Properties of the binary representation of integers

@[simp]

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

@[simp]

theorem PosNum.cast_one' {α : Type u_1} [One α] [Add α] : ↑one = 1

@[simp]

theorem PosNum.cast_bit0 {α : Type u_1} [One α] [Add α] (n : PosNum) : ↑n.bit0 = ↑n + ↑n

@[simp]

theorem PosNum.cast_bit1 {α : Type u_1} [One α] [Add α] (n : PosNum) : ↑n.bit1 = ↑n + ↑n + 1

@[simp]

theorem PosNum.cast_to_nat {α : Type u_1} [AddMonoidWithOne α] (n : PosNum) : ↑↑n = ↑n

theorem PosNum.to_nat_to_int (n : PosNum) : ↑↑n = ↑n

@[simp]

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

theorem PosNum.succ_to_nat (n : PosNum) : ↑n.succ = ↑n + 1

theorem PosNum.one_add (n : PosNum) : 1 + n = n.succ

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

theorem PosNum.add_to_nat (m n : PosNum) : ↑(m + n) = ↑m + ↑n

theorem PosNum.add_succ (m n : PosNum) : m + n.succ = (m + n).succ

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

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

theorem PosNum.mul_to_nat (m n : PosNum) : ↑(m * n) = ↑m * ↑n

theorem PosNum.to_nat_pos (n : PosNum) : 0 < ↑n

theorem PosNum.cmp_to_nat_lemma {m n : PosNum} : ↑m < ↑n → ↑m.bit1 < ↑n.bit0

theorem PosNum.cmp_swap (m n : PosNum) : (m.cmp n).swap = n.cmp m

theorem PosNum.cmp_to_nat (m n : PosNum) : Ordering.casesOn (m.cmp n) (↑m < ↑n) (m = n) (↑n < ↑m)

theorem PosNum.lt_to_nat {m n : PosNum} : ↑m < ↑n ↔ m < n

theorem PosNum.le_to_nat {m n : PosNum} : ↑m ≤ ↑n ↔ m ≤ n

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

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

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

theorem Num.add_succ (m n : Num) : m + n.succ = (m + n).succ

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

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

@[simp]

theorem Num.ofNat'_zero : ofNat' 0 = 0

theorem Num.ofNat'_bit (b : Bool) (n : ℕ) : ofNat' (Nat.bit b n) = (bif b then Num.bit1 else Num.bit0) (ofNat' n)

@[simp]

theorem Num.ofNat'_one : ofNat' 1 = 1

theorem Num.bit1_succ (n : Num) : n.bit1.succ = n.succ.bit0

theorem Num.ofNat'_succ {n : ℕ} : ofNat' (n + 1) = ofNat' n + 1

@[simp]

theorem Num.add_ofNat' (m n : ℕ) : ofNat' (m + n) = ofNat' m + ofNat' n

@[simp]

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

@[simp]

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

@[simp]

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

@[simp]

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

theorem Num.succ'_to_nat (n : Num) : ↑n.succ' = ↑n + 1

theorem Num.succ_to_nat (n : Num) : ↑n.succ = ↑n + 1

@[simp]

theorem Num.cast_to_nat {α : Type u_1} [AddMonoidWithOne α] (n : Num) : ↑↑n = ↑n

theorem Num.add_to_nat (m n : Num) : ↑(m + n) = ↑m + ↑n

theorem Num.mul_to_nat (m n : Num) : ↑(m * n) = ↑m * ↑n

theorem Num.cmp_to_nat (m n : Num) : Ordering.casesOn (m.cmp n) (↑m < ↑n) (m = n) (↑n < ↑m)

theorem Num.lt_to_nat {m n : Num} : ↑m < ↑n ↔ m < n

theorem Num.le_to_nat {m n : Num} : ↑m ≤ ↑n ↔ m ≤ n

@[simp]

theorem PosNum.of_to_nat' (n : PosNum) : Num.ofNat' ↑n = Num.pos n

@[simp]

theorem Num.of_to_nat' (n : Num) : ofNat' ↑n = n

theorem Num.toNat_injective : Function.Injective castNum

theorem Num.to_nat_inj {m n : Num} : ↑m = ↑n ↔ m = n

def Num.transfer_rw : Lean.ParserDescr

This tactic tries to turn an (in)equality about Nums to one about Nats by rewriting.

