mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: case odd.right
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n := ⋯
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := ⋯
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n := ⋯
cp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ n → (p ^ m).coeff (m * n) = p.coeff n ^ m := ⋯
m : ℕ
h₁ : (W.preΨ' (m + 1)).natDegree ≤ expDegree (m + 1) ∧ (W.preΨ' (m + 1)).coeff (expDegree (m + 1)) = ↑(expCoeff (m + 1))
h₂ : (W.preΨ' (m + 2)).natDegree ≤ expDegree (m + 2) ∧ (W.preΨ' (m + 2)).coeff (expDegree (m + 2)) = ↑(expCoeff (m + 2))
h₃ : (W.preΨ' (m + 3)).natDegree ≤ expDegree (m + 3) ∧ (W.preΨ' (m + 3)).coeff (expDegree (m + 3)) = ↑(expCoeff (m + 3))
h₄ : (W.preΨ' (m + 4)).natDegree ≤ expDegree (m + 4) ∧ (W.preΨ' (m + 4)).coeff (expDegree (m + 4)) = ↑(expCoeff (m + 4))
⊢ ((W.preΨ' (m + 4) * W.preΨ' (m + 2) ^ 3 * if Even m then W.Ψ₂Sq ^ 2 else 1) -
          W.preΨ' (m + 1) * W.preΨ' (m + 3) ^ 3 * if Even m then 1 else W.Ψ₂Sq ^ 2).coeff
      (expDegree (2 * (m + 2) + 1)) =
    ↑(expCoeff (2 * (m + 2) + 1))

Tactic:
coeff_sub,

Goal state after:
case odd.right
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
  fun {m n} {p q} ↦ natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n :=
  fun {m n} {p q} ↦ coeff_mul_add_eq_of_natDegree_le
cp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ n → (p ^ m).coeff (m * n) = p.coeff n ^ m :=
  fun {m n} {p} ↦ coeff_pow_of_natDegree_le
m : ℕ
h₁ : (W.preΨ' (m + 1)).natDegree ≤ expDegree (m + 1) ∧ (W.preΨ' (m + 1)).coeff (expDegree (m + 1)) = ↑(expCoeff (m + 1))
h₂ : (W.preΨ' (m + 2)).natDegree ≤ expDegree (m + 2) ∧ (W.preΨ' (m + 2)).coeff (expDegree (m + 2)) = ↑(expCoeff (m + 2))
h₃ : (W.preΨ' (m + 3)).natDegree ≤ expDegree (m + 3) ∧ (W.preΨ' (m + 3)).coeff (expDegree (m + 3)) = ↑(expCoeff (m + 3))
h₄ : (W.preΨ' (m + 4)).natDegree ≤ expDegree (m + 4) ∧ (W.preΨ' (m + 4)).coeff (expDegree (m + 4)) = ↑(expCoeff (m + 4))
⊢ (W.preΨ' (m + 4) * W.preΨ' (m + 2) ^ 3 * if Even m then W.Ψ₂Sq ^ 2 else 1).coeff (expDegree (2 * (m + 2) + 1)) -
      (W.preΨ' (m + 1) * W.preΨ' (m + 3) ^ 3 * if Even m then 1 else W.Ψ₂Sq ^ 2).coeff (expDegree (2 * (m + 2) + 1)) =
    ↑(expCoeff (2 * (m + 2) + 1))
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E✝ E'✝ : WeierstrassCurve F
inst✝⁴ : E✝.IsElliptic
inst✝³ : E'✝.IsElliptic
p : ℕ
inst✝² : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝³ : NeZero 2
this✝² : NeZero 4
this✝¹ : NeZero 6
this✝ : Invertible 2 := invertibleOfNonzero hchar2
this : Invertible 3 := invertibleOfNonzero hchar3
E : WeierstrassCurve F
inst✝¹ : E.IsElliptic
h✝¹ : E.IsShortNF
E' : WeierstrassCurve F
inst✝ : E'.IsElliptic
h✝ : E'.IsShortNF
heq :
  (-48 * E.a₄) ^ 3 * (-16 * (4 * E'.a₄ ^ 3 + 27 * E'.a₆ ^ 2)) =
    (-48 * E'.a₄) ^ 3 * (-16 * (4 * E.a₄ ^ 3 + 27 * E.a₆ ^ 2))
⊢ 47775744 ≠ 0

Tactic:
by
      have := mul_ne_zero (pow_ne_zero 16 hchar2) (pow_ne_zero 6 hchar3)
      norm_num1 at this
      exact this

