(*
Author: Jeremy Dawson, NICTA
*)
header {* Useful Numerical Lemmas *}
theory Misc_Numeric
imports Main Parity
begin
lemma the_elemI: "y = {x} ==> the_elem y = x"
by simp
lemmas split_split = prod.split
lemmas split_split_asm = prod.split_asm
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" by auto
lemma gt_or_eq_0: "0 < y \<or> 0 = (y::nat)" by arith
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)
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 arith
lemma nobm1:
"0 < (number_of w :: nat) ==>
number_of w - (1 :: nat) = number_of (Int.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 zless2: "0 < (2 :: int)" by arith
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" by arith
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: mod_mult_mult1)
done
lemmas eme1p = emep1 [simplified add_commute]
lemma le_diff_eq': "(a \<le> c - b) = (b + a \<le> (c::int))" by arith
lemma less_diff_eq': "(a < c - b) = (b + a < (c::int))" by arith
lemma diff_le_eq': "(a - b \<le> c) = (a \<le> b + (c::int))" by arith
lemma diff_less_eq': "(a - b < c) = (a < b + (c::int))" by arith
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 arith
lemma z1pdiv2:
"(2 * b + 1) div 2 = (b::int)" by arith
lemmas zdiv_le_dividend = xtr3 [OF div_by_1 [symmetric] zdiv_mono2,
simplified int_one_le_iff_zero_less, simplified, standard]
lemma axxbyy:
"a + m + m = b + n + n ==> (a = 0 | a = 1) ==> (b = 0 | b = 1) ==>
a = b & m = (n :: int)" by arith
lemma axxmod2:
"(1 + x + x) mod 2 = (1 :: int) & (0 + x + x) mod 2 = (0 :: int)" by arith
lemma axxdiv2:
"(1 + x + x) div 2 = (x :: int) & (0 + x + x) div 2 = (x :: int)" by arith
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 _ mod_add_eq [symmetric]])
apply (simp add: zmod_uminus mod_add_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 _ mod_add_right_eq [symmetric]])
apply (simp add : zmod_uminus mod_add_right_eq [symmetric])
done
lemma zmod_zsub_left_eq: "((a::int) - b) mod c = (a mod c - b) mod c"
by (rule mod_add_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 mod_add_left_eq
mod_add_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' = mod_add_eq [standard]
mod_mult_eq [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 =
mod_mult_self2_is_0 [THEN eq_reflection]
mod_mult_self1_is_0 [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
lemma int_mod_le': "0 <= b - n ==> 0 < (n :: int) ==> b mod n <= b - n"
by (rule int_mod_le [where a = "b - n" and n = 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
lemma int_mod_ge': "b < 0 ==> 0 < (n :: int) ==> b + n <= b mod n"
by (rule int_mod_ge [where a = "b + n" and n = 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)
done
(** Rep_Integ **)
lemma eqne: "equiv A r ==> X : A // r ==> X ~= {}"
unfolding equiv_def refl_on_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_minus)
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: dvd_eq_mod_eq_0 [symmetric])
apply (drule le_iff_add [THEN iffD1])
apply (force simp: power_add)
apply (rule mod_pos_pos_trivial)
apply (simp)
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 arith
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)" by arith
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 not_less [symmetric], 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 \<le> c ==> (w + 1) * c \<le> 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) \<le> j \<or> m < j'"
assumes R1: "i * k \<le> j * k \<Longrightarrow> R"
assumes R2: "Suc m * k' \<le> j' * k' \<Longrightarrow> 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" by arith
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