example (n : Num) (m : Num) : n ≤ n + m := by
  transfer_rw
  exact Nat.le_add_right _ _

Equations

• Num.transfer_rw = Lean.ParserDescr.node `Num.transfer_rw 1024 (Lean.ParserDescr.nonReservedSymbol "transfer_rw" false)

def Num.transfer : Lean.ParserDescr

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

example (n : Num) (m : Num) : n ≤ n + m := by transfer

Equations

• Num.transfer = Lean.ParserDescr.node `Num.transfer 1024 (Lean.ParserDescr.nonReservedSymbol "transfer" false)

instance Num.addMonoid : AddMonoid Num

Equations

• One or more equations did not get rendered due to their size.

instance Num.addMonoidWithOne : AddMonoidWithOne Num

Equations

• Num.addMonoidWithOne = { natCast := Num.ofNat', toAddMonoid := Num.addMonoid, one := 1, natCast_zero := Num.ofNat'_zero, natCast_succ := ⋯ }

instance Num.commSemiring : CommSemiring Num

Equations

• One or more equations did not get rendered due to their size.

instance Num.partialOrder : PartialOrder Num

Equations

• One or more equations did not get rendered due to their size.

instance Num.isOrderedCancelAddMonoid : IsOrderedCancelAddMonoid Num

instance Num.linearOrder : LinearOrder Num

Equations

• One or more equations did not get rendered due to their size.

instance Num.isStrictOrderedRing : IsStrictOrderedRing Num

theorem Num.add_of_nat (m n : ℕ) : ↑(m + n) = ↑m + ↑n

theorem Num.to_nat_to_int (n : Num) : ↑↑n = ↑n

@[simp]

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

theorem Num.to_of_nat (n : ℕ) : ↑↑n = n

@[simp]

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

theorem Num.of_nat_inj {m n : ℕ} : ↑m = ↑n ↔ m = n

@[simp]

theorem Num.of_to_nat (n : Num) : ↑↑n = n

theorem Num.dvd_to_nat (m n : Num) : ↑m ∣ ↑n ↔ m ∣ n

@[simp]

theorem PosNum.of_to_nat (n : PosNum) : ↑↑n = Num.pos n

theorem PosNum.to_nat_inj {m n : PosNum} : ↑m = ↑n ↔ m = n

theorem PosNum.pred'_to_nat (n : PosNum) : ↑n.pred' = (↑n).pred

@[simp]

theorem PosNum.pred'_succ' (n : Num) : n.succ'.pred' = n

@[simp]

theorem PosNum.succ'_pred' (n : PosNum) : n.pred'.succ' = n

instance PosNum.dvd : Dvd PosNum

Equations

• PosNum.dvd = { dvd := fun (m n : PosNum) => Num.pos m ∣ Num.pos n }

theorem PosNum.dvd_to_nat {m n : PosNum} : ↑m ∣ ↑n ↔ m ∣ n

theorem PosNum.size_to_nat (n : PosNum) : ↑n.size = (↑n).size

theorem PosNum.size_eq_natSize (n : PosNum) : ↑n.size = n.natSize

theorem PosNum.natSize_to_nat (n : PosNum) : n.natSize = (↑n).size

theorem PosNum.natSize_pos (n : PosNum) : 0 < n.natSize

def PosNum.transfer_rw : Lean.ParserDescr

This tactic tries to turn an (in)equality about PosNums to one about Nats by rewriting.

example (n : PosNum) (m : PosNum) : n ≤ n + m := by
  transfer_rw
  exact Nat.le_add_right _ _

