author | haftmann |
Fri, 17 Apr 2009 08:35:23 +0200 | |
changeset 30943 | eb3dbbe971f6 |
parent 30445 | 757ba2bb2b39 |
child 31018 | b8a8cf6e16f2 |
permissions | -rw-r--r-- |
24333 | 1 |
(* |
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Author: Jeremy Dawson, NICTA |
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*) |
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header {* Useful Numerical Lemmas *} |
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theory Num_Lemmas |
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imports Main Parity |
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begin |
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lemma contentsI: "y = {x} ==> contents y = x" |
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unfolding contents_def by auto -- {* FIXME move *} |
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lemmas split_split = prod.split [unfolded prod_case_split] |
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lemmas split_split_asm = prod.split_asm [unfolded prod_case_split] |
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lemmas split_splits = split_split split_split_asm |
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lemmas funpow_0 = funpow.simps(1) |
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lemmas funpow_Suc = funpow.simps(2) |
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lemma nonemptyE: "S ~= {} ==> (!!x. x : S ==> R) ==> R" by auto |
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lemma gt_or_eq_0: "0 < y \<or> 0 = (y::nat)" by arith |
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declare iszero_0 [iff] |
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||
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lemmas xtr1 = xtrans(1) |
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lemmas xtr2 = xtrans(2) |
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lemmas xtr3 = xtrans(3) |
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lemmas xtr4 = xtrans(4) |
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lemmas xtr5 = xtrans(5) |
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lemmas xtr6 = xtrans(6) |
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lemmas xtr7 = xtrans(7) |
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lemmas xtr8 = xtrans(8) |
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||
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lemmas nat_simps = diff_add_inverse2 diff_add_inverse |
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lemmas nat_iffs = le_add1 le_add2 |
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||
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lemma sum_imp_diff: "j = k + i ==> j - i = (k :: nat)" by arith |
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lemma nobm1: |
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"0 < (number_of w :: nat) ==> |
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number_of w - (1 :: nat) = number_of (Int.pred w)" |
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apply (unfold nat_number_of_def One_nat_def nat_1 [symmetric] pred_def) |
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apply (simp add: number_of_eq nat_diff_distrib [symmetric]) |
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done |
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lemma of_int_power: |
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"of_int (a ^ n) = (of_int a ^ n :: 'a :: {recpower, comm_ring_1})" |
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by (induct n) (auto simp add: power_Suc) |
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lemma zless2: "0 < (2 :: int)" by arith |
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lemmas zless2p [simp] = zless2 [THEN zero_less_power] |
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lemmas zle2p [simp] = zless2p [THEN order_less_imp_le] |
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||
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lemmas pos_mod_sign2 = zless2 [THEN pos_mod_sign [where b = "2::int"]] |
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lemmas pos_mod_bound2 = zless2 [THEN pos_mod_bound [where b = "2::int"]] |
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-- "the inverse(s) of @{text number_of}" |
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lemma nmod2: "n mod (2::int) = 0 | n mod 2 = 1" by arith |
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lemma emep1: |
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"even n ==> even d ==> 0 <= d ==> (n + 1) mod (d :: int) = (n mod d) + 1" |
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apply (simp add: add_commute) |
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apply (safe dest!: even_equiv_def [THEN iffD1]) |
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apply (subst pos_zmod_mult_2) |
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apply arith |
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apply (simp add: mod_mult_mult1) |
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done |
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lemmas eme1p = emep1 [simplified add_commute] |
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lemma le_diff_eq': "(a \<le> c - b) = (b + a \<le> (c::int))" by arith |
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lemma less_diff_eq': "(a < c - b) = (b + a < (c::int))" by arith |
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lemma diff_le_eq': "(a - b \<le> c) = (a \<le> b + (c::int))" by arith |
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lemma diff_less_eq': "(a - b < c) = (a < b + (c::int))" by arith |
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lemmas m1mod2k = zless2p [THEN zmod_minus1] |
