) = (b = -1)" |
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by (subst equation_minus_iff, auto) |
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lemma minus_less_iff_1 [simp,noatp]: |
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fixes a::"'b::{ordered_idom,number_ring}" |
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shows "(- a < 1) = (-1 < a)" |
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by auto |
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lemma minus_le_iff_1 [simp,noatp]: |
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fixes a::"'b::{ordered_idom,number_ring}" |
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shows "(- a \<le> 1) = (-1 \<le> a)" |
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by auto |
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lemma minus_equation_iff_1 [simp,noatp]: |
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fixes a::"'b::number_ring" |
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shows "(- a = 1) = (a = -1)" |
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by (subst minus_equation_iff, auto) |
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text {*Cancellation of constant factors in comparisons (@{text "<"} and @{text "\<le>"}) *} |
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lemmas mult_less_cancel_left_number_of = |
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mult_less_cancel_left [of "number_of v", standard] |
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declare mult_less_cancel_left_number_of [simp,noatp] |
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lemmas mult_less_cancel_right_number_of = |
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mult_less_cancel_right [of _ "number_of v", standard] |
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declare mult_less_cancel_right_number_of [simp,noatp] |
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lemmas mult_le_cancel_left_number_of = |
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mult_le_cancel_left [of "number_of v", standard] |
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declare mult_le_cancel_left_number_of [simp,noatp] |
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lemmas mult_le_cancel_right_number_of = |
|
324 |
mult_le_cancel_right [of _ "number_of v", standard] |
|
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|
325 |
declare mult_le_cancel_right_number_of [simp,noatp] |
23462 | 326 |
|
327 |
||
328 |
text {*Multiplying out constant divisors in comparisons (@{text "<"}, @{text "\<le>"} and @{text "="}) *} |
|
329 |
||
26314 | 330 |
lemmas le_divide_eq_number_of1 [simp] = le_divide_eq [of _ _ "number_of w", standard] |
331 |
lemmas divide_le_eq_number_of1 [simp] = divide_le_eq [of _ "number_of w", standard] |
|
332 |
lemmas less_divide_eq_number_of1 [simp] = less_divide_eq [of _ _ "number_of w", standard] |
|
333 |
lemmas divide_less_eq_number_of1 [simp] = divide_less_eq [of _ "number_of w", standard] |
|
334 |
lemmas eq_divide_eq_number_of1 [simp] = eq_divide_eq [of _ _ "number_of w", standard] |
|
335 |
lemmas divide_eq_eq_number_of1 [simp] = divide_eq_eq [of _ "number_of w", standard] |
|
23462 | 336 |
|
337 |
||
338 |
subsubsection{*Optional Simplification Rules Involving Constants*} |
|
339 |
||
340 |
text{*Simplify quotients that are compared with a literal constant.*} |
|
341 |
||
342 |
lemmas le_divide_eq_number_of = le_divide_eq [of "number_of w", standard] |
|
343 |
lemmas divide_le_eq_number_of = divide_le_eq [of _ _ "number_of w", standard] |
|
344 |
lemmas less_divide_eq_number_of = less_divide_eq [of "number_of w", standard] |
|
345 |
