mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.WittVector.MulCoeff

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ wittPolyProd p (n + 1) =
    -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
          ↑p ^ (n + 1) * X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) +
        ↑p ^ (n + 1) * X (1, n + 1) * (rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1)) +
      remainder p n

Tactic:
]

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ wittPolyProd p (n + 1) =
    -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
          ↑p ^ (n + 1) * X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) +
        ↑p ^ (n + 1) * X (1, n + 1) * (rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1)) +
      remainder p n
Module: Mathlib.RingTheory.WittVector.MulCoeff

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
            ↑p ^ (n + 1) * X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) +
          ↑p ^ (n + 1) * X (1, n + 1) * (rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1)) +
        (remainder p n - wittPolyProdRemainder p (n + 1)) +
      ↑p ^ (n + 1) * (↑p ^ (n + 1) * X (0, n + 1) * X (1, n + 1)) =
    ↑p ^ (n + 1) * (X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1))) +
      (↑p ^ (n + 1) * (X (1, n + 1) * (rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1))) +
        (remainder p n - wittPolyProdRemainder p (n + 1)))

Tactic:
]

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ -(↑p ^ (n + 1) * X (0, n + 1)) * (↑p ^ (n + 1) * X (1, n + 1)) +
            ↑p ^ (n + 1) * X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) +
          ↑p ^ (n + 1) * X (1, n + 1) * (rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1)) +
        (remainder p n - wittPolyProdRemainder p (n + 1)) +
      ↑p ^ (n + 1) * (↑p ^ (n + 1) * X (0, n + 1) * X (1, n + 1)) =
    ↑p ^ (n + 1) * (X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1))) +
      (↑p ^ (n + 1) * (X (1, n + 1) * (rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1))) +
        (remainder p n - wittPolyProdRemainder p (n + 1)))
Module: Mathlib.RingTheory.WittVector.MulCoeff

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ (remainder p n - wittPolyProdRemainder p (n + 1)).vars ⊆ univ ×ˢ range (n + 1)

Tactic:
]

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ (remainder p n - wittPolyProdRemainder p (n + 1)).vars ⊆ univ ×ˢ range (n + 1)
Module: Mathlib.RingTheory.WittVector.MulCoeff

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
this : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ (wittMul p (n + 1) + C (↑p ^ (n + 1)) * X (0, n + 1) * X (1, n + 1) -
          X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) -
        X (1, n + 1) * (rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1))).vars =
    (wittMul p (n + 1) + C (↑p ^ (n + 1)) * X (0, n + 1) * X (1, n + 1) -
          X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) -
        X (1, n + 1) * (rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1))).vars

pp: case ha
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
this : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ ↑p ^ (n + 1) ≠ 0

Tactic:
]

Goal state after:
case ha
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
this : ↑p ^ (n + 1) = C (↑p ^ (n + 1))
⊢ ↑p ^ (n + 1) ≠ 0
Module: Mathlib.RingTheory.WittVector.MulCoeff

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ (↑p ^ (n + 1) *
        (wittMul p (n + 1) + ↑p ^ (n + 1) * X (0, n + 1) * X (1, n + 1) -
            X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) -
          X (1, n + 1) * (rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1)))).vars ⊆
    univ ×ˢ range (n + 1)

Tactic:
]

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ (↑p ^ (n + 1) *
        (wittMul p (n + 1) + ↑p ^ (n + 1) * X (0, n + 1) * X (1, n + 1) -
            X (0, n + 1) * (rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1)) -
          X (1, n + 1) * (rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1)))).vars ⊆
    univ ×ˢ range (n + 1)
Module: Mathlib.RingTheory.WittVector.MulCoeff

