# Theory Misc_Arithmetic

theory Misc_Arithmetic
imports Bits_Int
```(*  Title:      HOL/Word/Misc_Arithmetic.thy  *)

section ‹Miscellaneous lemmas, mostly for arithmetic›

theory Misc_Arithmetic
imports Main Bits_Int
begin

lemma int_mod_lem: "0 < n ⟹ 0 ≤ b ∧ b < n ⟷ b mod n = b"
for b n :: int
apply safe
apply (erule (1) mod_pos_pos_trivial)
apply (erule_tac [!] subst)
apply auto
done

lemma int_mod_ge': "b < 0 ⟹ 0 < n ⟹ b + n ≤ b mod n"
for b n :: int

lemma int_mod_le': "0 ≤ b - n ⟹ b mod n ≤ b - n"
for b n :: int
by (metis minus_mod_self2 zmod_le_nonneg_dividend)

lemma emep1: "even n ⟹ even d ⟹ 0 ≤ d ⟹ (n + 1) mod d = (n mod d) + 1"
for n d :: int

lemma m1mod2k: "- 1 mod 2 ^ n = (2 ^ n - 1 :: int)"
by (rule zmod_minus1) simp

lemma sb_inc_lem: "a + 2^k < 0 ⟹ a + 2^k + 2^(Suc k) ≤ (a + 2^k) mod 2^(Suc k)"
for a :: int
using int_mod_ge' [where n = "2 ^ (Suc k)" and b = "a + 2 ^ k"]
by simp

lemma sb_inc_lem': "a < - (2^k) ⟹ a + 2^k + 2^(Suc k) ≤ (a + 2^k) mod 2^(Suc k)"
for a :: int
by (rule sb_inc_lem) simp

lemma sb_dec_lem: "0 ≤ - (2 ^ k) + a ⟹ (a + 2 ^ k) mod (2 * 2 ^ k) ≤ - (2 ^ k) + a"
for a :: int
using int_mod_le'[where n = "2 ^ (Suc k)" and b = "a + 2 ^ k"] by simp

lemma sb_dec_lem': "2 ^ k ≤ a ⟹ (a + 2 ^ k) mod (2 * 2 ^ k) ≤ - (2 ^ k) + a"
for a :: int
by (rule sb_dec_lem) simp

lemma one_mod_exp_eq_one [simp]:
"1 mod (2 * 2 ^ n) = (1::int)"
using power_gt1 [of 2 n] by (auto intro: mod_pos_pos_trivial)

lemma mod_2_neq_1_eq_eq_0: "k mod 2 ≠ 1 ⟷ k mod 2 = 0"
for k :: int
by (fact not_mod_2_eq_1_eq_0)

lemma z1pmod2: "(2 * b + 1) mod 2 = (1::int)"
for b :: int
by arith

lemma diff_le_eq': "a - b ≤ c ⟷ a ≤ b + c"
for a b c :: int
by arith

lemma zless2: "0 < (2 :: int)"
by (fact zero_less_numeral)

lemma zless2p: "0 < (2 ^ n :: int)"
by arith

lemma zle2p: "0 ≤ (2 ^ n :: int)"
by arith

lemma p1mod22k': "(1 + 2 * b) mod (2 * 2 ^ n) = 1 + 2 * (b mod 2 ^ n)"
for b :: int
using zle2p by (rule pos_zmod_mult_2)

lemma p1mod22k: "(2 * b + 1) mod (2 * 2 ^ n) = 2 * (b mod 2 ^ n) + 1"
for b :: int

lemma ex_eq_or: "(∃m. n = Suc m ∧ (m = k ∨ P m)) ⟷ n = Suc k ∨ (∃m. n = Suc m ∧ P m)"
by auto

lemma power_minus_simp: "0 < n ⟹ a ^ n = a * a ^ (n - 1)"
by (auto dest: gr0_implies_Suc)

lemma funpow_minus_simp: "0 < n ⟹ f ^^ n = f ∘ f ^^ (n - 1)"
by (auto dest: gr0_implies_Suc)

lemma power_numeral: "a ^ numeral k = a * a ^ (pred_numeral k)"

lemma funpow_numeral [simp]: "f ^^ numeral k = f ∘ f ^^ (pred_numeral k)"

lemma replicate_numeral [simp]: "replicate (numeral k) x = x # replicate (pred_numeral k) x"

lemma rco_alt: "(f ∘ g) ^^ n ∘ f = f ∘ (g ∘ f) ^^ n"
apply (rule ext)
apply (induct n)
done

lemma list_exhaust_size_gt0:
assumes "⋀a list. y = a # list ⟹ P"
shows "0 < length y ⟹ P"
apply (cases y)
apply simp
apply (rule assms)
apply fastforce
done

lemma list_exhaust_size_eq0:
assumes "y = [] ⟹ P"
shows "length y = 0 ⟹ P"
apply (cases y)
apply (rule assms)
apply simp
apply simp
done

lemma size_Cons_lem_eq: "y = xa # list ⟹ size y = Suc k ⟹ size list = k"
by auto

lemmas ls_splits = prod.split prod.split_asm if_split_asm

― ‹simplifications for specific word lengths›

lemmas s2n_ths = n2s_ths [symmetric]

lemma and_len: "xs = ys ⟹ xs = ys ∧ length xs = length ys"
by auto

lemma size_if: "size (if p then xs else ys) = (if p then size xs else size ys)"
by auto