lemmas divide_less_eq_number_of = divide_less_eq [of _ _ "number_of w", standard] |
|
346 |
lemmas eq_divide_eq_number_of = eq_divide_eq [of "number_of w", standard] |
|
347 |
lemmas divide_eq_eq_number_of = divide_eq_eq [of _ _ "number_of w", standard] |
|
348 |
||
349 |
||
350 |
text{*Not good as automatic simprules because they cause case splits.*} |
|
351 |
lemmas divide_const_simps = |
|
352 |
le_divide_eq_number_of divide_le_eq_number_of less_divide_eq_number_of |
|
353 |
divide_less_eq_number_of eq_divide_eq_number_of divide_eq_eq_number_of |
|
354 |
le_divide_eq_1 divide_le_eq_1 less_divide_eq_1 divide_less_eq_1 |
|
355 |
||
356 |
text{*Division By @{text "-1"}*} |
|
357 |
||
358 |
lemma divide_minus1 [simp]: |
|
359 |
"x/-1 = -(x::'a::{field,division_by_zero,number_ring})" |
|
360 |
by simp |
|
361 |
||
362 |
lemma minus1_divide [simp]: |
|
363 |
"-1 / (x::'a::{field,division_by_zero,number_ring}) = - (1/x)" |
|
364 |
by (simp add: divide_inverse inverse_minus_eq) |
|
365 |
||
366 |
lemma half_gt_zero_iff: |
|
367 |
"(0 < r/2) = (0 < (r::'a::{ordered_field,division_by_zero,number_ring}))" |
|
368 |
by auto |
|
369 |
||
370 |
lemmas half_gt_zero = half_gt_zero_iff [THEN iffD2, standard] |
|
371 |
declare half_gt_zero [simp] |
|
372 |
||
373 |
(* The following lemma should appear in Divides.thy, but there the proof |
|
374 |
doesn't work. *) |
|
375 |
||
376 |
lemma nat_dvd_not_less: |
|
377 |
"[| 0 < m; m < n |] ==> \<not> n dvd (m::nat)" |
|
378 |
by (unfold dvd_def) auto |
|
379 |
||
380 |
ML {* |
|
381 |
val divide_minus1 = @{thm divide_minus1}; |
|
382 |
val minus1_divide = @{thm minus1_divide}; |
|
383 |
*} |
|
384 |
||
385 |
||
386 |
subsection{* Groebner Bases for fields *} |
|
387 |
||
388 |
interpretation class_fieldgb: |
|
389 |
fieldgb["op +" "op *" "op ^" "0::'a::{field,recpower,number_ring}" "1" "op -" "uminus" "op /" "inverse"] apply (unfold_locales) by (simp_all add: divide_inverse) |
|
390 |
||
391 |
lemma divide_Numeral1: "(x::'a::{field,number_ring}) / Numeral1 = x" by simp |
|
392 |
lemma divide_Numeral0: "(x::'a::{field,number_ring, division_by_zero}) / Numeral0 = 0" |
|
393 |
by simp |
|
394 |
lemma mult_frac_frac: "((x::'a::{field,division_by_zero}) / y) * (z / w) = (x*z) / (y*w)" |
|
395 |
by simp |
|
396 |
lemma mult_frac_num: "((x::'a::{field, division_by_zero}) / y) * z = (x*z) / y" |
|
397 |
by simp |
|
398 |
lemma mult_num_frac: "((x::'a::{field, division_by_zero}) / y) * z = (x*z) / y" |
|
399 |
by simp |
|
400 |
||
401 |
lemma Numeral1_eq1_nat: "(1::nat) = Numeral1" by simp |
|
402 |
||
403 |
lemma add_frac_num: "y\<noteq> 0 \<Longrightarrow> (x::'a::{field, division_by_zero}) / y + z = (x + z*y) / y" |
|
404 |
by (simp add: add_divide_distrib) |
|
405 |
lemma add_num_frac: "y\<noteq> 0 \<Longrightarrow> z + (x::'a::{field, division_by_zero}) / y = (x + z*y) / y" |
|
406 |
by (simp add: add_divide_distrib) |
|
407 |
||
25249
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changeset
|
408 |
|
76b9892020d5
exported field_comp_conv: a numerical conversion over fields
chaieb
parents:
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diff
changeset
|
409 |
ML{* |
76b9892020d5
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diff
changeset
|
410 |
local |
23462 | 411 |
val zr = @{cpat "0"} |
412 |
val zT = ctyp_of_term zr |
|
413 |
val geq = @{cpat "op ="} |
|
414 |
val eqT = Thm.dest_ctyp (ctyp_of_term geq) |> hd |
|
415 |
val add_frac_eq = mk_meta_eq @{thm "add_frac_eq"} |
|
416 |
val add_frac_num = mk_meta_eq @{thm "add_frac_num"} |
|
417 |
val add_num_frac = mk_meta_eq @{thm "add_num_frac"} |
|
418 |
||
25249
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exported field_comp_conv: a numerical conversion over fields
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|