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: case neg
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
ih : ∀ (n : ℕ), (W.preΨ ↑n).coeff (((↑n).natAbs ^ 2 - if Even ↑n then 4 else 1) / 2) = ↑(if Even ↑n then ↑n / 2 else ↑n)
n : ℕ
⊢ -(W.preΨ ↑n).coeff (((↑n).natAbs ^ 2 - if Even ↑n then 4 else 1) / 2) = ↑(if Even ↑n then -↑n / 2 else -↑n)

Tactic:
rcases ih n, n.even_or_odd' with ⟨ih, ⟨n, rfl | rfl⟩⟩ <;>
      push_cast [even_two_mul, Int.not_even_two_mul_add_one, Int.neg_ediv_of_dvd ⟨n, rfl⟩] at * <;>
      rw [ih]

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
n : ℤ
⊢ (W.preΨ n).coeff ((n.natAbs ^ 2 - if Even n then 4 else 1) / 2) = ↑(if Even n then n / 2 else n)

Tactic:
by
  induction n using Int.negInduction with
  | nat n => exact_mod_cast W.preΨ_ofNat n ▸ W.coeff_preΨ' n
  | neg ih n =>
    simp_rw [preΨ_neg, coeff_neg, Int.natAbs_neg, even_neg]
    rcases ih n, n.even_or_odd' with ⟨ih, ⟨n, rfl | rfl⟩⟩ <;>
      push_cast [even_two_mul, Int.not_even_two_mul_add_one, Int.neg_ediv_of_dvd ⟨n, rfl⟩] at * <;>
      rw [ih]

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: case succ.succ.right
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n := ⋯
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := ⋯
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n := ⋯
h : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ expDegree n ∧ (W.preΨ' n).coeff (expDegree n) = ↑(expCoeff n) := ⋯
n : ℕ
hd : (n + 1 + 1) ^ 2 = 1 + 2 * expDegree (n + 2) + if Even (n + 1) then 0 else 3
hd' : (n + 1 + 1) ^ 2 = expDegree (n + 3) + expDegree (n + 1) + if Even (n + 1) then 3 else 0
hc :
  1 =
    (1 * expCoeff (n + 2) ^ 2 * if Even (n + 1) then 1 else 4) -
      expCoeff (n + 3) * expCoeff (n + 1) * if Even (n + 1) then 4 else 1
⊢ ((X * W.preΨ' (n + 1 + 1) ^ 2 * if Even (n + 1) then 1 else W.Ψ₂Sq) -
          W.preΨ' (n + 1 + 2) * W.preΨ' (n + 1) * if Even (n + 1) then W.Ψ₂Sq else 1).coeff
      ((n + 1 + 1) ^ 2) =
    1

Tactic:
coeff_sub,

Goal state after:
case succ.succ.right
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
dm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).natDegree ≤ m + n :=
  fun {m n} {p q} ↦ natDegree_mul_le_of_le
dp : ∀ {m n : ℕ} {p : R[X]}, p.natDegree ≤ m → (p ^ n).natDegree ≤ n * m := fun {m n} {p} ↦ natDegree_pow_le_of_le n
cm : ∀ {m n : ℕ} {p q : R[X]}, p.natDegree ≤ m → q.natDegree ≤ n → (p * q).coeff (m + n) = p.coeff m * q.coeff n :=
  fun {m n} {p q} ↦ coeff_mul_add_eq_of_natDegree_le
h : ∀ {n : ℕ}, (W.preΨ' n).natDegree ≤ expDegree n ∧ (W.preΨ' n).coeff (expDegree n) = ↑(expCoeff n) :=
  fun {n} ↦ natDegree_coeff_preΨ' W n
n : ℕ
hd : (n + 1 + 1) ^ 2 = 1 + 2 * expDegree (n + 2) + if Even (n + 1) then 0 else 3
hd' : (n + 1 + 1) ^ 2 = expDegree (n + 3) + expDegree (n + 1) + if Even (n + 1) then 3 else 0
hc :
  1 =
    (1 * expCoeff (n + 2) ^ 2 * if Even (n + 1) then 1 else 4) -
      expCoeff (n + 3) * expCoeff (n + 1) * if Even (n + 1) then 4 else 1
⊢ (X * W.preΨ' (n + 1 + 1) ^ 2 * if Even (n + 1) then 1 else W.Ψ₂Sq).coeff ((n + 1 + 1) ^ 2) -
      (W.preΨ' (n + 1 + 2) * W.preΨ' (n + 1) * if Even (n + 1) then W.Ψ₂Sq else 1).coeff ((n + 1 + 1) ^ 2) =
    1
Module: Mathlib.Topology.Algebra.Nonarchimedean.Bases

