mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Algebra.Order.Floor.Ring

Goal state before:
pp: R : Type u_2
inst✝³ : Ring R
inst✝² : LinearOrder R
inst✝¹ : FloorRing R
inst✝ : IsOrderedRing R
a b : R
z : ℤ
hz : a - b = ↑z
⊢ a - fract b = ↑(z + ⌊b⌋)

Tactic:
eq_add_of_sub_eq hz,

Goal state after:
R : Type u_2
inst✝³ : Ring R
inst✝² : LinearOrder R
inst✝¹ : FloorRing R
inst✝ : IsOrderedRing R
a b : R
z : ℤ
hz : a - b = ↑z
⊢ ↑z + b - fract b = ↑(z + ⌊b⌋)
Module: Mathlib.Algebra.Order.Floor.Ring

Goal state before:
pp: R : Type u_2
inst✝³ : Ring R
inst✝² : LinearOrder R
inst✝¹ : FloorRing R
inst✝ : IsOrderedRing R
x : R
hx : fract x ≠ 0
⊢ 0 ≤ 1 - fract x ∧ 1 - fract x < 1 ∧ ∃ z, -x - (1 - fract x) = ↑z

Tactic:
constructor

Goal state after:
case left
R : Type u_2
inst✝³ : Ring R
inst✝² : LinearOrder R
inst✝¹ : FloorRing R
inst✝ : IsOrderedRing R
x : R
hx : fract x ≠ 0
⊢ 0 ≤ 1 - fract x

case right
R : Type u_2
inst✝³ : Ring R
inst✝² : LinearOrder R
inst✝¹ : FloorRing R
inst✝ : IsOrderedRing R
x : R
hx : fract x ≠ 0
⊢ 1 - fract x < 1 ∧ ∃ z, -x - (1 - fract x) = ↑z
Module: Mathlib.Algebra.Order.Floor.Ring

Goal state before:
pp: R : Type u_2
inst✝³ : Ring R
inst✝² : LinearOrder R
inst✝¹ : FloorRing R
inst✝ : IsOrderedRing R
x : R
⊢ (∃ z, -x = ↑z) ↔ ∃ z, x = ↑z

Tactic:
constructor

Goal state after:
case mp
R : Type u_2
inst✝³ : Ring R
inst✝² : LinearOrder R
inst✝¹ : FloorRing R
inst✝ : IsOrderedRing R
x : R
⊢ (∃ z, -x = ↑z) → ∃ z, x = ↑z

case mpr
R : Type u_2
inst✝³ : Ring R
inst✝² : LinearOrder R
inst✝¹ : FloorRing R
inst✝ : IsOrderedRing R
x : R
⊢ (∃ z, x = ↑z) → ∃ z, -x = ↑z
Module: Mathlib.Algebra.Order.Floor.Ring

Goal state before:
pp: R : Type u_2
inst✝³ : Ring R
inst✝² : LinearOrder R
inst✝¹ : FloorRing R
inst✝ : IsOrderedRing R
s : Set R
x : R
⊢ (∃ x_1 ∈ s, fract x_1 = x) ↔ ∃ i, (∃ x_1 ∈ s, x_1 - ↑i = x) ∧ x ∈ Ico 0 1

Tactic:
constructor

Goal state after:
case mp
R : Type u_2
inst✝³ : Ring R
inst✝² : LinearOrder R
inst✝¹ : FloorRing R
inst✝ : IsOrderedRing R
s : Set R
x : R
⊢ (∃ x_1 ∈ s, fract x_1 = x) → ∃ i, (∃ x_1 ∈ s, x_1 - ↑i = x) ∧ x ∈ Ico 0 1

case mpr
R : Type u_2
inst✝³ : Ring R
inst✝² : LinearOrder R
inst✝¹ : FloorRing R
inst✝ : IsOrderedRing R
s : Set R
x : R
⊢ (∃ i, (∃ x_1 ∈ s, x_1 - ↑i = x) ∧ x ∈ Ico 0 1) → ∃ x_1 ∈ s, fract x_1 = x
Module: Mathlib.Data.Rat.Floor

Goal state before:
pp: n d : ℕ
⊢ ⌊↑n / ↑d⌋₊ = n / d

Tactic:
rw [← Int.ofNat_inj, Int.natCast_floor_eq_floor (by positivity)]

Goal state after:
n d : ℕ
⊢ ⌊↑n / ↑d⌋ = ↑(n / d)
Module: Mathlib.Data.Rat.Floor

Goal state before:
pp: α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : FloorRing α
n d : ℕ
inv : Invertible ↑d
⊢ IsNat ⌊↑n * ⅟↑d⌋ (n / d)

Tactic:
constructor

Goal state after:
α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : FloorRing α
n d : ℕ
inv : Invertible ↑d
⊢ ⌊↑n * ⅟↑d⌋ = ↑(n / d)
Module: Mathlib.Algebra.Order.Floor.Ring