419 |
fun prove_nz ss T t = |
23462 | 420 |
let |
421 |
val z = instantiate_cterm ([(zT,T)],[]) zr |
|
422 |
val eq = instantiate_cterm ([(eqT,T)],[]) geq |
|
423 |
val th = Simplifier.rewrite (ss addsimps simp_thms) |
|
424 |
(Thm.capply @{cterm "Trueprop"} (Thm.capply @{cterm "Not"} |
|
425 |
(Thm.capply (Thm.capply eq t) z))) |
|
426 |
in equal_elim (symmetric th) TrueI |
|
427 |
end |
|
428 |
||
25249
76b9892020d5
exported field_comp_conv: a numerical conversion over fields
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parents:
25128
diff
changeset
|
429 |
fun proc phi ss ct = |
23462 | 430 |
let |
431 |
val ((x,y),(w,z)) = |
|
432 |
(Thm.dest_binop #> (fn (a,b) => (Thm.dest_binop a, Thm.dest_binop b))) ct |
|
433 |
val _ = map (HOLogic.dest_number o term_of) [x,y,z,w] |
|
434 |
val T = ctyp_of_term x |
|
25249
76b9892020d5
exported field_comp_conv: a numerical conversion over fields
chaieb
parents:
25128
diff
changeset
|
435 |
val [y_nz, z_nz] = map (prove_nz ss T) [y, z] |
23462 | 436 |
val th = instantiate' [SOME T] (map SOME [y,z,x,w]) add_frac_eq |
437 |
in SOME (implies_elim (implies_elim th y_nz) z_nz) |
|
438 |
end |
|
439 |
handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE |
|
440 |
||
25249
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exported field_comp_conv: a numerical conversion over fields
chaieb
parents:
25128
diff
changeset
|
441 |
fun proc2 phi ss ct = |
23462 | 442 |
let |
443 |
val (l,r) = Thm.dest_binop ct |
|
444 |
val T = ctyp_of_term l |
|
445 |
in (case (term_of l, term_of r) of |
|
446 |
(Const(@{const_name "HOL.divide"},_)$_$_, _) => |
|
447 |
let val (x,y) = Thm.dest_binop l val z = r |
|
448 |
val _ = map (HOLogic.dest_number o term_of) [x,y,z] |
|
25249
76b9892020d5
exported field_comp_conv: a numerical conversion over fields
chaieb
parents:
25128
diff
changeset
|
449 |
val ynz = prove_nz ss T y |
23462 | 450 |
in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,x,z]) add_frac_num) ynz) |
451 |
end |
|
452 |
| (_, Const (@{const_name "HOL.divide"},_)$_$_) => |
|
453 |
let val (x,y) = Thm.dest_binop r val z = l |
|
454 |
val _ = map (HOLogic.dest_number o term_of) [x,y,z] |
|
25249
76b9892020d5
exported field_comp_conv: a numerical conversion over fields
chaieb
parents:
25128
diff
changeset
|
455 |
val ynz = prove_nz ss T y |
23462 | 456 |
in SOME (implies_elim (instantiate' [SOME T] (map SOME [y,z,x]) add_num_frac) ynz) |
457 |
end |
|
458 |
| _ => NONE) |
|
459 |
end |
|
460 |
handle CTERM _ => NONE | TERM _ => NONE | THM _ => NONE |
|
461 |
||
462 |
fun is_number (Const(@{const_name "HOL.divide"},_)$a$b) = is_number a andalso is_number b |
|
463 |
| is_number t = can HOLogic.dest_number t |
|
464 |
||
465 |
val is_number = is_number o term_of |
|
466 |
||
467 |
fun proc3 phi ss ct = |
|
468 |
(case term_of ct of |
|
23881 | 469 |
Const(@{const_name HOL.less},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ => |
23462 | 470 |
let |
471 |
val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop |
|
472 |
val _ = map is_number [a,b,c] |
|
473 |
val T = ctyp_of_term c |
|
474 |
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_less_eq"} |
|
475 |
in SOME (mk_meta_eq th) end |
|
23881 | 476 |
| Const(@{const_name HOL.less_eq},_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ => |
23462 | 477 |
let |
478 |
val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop |
|
479 |
val _ = map is_number [a,b,c] |
|
480 |
val T = ctyp_of_term c |
|
481 |
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_le_eq"} |
|
482 |
in SOME (mk_meta_eq th) end |
|
483 |
| Const("op =",_)$(Const(@{const_name "HOL.divide"},_)$_$_)$_ => |
|
484 |
let |
|
485 |
val ((a,b),c) = Thm.dest_binop ct |>> Thm.dest_binop |