lemma tl_if: "tl (if p then xs else ys) = (if p then tl xs else tl ys)"
by auto

lemma hd_if: "hd (if p then xs else ys) = (if p then hd xs else hd ys)"
by auto

lemma if_Not_x: "(if p then ¬ x else x) = (p = (¬ x))"
by auto

lemma if_x_Not: "(if p then x else ¬ x) = (p = x)"
by auto

lemma if_same_and: "(If p x y ∧ If p u v) = (if p then x ∧ u else y ∧ v)"
by auto

lemma if_same_eq: "(If p x y  = (If p u v)) = (if p then x = u else y = v)"
by auto

lemma if_same_eq_not: "(If p x y = (¬ If p u v)) = (if p then x = (¬ u) else y = (¬ v))"
by auto

― ‹note -- ‹if_Cons› can cause blowup in the size, if ‹p› is complex, so make a simproc›
lemma if_Cons: "(if p then x # xs else y # ys) = If p x y # If p xs ys"
by auto

lemma if_single: "(if xc then [xab] else [an]) = [if xc then xab else an]"
by auto

lemma if_bool_simps:
"If p True y = (p ∨ y) ∧ If p False y = (¬ p ∧ y) ∧
If p y True = (p ⟶ y) ∧ If p y False = (p ∧ y)"
by auto

lemmas if_simps =
if_x_Not if_Not_x if_cancel if_True if_False if_bool_simps

lemmas seqr = eq_reflection [where x = "size w"] for w (* FIXME: delete *)

lemma the_elemI: "y = {x} ⟹ the_elem y = x"
by simp

lemma nonemptyE: "S ≠ {} ⟹ (⋀x. x ∈ S ⟹ R) ⟹ R"
by auto

lemma gt_or_eq_0: "0 < y ∨ 0 = y"
for y :: nat
by arith

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 sum_imp_diff: "j = k + i ⟹ j - i = k"
for k :: nat
by arith

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"]]

lemma nmod2: "n mod 2 = 0 ∨ n mod 2 = 1"
for n :: int
by arith

lemma eme1p:
"even n ⟹ even d ⟹ 0 ≤ d ⟹ (1 + n) mod d = 1 + n mod d" for n d :: int
using emep1 [of n d] by (simp add: ac_simps)

lemma le_diff_eq': "a ≤ c - b ⟷ b + a ≤ c"
for a b c :: int
by arith

lemma less_diff_eq': "a < c - b ⟷ b + a < c"
for a b c :: int
by arith

lemma diff_less_eq': "a - b < c ⟷ a < b + c"
for a b c :: int
by arith

lemmas m1mod22k = mult_pos_pos [OF zless2 zless2p, THEN zmod_minus1]

lemma z1pdiv2: "(2 * b + 1) div 2 = b"
for b :: int
by arith

lemmas zdiv_le_dividend = xtr3 [OF div_by_1 [symmetric] zdiv_mono2,
simplified int_one_le_iff_zero_less, simplified]

lemma axxbyy: "a + m + m = b + n + n ⟹ a = 0 ∨ a = 1 ⟹ b = 0 ∨ b = 1 ⟹ a = b ∧ m = n"
for a b m n :: int
by arith

lemma axxmod2: "(1 + x + x) mod 2 = 1 ∧ (0 + x + x) mod 2 = 0"
for x :: int
by arith

lemma axxdiv2: "(1 + x + x) div 2 = x ∧ (0 + x + x) div 2 = x"
for x :: int
by arith

lemmas iszero_minus =
trans [THEN trans, OF iszero_def neg_equal_0_iff_equal iszero_def [symmetric]]

lemmas rdmods [symmetric] = mod_minus_eq

lemma mod_plus_right: "(a + x) mod m = (b + x) mod m ⟷ a mod m = b mod m"
for a b m x :: nat
by (induct x) (simp_all add: mod_Suc, arith)

lemma nat_minus_mod: "(n - n mod m) mod m = 0"
for m n :: 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], simplified]

mod_mult_eq mod_diff_eq
mod_minus_eq

lemmas push_mods = push_mods' [THEN eq_reflection]
lemmas pull_mods = push_mods [symmetric] rdmods [THEN eq_reflection]

lemma nat_mod_eq: "b < n ⟹ a mod n = b mod n ⟹ a mod n = b"
for a b n :: nat
by (induct a) auto

lemmas nat_mod_eq' = refl [THEN [2] nat_mod_eq]

lemma nat_mod_lem: "0 < n ⟹ b < n ⟷ b mod n = b"
for b n :: nat
apply safe
apply (erule nat_mod_eq')
apply (erule subst)
apply (erule mod_less_divisor)
done

lemma mod_nat_add: "x < z ⟹ y < z ⟹ (x + y) mod z = (if x + y < z then x + y else x + y - z)"