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lemmas m1mod22k = mult_pos_pos [OF zless2 zless2p, THEN zmod_minus1] |
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lemmas p1mod22k' = zless2p [THEN order_less_imp_le, THEN pos_zmod_mult_2] |
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lemmas z1pmod2' = zero_le_one [THEN pos_zmod_mult_2, simplified] |
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lemmas z1pdiv2' = zero_le_one [THEN pos_zdiv_mult_2, simplified] |
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lemma p1mod22k: |
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"(2 * b + 1) mod (2 * 2 ^ n) = 2 * (b mod 2 ^ n) + (1::int)" |
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by (simp add: p1mod22k' add_commute) |
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lemma z1pmod2: |
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"(2 * b + 1) mod 2 = (1::int)" by arith |
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lemma z1pdiv2: |
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"(2 * b + 1) div 2 = (b::int)" by arith |
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lemmas zdiv_le_dividend = xtr3 [OF div_by_1 [symmetric] zdiv_mono2, |
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simplified int_one_le_iff_zero_less, simplified, standard] |
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lemma axxbyy: |
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"a + m + m = b + n + n ==> (a = 0 | a = 1) ==> (b = 0 | b = 1) ==> |
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a = b & m = (n :: int)" by arith |
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lemma axxmod2: |
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"(1 + x + x) mod 2 = (1 :: int) & (0 + x + x) mod 2 = (0 :: int)" by arith |
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lemma axxdiv2: |
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"(1 + x + x) div 2 = (x :: int) & (0 + x + x) div 2 = (x :: int)" by arith |
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lemmas iszero_minus = trans [THEN trans, |
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OF iszero_def neg_equal_0_iff_equal iszero_def [symmetric], standard] |
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lemmas zadd_diff_inverse = trans [OF diff_add_cancel [symmetric] add_commute, |
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standard] |
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lemmas add_diff_cancel2 = add_commute [THEN diff_eq_eq [THEN iffD2], standard] |
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lemma zmod_uminus: "- ((a :: int) mod b) mod b = -a mod b" |
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by (simp add : zmod_zminus1_eq_if) |
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lemma zmod_zsub_distrib: "((a::int) - b) mod c = (a mod c - b mod c) mod c" |
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apply (unfold diff_int_def) |
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apply (rule trans [OF _ mod_add_eq [symmetric]]) |
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apply (simp add: zmod_uminus mod_add_eq [symmetric]) |
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done |
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lemma zmod_zsub_right_eq: "((a::int) - b) mod c = (a - b mod c) mod c" |
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apply (unfold diff_int_def) |
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apply (rule trans [OF _ mod_add_right_eq [symmetric]]) |
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apply (simp add : zmod_uminus mod_add_right_eq [symmetric]) |
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done |
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lemma zmod_zsub_left_eq: "((a::int) - b) mod c = (a mod c - b) mod c" |
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by (rule mod_add_left_eq [where b = "- b", simplified diff_int_def [symmetric]]) |
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lemma zmod_zsub_self [simp]: |
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"((b :: int) - a) mod a = b mod a" |
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by (simp add: zmod_zsub_right_eq) |
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lemma zmod_zmult1_eq_rev: |
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"b * a mod c = b mod c * a mod (c::int)" |
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apply (simp add: mult_commute) |
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apply (subst zmod_zmult1_eq) |
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apply simp |
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done |
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lemmas rdmods [symmetric] = zmod_uminus [symmetric] |
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zmod_zsub_left_eq zmod_zsub_right_eq mod_add_left_eq |
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mod_add_right_eq zmod_zmult1_eq zmod_zmult1_eq_rev |
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lemma mod_plus_right: |
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"((a + x) mod m = (b + x) mod m) = (a mod m = b mod (m :: nat))" |
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apply (induct x) |
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apply (simp_all add: mod_Suc) |
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apply arith |
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done |
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lemma nat_minus_mod: "(n - n mod m) mod m = (0 :: nat)" |
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by (induct n) (simp_all add : mod_Suc) |
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lemmas nat_minus_mod_plus_right = trans [OF nat_minus_mod mod_0 [symmetric], |