Goal state before:
pp: R : Type u_2
inst✝³ : Ring R
inst✝² : LinearOrder R
inst✝¹ : FloorRing R
inst✝ : IsOrderedRing R
n : ℕ
a : ℤ
⊢ ⌈↑n⌉ ≤ a ↔ ↑n ≤ a

Tactic:
by rw [ceil_le, ← cast_natCast, cast_le]

Goal state after:
Module: Mathlib.Data.Rat.Floor

Goal state before:
pp: α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : FloorRing α
n d : ℕ
inv : Invertible ↑d
⊢ IsInt ⌊↑(negOfNat n) * ⅟↑d⌋ (negOfNat (-(-↑n / ↑d)).toNat)

Tactic:
constructor

Goal state after:
α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : FloorRing α
n d : ℕ
inv : Invertible ↑d
⊢ ⌊↑(negOfNat n) * ⅟↑d⌋ = ↑(negOfNat (-(-↑n / ↑d)).toNat)
Module: Mathlib.Data.Rat.Floor

Goal state before:
pp: α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : FloorRing α
n d : ℕ
inv : Invertible ↑d
⊢ IsNat ⌈↑n * ⅟↑d⌉ (-(-↑n / ↑d)).toNat

Tactic:
constructor

Goal state after:
α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : FloorRing α
n d : ℕ
inv : Invertible ↑d
⊢ ⌈↑n * ⅟↑d⌉ = ↑(-(-↑n / ↑d)).toNat
Module: Mathlib.Data.Rat.Floor

Goal state before:
pp: α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : FloorRing α
n d : ℕ
inv : Invertible ↑d
⊢ IsInt ⌈↑(negOfNat n) * ⅟↑d⌉ (negOfNat (n / d))

Tactic:
constructor

Goal state after:
α : Type u_1
inst✝³ : Field α
inst✝² : LinearOrder α
inst✝¹ : IsStrictOrderedRing α
inst✝ : FloorRing α
n d : ℕ
inv : Invertible ↑d
⊢ ⌈↑(negOfNat n) * ⅟↑d⌉ = ↑(negOfNat (n / d))
Module: Mathlib.Algebra.Order.Floor.Ring

Goal state before:
pp: R : Type u_2
inst✝³ : Ring R
inst✝² : LinearOrder R
inst✝¹ : FloorRing R
inst✝ : IsOrderedRing R
a : R
ha : fract a ≠ 0
this : ↑⌈a⌉ = ↑⌊a⌋ + 1
⊢ fract a = a + 1 - (↑⌊a⌋ + 1)

Tactic:
← self_sub_fract

Goal state after:
R : Type u_2
inst✝³ : Ring R
inst✝² : LinearOrder R
inst✝¹ : FloorRing R
inst✝ : IsOrderedRing R
a : R
ha : fract a ≠ 0
this : ↑⌈a⌉ = ↑⌊a⌋ + 1
⊢ fract a = a + 1 - (a - fract a + 1)
Module: Mathlib.Algebra.AddConstMap.Basic

Goal state before:
pp: F : Type u_1
G : Type u_2
H : Type u_3
inst✝³ : FunLike F G H
b : H
inst✝² : AddGroupWithOne G
inst✝¹ : AddGroup H
inst✝ : AddConstMapClass F G H 1 b
f : F
x : G
n : ℤ
⊢ f (x + ↑n) = f (x + n • 1)

Tactic:
zsmul_one

Goal state after:
F : Type u_1
G : Type u_2
H : Type u_3
inst✝³ : FunLike F G H
b : H
inst✝² : AddGroupWithOne G
inst✝¹ : AddGroup H
inst✝ : AddConstMapClass F G H 1 b
f : F
x : G
n : ℤ
⊢ f (x + ↑n) = f (x + ↑n)
Module: Mathlib.Data.Rat.Floor

Goal state before:
pp: R : Type u_3
inst✝³ : Field R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : FloorRing R
n : ℤ
d : ℕ
inv : Invertible ↑d
⊢ IsInt (round (↑n * ⅟↑d)) (round (↑n / ↑d))

Tactic:
constructor

Goal state after:
R : Type u_3
inst✝³ : Field R
inst✝² : LinearOrder R
inst✝¹ : IsStrictOrderedRing R
inst✝ : FloorRing R
n : ℤ
d : ℕ
inv : Invertible ↑d
⊢ round (↑n * ⅟↑d) = ↑(round (↑n / ↑d))
Module: Mathlib.Algebra.AddConstMap.Basic

Goal state before:
pp: F : Type u_1
G : Type u_2
H : Type u_3
inst✝³ : FunLike F G H
b : H
inst✝² : AddGroupWithOne G
inst✝¹ : AddGroup H
inst✝ : AddConstMapClass F G H 1 b
f : F
x : G
n : ℤ
⊢ f (x - ↑n) = f (x - n • 1)