for x y z :: nat
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 < z ⟹ (x - y) mod z = x - y"
for x y :: nat
by (rule nat_mod_eq') arith

lemma int_mod_eq: "0 ≤ b ⟹ b < n ⟹ a mod n = b mod n ⟹ a mod n = b"
for a b n :: int
by (metis mod_pos_pos_trivial)

lemmas int_mod_eq' = mod_pos_pos_trivial (* FIXME delete *)

lemmas int_mod_le = zmod_le_nonneg_dividend (* FIXME: delete *)

"x < z ⟹ y < z ⟹ 0 ≤ y ⟹ 0 ≤ x ⟹ 0 ≤ z ⟹
(x + y) mod z = (if x + y < z then x + y else x + y - z)"
for x y z :: int
by (auto intro: int_mod_eq)

lemma mod_sub_if_z:
"x < z ⟹ y < z ⟹ 0 ≤ y ⟹ 0 ≤ x ⟹ 0 ≤ z ⟹
(x - y) mod z = (if y ≤ x then x - y else x - y + z)"
for x y z :: int
by (auto intro: int_mod_eq)

lemmas zmde = mult_div_mod_eq [symmetric, 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 ⟹ (a + m * n) div m = a div m + n"
for a m n :: int
apply (rule mcl)
prefer 2
apply (erule asm_rl)
done

lemma mod_power_lem: "a > 1 ⟹ a ^ n mod a ^ m = (if m ≤ n then 0 else a ^ n)"
for a :: int

lemma pl_pl_rels: "a + b = c + d ⟹ a ≥ c ∧ b ≤ d ∨ a ≤ c ∧ b ≥ d"
for a b c 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"
for k m :: nat
by arith

lemma pl_pl_mm: "a + b = c + d ⟹ a - c = d - b"
for a b c d :: nat
by arith

lemmas pl_pl_mm' = add.commute [THEN [2] trans, THEN pl_pl_mm]

lemmas dme = div_mult_mod_eq
lemmas dtle = div_times_less_eq_dividend
lemmas th2 = order_trans [OF order_refl [THEN [2] mult_le_mono] div_times_less_eq_dividend]

lemma td_gal: "0 < c ⟹ a ≥ b * c ⟷ a div c ≥ b"
for a b c :: 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]

lemmas div_mult_le = div_times_less_eq_dividend

lemmas sdl = div_nat_eqI

lemma given_quot: "f > 0 ⟹ (f * l + (f - 1)) div f = l"
for f l :: nat
by (rule div_nat_eqI) (simp_all)

lemma given_quot_alt: "f > 0 ⟹ (l * f + f - Suc 0) div f = l"
for f l :: nat
apply (frule given_quot)
apply (rule trans)
prefer 2
apply (erule asm_rl)
apply (rule_tac f="λn. n div f" in arg_cong)
done

lemma diff_mod_le: "a < d ⟹ b dvd d ⟹ a - a mod b ≤ d - b"
for a b d :: nat
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 minus_mod_eq_mult_div [symmetric, 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"
for b c w :: int
apply (rule mult_right_mono)
apply (erule (1) mult_right_less_imp_less)
apply assumption
done

lemma less_le_mult: "w * c < b * c ⟹ 0 ≤ c ⟹ w * c + c ≤ b * c"
for b c w :: int
using less_le_mult' [of w c b] by (simp add: algebra_simps)

lemmas less_le_mult_minus = iffD2 [OF le_diff_eq less_le_mult,
simplified left_diff_distrib]

lemma gen_minus: "0 < n ⟹ f n = f (Suc (n - 1))"
by auto

lemma mpl_lem: "j ≤ i ⟹ k < j ⟹ i - j + k < i"
for i j k :: nat
by arith

lemma nonneg_mod_div: "0 ≤ a ⟹ 0 ≤ b ⟹ 0 ≤ (a mod b) ∧ 0 ≤ a div b"
for a b :: int
by (cases "b = 0") (auto intro: pos_imp_zdiv_nonneg_iff [THEN iffD2])

declare iszero_0 [intro]

lemma min_pm [simp]: "min a b + (a - b) = a"
for a b :: nat
by arith

lemma min_pm1 [simp]: "a - b + min a b = a"
for a b :: nat
by arith

lemma rev_min_pm [simp]: "min b a + (a - b) = a"
for a b :: nat
by arith

lemma rev_min_pm1 [simp]: "a - b + min b a = a"
for a b :: nat
by arith

lemma min_minus [simp]: "min m (m - k) = m - k"
for m k :: nat
by arith

lemma min_minus' [simp]: "min (m - k) m = m - k"
for m k :: nat
by arith

lemmas m2pths = pos_mod_sign pos_mod_bound [OF zless2p]

end
```