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THEN mod_plus_right [THEN iffD2], standard, simplified] |
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lemmas push_mods' = mod_add_eq [standard] |
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mod_mult_eq [standard] zmod_zsub_distrib [standard] |
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zmod_uminus [symmetric, standard] |
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lemmas push_mods = push_mods' [THEN eq_reflection, standard] |
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lemmas pull_mods = push_mods [symmetric] rdmods [THEN eq_reflection, standard] |
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lemmas mod_simps = |
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mod_mult_self2_is_0 [THEN eq_reflection] |
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mod_mult_self1_is_0 [THEN eq_reflection] |
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mod_mod_trivial [THEN eq_reflection] |
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lemma nat_mod_eq: |
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"!!b. b < n ==> a mod n = b mod n ==> a mod n = (b :: nat)" |
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by (induct a) auto |
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lemmas nat_mod_eq' = refl [THEN [2] nat_mod_eq] |
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lemma nat_mod_lem: |
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"(0 :: nat) < n ==> b < n = (b mod n = b)" |
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apply safe |
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apply (erule nat_mod_eq') |
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apply (erule subst) |
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apply (erule mod_less_divisor) |
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done |
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lemma mod_nat_add: |
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"(x :: nat) < z ==> y < z ==> |
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(x + y) mod z = (if x + y < z then x + y else x + y - z)" |
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apply (rule nat_mod_eq) |
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apply auto |
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apply (rule trans) |
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apply (rule le_mod_geq) |
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apply simp |
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apply (rule nat_mod_eq') |
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apply arith |
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done |
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lemma mod_nat_sub: |
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"(x :: nat) < z ==> (x - y) mod z = x - y" |
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by (rule nat_mod_eq') arith |
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lemma int_mod_lem: |
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"(0 :: int) < n ==> (0 <= b & b < n) = (b mod n = b)" |
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apply safe |
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apply (erule (1) mod_pos_pos_trivial) |
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apply (erule_tac [!] subst) |
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apply auto |
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done |
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lemma int_mod_eq: |
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"(0 :: int) <= b ==> b < n ==> a mod n = b mod n ==> a mod n = b" |
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by clarsimp (rule mod_pos_pos_trivial) |
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lemmas int_mod_eq' = refl [THEN [3] int_mod_eq] |
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lemma int_mod_le: "0 <= a ==> 0 < (n :: int) ==> a mod n <= a" |
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apply (cases "a < n") |
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apply (auto dest: mod_pos_pos_trivial pos_mod_bound [where a=a]) |
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done |
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lemma int_mod_le': "0 <= b - n ==> 0 < (n :: int) ==> b mod n <= b - n" |
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by (rule int_mod_le [where a = "b - n" and n = n, simplified]) |
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lemma int_mod_ge: "a < n ==> 0 < (n :: int) ==> a <= a mod n" |
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apply (cases "0 <= a") |
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apply (drule (1) mod_pos_pos_trivial) |
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apply simp |
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apply (rule order_trans [OF _ pos_mod_sign]) |
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apply simp |
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apply assumption |
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done |
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lemma int_mod_ge': "b < 0 ==> 0 < (n :: int) ==> b + n <= b mod n" |
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by (rule int_mod_ge [where a = "b + n" and n = n, simplified]) |
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lemma mod_add_if_z: |
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"(x :: int) < z ==> y < z ==> 0 <= y ==> 0 <= x ==> 0 <= z ==> |
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(x + y) mod z = (if x + y < z then x + y else x + y - z)" |
