(* ID: $Id$ Author: Jeremy Dawson, NICTA useful numerical lemmas *) theory Num_Lemmas imports Parity begin lemma contentsI: "y = {x} ==> contents y = x" unfolding contents_def by auto lemma prod_case_split: "prod_case = split" by (rule ext)+ auto lemmas split_split = prod.split [unfolded prod_case_split] lemmas split_split_asm = prod.split_asm [unfolded prod_case_split] lemmas "split.splits" = split_split split_split_asm lemmas funpow_0 = funpow.simps(1) lemmas funpow_Suc = funpow.simps(2) lemma nonemptyE: "S ~= {} ==> (!!x. x : S ==> R) ==> R" apply (erule contrapos_np) apply (rule equals0I) apply auto done lemma int_number_of: "number_of (y::int) = y" by (simp add: number_of_eq) lemma gt_or_eq_0: "0 < y \ 0 = (y::nat)" by auto constdefs mod_alt :: "'a => 'a => 'a :: Divides.div" "mod_alt n m == n mod m" -- "alternative way of defining @{text bin_last}, @{text bin_rest}" bin_rl :: "int => int * bit" "bin_rl w == SOME (r, l). w = r BIT l" declare iszero_0 [iff] lemmas xtr1 = xtrans(1) lemmas xtr2 = xtrans(2) lemmas xtr3 = xtrans(3) lemmas xtr4 = xtrans(4) lemmas xtr5 = xtrans(5) lemmas xtr6 = xtrans(6) lemmas xtr7 = xtrans(7) lemmas xtr8 = xtrans(8) lemma Min_ne_Pls [iff]: "Numeral.Min ~= Numeral.Pls" unfolding Min_def Pls_def by auto lemmas Pls_ne_Min [iff] = Min_ne_Pls [symmetric] lemmas PlsMin_defs [intro!] = Pls_def Min_def Pls_def [symmetric] Min_def [symmetric] lemmas PlsMin_simps [simp] = PlsMin_defs [THEN Eq_TrueI] lemma number_of_False_cong: "False ==> number_of x = number_of y" by auto lemmas nat_simps = diff_add_inverse2 diff_add_inverse lemmas nat_iffs = le_add1 le_add2 lemma sum_imp_diff: "j = k + i ==> j - i = (k :: nat)" by (clarsimp simp add: nat_simps) lemma nobm1: "0 < (number_of w :: nat) ==> number_of w - (1 :: nat) = number_of (Numeral.pred w)" apply (unfold nat_number_of_def One_nat_def nat_1 [symmetric] pred_def) apply (simp add: number_of_eq nat_diff_distrib [symmetric]) done lemma of_int_power: "of_int (a ^ n) = (of_int a ^ n :: 'a :: {recpower, comm_ring_1})" by (induct n) (auto simp add: power_Suc) lemma zless2: "0 < (2 :: int)" by auto lemmas zless2p [simp] = zless2 [THEN zero_less_power] lemmas zle2p [simp] = zless2p [THEN order_less_imp_le] lemmas pos_mod_sign2 = zless2 [THEN pos_mod_sign [where b = "2::int"]] lemmas pos_mod_bound2 = zless2 [THEN pos_mod_bound [where b = "2::int"]] -- "the inverse(s) of @{text number_of}" lemma nmod2: "n mod (2::int) = 0 | n mod 2 = 1" using pos_mod_sign2 [of n] pos_mod_bound2 [of n] unfolding mod_alt_def [symmetric] by auto lemma emep1: "even n ==> even d ==> 0 <= d ==> (n + 1) mod (d :: int) = (n mod d) + 1" apply (simp add: add_commute) apply (safe dest!: even_equiv_def [THEN iffD1]) apply (subst pos_zmod_mult_2) apply arith apply (simp add: zmod_zmult_zmult1) done lemmas eme1p = emep1 [simplified add_commute] lemma