Properties of the ZNum representation of integers

This file was split from Mathlib/Data/Num/Lemmas.lean to keep the former under 1500 lines.

@[simp]

theorem ZNum.cast_zero {α : Type u_1} [Zero α] [One α] [Add α] [Neg α] : ↑0 = 0

@[simp]

theorem ZNum.cast_zero' {α : Type u_1} [Zero α] [One α] [Add α] [Neg α] : ↑zero = 0

@[simp]

theorem ZNum.cast_one {α : Type u_1} [Zero α] [One α] [Add α] [Neg α] : ↑1 = 1

@[simp]

theorem ZNum.cast_pos {α : Type u_1} [Zero α] [One α] [Add α] [Neg α] (n : PosNum) : ↑(pos n) = ↑n

@[simp]

theorem ZNum.cast_neg {α : Type u_1} [Zero α] [One α] [Add α] [Neg α] (n : PosNum) : ↑(neg n) = -↑n

@[simp]

theorem ZNum.cast_zneg {α : Type u_1} [SubtractionMonoid α] [One α] (n : ZNum) : ↑(-n) = -↑n

theorem ZNum.neg_zero : -0 = 0

theorem ZNum.zneg_pos (n : PosNum) : -pos n = neg n

theorem ZNum.zneg_neg (n : PosNum) : -neg n = pos n

theorem ZNum.zneg_zneg (n : ZNum) : - -n = n

theorem ZNum.zneg_bit1 (n : ZNum) : -n.bit1 = (-n).bitm1

theorem ZNum.zneg_bitm1 (n : ZNum) : -n.bitm1 = (-n).bit1

theorem ZNum.zneg_succ (n : ZNum) : -n.succ = (-n).pred

theorem ZNum.zneg_pred (n : ZNum) : -n.pred = (-n).succ

@[simp]

theorem ZNum.abs_to_nat (n : ZNum) : ↑n.abs = (↑n).natAbs

@[simp]

theorem ZNum.abs_toZNum (n : Num) : n.toZNum.abs = n

@[simp]

theorem ZNum.cast_to_int {α : Type u_1} [AddGroupWithOne α] (n : ZNum) : ↑↑n = ↑n

theorem ZNum.bit0_of_bit0 (n : ZNum) : n + n = n.bit0

theorem ZNum.bit1_of_bit1 (n : ZNum) : n + n + 1 = n.bit1

@[simp]

theorem ZNum.cast_bit0 {α : Type u_1} [AddGroupWithOne α] (n : ZNum) : ↑n.bit0 = ↑n + ↑n

@[simp]

theorem ZNum.cast_bit1 {α : Type u_1} [AddGroupWithOne α] (n : ZNum) : ↑n.bit1 = ↑n + ↑n + 1

@[simp]

theorem ZNum.cast_bitm1 {α : Type u_1} [AddGroupWithOne α] (n : ZNum) : ↑n.bitm1 = ↑n + ↑n - 1

theorem ZNum.add_zero (n : ZNum) : n + 0 = n

theorem ZNum.zero_add (n : ZNum) : 0 + n = n

theorem ZNum.add_one (n : ZNum) : n + 1 = n.succ

theorem PosNum.cast_to_znum (n : PosNum) : ↑n = ZNum.pos n

theorem PosNum.cast_sub' {α : Type u_1} [AddGroupWithOne α] (m n : PosNum) : ↑(m.sub' n) = ↑m - ↑n

theorem PosNum.to_nat_eq_succ_pred (n : PosNum) : ↑n = ↑n.pred' + 1

theorem PosNum.to_int_eq_succ_pred (n : PosNum) : ↑n = ↑↑n.pred' + 1

@[simp]

theorem Num.cast_sub' {α : Type u_1} [AddGroupWithOne α] (m n : Num) : ↑(m.sub' n) = ↑m - ↑n

theorem Num.toZNum_succ (n : Num) : n.succ.toZNum = n.toZNum.succ

theorem Num.toZNumNeg_succ (n : Num) : n.succ.toZNumNeg = n.toZNumNeg.pred

@[simp]