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
x y : 𝕎 k
⊢ (aeval (Function.uncurry ![fun i ↦ x.coeff i, fun i ↦ y.coeff i]))
          (wittMul p (n + 1) + ↑p ^ (n + 1) * X (0, n + 1) * X (1, n + 1)) -
        y.coeff (n + 1) *
          (aeval (Function.uncurry ![fun i ↦ x.coeff i, fun i ↦ y.coeff i]))
            ((rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1))) -
      x.coeff (n + 1) *
        (aeval (Function.uncurry ![fun i ↦ x.coeff i, fun i ↦ y.coeff i]))
          ((rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1))) =
    (x * y).coeff (n + 1) + ↑p ^ (n + 1) * x.coeff (n + 1) * y.coeff (n + 1) -
        y.coeff (n + 1) * ∑ i ∈ range (n + 1 + 1), ↑p ^ i * x.coeff i ^ p ^ (n + 1 - i) -
      x.coeff (n + 1) * ∑ i ∈ range (n + 1 + 1), ↑p ^ i * y.coeff i ^ p ^ (n + 1 - i)

Tactic:
]

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝ : CommRing k
n : ℕ
x y : 𝕎 k
⊢ (aeval (Function.uncurry ![fun i ↦ x.coeff i, fun i ↦ y.coeff i]))
          (wittMul p (n + 1) + ↑p ^ (n + 1) * X (0, n + 1) * X (1, n + 1)) -
        y.coeff (n + 1) *
          (aeval (Function.uncurry ![fun i ↦ x.coeff i, fun i ↦ y.coeff i]))
            ((rename (Prod.mk 0)) (wittPolynomial p ℤ (n + 1))) -
      x.coeff (n + 1) *
        (aeval (Function.uncurry ![fun i ↦ x.coeff i, fun i ↦ y.coeff i]))
          ((rename (Prod.mk 1)) (wittPolynomial p ℤ (n + 1))) =
    (x * y).coeff (n + 1) + ↑p ^ (n + 1) * x.coeff (n + 1) * y.coeff (n + 1) -
        y.coeff (n + 1) * ∑ i ∈ range (n + 1 + 1), ↑p ^ i * x.coeff i ^ p ^ (n + 1 - i) -
      x.coeff (n + 1) * ∑ i ∈ range (n + 1 + 1), ↑p ^ i * y.coeff i ^ p ^ (n + 1 - i)
Module: Mathlib.RingTheory.WittVector.MulCoeff

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
x y : 𝕎 k
⊢ (x * y).coeff (n + 1) + ↑p ^ (n + 1) * x.coeff (n + 1) * y.coeff (n + 1) -
        y.coeff (n + 1) * ∑ i ∈ range (n + 1 + 1), ↑p ^ i * x.coeff i ^ p ^ (n + 1 - i) -
      x.coeff (n + 1) * ∑ i ∈ range (n + 1 + 1), ↑p ^ i * y.coeff i ^ p ^ (n + 1 - i) =
    (x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) - x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1)

Tactic:
]

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
x y : 𝕎 k
⊢ (x * y).coeff (n + 1) + ↑p ^ (n + 1) * x.coeff (n + 1) * y.coeff (n + 1) -
        y.coeff (n + 1) * ∑ i ∈ range (n + 1 + 1), ↑p ^ i * x.coeff i ^ p ^ (n + 1 - i) -
      x.coeff (n + 1) * ∑ i ∈ range (n + 1 + 1), ↑p ^ i * y.coeff i ^ p ^ (n + 1 - i) =
    (x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) - x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1)
Module: Mathlib.RingTheory.WittVector.MulCoeff

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
x y✝ : 𝕎 k
this : ↑p = 0
y : 𝕎 k
⊢ 0 ^ 0 * y.coeff 0 ^ p ^ (n + 1 - 0) = y.coeff 0 ^ p ^ (n + 1)

pp: case h
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
x y✝ : 𝕎 k
this : ↑p = 0
y : 𝕎 k
⊢ 0 ∈ range (n + 1 + 1)

pp: case h₀
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
x y✝ : 𝕎 k
this : ↑p = 0
y : 𝕎 k
⊢ ∀ b ∈ range (n + 1 + 1), b ≠ 0 → 0 ^ b * y.coeff b ^ p ^ (n + 1 - b) = 0

Tactic:
]