le_diff_eq': "(a \ c - b) = (b + a \ (c::int))" by (simp add: le_diff_eq add_commute) lemma less_diff_eq': "(a < c - b) = (b + a < (c::int))" by (simp add: less_diff_eq add_commute) lemma diff_le_eq': "(a - b \ c) = (a \ b + (c::int))" by (simp add: diff_le_eq add_commute) lemma diff_less_eq': "(a - b < c) = (a < b + (c::int))" by (simp add: diff_less_eq add_commute) lemmas m1mod2k = zless2p [THEN zmod_minus1] lemmas m1mod22k = mult_pos_pos [OF zless2 zless2p, THEN zmod_minus1] lemmas p1mod22k' = zless2p [THEN order_less_imp_le, THEN pos_zmod_mult_2] lemmas z1pmod2' = zero_le_one [THEN pos_zmod_mult_2, simplified] lemmas z1pdiv2' = zero_le_one [THEN pos_zdiv_mult_2, simplified] lemma p1mod22k: "(2 * b + 1) mod (2 * 2 ^ n) = 2 * (b mod 2 ^ n) + (1::int)" by (simp add: p1mod22k' add_commute) lemma z1pmod2: "(2 * b + 1) mod 2 = (1::int)" by (simp add: z1pmod2' add_commute) lemma z1pdiv2: "(2 * b + 1) div 2 = (b::int)" by (simp add: z1pdiv2' add_commute) lemmas zdiv_le_dividend = xtr3 [OF zdiv_1 [symmetric] zdiv_mono2, simplified int_one_le_iff_zero_less, simplified, standard] (** ways in which type Bin resembles a datatype **) lemma BIT_eq: "u BIT b = v BIT c ==> u = v & b = c" apply (unfold Bit_def) apply (simp (no_asm_use) split: bit.split_asm) apply simp_all apply (drule_tac f=even in arg_cong, clarsimp)+ done lemmas BIT_eqE [elim!] = BIT_eq [THEN conjE, standard] lemma BIT_eq_iff [simp]: "(u BIT b = v BIT c) = (u = v \ b = c)" by (rule iffI) auto lemmas BIT_eqI [intro!] = conjI [THEN BIT_eq_iff [THEN iffD2]] lemma less_Bits: "(v BIT b < w BIT c) = (v < w | v <= w & b = bit.B0 & c = bit.B1)" unfolding Bit_def by (auto split: bit.split) lemma le_Bits: "(v BIT b <= w BIT c) = (v < w | v <= w & (b ~= bit.B1 | c ~= bit.B0))" unfolding Bit_def by (auto split: bit.split) lemma neB1E [elim!]: assumes ne: "y \ bit.B1" assumes y: "y = bit.B0 \ P" shows "P" apply (rule y) apply (cases y rule: bit.exhaust, simp) apply (simp add: ne) done lemma bin_ex_rl: "EX w b. w BIT b = bin" apply (unfold Bit_def) apply (cases "even bin") apply (clarsimp simp: even_equiv_def) apply (auto simp: odd_equiv_def split: bit.split) done lemma bin_exhaust: assumes Q: "\x b. bin = x BIT b \ Q" shows "Q" apply (insert bin_ex_rl [of bin]) apply (erule exE)+ apply (rule Q) apply force done lemma bin_rl_char: "(bin_rl w = (r, l)) = (r BIT l = w)" apply (unfold bin_rl_def) apply safe apply (cases w rule: bin_exhaust) apply auto done lemmas bin_rl_simps [THEN bin_rl_char [THEN iffD2], standard, simp] = Pls_0_eq Min_1_eq refl lemma bin_abs_lem: "bin = (w BIT b) ==> ~ bin = Numeral.Min --> ~ bin = Numeral.Pls --> nat (abs w) < nat (abs bin)" apply (clarsimp simp add: bin_rl_char) apply (unfold Pls_def Min_def Bit_def) apply (cases b) apply (clarsimp, arith) apply (clarsimp, arith) done lemma bin_induct: assumes