Goal state before:
pp: A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
⊢ ∀ {x y : Set A}, x ∈ {U | ∃ i, U = ↑(B i)} → y ∈ {U | ∃ i, U = ↑(B i)} → ∃ z ∈ {U | ∃ i, U = ↑(B i)}, z ⊆ x ∩ y

Tactic:
by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    obtain ⟨k, hk⟩ := hB.inter i j
    use B k
    constructor
    · use k
    · exact hk

Goal state after:
Module: Mathlib.Topology.Algebra.Nonarchimedean.Bases

Goal state before:
pp: ι : Type u_1
R : Type u_2
A : Type u_3
inst✝⁶ : CommRing R
inst✝⁵ : CommRing A
inst✝⁴ : Algebra R A
M : Type u_4
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : TopologicalSpace R
inst✝ : Nonempty ι
B : ι → Submodule R M
hB : SubmodulesBasis B
⊢ ∀ {x y : Set M}, x ∈ {U | ∃ i, U = ↑(B i)} → y ∈ {U | ∃ i, U = ↑(B i)} → ∃ z ∈ {U | ∃ i, U = ↑(B i)}, z ⊆ x ∩ y

Tactic:
by
    rintro _ _ ⟨i, rfl⟩ ⟨j, rfl⟩
    obtain ⟨k, hk⟩ := hB.inter i j
    use B k
    constructor
    · use k
    · exact hk

Goal state after:
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: case refine_1
R : Type u_1
inst✝⁷ : Ring R
inst✝⁶ : ValuativeRel R
K : Type u_2
inst✝⁵ : DivisionRing K
inst✝⁴ : ValuativeRel K
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsValuativeTopology R
cts_add : ContinuousConstVAdd R R
basis : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | (valuation R).restrict x < ↑γ}
⊢ Tendsto (Function.uncurry fun x1 x2 ↦ x1 + x2) (𝓝 0 ×ˢ 𝓝 0) (𝓝 0)

Tactic:
exact (basis.prod_self.tendsto_iff basis).2 fun γ _ ↦
      ⟨γ, trivial, fun ⟨_, _⟩ hx ↦ (valuation R).restrict.map_add_lt hx.left hx.right⟩

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case mk
R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
a : R
b : ↥(posSubmonoid R)
⊢ ∃ a_1 b_1, (valuation R) a_1 / (valuation R) ↑b_1 = ValueGroupWithZero.mk a b

Tactic:
| mk a b
   => 

Goal state after:
case mk
R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
a : R
b : ↥(posSubmonoid R)
⊢ ∃ a_1 b_1, (valuation R) a_1 / (valuation R) ↑b_1 = ValueGroupWithZero.mk a b
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : Semiring R
inst✝² : ValuativeRel R
inst✝¹ : IsDiscrete R
inst✝ : IsNontrivial R
γ : ValueGroupWithZero R
hγ : 0 < γ
hγ' : γ < 1
⊢ 0 < uniformizer R

Tactic:
exact hγ.trans_le (le_uniformizer_iff.mpr hγ')

Goal state after:
Module: Mathlib.Topology.Algebra.UniformField

Goal state before:
pp: K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : IsTopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : IsUniformAddGroup K
x : hat K
x_ne : x ≠ 0
this : T1Space (hat K)
⊢ x * x.hatInv = 1

Tactic:
let f := fun x : hat K => x * hatInv x

Goal state after:
K : Type u_1
inst✝⁴ : Field K
inst✝³ : UniformSpace K
inst✝² : IsTopologicalDivisionRing K
inst✝¹ : CompletableTopField K
inst✝ : IsUniformAddGroup K
x : hat K
x_ne : x ≠ 0
this : T1Space (hat K)
f : hat K → hat K := fun x ↦ x * x.hatInv
⊢ x * x.hatInv = 1
Module: Mathlib.RingTheory.Valuation.Discrete.Basic

Goal state before:
pp: Γ : Type u_1
inst✝² : LinearOrderedCommGroupWithZero Γ
A : Type u_2
inst✝¹ : Ring A
v : Valuation A Γ
inst✝ : v.IsRankOneDiscrete
⊢ generator v ∈ (MonoidWithZeroHom.ofClass v).valueGroup

Tactic:
rw [← IsRankOneDiscrete.generator_zpowers_eq_valueGroup]
  exact mem_zpowers (IsRankOneDiscrete.generator v)