Tactic:
zsmul_one

Goal state after:
F : Type u_1
G : Type u_2
H : Type u_3
inst✝³ : FunLike F G H
b : H
inst✝² : AddGroupWithOne G
inst✝¹ : AddGroup H
inst✝ : AddConstMapClass F G H 1 b
f : F
x : G
n : ℤ
⊢ f (x - ↑n) = f (x - ↑n)
Module: Mathlib.Algebra.Order.Floor.Ring

Goal state before:
pp: k : Type u_4
inst✝³ : Field k
inst✝² : LinearOrder k
inst✝¹ : IsOrderedRing k
inst✝ : FloorRing k
a b : k
hb₀ : 0 < b
hb : b < 1
hba : ↑⌈b / (1 - b)⌉ ≤ a
⊢ b * (↑⌊a⌋ + 1) ≤ ↑⌊a⌋

Tactic:
by
      rwa [_root_.mul_add_one, ← le_sub_iff_add_le', ← one_sub_mul, ← div_le_iff₀' (by linarith),
        ← ceil_le, le_floor]

Goal state after:
Module: Mathlib.Algebra.AddConstMap.Basic

Goal state before:
pp: F : Type u_1
G : Type u_2
H : Type u_3
inst✝³ : FunLike F G H
b : H
inst✝² : AddCommGroupWithOne G
inst✝¹ : AddGroup H
inst✝ : AddConstMapClass F G H 1 b
f : F
n : ℤ
x : G
⊢ f (↑n + x) = f (n • 1 + x)

Tactic:
zsmul_one

Goal state after:
F : Type u_1
G : Type u_2
H : Type u_3
inst✝³ : FunLike F G H
b : H
inst✝² : AddCommGroupWithOne G
inst✝¹ : AddGroup H
inst✝ : AddConstMapClass F G H 1 b
f : F
n : ℤ
x : G
⊢ f (↑n + x) = f (↑n + x)
Module: Mathlib.Data.Rat.Floor

Goal state before:
pp: q : ℚ
⊢ 0 < 1 - fract q

Tactic:
have : fract q < 1 := fract_lt_one q

Goal state after:
q : ℚ
this : fract q < 1
⊢ 0 < 1 - fract q
Module: Mathlib.Data.Rat.Floor

Goal state before:
pp: q : ℚ
q_pos : 0 < q
q_num_pos : 0 < q.num
q_num_abs_eq_q_num : ↑q.num.natAbs = q.num
q_inv : ℚ := ↑q.den / ↑q.num
q_inv_def : q_inv = ↑q.den / ↑q.num
q_inv_eq : q⁻¹ = q_inv
q_inv_num_denom_ineq : q⁻¹.num - ⌊q⁻¹⌋ * ↑q⁻¹.den < ↑q⁻¹.den
coprime_q_denom_q_num : (↑q.den).natAbs.Coprime q.num.natAbs
this : (↑q.den).natAbs = q.den
⊢ (↑q.den / ↑q.num).num = ↑q.den ∧ (↑q.den / ↑q.num).den = q.num.natAbs

Tactic:
constructor

Goal state after:
case left
q : ℚ
q_pos : 0 < q
q_num_pos : 0 < q.num
q_num_abs_eq_q_num : ↑q.num.natAbs = q.num
q_inv : ℚ := ⋯
q_inv_def : q_inv = ↑q.den / ↑q.num
q_inv_eq : q⁻¹ = q_inv
q_inv_num_denom_ineq : q⁻¹.num - ⌊q⁻¹⌋ * ↑q⁻¹.den < ↑q⁻¹.den
coprime_q_denom_q_num : (↑q.den).natAbs.Coprime q.num.natAbs
this : (↑q.den).natAbs = q.den
⊢ (↑q.den / ↑q.num).num = ↑q.den

case right
q : ℚ
q_pos : 0 < q
q_num_pos : 0 < q.num
q_num_abs_eq_q_num : ↑q.num.natAbs = q.num
q_inv : ℚ := ⋯
q_inv_def : q_inv = ↑q.den / ↑q.num
q_inv_eq : q⁻¹ = q_inv
q_inv_num_denom_ineq : q⁻¹.num - ⌊q⁻¹⌋ * ↑q⁻¹.den < ↑q⁻¹.den
coprime_q_denom_q_num : (↑q.den).natAbs.Coprime q.num.natAbs
this : (↑q.den).natAbs = q.den
⊢ (↑q.den / ↑q.num).den = q.num.natAbs
Module: Mathlib.Algebra.Order.Floor.Ring

Goal state before:
pp: R : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : FloorRing R
a b : R
⊢ Int.cast ⁻¹' Ioo a b = Ioo ⌊a⌋ ⌈b⌉

Tactic:
ext

Goal state after:
R : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : FloorRing R
a b : R
x✝ : ℤ
⊢ x✝ ∈ Int.cast ⁻¹' Ioo a b ↔ x✝ ∈ Ioo ⌊a⌋ ⌈b⌉
Module: Mathlib.Algebra.Order.Floor.Ring