Goal state after:
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
x y✝ : 𝕎 k
this : ↑p = 0
y : 𝕎 k
⊢ 0 ^ 0 * y.coeff 0 ^ p ^ (n + 1 - 0) = y.coeff 0 ^ p ^ (n + 1)

case h
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
x y✝ : 𝕎 k
this : ↑p = 0
y : 𝕎 k
⊢ 0 ∈ range (n + 1 + 1)

case h₀
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
x y✝ : 𝕎 k
this : ↑p = 0
y : 𝕎 k
⊢ ∀ b ∈ range (n + 1 + 1), b ≠ 0 → 0 ^ b * y.coeff b ^ p ^ (n + 1 - b) = 0
Module: Mathlib.RingTheory.WittVector.Isocrystal

Goal state before:
pp: p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
k : Type u_2
inst✝⁴ : Field k
inst✝³ : IsAlgClosed k
inst✝² : CharP k p
V : Type u_3
inst✝¹ : AddCommGroup V
inst✝ : Isocrystal p k V
this✝ : Nontrivial V
x : V
h_dim : ∀ (w : V), ∃ c, c • x = w
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
⊢ ∃ a, a ≠ 0 ∧ Φ(p, k) x = a • x

Tactic:
]

Goal state after:
p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
k : Type u_2
inst✝⁴ : Field k
inst✝³ : IsAlgClosed k
inst✝² : CharP k p
V : Type u_3
inst✝¹ : AddCommGroup V
inst✝ : Isocrystal p k V
this✝ : Nontrivial V
x : V
h_dim : ∀ (w : V), ∃ c, c • x = w
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
⊢ ∃ a, a ≠ 0 ∧ Φ(p, k) x = a • x
Module: Mathlib.RingTheory.WittVector.Isocrystal

Goal state before:
pp: case refine_1
p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
k : Type u_2
inst✝⁴ : Field k
inst✝³ : IsAlgClosed k
inst✝² : CharP k p
V : Type u_3
inst✝¹ : AddCommGroup V
inst✝ : Isocrystal p k V
h_dim : finrank K(p, k) V = 1
this✝ : Nontrivial V
x : V
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
a : K(p, k)
ha : a ≠ 0
hax : Φ(p, k) x = a • x
b : K(p, k)
hb : b ≠ 0
m : ℤ
hmb : φ(p, k) b * a = ↑p ^ m * b
F₀ : StandardOneDimIsocrystal p k m →ₗ[K(p, k)] V := LinearMap.toSpanSingleton K(p, k) V x
⊢ F₀.ker = ⊥

Tactic:
]

Goal state after:
case refine_1
p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
k : Type u_2
inst✝⁴ : Field k
inst✝³ : IsAlgClosed k
inst✝² : CharP k p
V : Type u_3
inst✝¹ : AddCommGroup V
inst✝ : Isocrystal p k V
h_dim : finrank K(p, k) V = 1
this✝ : Nontrivial V
x : V
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
a : K(p, k)
ha : a ≠ 0
hax : Φ(p, k) x = a • x
b : K(p, k)
hb : b ≠ 0
m : ℤ
hmb : φ(p, k) b * a = ↑p ^ m * b
F₀ : StandardOneDimIsocrystal p k m →ₗ[K(p, k)] V := LinearMap.toSpanSingleton K(p, k) V x
⊢ F₀.ker = ⊥
Module: Mathlib.RingTheory.WittVector.Isocrystal

Goal state before:
pp: case refine_2
p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
k : Type u_2
inst✝⁴ : Field k
inst✝³ : IsAlgClosed k
inst✝² : CharP k p
V : Type u_3
inst✝¹ : AddCommGroup V
inst✝ : Isocrystal p k V
h_dim : finrank K(p, k) V = 1
this✝ : Nontrivial V
x : V
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
a : K(p, k)
ha : a ≠ 0
hax : Φ(p, k) x = a • x
b : K(p, k)
hb : b ≠ 0
m : ℤ
hmb : φ(p, k) b * a = ↑p ^ m * b
F₀ : StandardOneDimIsocrystal p k m →ₗ[K(p, k)] V := LinearMap.toSpanSingleton K(p, k) V x
⊢ F₀.range = ⊤

Tactic:
]