PPls: "P Numeral.Pls" and PMin: "P Numeral.Min" and PBit: "!!bin bit. P bin ==> P (bin BIT bit)" shows "P bin" apply (rule_tac P=P and a=bin and f1="nat o abs" in wf_measure [THEN wf_induct]) apply (simp add: measure_def inv_image_def) apply (case_tac x rule: bin_exhaust) apply (frule bin_abs_lem) apply (auto simp add : PPls PMin PBit) done lemma no_no [simp]: "number_of (number_of i) = i" unfolding number_of_eq by simp lemma Bit_B0: "k BIT bit.B0 = k + k" by (unfold Bit_def) simp lemma Bit_B1: "k BIT bit.B1 = k + k + 1" by (unfold Bit_def) simp lemma Bit_B0_2t: "k BIT bit.B0 = 2 * k" by (rule trans, rule Bit_B0) simp lemma Bit_B1_2t: "k BIT bit.B1 = 2 * k + 1" by (rule trans, rule Bit_B1) simp lemma B_mod_2': "X = 2 ==> (w BIT bit.B1) mod X = 1 & (w BIT bit.B0) mod X = 0" apply (simp (no_asm) only: Bit_B0 Bit_B1) apply (simp add: z1pmod2) done lemmas B1_mod_2 [simp] = B_mod_2' [OF refl, THEN conjunct1, standard] lemmas B0_mod_2 [simp] = B_mod_2' [OF refl, THEN conjunct2, standard] lemma axxbyy: "a + m + m = b + n + n ==> (a = 0 | a = 1) ==> (b = 0 | b = 1) ==> a = b & m = (n :: int)" apply auto apply (drule_tac f="%n. n mod 2" in arg_cong) apply (clarsimp simp: z1pmod2) apply (drule_tac f="%n. n mod 2" in arg_cong) apply (clarsimp simp: z1pmod2) done lemma axxmod2: "(1 + x + x) mod 2 = (1 :: int) & (0 + x + x) mod 2 = (0 :: int)" by simp (rule z1pmod2) lemma axxdiv2: "(1 + x + x) div 2 = (x :: int) & (0 + x + x) div 2 = (x :: int)" by simp (rule z1pdiv2) lemmas iszero_minus = trans [THEN trans, OF iszero_def neg_equal_0_iff_equal iszero_def [symmetric], standard] lemmas zadd_diff_inverse = trans [OF diff_add_cancel [symmetric] add_commute, standard] lemmas add_diff_cancel2 = add_commute [THEN diff_eq_eq [THEN iffD2], standard] lemma zmod_uminus: "- ((a :: int) mod b) mod b = -a mod b" by (simp add : zmod_zminus1_eq_if) lemma zmod_zsub_distrib: "((a::int) - b) mod c = (a mod c - b mod c) mod c" apply (unfold diff_int_def) apply (rule trans [OF _ zmod_zadd1_eq [symmetric]]) apply (simp add: zmod_uminus zmod_zadd1_eq [symmetric]) done lemma zmod_zsub_right_eq: "((a::int) - b) mod c = (a - b mod c) mod c" apply (unfold diff_int_def) apply (rule trans [OF _ zmod_zadd_right_eq [symmetric]]) apply (simp add : zmod_uminus zmod_zadd_right_eq [symmetric]) done lemmas zmod_zsub_left_eq = zmod_zadd_left_eq [where b = "- ?b", simplified diff_int_def [symmetric]] lemma zmod_zsub_self [simp]: "((b :: int) - a) mod a = b mod a" by (simp add: zmod_zsub_right_eq) lemma zmod_zmult1_eq_rev: "b * a mod c = b mod c * a mod (c::int)" apply (simp add: mult_commute) apply (subst zmod_zmult1_eq) apply simp done lemmas rdmods [symmetric] = zmod_uminus [symmetric] zmod_zsub_left_eq zmod_zsub_right_eq zmod_zadd_left_eq zmod_zadd_right_eq zmod_zmult1_eq zmod_zmult1_eq_rev lemma mod_plus_right: "((a + x) mod m = (b + x) mod m) = (a mod m = b mod (m :: nat))" apply (induct x) apply (simp_all add: mod_Suc) apply arith done lemma nat_minus_mod: "(n - n mod m) mod m = (0 :: nat)" by (induct n) (simp_all add : mod_Suc) lemmas nat_minus_mod_plus_right = trans [OF nat_minus_mod mod_0 [symmetric], THEN mod_plus_right [THEN iffD2], standard, simplified] lemmas push_mods' = zmod_zadd1_eq [standard] zmod_zmult_distrib [standard] zmod_zsub_distrib [standard] zmod_uminus [symmetric, standard] lemmas push_mods = push_mods' [THEN eq_reflection, standard] lemmas pull_mods = push_mods [symmetric] rdmods [THEN eq_reflection, standard] lemmas mod_simps = zmod_zmult_self1 [THEN eq_reflection] zmod_zmult_self2 [THEN eq_reflection] mod_mod_trivial [THEN eq_reflection] lemma nat_mod_eq: "!!b. b < n ==> a mod n = b mod n ==> a mod n = (b :: nat)" by (induct a) auto lemmas nat_mod_eq' = refl [THEN [2] nat_mod_eq] lemma nat_mod_lem: "(0 :: nat) < n ==> b < n = (b mod n = b)" apply safe apply (erule nat_mod_eq') apply (erule subst) apply (erule mod_less_divisor) done lemma mod_nat_add: "(x :: nat) < z ==> y < z ==> (x + y) mod z = (if x + y < z then x + y else x + y - z)" apply (rule nat_mod_eq) apply auto apply (rule trans) apply (rule le_mod_geq) apply simp apply (rule nat_mod_eq') apply arith done lemma mod_nat_sub: "(x :: nat) < z ==> (x - y) mod z = x - y" by (rule nat_mod_eq') arith lemma int_mod_lem: "(0 :: int) < n ==> (0 <= b & b < n) = (b mod n = b)" apply safe apply (erule (1) mod_pos_pos_trivial) apply (erule_tac [!] subst) apply auto done lemma int_mod_eq: "(0 :: int) <= b ==> b < n ==> a mod n = b mod n ==> a mod n = b" by clarsimp (rule mod_pos_pos_trivial) lemmas int_mod_eq' = refl [THEN [3] int_mod_eq] lemma int_mod_le: "0 <= a ==> 0 < (n :: int) ==> a mod n <= a" apply (cases "a < n") apply (auto dest: mod_pos_pos_trivial pos_mod_bound [where a=a]) done lemmas int_mod_le' = int_mod_le [where a = "?b - ?n", simplified] lemma int_mod_ge: "a < n ==> 0 < (n :: int) ==> a <= a mod n" apply (cases "0 <= a") apply (drule (1) mod_pos_pos_trivial) apply simp apply (rule order_trans [OF _ pos_mod_sign]) apply simp apply assumption done lemmas int_mod_ge' = int_mod_ge [where a = "?b + ?n", simplified] lemma mod_add_if_z: "(x :: int) < z ==> y < z ==> 0 <= y ==> 0 <= x ==> 0 <= z ==> (x + y) mod z = (if x + y < z then x + y else x + y - z)" by (auto intro: int_mod_eq) lemma mod_sub_if_z: "(x :: int) < z ==> y < z ==> 0 <= y ==> 0 <= x ==> 0 <= z ==> (x - y) mod z = (if y <= x then x - y else x - y + z)" by (auto intro: int_mod_eq) lemmas zmde = zmod_zdiv_equality [THEN diff_eq_eq [THEN iffD2], symmetric] lemmas mcl = mult_cancel_left [THEN iffD1, THEN make_pos_rule] (* already have this for naturals, div_mult_self1/2, but not for ints *) lemma zdiv_mult_self: "m ~= (0 :: int) ==> (a + m * n) div