Goal state before:
pp: R : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : FloorRing R
a b : R
⊢ Int.cast ⁻¹' Ico a b = Ico ⌈a⌉ ⌈b⌉

Tactic:
ext

Goal state after:
R : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : FloorRing R
a b : R
x✝ : ℤ
⊢ x✝ ∈ Int.cast ⁻¹' Ico a b ↔ x✝ ∈ Ico ⌈a⌉ ⌈b⌉
Module: Mathlib.Algebra.Order.Floor.Ring

Goal state before:
pp: R : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : FloorRing R
a b : R
⊢ Int.cast ⁻¹' Ioc a b = Ioc ⌊a⌋ ⌊b⌋

Tactic:
ext

Goal state after:
R : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : FloorRing R
a b : R
x✝ : ℤ
⊢ x✝ ∈ Int.cast ⁻¹' Ioc a b ↔ x✝ ∈ Ioc ⌊a⌋ ⌊b⌋
Module: Mathlib.Algebra.Order.Floor.Ring

Goal state before:
pp: R : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : FloorRing R
a b : R
⊢ Int.cast ⁻¹' Icc a b = Icc ⌈a⌉ ⌊b⌋

Tactic:
ext

Goal state after:
R : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : FloorRing R
a b : R
x✝ : ℤ
⊢ x✝ ∈ Int.cast ⁻¹' Icc a b ↔ x✝ ∈ Icc ⌈a⌉ ⌊b⌋
Module: Mathlib.Algebra.Order.Floor.Ring

Goal state before:
pp: R : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : FloorRing R
a : R
⊢ Int.cast ⁻¹' Ioi a = Ioi ⌊a⌋

Tactic:
ext

Goal state after:
R : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : FloorRing R
a : R
x✝ : ℤ
⊢ x✝ ∈ Int.cast ⁻¹' Ioi a ↔ x✝ ∈ Ioi ⌊a⌋
Module: Mathlib.Algebra.Order.Floor.Ring

Goal state before:
pp: R : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : FloorRing R
a : R
⊢ Int.cast ⁻¹' Ici a = Ici ⌈a⌉

Tactic:
ext

Goal state after:
R : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : FloorRing R
a : R
x✝ : ℤ
⊢ x✝ ∈ Int.cast ⁻¹' Ici a ↔ x✝ ∈ Ici ⌈a⌉
Module: Mathlib.Algebra.Order.Floor.Ring

Goal state before:
pp: R : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : FloorRing R
a : R
⊢ Int.cast ⁻¹' Iio a = Iio ⌈a⌉

Tactic:
ext

Goal state after:
R : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : FloorRing R
a : R
x✝ : ℤ
⊢ x✝ ∈ Int.cast ⁻¹' Iio a ↔ x✝ ∈ Iio ⌈a⌉
Module: Mathlib.Algebra.Order.Floor.Ring

Goal state before:
pp: R : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : FloorRing R
a : R
⊢ Int.cast ⁻¹' Iic a = Iic ⌊a⌋

Tactic:
ext

Goal state after:
R : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrder R
inst✝ : FloorRing R
a : R
x✝ : ℤ
⊢ x✝ ∈ Int.cast ⁻¹' Iic a ↔ x✝ ∈ Iic ⌊a⌋
Module: Mathlib.Algebra.Group.Action.Units

Goal state before:
pp: G : Type u_1
H : Type u_2
M✝ : Type u_3
N : Type u_4
α : Type u_5
M : Type u_6
inst✝ : CommMonoid M
⊢ mulAction'.toSMul = instSMulOfMul

Tactic:
ext

Goal state after:
G : Type u_1
H : Type u_2
M✝ : Type u_3
N : Type u_4
α : Type u_5
M : Type u_6
inst✝ : CommMonoid M
x✝¹ x✝ : Mˣ
⊢ ↑(SMul.smul x✝¹ x✝) = ↑(SMul.smul x✝¹ x✝)
Module: Mathlib.Algebra.Group.Action.Units

Goal state before:
pp: M : Type u_6
inst✝ : CommMonoid M
u₁ u₂ : Mˣ
⊢ u₁ • u₂ = u₁ * u₂

Tactic:
ext

Goal state after:
M : Type u_6
inst✝ : CommMonoid M
u₁ u₂ : Mˣ
⊢ ↑(u₁ • u₂) = ↑(u₁ * u₂)
Module: Mathlib.Algebra.Order.Archimedean.Basic

Goal state before:
pp: K : Type u_4
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
h : ∀ (x : K), ∃ n, x < ↑n
x : K
⊢ ∃ n, x < ↑n

Tactic:
obtain ⟨n, h⟩ := h x

Goal state after:
K : Type u_4
inst✝² : Field K
inst✝¹ : LinearOrder K
inst✝ : IsStrictOrderedRing K
h✝ : ∀ (x : K), ∃ n, x < ↑n
x : K
n : ℤ
h : x < ↑n
⊢ ∃ n, x < ↑n
Module: Mathlib.Algebra.AddConstMap.Basic