Goal state after:
case refine_2
p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
k : Type u_2
inst✝⁴ : Field k
inst✝³ : IsAlgClosed k
inst✝² : CharP k p
V : Type u_3
inst✝¹ : AddCommGroup V
inst✝ : Isocrystal p k V
h_dim : finrank K(p, k) V = 1
this✝ : Nontrivial V
x : V
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
a : K(p, k)
ha : a ≠ 0
hax : Φ(p, k) x = a • x
b : K(p, k)
hb : b ≠ 0
m : ℤ
hmb : φ(p, k) b * a = ↑p ^ m * b
F₀ : StandardOneDimIsocrystal p k m →ₗ[K(p, k)] V := LinearMap.toSpanSingleton K(p, k) V x
⊢ F₀.range = ⊤
Module: Mathlib.RingTheory.WittVector.Isocrystal

Goal state before:
pp: case refine_2
p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
k : Type u_2
inst✝⁴ : Field k
inst✝³ : IsAlgClosed k
inst✝² : CharP k p
V : Type u_3
inst✝¹ : AddCommGroup V
inst✝ : Isocrystal p k V
h_dim : finrank K(p, k) V = 1
this✝ : Nontrivial V
x : V
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
a : K(p, k)
ha : a ≠ 0
hax : Φ(p, k) x = a • x
b : K(p, k)
hb : b ≠ 0
m : ℤ
hmb : φ(p, k) b * a = ↑p ^ m * b
F₀ : StandardOneDimIsocrystal p k m →ₗ[K(p, k)] V := LinearMap.toSpanSingleton K(p, k) V x
⊢ F₀.range = K(p, k) ∙ x

Tactic:
]

Goal state after:
case refine_2
p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
k : Type u_2
inst✝⁴ : Field k
inst✝³ : IsAlgClosed k
inst✝² : CharP k p
V : Type u_3
inst✝¹ : AddCommGroup V
inst✝ : Isocrystal p k V
h_dim : finrank K(p, k) V = 1
this✝ : Nontrivial V
x : V
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
a : K(p, k)
ha : a ≠ 0
hax : Φ(p, k) x = a • x
b : K(p, k)
hb : b ≠ 0
m : ℤ
hmb : φ(p, k) b * a = ↑p ^ m * b
F₀ : StandardOneDimIsocrystal p k m →ₗ[K(p, k)] V := LinearMap.toSpanSingleton K(p, k) V x
⊢ F₀.range = K(p, k) ∙ x
Module: Mathlib.SetTheory.Cardinal.Cofinality.Club

Goal state before:
pp: α : Type v
inst✝¹ : LinearOrder α
s : Set (Set α)
inst✝ : OrderTop α
hs : ∀ x ∈ s, IsClub x
⊢ ∀ t ∈ s, ⊤ ∈ t

Tactic:
]

Goal state after:
α : Type v
inst✝¹ : LinearOrder α
s : Set (Set α)
inst✝ : OrderTop α
hs : ∀ x ∈ s, IsClub x
⊢ ∀ t ∈ s, ⊤ ∈ t
Module: Mathlib.RingTheory.WittVector.Isocrystal

Goal state before:
pp: case refine_2
p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
k : Type u_2
inst✝⁴ : Field k
inst✝³ : IsAlgClosed k
inst✝² : CharP k p
V : Type u_3
inst✝¹ : AddCommGroup V
inst✝ : Isocrystal p k V
h_dim : finrank K(p, k) V = 1
this✝ : Nontrivial V
x : V
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
a : K(p, k)
ha : a ≠ 0
hax : Φ(p, k) x = a • x
b : K(p, k)
hb : b ≠ 0
m : ℤ
hmb : φ(p, k) b * a = ↑p ^ m * b
F₀ : StandardOneDimIsocrystal p k m →ₗ[K(p, k)] V := LinearMap.toSpanSingleton K(p, k) V x
⊢ F₀.range = (LinearMap.toSpanSingleton K(p, k) V x).range

Tactic:
]

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Cofinality.Club

Goal state before:
pp: α : Type v
inst✝¹ : LinearOrder α
ι : Type u_1
f : ι → Set α
inst✝ : OrderTop α
hs : ∀ (i : ι), IsClub (f i)
⊢ IsClub (⋂₀ range f)