m = a div m + n" apply (rule mcl) prefer 2 apply (erule asm_rl) apply (simp add: zmde ring_distribs) apply (simp add: push_mods) done (** Rep_Integ **) lemma eqne: "equiv A r ==> X : A // r ==> X ~= {}" unfolding equiv_def refl_def quotient_def Image_def by auto lemmas Rep_Integ_ne = Integ.Rep_Integ [THEN equiv_intrel [THEN eqne, simplified Integ_def [symmetric]], standard] lemmas riq = Integ.Rep_Integ [simplified Integ_def] lemmas intrel_refl = refl [THEN equiv_intrel_iff [THEN iffD1], standard] lemmas Rep_Integ_equiv = quotient_eq_iff [OF equiv_intrel riq riq, simplified Integ.Rep_Integ_inject, standard] lemmas Rep_Integ_same = Rep_Integ_equiv [THEN intrel_refl [THEN rev_iffD2], standard] lemma RI_int: "(a, 0) : Rep_Integ (int a)" unfolding int_def by auto lemmas RI_intrel [simp] = UNIV_I [THEN quotientI, THEN Integ.Abs_Integ_inverse [simplified Integ_def], standard] lemma RI_minus: "(a, b) : Rep_Integ x ==> (b, a) : Rep_Integ (- x)" apply (rule_tac z=x in eq_Abs_Integ) apply (clarsimp simp: minus) done lemma RI_add: "(a, b) : Rep_Integ x ==> (c, d) : Rep_Integ y ==> (a + c, b + d) : Rep_Integ (x + y)" apply (rule_tac z=x in eq_Abs_Integ) apply (rule_tac z=y in eq_Abs_Integ) apply (clarsimp simp: add) done lemma mem_same: "a : S ==> a = b ==> b : S" by fast (* two alternative proofs of this *) lemma RI_eq_diff': "(a, b) : Rep_Integ (int a - int b)" apply (unfold diff_def) apply (rule mem_same) apply (rule RI_minus RI_add RI_int)+ apply simp done lemma RI_eq_diff: "((a, b) : Rep_Integ x) = (int a - int b = x)" apply safe apply (rule Rep_Integ_same) prefer 2 apply (erule asm_rl) apply (rule RI_eq_diff')+ done lemma mod_power_lem: "a > 1 ==> a ^ n mod a ^ m = (if m <= n then 0 else (a :: int) ^ n)" apply clarsimp apply safe apply (simp add: zdvd_iff_zmod_eq_0 [symmetric]) apply (drule le_iff_add [THEN iffD1]) apply (force simp: zpower_zadd_distrib) apply (rule mod_pos_pos_trivial) apply (simp add: zero_le_power) apply (rule power_strict_increasing) apply auto done lemma min_pm [simp]: "min a b + (a - b) = (a :: nat)" by arith lemmas min_pm1 [simp] = trans [OF add_commute min_pm] lemma rev_min_pm [simp]: "min b a + (a - b) = (a::nat)" by simp lemmas rev_min_pm1 [simp] = trans [OF add_commute rev_min_pm] lemma pl_pl_rels: "a + b = c + d ==> a >= c & b <= d | a <= c & b >= (d :: nat)" apply (cut_tac n=a and m=c in nat_le_linear) apply (safe dest!: le_iff_add [THEN iffD1]) apply arith+ done lemmas pl_pl_rels' = add_commute [THEN [2] trans, THEN pl_pl_rels] lemma minus_eq: "(m - k = m) = (k = 0 | m = (0 :: nat))" by arith lemma pl_pl_mm: "(a :: nat) + b = c + d ==> a - c = d - b" by arith lemmas pl_pl_mm' = add_commute [THEN [2] trans, THEN pl_pl_mm] lemma min_minus [simp] : "min m (m - k) = (m - k :: nat)" by arith lemmas min_minus' [simp] = trans [OF min_max.inf_commute