|
486 |
val _ = map is_number [a,b,c] |
|
487 |
val T = ctyp_of_term c |
|
488 |
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "divide_eq_eq"} |
|
489 |
in SOME (mk_meta_eq th) end |
|
23881 | 490 |
| Const(@{const_name HOL.less},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) => |
23462 | 491 |
let |
492 |
val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop |
|
493 |
val _ = map is_number [a,b,c] |
|
494 |
val T = ctyp_of_term c |
|
495 |
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "less_divide_eq"} |
|
496 |
in SOME (mk_meta_eq th) end |
|
23881 | 497 |
| Const(@{const_name HOL.less_eq},_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) => |
23462 | 498 |
let |
499 |
val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop |
|
500 |
val _ = map is_number [a,b,c] |
|
501 |
val T = ctyp_of_term c |
|
502 |
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "le_divide_eq"} |
|
503 |
in SOME (mk_meta_eq th) end |
|
504 |
| Const("op =",_)$_$(Const(@{const_name "HOL.divide"},_)$_$_) => |
|
505 |
let |
|
506 |
val (a,(b,c)) = Thm.dest_binop ct ||> Thm.dest_binop |
|
507 |
val _ = map is_number [a,b,c] |
|
508 |
val T = ctyp_of_term c |
|
509 |
val th = instantiate' [SOME T] (map SOME [a,b,c]) @{thm "eq_divide_eq"} |
|
510 |
in SOME (mk_meta_eq th) end |
|
511 |
| _ => NONE) |
|
512 |
handle TERM _ => NONE | CTERM _ => NONE | THM _ => NONE |
|
513 |
||
25249
76b9892020d5
exported field_comp_conv: a numerical conversion over fields
chaieb
parents:
25128
diff
changeset
|
514 |
val add_frac_frac_simproc = |
23462 | 515 |
make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + (?w::?'a::field)/?z"}], |
516 |
name = "add_frac_frac_simproc", |
|
25249
76b9892020d5
exported field_comp_conv: a numerical conversion over fields
chaieb
parents:
25128
diff
changeset
|
517 |
proc = proc, identifier = []} |
23462 | 518 |
|
25249
76b9892020d5
exported field_comp_conv: a numerical conversion over fields
chaieb
parents:
25128
diff
changeset
|
519 |
val add_frac_num_simproc = |
23462 | 520 |
make_simproc {lhss = [@{cpat "(?x::?'a::field)/?y + ?z"}, @{cpat "?z + (?x::?'a::field)/?y"}], |
521 |
name = "add_frac_num_simproc", |
|
25249
76b9892020d5
exported field_comp_conv: a numerical conversion over fields
chaieb
parents:
25128
diff
changeset
|
522 |
proc = proc2, identifier = []} |
23462 | 523 |
|
524 |
val ord_frac_simproc = |
|
525 |
make_simproc |
|
526 |
{lhss = [@{cpat "(?a::(?'a::{field, ord}))/?b < ?c"}, |
|
527 |
@{cpat "(?a::(?'a::{field, ord}))/?b \<le> ?c"}, |
|
528 |
@{cpat "?c < (?a::(?'a::{field, ord}))/?b"}, |
|
529 |
@{cpat "?c \<le> (?a::(?'a::{field, ord}))/?b"}, |
|
530 |
@{cpat "?c = ((?a::(?'a::{field, ord}))/?b)"}, |
|
531 |
@{cpat "((?a::(?'a::{field, ord}))/ ?b) = ?c"}], |
|
532 |
name = "ord_frac_simproc", proc = proc3, identifier = []} |
|
533 |
||
534 |
val nat_arith = map thm ["add_nat_number_of", "diff_nat_number_of", |
|
535 |
"mult_nat_number_of", "eq_nat_number_of", "less_nat_number_of"] |
|
536 |
||
537 |
val comp_arith = (map thm ["Let_def", "if_False", "if_True", "add_0", |
|
538 |
"add_Suc", "add_number_of_left", "mult_number_of_left", |
|
539 |
"Suc_eq_add_numeral_1"])@ |
|
540 |
(map (fn s => thm s RS sym) ["numeral_1_eq_1", "numeral_0_eq_0"]) |
|
25481 | 541 |
@ @{thms arith_simps} @ nat_arith @ @{thms rel_simps} |
23462 | 542 |
val ths = [@{thm "mult_numeral_1"}, @{thm "mult_numeral_1_right"}, |
543 |
@{thm "divide_Numeral1"}, |
|
544 |
@{thm "Ring_and_Field.divide_zero"}, @{thm "divide_Numeral0"}, |
|
545 |
@{thm "divide_divide_eq_left"}, @{thm "mult_frac_frac"}, |
|
546 |
@{thm "mult_num_frac"}, @{thm "mult_frac_num"}, |
|
547 |
@{thm "mult_frac_frac"}, @{thm "times_divide_eq_right"}, |