Tactic:
]

Goal state after:
α : Type v
inst✝¹ : LinearOrder α
ι : Type u_1
f : ι → Set α
inst✝ : OrderTop α
hs : ∀ (i : ι), IsClub (f i)
⊢ IsClub (⋂₀ range f)
Module: Mathlib.RingTheory.WittVector.MulCoeff

Goal state before:
pp: case h
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : (↑↑(univ ×ˢ range (n + 1)) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=
  fun x y ↦
    f₀ fun a ↦
      Subtype.casesOn a fun a ha ↦
        Function.uncurry ![x, y]
          (let S := {a | a.2 = n ∨ a.2 < n};
          have ha' := ⋯;
          (a.1, ⟨a.2, ⋯⟩))
x y : 𝕎 k
⊢ (f₀ fun a ↦ ![truncateFun (n + 1) x, truncateFun (n + 1) y] (↑a).1 ⟨(↑a).2, ⋯⟩) =
    f₀ (Function.uncurry ![fun i ↦ x.coeff i, fun i ↦ y.coeff i] ∘ Subtype.val)

Tactic:
]

Goal state after:
case h
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f₀ : (↑↑(univ ×ˢ range (n + 1)) → k) → k
hf₀ : ∀ (x : Fin 2 × ℕ → k), f₀ (x ∘ Subtype.val) = (aeval x) (polyOfInterest p n)
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k :=
  fun x y ↦
    f₀ fun a ↦
      Subtype.casesOn a fun a ha ↦
        Function.uncurry ![x, y]
          (let S := {a | a.2 = n ∨ a.2 < n};
          have ha' := ⋯;
          (a.1, ⟨a.2, ⋯⟩))
x y : 𝕎 k
⊢ (f₀ fun a ↦ ![truncateFun (n + 1) x, truncateFun (n + 1) y] (↑a).1 ⟨(↑a).2, ⋯⟩) =
    f₀ (Function.uncurry ![fun i ↦ x.coeff i, fun i ↦ y.coeff i] ∘ Subtype.val)
Module: Mathlib.SetTheory.Cardinal.Cofinality.Club

Goal state before:
pp: α : Type v
inst✝ : LinearOrder α
ι : Type u_1
f : ι → Set α
hα : cof α ≤ 1
hs : ∀ (i : ι), IsClub (f i)
⊢ IsClub (⋂₀ range f)

Tactic:
]

Goal state after:
α : Type v
inst✝ : LinearOrder α
ι : Type u_1
f : ι → Set α
hα : cof α ≤ 1
hs : ∀ (i : ι), IsClub (f i)
⊢ IsClub (⋂₀ range f)
Module: Mathlib.RingTheory.WittVector.Isocrystal

Goal state before:
pp: case h
p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
k : Type u_2
inst✝⁴ : Field k
inst✝³ : IsAlgClosed k
inst✝² : CharP k p
V : Type u_3
inst✝¹ : AddCommGroup V
inst✝ : Isocrystal p k V
h_dim : finrank K(p, k) V = 1
this✝ : Nontrivial V
x : V
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
a : K(p, k)
ha : a ≠ 0
hax : Φ(p, k) x = a • x
b : K(p, k)
hb : b ≠ 0
m : ℤ
hmb : φ(p, k) b * a = ↑p ^ m * b
F₀ : StandardOneDimIsocrystal p k m →ₗ[K(p, k)] V := LinearMap.toSpanSingleton K(p, k) V x
F : StandardOneDimIsocrystal p k m ≃ₗ[K(p, k)] V := LinearEquiv.ofBijective F₀ ⋯
c : StandardOneDimIsocrystal p k m
⊢ Φ(p, k) (b • F₀ c) = b • F₀ (↑p ^ m • φ(p, k) c)

Tactic:
]