theorem Num.pred_succ (n : ZNum) : n.pred.succ = n

theorem Num.succ_ofInt' (n : ℤ) : ZNum.ofInt' (n + 1) = ZNum.ofInt' n + 1

theorem Num.ofInt'_toZNum (n : ℕ) : (↑n).toZNum = ZNum.ofInt' ↑n

theorem Num.mem_ofZNum' {m : Num} {n : ZNum} : m ∈ ofZNum' n ↔ n = m.toZNum

theorem Num.ofZNum'_toNat (n : ZNum) : castNum <$> ofZNum' n = (↑n).toNat?

theorem Num.ofZNum_toNat (n : ZNum) : ↑(ofZNum n) = (↑n).toNat

@[simp]

theorem Num.cast_ofZNum {α : Type u_1} [AddMonoidWithOne α] (n : ZNum) : ↑(ofZNum n) = ↑(↑n).toNat

@[simp]

theorem Num.sub_to_nat (m n : Num) : ↑(m - n) = ↑m - ↑n

@[simp]

theorem ZNum.cast_add {α : Type u_1} [AddGroupWithOne α] (m n : ZNum) : ↑(m + n) = ↑m + ↑n

@[simp]

theorem ZNum.cast_succ {α : Type u_1} [AddGroupWithOne α] (n : ZNum) : ↑n.succ = ↑n + 1

@[simp]

theorem ZNum.mul_to_int (m n : ZNum) : ↑(m * n) = ↑m * ↑n

theorem ZNum.cast_mul {α : Type u_1} [NonAssocRing α] (m n : ZNum) : ↑(m * n) = ↑m * ↑n

theorem ZNum.ofInt'_neg (n : ℤ) : ofInt' (-n) = -ofInt' n

theorem ZNum.of_to_int' (n : ZNum) : ofInt' ↑n = n

theorem ZNum.to_int_inj {m n : ZNum} : ↑m = ↑n ↔ m = n

theorem ZNum.cmp_to_int (m n : ZNum) : Ordering.casesOn (m.cmp n) (↑m < ↑n) (m = n) (↑n < ↑m)

theorem ZNum.lt_to_int {m n : ZNum} : ↑m < ↑n ↔ m < n

theorem ZNum.le_to_int {m n : ZNum} : ↑m ≤ ↑n ↔ m ≤ n

@[simp]

theorem ZNum.cast_lt {α : Type u_1} [Ring α] [PartialOrder α] [IsStrictOrderedRing α] {m n : ZNum} : ↑m < ↑n ↔ m < n

@[simp]

theorem ZNum.cast_le {α : Type u_1} [Ring α] [LinearOrder α] [IsStrictOrderedRing α] {m n : ZNum} : ↑m ≤ ↑n ↔ m ≤ n

@[simp]

theorem ZNum.cast_inj {α : Type u_1} [Ring α] [PartialOrder α] [IsStrictOrderedRing α] {m n : ZNum} : ↑m = ↑n ↔ m = n

def ZNum.transfer_rw : Lean.ParserDescr

This tactic tries to turn an (in)equality about ZNums to one about Ints by rewriting.

example (n : ZNum) (m : ZNum) : n ≤ n + m * m := by
  transfer_rw
  exact le_add_of_nonneg_right (mul_self_nonneg _)

def ZNum.transfer : Lean.ParserDescr

This tactic tries to prove (in)equalities about ZNums by transferring them to the Int world and then trying to call simp.