Equations

• PosNum.transfer_rw = Lean.ParserDescr.node `PosNum.transfer_rw 1024 (Lean.ParserDescr.nonReservedSymbol "transfer_rw" false)

def PosNum.transfer : Lean.ParserDescr

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

example (n : PosNum) (m : PosNum) : n ≤ n + m := by transfer

Equations

• PosNum.transfer = Lean.ParserDescr.node `PosNum.transfer 1024 (Lean.ParserDescr.nonReservedSymbol "transfer" false)

instance PosNum.addCommSemigroup : AddCommSemigroup PosNum

Equations

• PosNum.addCommSemigroup = { add := fun (x1 x2 : PosNum) => x1 + x2, add_assoc := PosNum.addCommSemigroup._proof_43, add_comm := PosNum.addCommSemigroup._proof_44 }

instance PosNum.commMonoid : CommMonoid PosNum

Equations

• One or more equations did not get rendered due to their size.

instance PosNum.distrib : Distrib PosNum

Equations

• PosNum.distrib = { mul := fun (x1 x2 : PosNum) => x1 * x2, add := fun (x1 x2 : PosNum) => x1 + x2, left_distrib := PosNum.distrib._proof_51, right_distrib := PosNum.distrib._proof_52 }

instance PosNum.linearOrder : LinearOrder PosNum

Equations

• One or more equations did not get rendered due to their size.

@[simp]

theorem PosNum.cast_to_num (n : PosNum) : ↑n = Num.pos n

@[simp]

theorem PosNum.bit_to_nat (b : Bool) (n : PosNum) : ↑(bit b n) = Nat.bit b ↑n

@[simp]

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

@[simp]

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

@[simp]

theorem PosNum.cast_inj {α : Type u_1} [AddMonoidWithOne α] [CharZero α] {m n : PosNum} : ↑m = ↑n ↔ m = n

@[simp]

theorem PosNum.one_le_cast {α : Type u_1} [Semiring α] [PartialOrder α] [IsStrictOrderedRing α] (n : PosNum) : 1 ≤ ↑n

@[simp]

theorem PosNum.cast_pos {α : Type u_1} [Semiring α] [PartialOrder α] [IsStrictOrderedRing α] (n : PosNum) : 0 < ↑n

@[simp]

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

@[simp]

theorem PosNum.cmp_eq (m n : PosNum) : m.cmp n = Ordering.eq ↔ m = n

@[simp]

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

@[simp]

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

theorem Num.bit_to_nat (b : Bool) (n : Num) : ↑(bit b n) = Nat.bit b ↑n

theorem Num.cast_succ' {α : Type u_1} [AddMonoidWithOne α] (n : Num) : ↑n.succ' = ↑n + 1

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

@[simp]

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

@[simp]

theorem Num.cast_bit0 {α : Type u_1} [NonAssocSemiring α] (n : Num) : ↑n.bit0 = 2 * ↑n

@[simp]

theorem Num.cast_bit1 {α : Type u_1} [NonAssocSemiring α] (n : Num) : ↑n.bit1 = 2 * ↑n + 1

@[simp]

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

theorem Num.size_to_nat (n : Num) : ↑n.size = (↑n).size

theorem Num.size_eq_natSize (n : Num) : ↑n.size = n.natSize

theorem Num.natSize_to_nat (n : Num) : n.natSize = (↑n).size

@[simp]

theorem Num.ofNat'_eq (n : ℕ) : ofNat' n = ↑n

theorem Num.zneg_toZNum (n : Num) : -n.toZNum = n.toZNumNeg

theorem Num.zneg_toZNumNeg (n : Num) : -n.toZNumNeg = n.toZNum

theorem Num.toZNum_inj {m n : Num} : m.toZNum = n.toZNum ↔ m = n

@[simp]

theorem Num.cast_toZNum {α : Type u_1} [Zero α] [One α] [Add α] [Neg α] (n : Num) : ↑n.toZNum = ↑n

@[simp]

theorem Num.cast_toZNumNeg {α : Type u_1} [SubtractionMonoid α] [One α] (n : Num) : ↑n.toZNumNeg = -↑n