Goal state after:
case h
p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
k : Type u_2
inst✝⁴ : Field k
inst✝³ : IsAlgClosed k
inst✝² : CharP k p
V : Type u_3
inst✝¹ : AddCommGroup V
inst✝ : Isocrystal p k V
h_dim : finrank K(p, k) V = 1
this✝ : Nontrivial V
x : V
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
a : K(p, k)
ha : a ≠ 0
hax : Φ(p, k) x = a • x
b : K(p, k)
hb : b ≠ 0
m : ℤ
hmb : φ(p, k) b * a = ↑p ^ m * b
F₀ : StandardOneDimIsocrystal p k m →ₗ[K(p, k)] V := LinearMap.toSpanSingleton K(p, k) V x
F : StandardOneDimIsocrystal p k m ≃ₗ[K(p, k)] V := LinearEquiv.ofBijective F₀ ⋯
c : StandardOneDimIsocrystal p k m
⊢ Φ(p, k) (b • F₀ c) = b • F₀ (↑p ^ m • φ(p, k) c)
Module: Mathlib.SetTheory.Cardinal.Cofinality.Club

Goal state before:
pp: α : Type v
inst✝¹ : LinearOrder α
inst✝ : WellFoundedLT α
s : Set (Set α)
hα : cof α ≠ ℵ₀
hsα : #↑s < cof α
hs : ∀ x ∈ s, IsClub x
⊢ IsClub (⋂₀ s)

Tactic:
by
  cases isEmpty_or_nonempty α; · simp
  obtain hα | hα := hα.lt_or_gt
  · exact .sInter_of_cof_le_one (cof_lt_aleph0_iff.1 hα) hs
  refine ⟨.sInter fun x hx ↦ (hs x hx).dirSupClosed, fun a ↦ ?_⟩
  choose f hf using fun x : s ↦ (hs _ x.2).isCofinal
  let g : ℕ → α := Nat.rec a fun _ IH ↦ sSup (.range (f · IH))
  have hg : BddAbove (.range g) := by
    refine .of_not_isCofinal fun hg ↦ (cof_le hg).not_gt (hα.trans_le' ?_)
    simpa using mk_range_le_lift (f := g)
  refine ⟨_, fun t ht ↦ ?_, le_csSup hg ⟨0, rfl⟩⟩
  apply (hs t ht).isLUB_mem (t := .range fun n ↦ f ⟨t, ht⟩ (g n)) _ (range_nonempty _)
  · refine ⟨?_, fun b hb ↦ csSup_le' ?_⟩ <;> rintro _ ⟨n, rfl⟩
    · apply (le_csSup (.of_not_isCofinal _) _).trans (le_csSup hg ⟨n + 1, rfl⟩)
      · exact fun hg' ↦ (cof_le hg').not_gt (mk_range_le.trans_lt hsα)
      · use ⟨t, ht⟩
    · exact (hf ⟨t, ht⟩ _).2.trans <| hb ⟨_, rfl⟩
  · grind

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Cofinality.Club

Goal state before:
pp: α : Type v
inst✝¹ : LinearOrder α
inst✝ : WellFoundedLT α
ι : Type u
f : ι → Set α
hα : cof α ≠ ℵ₀
hι : lift.{v, u} #ι < lift.{u, v} (cof α)
hf : ∀ (i : ι), IsClub (f i)
⊢ IsClub (⋂₀ range f)

Tactic:
]

Goal state after:
α : Type v
inst✝¹ : LinearOrder α
inst✝ : WellFoundedLT α
ι : Type u
f : ι → Set α
hα : cof α ≠ ℵ₀
hι : lift.{v, u} #ι < lift.{u, v} (cof α)
hf : ∀ (i : ι), IsClub (f i)
⊢ IsClub (⋂₀ range f)
Module: Mathlib.RingTheory.WittVector.Isocrystal