Goal state after:
Module: Mathlib.RingTheory.Valuation.Discrete.Basic

Goal state before:
pp: Γ : Type u_1
inst✝¹ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝ : Field K
v : Valuation K Γ
hv : v.IsRankOneDiscrete
π₁ π₂ : ↥v.valuationSubring
h1 : v.IsUniformizer ↑π₁
u : (↥v.valuationSubring)ˣ
hu : π₁ * ↑u = π₂
this : v ↑↑u = 1
⊢ v ↑π₂ = ↑(generator v)

Tactic:
← hu,

Goal state after:
Γ : Type u_1
inst✝¹ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝ : Field K
v : Valuation K Γ
hv : v.IsRankOneDiscrete
π₁ π₂ : ↥v.valuationSubring
h1 : v.IsUniformizer ↑π₁
u : (↥v.valuationSubring)ˣ
hu : π₁ * ↑u = π₂
this : v ↑↑u = 1
⊢ v ↑(π₁ * ↑u) = ↑(generator v)
Module: Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing

Goal state before:
pp: case h.right
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
v : Valuation K (WithZero (Multiplicative ℤ)) := ⋯
π : K := ⋯
hπ : v π = ↑(ofAdd (-1))
⊢ Units.mk0 (v π) ⋯ ≠ 1

Tactic:
simpa [hπ] using not_eq_of_beq_eq_false rfl

Goal state after:
Module: Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing

Goal state before:
pp: case neg
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
H : (valuation K (maximalIdeal A)) x ≤ 1
π : A
hπ : Irreducible π
a b : A
hb : b ∈ nonZeroDivisors A
h_frac : (algebraMap A K) a / (algebraMap A K) b = x
ha : ¬a = 0
⊢ ∃ a, (algebraMap A K) a = x

Tactic:
rw [← h_frac] at H

Goal state after:
case neg
A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
x : K
π : A
hπ : Irreducible π
a b : A
H : (valuation K (maximalIdeal A)) ((algebraMap A K) a / (algebraMap A K) b) ≤ 1
hb : b ∈ nonZeroDivisors A
h_frac : (algebraMap A K) a / (algebraMap A K) b = x
ha : ¬a = 0
⊢ ∃ a, (algebraMap A K) a = x
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_2
G : Type u_1
G' : Type u_2
inst✝⁵ : CommGroup G
inst✝⁴ : LinearOrder G
inst✝³ : IsOrderedMonoid G
inst✝² : CommGroup G'
inst✝¹ : LinearOrder G'
inst✝ : IsOrderedMonoid G'
x : G
y : G'
hxy : x = 1 ↔ y = 1
hx : ¬x = 1
x' : G := max x x⁻¹
hx' : x' = max x x⁻¹
xpos : 1 < x'
y' : G' := max y y⁻¹
hy' : y' = max y y⁻¹
ypos : 1 < y'
hxc : closure {x} = closure {x'}
hyc : closure {y} = closure {y'}
Hx : ∀ (z : ↥(closure {x})), z ∈ zpowers ⟨x', ⋯⟩
Hy : ∀ (z : ↥(closure {y})), z ∈ zpowers ⟨y', ⋯⟩
⊢ ∀ {a b : ↥(closure {x})}, (mulEquivOfOrderOfEq Hx Hy ⋯).toFun a ≤ (mulEquivOfOrderOfEq Hx Hy ⋯).toFun b ↔ a ≤ b

Tactic:
intro a b
      obtain ⟨m, rfl⟩ := Hx a
      obtain ⟨n, rfl⟩ := Hx b
      simp only [MulEquiv.toEquiv_eq_coe, Equiv.toFun_as_coe, EquivLike.coe_coe, map_zpow,
        mulEquivOfOrderOfEq_apply_gen]
      simp_all [zpow_le_zpow_iff_right, ← Subtype.coe_lt_coe]

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case pos
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Nontrivial G
g : G
h : ∀ (s : Set ↑{x | 0 ≤ x}), s.Nonempty → ∃ m ∈ s, ∀ x ∈ s, ¬x < m
H : ∀ (x : G) {y : G}, 0 < y → ∃ n, x ≤ n • y
⊢ Nonempty (G ≃+o ℤ)

pp: case neg
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Nontrivial G
g : G
h : ∀ (s : Set ↑{x | 0 ≤ x}), s.Nonempty → ∃ m ∈ s, ∀ x ∈ s, ¬x < m
H : ∃ x y, 0 < y ∧ ∀ (n : ℕ), n • y < x
⊢ Nonempty (G ≃+o ℤ)