Goal state before:
pp: G : Type u_1
inst✝ : AddMonoid G
a : G
⊢ Multiplicative.toAdd 1 +ᵥ AddConstMap.id = 1

Tactic:
ext

Goal state after:
G : Type u_1
inst✝ : AddMonoid G
a x✝ : G
⊢ (Multiplicative.toAdd 1 +ᵥ AddConstMap.id) x✝ = 1 x✝
Module: Mathlib.Algebra.AddConstMap.Basic

Goal state before:
pp: G : Type u_1
inst✝ : AddMonoid G
a : G
x✝¹ x✝ : Multiplicative G
⊢ Multiplicative.toAdd (x✝¹ * x✝) +ᵥ AddConstMap.id =
    (Multiplicative.toAdd x✝¹ +ᵥ AddConstMap.id) * (Multiplicative.toAdd x✝ +ᵥ AddConstMap.id)

Tactic:
ext

Goal state after:
G : Type u_1
inst✝ : AddMonoid G
a : G
x✝² x✝¹ : Multiplicative G
x✝ : G
⊢ (Multiplicative.toAdd (x✝² * x✝¹) +ᵥ AddConstMap.id) x✝ =
    ((Multiplicative.toAdd x✝² +ᵥ AddConstMap.id) * (Multiplicative.toAdd x✝¹ +ᵥ AddConstMap.id)) x✝
Module: Mathlib.Algebra.Order.Archimedean.Basic

Goal state before:
pp: K : Type u_4
inst✝³ : Field K
inst✝² : LinearOrder K
inst✝¹ : IsStrictOrderedRing K
inst✝ : Archimedean K
x : K
x0 : 0 < x
⊢ ∃ q, 0 < q ∧ ↑q < x

Tactic:
by
  simpa only [Rat.cast_pos] using exists_rat_btwn x0

Goal state after:
Module: Mathlib.Algebra.Order.Archimedean.Basic

Goal state before:
pp: G : Type u_1
M✝ : Type u_2
R : Type u_3
K : Type u_4
M : Type u_5
inst✝² : AddCommMonoid M
inst✝¹ : PartialOrder M
inst✝ : Archimedean M
⊢ Archimedean (WithBot M)

Tactic:
constructor

Goal state after:
G : Type u_1
M✝ : Type u_2
R : Type u_3
K : Type u_4
M : Type u_5
inst✝² : AddCommMonoid M
inst✝¹ : PartialOrder M
inst✝ : Archimedean M
⊢ ∀ (x : WithBot M) {y : WithBot M}, 0 < y → ∃ n, x ≤ n • y
Module: Mathlib.Algebra.Order.Archimedean.Basic

Goal state before:
pp: G : Type u_1
M✝ : Type u_2
R : Type u_3
K : Type u_4
M : Type u_5
inst✝² : CommMonoid M
inst✝¹ : PartialOrder M
inst✝ : MulArchimedean M
⊢ MulArchimedean (WithZero M)

Tactic:
constructor

Goal state after:
G : Type u_1
M✝ : Type u_2
R : Type u_3
K : Type u_4
M : Type u_5
inst✝² : CommMonoid M
inst✝¹ : PartialOrder M
inst✝ : MulArchimedean M
⊢ ∀ (x : WithZero M) {y : WithZero M}, 1 < y → ∃ n, x ≤ y ^ n
Module: Mathlib.Data.List.Perm.Subperm

Goal state before:
pp: α : Type u_1
l₁ l₂ : List α
⊢ l₁ <+~ l₂ → ∃ l, l ~ l₂ ∧ l₁ <+ l

Tactic:
rintro ⟨l, h₁, h₂⟩

Goal state after:
α : Type u_1
l₁ l₂ l : List α
h₁ : l ~ l₁
h₂ : l <+ l₂
⊢ ∃ l, l ~ l₂ ∧ l₁ <+ l
Module: Mathlib.Data.List.Perm.Subperm

Goal state before:
pp: α : Type u_1
l : List α
a : α
⊢ l <+~ [a] ↔ l = [] ∨ l = [a]

Tactic:
constructor

Goal state after:
case mp
α : Type u_1
l : List α
a : α
⊢ l <+~ [a] → l = [] ∨ l = [a]

case mpr
α : Type u_1
l : List α
a : α
⊢ l = [] ∨ l = [a] → l <+~ [a]
Module: Mathlib.Data.List.Perm.Subperm

Goal state before:
pp: case h
α : Type u_2
β : Type u_3
l₁ l₂ : List α
f : α → β
hf : Function.Injective f
w : List β
perm : w ~ map f l₁
sublist : w <+ map f l₂
x : List α
sublistₓ : x <+ l₂
mapₓ : w = map f x
⊢ x ~ l₁ ∧ x <+ l₂