Goal state before:
pp: case h
p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
k : Type u_2
inst✝⁴ : Field k
inst✝³ : IsAlgClosed k
inst✝² : CharP k p
V : Type u_3
inst✝¹ : AddCommGroup V
inst✝ : Isocrystal p k V
h_dim : finrank K(p, k) V = 1
this✝ : Nontrivial V
x : V
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
a : K(p, k)
ha : a ≠ 0
hax : Φ(p, k) x = a • x
b : K(p, k)
hb : b ≠ 0
m : ℤ
hmb : φ(p, k) b * a = ↑p ^ m * b
F₀ : StandardOneDimIsocrystal p k m →ₗ[K(p, k)] V := LinearMap.toSpanSingleton K(p, k) V x
F : StandardOneDimIsocrystal p k m ≃ₗ[K(p, k)] V := LinearEquiv.ofBijective F₀ ⋯
c : StandardOneDimIsocrystal p k m
⊢ Φ(p, k) (b • c • x) = b • (↑p ^ m • φ(p, k) c) • x

Tactic:
]

Goal state after:
case h
p : ℕ
inst✝⁵ : Fact (Nat.Prime p)
k : Type u_2
inst✝⁴ : Field k
inst✝³ : IsAlgClosed k
inst✝² : CharP k p
V : Type u_3
inst✝¹ : AddCommGroup V
inst✝ : Isocrystal p k V
h_dim : finrank K(p, k) V = 1
this✝ : Nontrivial V
x : V
hx : x ≠ 0
this : Φ(p, k) x ≠ 0
a : K(p, k)
ha : a ≠ 0
hax : Φ(p, k) x = a • x
b : K(p, k)
hb : b ≠ 0
m : ℤ
hmb : φ(p, k) b * a = ↑p ^ m * b
F₀ : StandardOneDimIsocrystal p k m →ₗ[K(p, k)] V := LinearMap.toSpanSingleton K(p, k) V x
F : StandardOneDimIsocrystal p k m ≃ₗ[K(p, k)] V := LinearEquiv.ofBijective F₀ ⋯
c : StandardOneDimIsocrystal p k m
⊢ Φ(p, k) (b • c • x) = b • (↑p ^ m • φ(p, k) c) • x
Module: Mathlib.RingTheory.WittVector.MulCoeff

Goal state before:
pp: case h
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k
hf :
  ∀ (x y : 𝕎 k),
    f (truncateFun (n + 1) x) (truncateFun (n + 1) y) =
      (x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) - x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1)
x y : 𝕎 k
⊢ (x * y).coeff (n + 1) =
    x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1) + y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) +
      ((x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) - x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1))

Tactic:
]

Goal state after:
case h
p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
n : ℕ
f : TruncatedWittVector p (n + 1) k → TruncatedWittVector p (n + 1) k → k
hf :
  ∀ (x y : 𝕎 k),
    f (truncateFun (n + 1) x) (truncateFun (n + 1) y) =
      (x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) - x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1)
x y : 𝕎 k
⊢ (x * y).coeff (n + 1) =
    x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1) + y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) +
      ((x * y).coeff (n + 1) - y.coeff (n + 1) * x.coeff 0 ^ p ^ (n + 1) - x.coeff (n + 1) * y.coeff 0 ^ p ^ (n + 1))
Module: Mathlib.SetTheory.Cardinal.Cofinality.Club

Goal state before:
pp: α : Type v
inst✝¹ : LinearOrder α
inst✝ : WellFoundedLT α
ι : Type u
f : ι → Set α
hα : cof α ≠ ℵ₀
hι : lift.{v, u} #ι < lift.{u, v} (cof α)
hf : ∀ (i : ι), IsClub (f i)
⊢ lift.{?u.38, v} #↑(range f) < lift.{?u.38, v} (cof α)

Tactic:
]

Goal state after:
α : Type v
inst✝¹ : LinearOrder α
inst✝ : WellFoundedLT α
ι : Type u
f : ι → Set α
hα : cof α ≠ ℵ₀
hι : lift.{v, u} #ι < lift.{u, v} (cof α)
hf : ∀ (i : ι), IsClub (f i)
⊢ lift.{?u.38, v} #↑(range f) < lift.{?u.38, v} (cof α)
Module: Mathlib.SetTheory.Cardinal.EventuallyConst

Goal state before:
pp: α : Type u
β : Type v
inst✝² : LinearOrder α
inst✝¹ : PartialOrder β
f : α → β
inst✝ : Nonempty α
hf : Monotone f
hf' : ¬IsCofinal (range (rangeSplitting f))
⊢ ∃ i, ∀ (j : α), i ≤ j → f j = f i