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by (auto intro: int_mod_eq) |
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lemma mod_sub_if_z: |
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"(x :: int) < z ==> y < z ==> 0 <= y ==> 0 <= x ==> 0 <= z ==> |
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(x - y) mod z = (if y <= x then x - y else x - y + z)" |
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by (auto intro: int_mod_eq) |
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lemmas zmde = zmod_zdiv_equality [THEN diff_eq_eq [THEN iffD2], symmetric] |
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lemmas mcl = mult_cancel_left [THEN iffD1, THEN make_pos_rule] |
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(* already have this for naturals, div_mult_self1/2, but not for ints *) |
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lemma zdiv_mult_self: "m ~= (0 :: int) ==> (a + m * n) div m = a div m + n" |
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apply (rule mcl) |
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prefer 2 |
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apply (erule asm_rl) |
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apply (simp add: zmde ring_distribs) |
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done |
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(** Rep_Integ **) |
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lemma eqne: "equiv A r ==> X : A // r ==> X ~= {}" |
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unfolding equiv_def refl_on_def quotient_def Image_def by auto |
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lemmas Rep_Integ_ne = Integ.Rep_Integ |
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[THEN equiv_intrel [THEN eqne, simplified Integ_def [symmetric]], standard] |
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lemmas riq = Integ.Rep_Integ [simplified Integ_def] |
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lemmas intrel_refl = refl [THEN equiv_intrel_iff [THEN iffD1], standard] |
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lemmas Rep_Integ_equiv = quotient_eq_iff |
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[OF equiv_intrel riq riq, simplified Integ.Rep_Integ_inject, standard] |
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lemmas Rep_Integ_same = |
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Rep_Integ_equiv [THEN intrel_refl [THEN rev_iffD2], standard] |
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lemma RI_int: "(a, 0) : Rep_Integ (int a)" |
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unfolding int_def by auto |
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lemmas RI_intrel [simp] = UNIV_I [THEN quotientI, |
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THEN Integ.Abs_Integ_inverse [simplified Integ_def], standard] |
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lemma RI_minus: "(a, b) : Rep_Integ x ==> (b, a) : Rep_Integ (- x)" |
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apply (rule_tac z=x in eq_Abs_Integ) |
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apply (clarsimp simp: minus) |
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done |
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lemma RI_add: |
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"(a, b) : Rep_Integ x ==> (c, d) : Rep_Integ y ==> |
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(a + c, b + d) : Rep_Integ (x + y)" |
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apply (rule_tac z=x in eq_Abs_Integ) |
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apply (rule_tac z=y in eq_Abs_Integ) |
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apply (clarsimp simp: add) |
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292 |
done |
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lemma mem_same: "a : S ==> a = b ==> b : S" |
|
295 |
by fast |
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296 |
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(* two alternative proofs of this *) |
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298 |
lemma RI_eq_diff': "(a, b) : Rep_Integ (int a - int b)" |
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299 |
apply (unfold diff_def) |
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300 |
apply (rule mem_same) |
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apply (rule RI_minus RI_add RI_int)+ |
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apply simp |
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303 |
done |
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lemma RI_eq_diff: "((a, b) : Rep_Integ x) = (int a - int b = x)" |
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306 |
apply safe |
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307 |
apply (rule Rep_Integ_same) |
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prefer 2 |
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309 |
apply (erule asm_rl) |
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310 |
apply (rule RI_eq_diff')+ |
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311 |
done |
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lemma mod_power_lem: |
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314 |
"a > 1 ==> a ^ n mod a ^ m = (if m <= n then 0 else (a :: int) ^ n)" |
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315 |
apply clarsimp |
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316 |
apply safe |
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apply (simp add: dvd_eq_mod_eq_0 [symmetric]) |
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apply (drule le_iff_add [THEN iffD1]) |
319 |
apply (force simp: zpower_zadd_distrib) |
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320 |