@[simp]

theorem Num.add_toZNum (m n : Num) : (m + n).toZNum = m.toZNum + n.toZNum

theorem PosNum.pred_to_nat {n : PosNum} (h : 1 < n) : ↑n.pred = (↑n).pred

theorem PosNum.sub'_one (a : PosNum) : a.sub' 1 = a.pred'.toZNum

theorem PosNum.one_sub' (a : PosNum) : sub' 1 a = a.pred'.toZNumNeg

theorem PosNum.lt_iff_cmp {m n : PosNum} : m < n ↔ m.cmp n = Ordering.lt

theorem PosNum.le_iff_cmp {m n : PosNum} : m ≤ n ↔ m.cmp n ≠ Ordering.gt

theorem Num.pred_to_nat (n : Num) : ↑n.pred = (↑n).pred

theorem Num.ppred_to_nat (n : Num) : castNum <$> n.ppred = (↑n).ppred

theorem Num.cmp_swap (m n : Num) : (m.cmp n).swap = n.cmp m

theorem Num.cmp_eq (m n : Num) : m.cmp n = Ordering.eq ↔ m = n

@[simp]

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

@[simp]

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

@[simp]

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

theorem Num.lt_iff_cmp {m n : Num} : m < n ↔ m.cmp n = Ordering.lt

theorem Num.le_iff_cmp {m n : Num} : m ≤ n ↔ m.cmp n ≠ Ordering.gt

theorem Num.castNum_eq_bitwise {f : Num → Num → Num} {g : Bool → Bool → Bool} (p : PosNum → PosNum → Num) (gff : g false false = false) (f00 : f 0 0 = 0) (f0n : ∀ (n : PosNum), f 0 (pos n) = bif g false true then pos n else 0) (fn0 : ∀ (n : PosNum), f (pos n) 0 = bif g true false then pos n else 0) (fnn : ∀ (m n : PosNum), f (pos m) (pos n) = p m n) (p11 : p 1 1 = bif g true true then 1 else 0) (p1b : ∀ (b : Bool) (n : PosNum), p 1 (PosNum.bit b n) = bit (g true b) (bif g false true then pos n else 0)) (pb1 : ∀ (a : Bool) (m : PosNum), p (PosNum.bit a m) 1 = bit (g a true) (bif g true false then pos m else 0)) (pbb : ∀ (a b : Bool) (m n : PosNum), p (PosNum.bit a m) (PosNum.bit b n) = bit (g a b) (p m n)) (m n : Num) : ↑(f m n) = Nat.bitwise g ↑m ↑n

@[simp]

theorem Num.castNum_or (m n : Num) : ↑(m ||| n) = ↑m ||| ↑n

@[simp]

theorem Num.castNum_and (m n : Num) : ↑(m &&& n) = ↑m &&& ↑n

@[simp]

theorem Num.castNum_ldiff (m n : Num) : ↑(m.ldiff n) = (↑m).ldiff ↑n

@[simp]

theorem Num.castNum_xor (m n : Num) : ↑(m ^^^ n) = ↑m ^^^ ↑n

@[simp]

theorem Num.castNum_shiftLeft (m : Num) (n : ℕ) : ↑(m <<< n) = ↑m <<< n

@[simp]

theorem Num.castNum_shiftRight (m : Num) (n : ℕ) : ↑(m >>> n) = ↑m >>> n

@[simp]

theorem Num.castNum_testBit (m : Num) (n : ℕ) : m.testBit n = (↑m).testBit n

def Int.ofSnum : SNum → ℤ

Cast a SNum to the corresponding integer.

Equations

• Int.ofSnum = SNum.rec' (fun (a : Bool) => bif a then -1 else 0) fun (a : Bool) (_p : SNum) (IH : ℤ) => bif a then 2 * IH + 1 else 2 * IH

instance Int.snumCoe : Coe SNum ℤ

Equations

• Int.snumCoe = { coe := Int.ofSnum }

instance SNum.lt : LT SNum

Equations

• SNum.lt = { lt := fun (a b : SNum) => Int.ofSnum a < Int.ofSnum b }

instance SNum.le : LE SNum

Equations

• SNum.le = { le := fun (a b : SNum) => Int.ofSnum a ≤ Int.ofSnum b }