Tactic:
]

Goal state after:
α : Type u
β : Type v
inst✝² : LinearOrder α
inst✝¹ : PartialOrder β
f : α → β
inst✝ : Nonempty α
hf : Monotone f
hf' : ¬IsCofinal (range (rangeSplitting f))
⊢ ∃ i, ∀ (j : α), i ≤ j → f j = f i
Module: Mathlib.SetTheory.Cardinal.EventuallyConst

Goal state before:
pp: α : Type u
β : Type v
inst✝² : LinearOrder α
inst✝¹ : PartialOrder β
f : α → β
inst✝ : Nonempty α
hf : Monotone f
hf' : ¬IsCofinal (range (rangeSplitting f))
i : α
hi : ∀ y ∈ range (rangeSplitting f), y < i
j : α
hij : i ≤ j
⊢ f j ≤ ↑⟨f j, ⋯⟩

Tactic:
]

Goal state after:
Module: Mathlib.SetTheory.Cardinal.EventuallyConst

Goal state before:
pp: α : Type u
β : Type v
inst✝² : LinearOrder α
inst✝¹ : PartialOrder β
f : α → β
inst✝ : Nonempty α
hf : Monotone f
hf' : ¬IsCofinal (range (rangeSplitting f))
⊢ EventuallyConst f atTop

Tactic:
rw [eventuallyConst_atTop]
  obtain ⟨i, hi⟩ := not_isCofinal_iff.1 hf'
  refine ⟨i, fun j hij ↦ (hf hij).antisymm' <| (hf (hi _ ⟨⟨f j, j, rfl⟩, rfl⟩).le).trans' ?_⟩
  rw [apply_rangeSplitting f]

Goal state after:
Module: Mathlib.SetTheory.Cardinal.EventuallyConst

Goal state before:
pp: α : Type u
β : Type v
inst✝¹ : LinearOrder α
inst✝ : PartialOrder β
f : α → β
hf : Monotone f
this✝ : Nonempty α
hα : IsCofinal (range (rangeSplitting f))
this : LinearOrder ↑(range f) := ⋯.linearOrder
⊢ cof ↑(range f) ≤ #β

Tactic:
]

Goal state after:
α : Type u
β : Type v
inst✝¹ : LinearOrder α
inst✝ : PartialOrder β
f : α → β
hf : Monotone f
this✝ : Nonempty α
hα : IsCofinal (range (rangeSplitting f))
this : LinearOrder ↑(range f) := ⋯.linearOrder
⊢ cof ↑(range f) ≤ #β
Module: Mathlib.SetTheory.Cardinal.Cofinality.Club

Goal state before:
pp: α : Type v
inst✝¹ : LinearOrder α
inst✝ : WellFoundedLT α
s : Set (Set α)
hα✝ : cof α ≠ ℵ₀
hsα : s.Countable
hs : ∀ x ∈ s, IsClub x
hα : cof α < ℵ₀
⊢ cof α < ℵ₀

Tactic:
]

Goal state after:
α : Type v
inst✝¹ : LinearOrder α
inst✝ : WellFoundedLT α
s : Set (Set α)
hα✝ : cof α ≠ ℵ₀
hsα : s.Countable
hs : ∀ x ∈ s, IsClub x
hα : cof α < ℵ₀
⊢ cof α < ℵ₀
Module: Mathlib.SetTheory.Cardinal.Cofinality.Club

Goal state before:
pp: α : Type v
inst✝¹ : LinearOrder α
inst✝ : WellFoundedLT α
s : Set (Set α)
hα✝ : cof α ≠ ℵ₀
hsα : s.Countable
hs : ∀ x ∈ s, IsClub x
hα : ℵ₀ < cof α
⊢ s.Countable

Tactic:
]

Goal state after:
α : Type v
inst✝¹ : LinearOrder α
inst✝ : WellFoundedLT α
s : Set (Set α)
hα✝ : cof α ≠ ℵ₀
hsα : s.Countable
hs : ∀ x ∈ s, IsClub x
hα : ℵ₀ < cof α
⊢ s.Countable