Tactic:
constructor

Goal state after:
case h.left
α : Type u_2
β : Type u_3
l₁ l₂ : List α
f : α → β
hf : Function.Injective f
w : List β
perm : w ~ map f l₁
sublist : w <+ map f l₂
x : List α
sublistₓ : x <+ l₂
mapₓ : w = map f x
⊢ x ~ l₁

case h.right
α : Type u_2
β : Type u_3
l₁ l₂ : List α
f : α → β
hf : Function.Injective f
w : List β
perm : w ~ map f l₁
sublist : w <+ map f l₂
x : List α
sublistₓ : x <+ l₂
mapₓ : w = map f x
⊢ x <+ l₂
Module: Mathlib.Algebra.Group.Pointwise.Set.Scalar

Goal state before:
pp: α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : SMul αᵐᵒᵖ β
inst✝¹ : SMul β γ
inst✝ : SMul α γ
a : α
s : Set β
t : Set γ
h : ∀ (a : α) (b : β) (c : γ), (op a • b) • c = b • a • c
⊢ (op a • s) • t = s • a • t

Tactic:
ext

Goal state after:
α : Type u_2
β : Type u_3
γ : Type u_4
inst✝² : SMul αᵐᵒᵖ β
inst✝¹ : SMul β γ
inst✝ : SMul α γ
a : α
s : Set β
t : Set γ
h : ∀ (a : α) (b : β) (c : γ), (op a • b) • c = b • a • c
x✝ : γ
⊢ x✝ ∈ (op a • s) • t ↔ x✝ ∈ s • a • t
Module: Mathlib.Algebra.Torsor.Basic

Goal state before:
pp: G : Type u_1
P : Type u_2
inst✝¹ : CommGroup G
inst✝ : Torsor G P
v : G
s t : Set P
⊢ v • s /ₛ v • t = s /ₛ t

Tactic:
ext

Goal state after:
G : Type u_1
P : Type u_2
inst✝¹ : CommGroup G
inst✝ : Torsor G P
v : G
s t : Set P
x✝ : G
⊢ x✝ ∈ v • s /ₛ v • t ↔ x✝ ∈ s /ₛ t
Module: Mathlib.Algebra.Torsor.Basic

Goal state before:
pp: G : Type u_5
inst✝ : AddCommGroup G
x : G
⊢ pointReflection x = Equiv.subLeft (2 • x)

Tactic:
ext

Goal state after:
G : Type u_5
inst✝ : AddCommGroup G
x x✝ : G
⊢ (pointReflection x) x✝ = (Equiv.subLeft (2 • x)) x✝
Module: Mathlib.Data.Multiset.ZeroCons

Goal state before:
pp: α : Type u_1
a b : α
as bs : Multiset α
this : DecidableEq α
⊢ a ::ₘ as = b ::ₘ bs ↔ a = b ∧ as = bs ∨ a ≠ b ∧ ∃ cs, as = b ::ₘ cs ∧ bs = a ::ₘ cs

Tactic:
constructor

Goal state after:
case mp
α : Type u_1
a b : α
as bs : Multiset α
this : DecidableEq α
⊢ a ::ₘ as = b ::ₘ bs → a = b ∧ as = bs ∨ a ≠ b ∧ ∃ cs, as = b ::ₘ cs ∧ bs = a ::ₘ cs

case mpr
α : Type u_1
a b : α
as bs : Multiset α
this : DecidableEq α
⊢ (a = b ∧ as = bs ∨ a ≠ b ∧ ∃ cs, as = b ::ₘ cs ∧ bs = a ::ₘ cs) → a ::ₘ as = b ::ₘ bs
Module: Mathlib.Data.Multiset.ZeroCons

Goal state before:
pp: case neg
α : Type u_1
a b : α
as bs : Multiset α
this✝¹ : DecidableEq α
eq : a ::ₘ as = b ::ₘ bs
h : ¬a = b
this✝ : a ∈ b ::ₘ bs
this : a ∈ bs
⊢ a = b ∧ as = bs ∨ a ≠ b ∧ ∃ cs, as = b ::ₘ cs ∧ bs = a ::ₘ cs

Tactic:
rcases exists_cons_of_mem this with ⟨cs, hcs⟩

Goal state after:
case neg
α : Type u_1
a b : α
as bs : Multiset α
this✝¹ : DecidableEq α
eq : a ::ₘ as = b ::ₘ bs
h : ¬a = b
this✝ : a ∈ b ::ₘ bs
this : a ∈ bs
cs : Multiset α
hcs : bs = a ::ₘ cs
⊢ a = b ∧ as = bs ∨ a ≠ b ∧ ∃ cs, as = b ::ₘ cs ∧ bs = a ::ₘ cs
Module: Mathlib.Data.Multiset.Replicate

Goal state before:
pp: α : Type u_1
m : Multiset α
x : α
n : ℕ
⊢ (∃ a, a ::ₘ m ≤ replicate (n + 1) x) ↔ m ≤ replicate n x