apply (rule mod_pos_pos_trivial) |
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apply (simp) |
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apply (rule power_strict_increasing) |
323 |
apply auto |
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324 |
done |
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lemma min_pm [simp]: "min a b + (a - b) = (a :: nat)" by arith |
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|
328 |
lemmas min_pm1 [simp] = trans [OF add_commute min_pm] |
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lemma rev_min_pm [simp]: "min b a + (a - b) = (a::nat)" by arith |
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332 |
lemmas rev_min_pm1 [simp] = trans [OF add_commute rev_min_pm] |
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lemma pl_pl_rels: |
335 |
"a + b = c + d ==> |
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a >= c & b <= d | a <= c & b >= (d :: nat)" by arith |
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338 |
lemmas pl_pl_rels' = add_commute [THEN [2] trans, THEN pl_pl_rels] |
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lemma minus_eq: "(m - k = m) = (k = 0 | m = (0 :: nat))" by arith |
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lemma pl_pl_mm: "(a :: nat) + b = c + d ==> a - c = d - b" by arith |
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lemmas pl_pl_mm' = add_commute [THEN [2] trans, THEN pl_pl_mm] |
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lemma min_minus [simp] : "min m (m - k) = (m - k :: nat)" by arith |
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lemmas min_minus' [simp] = trans [OF min_max.inf_commute min_minus] |
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lemma nat_no_eq_iff: |
351 |
"(number_of b :: int) >= 0 ==> (number_of c :: int) >= 0 ==> |
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(number_of b = (number_of c :: nat)) = (b = c)" |
353 |
apply (unfold nat_number_of_def) |
|
24465 | 354 |
apply safe |
355 |
apply (drule (2) eq_nat_nat_iff [THEN iffD1]) |
|
356 |
apply (simp add: number_of_eq) |
|
357 |
done |
|
358 |
||
24333 | 359 |
lemmas dme = box_equals [OF div_mod_equality add_0_right add_0_right] |
360 |
lemmas dtle = xtr3 [OF dme [symmetric] le_add1] |
|
361 |
lemmas th2 = order_trans [OF order_refl [THEN [2] mult_le_mono] dtle] |
|
362 |
||
363 |
lemma td_gal: |
|
364 |
"0 < c ==> (a >= b * c) = (a div c >= (b :: nat))" |
|
365 |
apply safe |
|
366 |
apply (erule (1) xtr4 [OF div_le_mono div_mult_self_is_m]) |
|
367 |
apply (erule th2) |
|
368 |
done |
|
369 |
||
26072
f65a7fa2da6c
<= and < on nat no longer depend on wellfounded relations
haftmann
parents:
25937
diff
changeset
|
370 |
lemmas td_gal_lt = td_gal [simplified not_less [symmetric], simplified] |
24333 | 371 |
|
372 |
lemma div_mult_le: "(a :: nat) div b * b <= a" |
|
373 |
apply (cases b) |
|
374 |
prefer 2 |
|
375 |
apply (rule order_refl [THEN th2]) |
|
376 |
apply auto |
|
377 |
done |
|
378 |
||
379 |
lemmas sdl = split_div_lemma [THEN iffD1, symmetric] |
|
380 |
||
381 |
lemma given_quot: "f > (0 :: nat) ==> (f * l + (f - 1)) div f = l" |
|
382 |
by (rule sdl, assumption) (simp (no_asm)) |
|
383 |
||
384 |
lemma given_quot_alt: "f > (0 :: nat) ==> (l * f + f - Suc 0) div f = l" |
|
385 |
apply (frule given_quot) |
|
386 |
apply (rule trans) |
|
387 |
prefer 2 |
|
388 |
apply (erule asm_rl) |
|
389 |
apply (rule_tac f="%n. n div f" in arg_cong) |
|
390 |
apply (simp add : mult_ac) |
|
391 |
done |
|
392 |
||
24465 | 393 |
lemma diff_mod_le: "(a::nat) < d ==> b dvd d ==> a - a mod b <= d - b" |
394 |
apply (unfold dvd_def) |
|
395 |
apply clarify |
|
396 |
apply (case_tac k) |
|
397 |
apply clarsimp |
|
398 |
apply clarify |
|
399 |
apply (cases "b > 0") |
|
400 |
apply (drule mult_commute [THEN xtr1]) |
|
401 |
apply (frule (1) td_gal_lt [THEN iffD1]) |
|
402 |
apply (clarsimp simp: le_simps) |
|
403 |
apply (rule mult_div_cancel [THEN [2] xtr4]) |
|
404 |
apply (rule mult_mono) |
|
405 |
apply auto |
|
406 |
done |
|
407 |
||
24333 | 408 |
lemma less_le_mult': |
409 |
"w * c < b * c ==> 0 \<le> c ==> (w + 1) * c \<le> b * (c::int)" |
|
410 |
apply (rule mult_right_mono) |
|
411 |
apply (rule zless_imp_add1_zle) |
|
412 |
apply (erule (1) mult_right_less_imp_less) |
|
413 |
apply assumption |
|
414 |
done |
|
415 |
||
416 |
lemmas less_le_mult = less_le_mult' [simplified left_distrib, simplified] |
|
24465 | 417 |
|
418 |
lemmas less_le_mult_minus = iffD2 [OF le_diff_eq less_le_mult, |
|
419 |
simplified left_diff_distrib, standard] |
|
24333 | 420 |
|
421 |
lemma lrlem': |
|
422 |
assumes d: "(i::nat) \<le> j \<or> m < j'" |
|
423 |
assumes R1: "i * k \<le> j * k \<Longrightarrow> R" |
|
424 |
assumes R2: "Suc m * k' \<le> j' * k' \<Longrightarrow> R" |
|
425 |
shows "R" using d |
|
426 |
apply safe |
|
427 |
apply (rule R1, erule mult_le_mono1) |
|
428 |
apply (rule R2, erule Suc_le_eq [THEN iffD2 [THEN mult_le_mono1]]) |
|
429 |
done |
|
430 |
||
431 |
lemma lrlem: "(0::nat) < sc ==> |
|
432 |
(sc - n + (n + lb * n) <= m * n) = (sc + lb * n <= m * n)" |
|
433 |
apply safe |
|
434 |
apply arith |
|
435 |
apply (case_tac "sc >= n") |
|
436 |
apply arith |
|
437 |
apply (insert linorder_le_less_linear [of m lb]) |
|
438 |
apply (erule_tac k=n and k'=n in lrlem') |
|
439 |
apply arith |
|
440 |
apply simp |
|
441 |
done |
|
442 |
||
443 |
lemma gen_minus: "0 < n ==> f n = f (Suc (n - 1))" |
|
444 |
by auto |
|
445 |
||
27570 | 446 |
lemma mpl_lem: "j <= (i :: nat) ==> k < j ==> i - j + k < i" by arith |
24333 | 447 |
|
24465 | 448 |
lemma nonneg_mod_div: |
449 |
"0 <= a ==> 0 <= b ==> 0 <= (a mod b :: int) & 0 <= a div b" |
|
450 |
apply (cases "b = 0", clarsimp) |
|
451 |
apply (auto intro: pos_imp_zdiv_nonneg_iff [THEN iffD2]) |
|
452 |
done |
|
24399 | 453 |
|
24333 | 454 |
end |