min_minus] lemma nat_no_eq_iff: "(number_of b :: int) >= 0 ==> (number_of c :: int) >= 0 ==> (number_of b = (number_of c :: nat)) = (b = c)" apply (unfold nat_number_of_def) apply safe apply (drule (2) eq_nat_nat_iff [THEN iffD1]) apply (simp add: number_of_eq) done lemmas dme = box_equals [OF div_mod_equality add_0_right add_0_right] lemmas dtle = xtr3 [OF dme [symmetric] le_add1] lemmas th2 = order_trans [OF order_refl [THEN [2] mult_le_mono] dtle] lemma td_gal: "0 < c ==> (a >= b * c) = (a div c >= (b :: nat))" apply safe apply (erule (1) xtr4 [OF div_le_mono div_mult_self_is_m]) apply (erule th2) done lemmas td_gal_lt = td_gal [simplified le_def, simplified] lemma div_mult_le: "(a :: nat) div b * b <= a" apply (cases b) prefer 2 apply (rule order_refl [THEN th2]) apply auto done lemmas sdl = split_div_lemma [THEN iffD1, symmetric] lemma given_quot: "f > (0 :: nat) ==> (f * l + (f - 1)) div f = l" by (rule sdl, assumption) (simp (no_asm)) lemma given_quot_alt: "f > (0 :: nat) ==> (l * f + f - Suc 0) div f = l" apply (frule given_quot) apply (rule trans) prefer 2 apply (erule asm_rl) apply (rule_tac f="%n. n div f" in arg_cong) apply (simp add : mult_ac) done lemma diff_mod_le: "(a::nat) < d ==> b dvd d ==> a - a mod b <= d - b" apply (unfold dvd_def) apply clarify apply (case_tac k) apply clarsimp apply clarify apply (cases "b > 0") apply (drule mult_commute [THEN xtr1]) apply (frule (1) td_gal_lt [THEN iffD1]) apply (clarsimp simp: le_simps) apply (rule mult_div_cancel [THEN [2] xtr4]) apply (rule mult_mono) apply auto done lemma less_le_mult': "w * c < b * c ==> 0 \ c ==> (w + 1) * c \ b * (c::int)" apply (rule mult_right_mono) apply (rule zless_imp_add1_zle) apply (erule (1) mult_right_less_imp_less) apply assumption done lemmas less_le_mult = less_le_mult' [simplified left_distrib, simplified] lemmas less_le_mult_minus = iffD2 [OF le_diff_eq less_le_mult, simplified left_diff_distrib, standard] lemma lrlem': assumes d: "(i::nat) \ j \ m < j'" assumes R1: "i * k \ j * k \ R" assumes R2: "Suc m * k' \ j' * k' \ R" shows "R" using d apply safe apply (rule R1, erule mult_le_mono1) apply (rule R2, erule Suc_le_eq [THEN iffD2 [THEN mult_le_mono1]]) done lemma lrlem: "(0::nat) < sc ==> (sc - n + (n + lb * n) <= m * n) = (sc + lb * n <= m * n)" apply safe apply arith apply (case_tac "sc >= n") apply arith apply (insert linorder_le_less_linear [of m lb]) apply (erule_tac k=n and k'=n in lrlem') apply arith apply simp done lemma gen_minus: "0 < n ==> f n = f (Suc (n - 1))" by auto lemma mpl_lem: "j <= (i :: nat) ==> k < j ==> i - j + k < i" apply (induct i, clarsimp) apply (cases j, clarsimp) apply arith done lemma nonneg_mod_div: "0 <= a ==> 0 <= b ==> 0 <= (a mod b :: int) & 0 <= a div b" apply (cases "b = 0", clarsimp) apply (auto intro: pos_imp_zdiv_nonneg_iff [THEN iffD2]) done end