Tactic:
constructor

Goal state after:
case mp
α : Type u_1
m : Multiset α
x : α
n : ℕ
⊢ (∃ a, a ::ₘ m ≤ replicate (n + 1) x) → m ≤ replicate n x

case mpr
α : Type u_1
m : Multiset α
x : α
n : ℕ
⊢ m ≤ replicate n x → ∃ a, a ::ₘ m ≤ replicate (n + 1) x
Module: Mathlib.Data.Multiset.ZeroCons

Goal state before:
pp: α : Type u_1
s t : Multiset α
⊢ Rel (fun x1 x2 ↦ x1 = x2) s t ↔ s = t

Tactic:
constructor

Goal state after:
case mp
α : Type u_1
s t : Multiset α
⊢ Rel (fun x1 x2 ↦ x1 = x2) s t → s = t

case mpr
α : Type u_1
s t : Multiset α
⊢ s = t → Rel (fun x1 x2 ↦ x1 = x2) s t
Module: Mathlib.Data.List.Nodup

Goal state before:
pp: α : Type u
β : Type v
r : α → β → Prop
hr : Relator.BiUnique r
⊢ (fun x1 x2 ↦ x1 ↔ x2) [].Nodup [].Nodup

Tactic:
by simp only [nodup_nil]

Goal state after:
Module: Mathlib.Data.Multiset.ZeroCons

Goal state before:
pp: α : Type u_1
β : Type v
r : α → β → Prop
a : α
as : Multiset α
bs : Multiset β
⊢ Rel r (a ::ₘ as) bs ↔ ∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs'

Tactic:
constructor

Goal state after:
case mp
α : Type u_1
β : Type v
r : α → β → Prop
a : α
as : Multiset α
bs : Multiset β
⊢ Rel r (a ::ₘ as) bs → ∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs'

case mpr
α : Type u_1
β : Type v
r : α → β → Prop
a : α
as : Multiset α
bs : Multiset β
⊢ (∃ b bs', r a b ∧ Rel r as bs' ∧ bs = b ::ₘ bs') → Rel r (a ::ₘ as) bs
Module: Mathlib.Data.Multiset.AddSub

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s t : Multiset α
a : α
⊢ a ::ₘ s = t ↔ a ∈ t ∧ s = t.erase a

Tactic:
constructor

Goal state after:
case mp
α : Type u_1
inst✝ : DecidableEq α
s t : Multiset α
a : α
⊢ a ::ₘ s = t → a ∈ t ∧ s = t.erase a

case mpr
α : Type u_1
inst✝ : DecidableEq α
s t : Multiset α
a : α
⊢ a ∈ t ∧ s = t.erase a → a ::ₘ s = t
Module: Mathlib.Data.Multiset.AddSub

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
s t u : Multiset α
⊢ s - (t + u) = s - t - u

Tactic:
ext

Goal state after:
α : Type u_1
inst✝ : DecidableEq α
s t u : Multiset α
a✝ : α
⊢ count a✝ (s - (t + u)) = count a✝ (s - t - u)
Module: Mathlib.Data.Multiset.AddSub

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
a : α
s : Multiset α
⊢ s - {a} = s.erase a

Tactic:
ext

Goal state after:
α : Type u_1
inst✝ : DecidableEq α
a : α
s : Multiset α
a✝ : α
⊢ count a✝ (s - {a}) = count a✝ (s.erase a)
Module: Mathlib.Data.Multiset.AddSub

Goal state before:
pp: α : Type u_1
β : Type v
r : α → β → Prop
as₀ as₁ : Multiset α
a : α
s : Multiset α
ih : ∀ {bs : Multiset β}, Rel r (s + as₁) bs ↔ ∃ bs₀ bs₁, Rel r s bs₀ ∧ Rel r as₁ bs₁ ∧ bs = bs₀ + bs₁
bs : Multiset β
⊢ (∃ b bs', r a b ∧ (∃ bs₀ bs₁, Rel r s bs₀ ∧ Rel r as₁ bs₁ ∧ bs' = bs₀ + bs₁) ∧ bs = b ::ₘ bs') ↔
    ∃ bs₀ bs₁, (∃ b bs', r a b ∧ Rel r s bs' ∧ bs₀ = b ::ₘ bs') ∧ Rel r as₁ bs₁ ∧ bs = bs₀ + bs₁