|
548 |
@{thm "times_divide_eq_left"}, @{thm "divide_divide_eq_right"}, |
|
549 |
@{thm "diff_def"}, @{thm "minus_divide_left"}, |
|
550 |
@{thm "Numeral1_eq1_nat"}, @{thm "add_divide_distrib"} RS sym] |
|
551 |
||
552 |
local |
|
553 |
open Conv |
|
554 |
in |
|
25249
76b9892020d5
exported field_comp_conv: a numerical conversion over fields
chaieb
parents:
25128
diff
changeset
|
555 |
val comp_conv = (Simplifier.rewrite |
23462 | 556 |
(HOL_basic_ss addsimps @{thms "Groebner_Basis.comp_arith"} |
557 |
addsimps ths addsimps comp_arith addsimps simp_thms |
|
558 |
addsimprocs field_cancel_numeral_factors |
|
25249
76b9892020d5
exported field_comp_conv: a numerical conversion over fields
chaieb
parents:
25128
diff
changeset
|
559 |
addsimprocs [add_frac_frac_simproc, add_frac_num_simproc, |
23462 | 560 |
ord_frac_simproc] |
561 |
addcongs [@{thm "if_weak_cong"}])) |
|
562 |
then_conv (Simplifier.rewrite (HOL_basic_ss addsimps |
|
563 |
[@{thm numeral_1_eq_1},@{thm numeral_0_eq_0}] @ @{thms numerals(1-2)})) |
|
564 |
end |
|
565 |
||
566 |
fun numeral_is_const ct = |
|
567 |
case term_of ct of |
|
568 |
Const (@{const_name "HOL.divide"},_) $ a $ b => |
|
569 |
numeral_is_const (Thm.dest_arg1 ct) andalso numeral_is_const (Thm.dest_arg ct) |
|
570 |
| Const (@{const_name "HOL.uminus"},_)$t => numeral_is_const (Thm.dest_arg ct) |
|
571 |
| t => can HOLogic.dest_number t |
|
572 |
||
25128
962e4f4142fa
Fixed Bug in instantiation of Groebner Bases to field: dest_const used to raise TERM where the tactic handles ERROR
chaieb
parents:
24286
diff
changeset
|
573 |
fun dest_const ct = ((case term_of ct of |
23462 | 574 |
Const (@{const_name "HOL.divide"},_) $ a $ b=> |
575 |
Rat.rat_of_quotient (snd (HOLogic.dest_number a), snd (HOLogic.dest_number b)) |
|
25128
962e4f4142fa
Fixed Bug in instantiation of Groebner Bases to field: dest_const used to raise TERM where the tactic handles ERROR
chaieb
parents:
24286
diff
changeset
|
576 |
| t => Rat.rat_of_int (snd (HOLogic.dest_number t))) |
962e4f4142fa
Fixed Bug in instantiation of Groebner Bases to field: dest_const used to raise TERM where the tactic handles ERROR
chaieb
parents:
24286
diff
changeset
|
577 |
handle TERM _ => error "ring_dest_const") |
23462 | 578 |
|
579 |
fun mk_const phi cT x = |
|
580 |
let val (a, b) = Rat.quotient_of_rat x |
|
23572 | 581 |
in if b = 1 then Numeral.mk_cnumber cT a |
23462 | 582 |
else Thm.capply |
583 |
(Thm.capply (Drule.cterm_rule (instantiate' [SOME cT] []) @{cpat "op /"}) |
|
23572 | 584 |
(Numeral.mk_cnumber cT a)) |
585 |
(Numeral.mk_cnumber cT b) |
|
23462 | 586 |
end |
587 |
||
588 |
in |
|
25249
76b9892020d5
exported field_comp_conv: a numerical conversion over fields
chaieb
parents:
25128
diff
changeset
|
589 |
val field_comp_conv = comp_conv; |
76b9892020d5
exported field_comp_conv: a numerical conversion over fields
chaieb
parents:
25128
diff
changeset
|
590 |
val fieldgb_declaration = |
26462 | 591 |
NormalizerData.funs @{thm class_fieldgb.fieldgb_axioms'} |
23462 | 592 |
{is_const = K numeral_is_const, |
593 |
dest_const = K dest_const, |
|
594 |
mk_const = mk_const, |
|
25249
76b9892020d5
exported field_comp_conv: a numerical conversion over fields
chaieb
parents:
25128
diff
changeset
|
595 |
conv = K (K comp_conv)} |
76b9892020d5
exported field_comp_conv: a numerical conversion over fields
chaieb
parents:
25128
diff
changeset
|
596 |
end; |
76b9892020d5
exported field_comp_conv: a numerical conversion over fields
chaieb
parents:
25128
diff
changeset
|
597 |
*} |
23462 | 598 |
|
25249
76b9892020d5
exported field_comp_conv: a numerical conversion over fields
chaieb
parents:
25128
diff
changeset
|
599 |
declaration{* fieldgb_declaration *} |
23462 | 600 |
end |