example (n : ZNum) (m : ZNum) : n ≤ n + m * m := by
  transfer
  exact mul_self_nonneg _

instance ZNum.linearOrder : LinearOrder ZNum

instance ZNum.addMonoid : AddMonoid ZNum

instance ZNum.addCommGroup : AddCommGroup ZNum

instance ZNum.addMonoidWithOne : AddMonoidWithOne ZNum

instance ZNum.commRing : CommRing ZNum

instance ZNum.nontrivial : Nontrivial ZNum

instance ZNum.zeroLEOneClass : ZeroLEOneClass ZNum

instance ZNum.isOrderedAddMonoid : IsOrderedAddMonoid ZNum

instance ZNum.isStrictOrderedRing : IsStrictOrderedRing ZNum

@[simp]

theorem ZNum.cast_sub {α : Type u_1} [AddCommGroupWithOne α] (m n : ZNum) : ↑(m - n) = ↑m - ↑n

theorem ZNum.neg_of_int (n : ℤ) : ↑(-n) = -↑n

@[simp]

theorem ZNum.ofInt'_eq (n : ℤ) : ofInt' n = ↑n

@[simp]

theorem ZNum.of_nat_toZNum (n : ℕ) : (↑n).toZNum = ↑n

@[simp]

theorem ZNum.of_to_int (n : ZNum) : ↑↑n = n

theorem ZNum.to_of_int (n : ℤ) : ↑↑n = n

@[simp]

theorem ZNum.of_nat_toZNumNeg (n : ℕ) : (↑n).toZNumNeg = -↑n

@[simp]

theorem ZNum.of_intCast {α : Type u_1} [AddGroupWithOne α] (n : ℤ) : ↑↑n = ↑n

@[simp]

theorem ZNum.of_natCast {α : Type u_1} [AddGroupWithOne α] (n : ℕ) : ↑↑n = ↑n

@[simp]

theorem ZNum.dvd_to_int (m n : ZNum) : ↑m ∣ ↑n ↔ m ∣ n

theorem PosNum.divMod_to_nat_aux {n d : PosNum} {q r : Num} (h₁ : ↑r + ↑d * (↑q + ↑q) = ↑n) (h₂ : ↑r < 2 * ↑d) : ↑(d.divModAux q r).2 + ↑d * ↑(d.divModAux q r).1 = ↑n ∧ ↑(d.divModAux q r).2 < ↑d

theorem PosNum.divMod_to_nat (d n : PosNum) : ↑n / ↑d = ↑(d.divMod n).1 ∧ ↑n % ↑d = ↑(d.divMod n).2

@[simp]

theorem PosNum.div'_to_nat (n d : PosNum) : ↑(n.div' d) = ↑n / ↑d

@[simp]

theorem PosNum.mod'_to_nat (n d : PosNum) : ↑(n.mod' d) = ↑n % ↑d

@[simp]

theorem Num.div_zero (n : Num) : n / 0 = 0

@[simp]

theorem Num.div_to_nat (n d : Num) : ↑(n / d) = ↑n / ↑d

@[simp]

theorem Num.mod_zero (n : Num) : n % 0 = n

@[simp]

theorem Num.mod_to_nat (n d : Num) : ↑(n % d) = ↑n % ↑d

theorem Num.gcd_to_nat_aux {n : ℕ} {a b : Num} : a ≤ b → (a * b).natSize ≤ n → ↑(gcdAux n a b) = (↑a).gcd ↑b

@[simp]

theorem Num.gcd_to_nat (a b : Num) : ↑(a.gcd b) = (↑a).gcd ↑b

theorem Num.dvd_iff_mod_eq_zero {m n : Num} : m ∣ n ↔ n % m = 0

instance Num.decidableDvd : DecidableRel fun (x1 x2 : Num) => x1 ∣ x2

instance PosNum.decidableDvd : DecidableRel fun (x1 x2 : PosNum) => x1 ∣ x2

@[simp]

theorem ZNum.div_zero (n : ZNum) : n / 0 = 0

@[simp]

theorem ZNum.div_to_int (n d : ZNum) : ↑(n / d) = ↑n / ↑d

@[simp]

theorem ZNum.mod_to_int (n d : ZNum) : ↑(n % d) = ↑n % ↑d

@[simp]

theorem ZNum.gcd_to_nat (a b : ZNum) : ↑(a.gcd b) = (↑a).gcd ↑b

theorem ZNum.dvd_iff_mod_eq_zero {m n : ZNum} : m ∣ n ↔ n % m = 0

instance ZNum.decidableDvd : DecidableRel fun (x1 x2 : ZNum) => x1 ∣ x2