Tactic:
constructor

Goal state after:
case mp
α : Type u_1
β : Type v
r : α → β → Prop
as₀ as₁ : Multiset α
a : α
s : Multiset α
ih : ∀ {bs : Multiset β}, Rel r (s + as₁) bs ↔ ∃ bs₀ bs₁, Rel r s bs₀ ∧ Rel r as₁ bs₁ ∧ bs = bs₀ + bs₁
bs : Multiset β
⊢ (∃ b bs', r a b ∧ (∃ bs₀ bs₁, Rel r s bs₀ ∧ Rel r as₁ bs₁ ∧ bs' = bs₀ + bs₁) ∧ bs = b ::ₘ bs') →
    ∃ bs₀ bs₁, (∃ b bs', r a b ∧ Rel r s bs' ∧ bs₀ = b ::ₘ bs') ∧ Rel r as₁ bs₁ ∧ bs = bs₀ + bs₁

case mpr
α : Type u_1
β : Type v
r : α → β → Prop
as₀ as₁ : Multiset α
a : α
s : Multiset α
ih : ∀ {bs : Multiset β}, Rel r (s + as₁) bs ↔ ∃ bs₀ bs₁, Rel r s bs₀ ∧ Rel r as₁ bs₁ ∧ bs = bs₀ + bs₁
bs : Multiset β
⊢ (∃ bs₀ bs₁, (∃ b bs', r a b ∧ Rel r s bs' ∧ bs₀ = b ::ₘ bs') ∧ Rel r as₁ bs₁ ∧ bs = bs₀ + bs₁) →
    ∃ b bs', r a b ∧ (∃ bs₀ bs₁, Rel r s bs₀ ∧ Rel r as₁ bs₁ ∧ bs' = bs₀ + bs₁) ∧ bs = b ::ₘ bs'
Module: Mathlib.Data.Finset.Attach

Goal state before:
pp: α : Type u_1
s : Finset α
⊢ ↑s.attach = Set.univ

Tactic:
ext

Goal state after:
α : Type u_1
s : Finset α
x✝ : ↥s
⊢ x✝ ∈ ↑s.attach ↔ x✝ ∈ Set.univ
Module: Mathlib.Data.List.Nodup

Goal state before:
pp: case cons.succ
α : Type u
inst✝¹ : BEq α
inst✝ : LawfulBEq α
a : α
l : List α
IH : l.Nodup → ∀ (i : ℕ) (h : i < l.length), l.erase l[i] = l.eraseIdx i
hl : (a :: l).Nodup
i : ℕ
h : i + 1 < (a :: l).length
⊢ (a :: l).erase (a :: l)[i + 1] = (a :: l).eraseIdx (i + 1)

Tactic:
| succ i =>

Goal state after:
Module: Mathlib.Data.Multiset.MapFold

Goal state before:
pp: α : Type u_1
β : Type v
f : α → β
t : α
⊢ map f ∘ cons t = cons (f t) ∘ map f

Tactic:
ext

Goal state after:
α : Type u_1
β : Type v
f : α → β
t : α
x✝ : Multiset α
⊢ (map f ∘ cons t) x✝ = (cons (f t) ∘ map f) x✝
Module: Mathlib.Data.List.Nodup

Goal state before:
pp: α : Type u
l : List α
a : α
inst✝¹ : BEq α
inst✝ : LawfulBEq α
h : l.Nodup
h' : ¬a ∈ l
⊢ ¬a ∈ l ∧ l.Nodup

Tactic:
constructor

Goal state after:
case left
α : Type u
l : List α
a : α
inst✝¹ : BEq α
inst✝ : LawfulBEq α
h : l.Nodup
h' : ¬a ∈ l
⊢ ¬a ∈ l

case right
α : Type u
l : List α
a : α
inst✝¹ : BEq α
inst✝ : LawfulBEq α
h : l.Nodup
h' : ¬a ∈ l
⊢ l.Nodup
Module: Mathlib.Data.Multiset.MapFold

Goal state before:
pp: α : Type u_1
β : Type v
f : α → β
s : Multiset α
b : β
⊢ map f s = {b} ↔ ∃ a, s = {a} ∧ f a = b

Tactic:
constructor

Goal state after:
case mp
α : Type u_1
β : Type v
f : α → β
s : Multiset α
b : β
⊢ map f s = {b} → ∃ a, s = {a} ∧ f a = b

case mpr
α : Type u_1
β : Type v
f : α → β
s : Multiset α
b : β
⊢ (∃ a, s = {a} ∧ f a = b) → map f s = {b}
Module: Mathlib.Data.Multiset.MapFold

Goal state before:
pp: α : Type u_1
β : Type v
inst✝ : DecidableEq α
f : α → β
s : Multiset α
t : Multiset β
b : β
⊢ (∃ a, a ∈ s ∧ f a = b ∧ map f (s.erase a) = t) ↔ map f s = b ::ₘ t

Tactic:
constructor

Goal state after:
case mp
α : Type u_1
β : Type v
inst✝ : DecidableEq α
f : α → β
s : Multiset α
t : Multiset β
b : β
⊢ (∃ a, a ∈ s ∧ f a = b ∧ map f (s.erase a) = t) → map f s = b ::ₘ t

case mpr
α : Type u_1
β : Type v
inst✝ : DecidableEq α
f : α → β
s : Multiset α
t : Multiset β
b : β
⊢ map f s = b ::ₘ t → ∃ a, a ∈ s ∧ f a = b ∧ map f (s.erase a) = t