Tactic:
· replace H : Archimedean G := ⟨H⟩
      rw [LinearOrderedAddCommGroup.discrete_iff_not_denselyOrdered]
      intro hd
      obtain ⟨y, hy⟩ := exists_ne (0 : G)
      wlog! hy' : 0 < y generalizing y
      · refine this (-y) ?_ ?_
        · simp [hy]
        · simp [lt_of_le_of_ne hy' hy]
      obtain ⟨⟨z, hz⟩, hz', hz''⟩ := h ({x | ⟨0, le_rfl⟩ < x}) ⟨⟨y, hy'.le⟩, hy'⟩
      obtain ⟨w, hw, hw'⟩ := exists_between hz'
      exact hz'' ⟨w, hw.le⟩ hw hw'

Goal state after:
case neg
G : Type u_2
inst✝³ : AddCommGroup G
inst✝² : LinearOrder G
inst✝¹ : IsOrderedAddMonoid G
inst✝ : Nontrivial G
g : G
h : ∀ (s : Set ↑{x | 0 ≤ x}), s.Nonempty → ∃ m ∈ s, ∀ x ∈ s, ¬x < m
H : ∃ x y, 0 < y ∧ ∀ (n : ℕ), n • y < x
⊢ Nonempty (G ≃+o ℤ)
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
x : hat K
this : extensionValuation x = 0 ↔ x = 0
⊢ extension x = 0 ↔ x = 0

Tactic:
by
    simpa only [extensionValuation_toFun, map_eq_zero]

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
⊢ extensionValuation.restrict ↑⋯.choose = extensionValuation.restrict ↑x * extensionValuation.restrict ↑⋯.choose

Tactic:
have hx := (restrict₀_surjective (.ofClass hv.v) a).choose_spec

Goal state after:
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
x : K := ⋯.choose
hx_def : x = ⋯.choose
hx : (restrict₀ (ofClass v)) ⋯.choose = a
⊢ extensionValuation.restrict ↑⋯.choose = extensionValuation.restrict ↑x * extensionValuation.restrict ↑⋯.choose
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: G₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero G₀
⊢ MulArchimedean G₀ˣ ↔ MulArchimedean G₀

Tactic:
constructor <;> intro _
  · exact OrderMonoidIso.withZeroUnits.mulArchimedean
  · infer_instance

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
⊢ exp (-↑((Associates.mk v.asIdeal).count (Associates.mk 1).factors)) = 1

Tactic:
Associates.mk_one,

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
⊢ exp (-↑((Associates.mk v.asIdeal).count (Associates.factors 1))) = 1
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: case succ
R : Type u_1
inst✝ : Semiring R
φ : R⟦X⟧
n : ℕ
⊢ (coeff (n + 1)) φ = (coeff (n + 1)) ((X * mk fun p ↦ (coeff (p + 1)) φ) + C (constantCoeff φ))

Tactic:
· simp only [coeff_succ_X_mul, coeff_mk, map_add, coeff_C, n.succ_ne_zero,
      if_false, add_zero]

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case refine_1
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
hab : valueGroup₀_hom_extensionValuation a = valueGroup₀_hom_extensionValuation b
x : K := ⋯.choose
hx_def : x = ⋯.choose
⊢ a = b

Tactic:
have hx := (restrict₀_surjective (.ofClass hv.v) a).choose_spec

Goal state after:
case refine_1
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
hab : valueGroup₀_hom_extensionValuation a = valueGroup₀_hom_extensionValuation b
x : K := ⋯.choose
hx_def : x = ⋯.choose
hx : (restrict₀ (ofClass v)) ⋯.choose = a
⊢ a = b
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: case neg
σ : Type u_1
R : Type u_2
inst✝¹ : Semiring R
inst✝ : Nontrivial R
s t : σ
H : ¬single s 1 = single t 1
h : 1 = 0
⊢ s = t

Tactic:
exfalso
      exact one_ne_zero h

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: σ : Type u_1
R : Type u_2
S : Type u_3
T : Type u_4
inst✝² : Semiring R
inst✝¹ : Semiring S
inst✝ : Semiring T
f : R →+* S
g : S →+* T
n : σ →₀ ℕ
⊢ f (if n = 0 then 1 else 0) = (coeff n) 1

Tactic:
coeff_one

Goal state after:
σ : Type u_1
R : Type u_2
S : Type u_3
T : Type u_4
inst✝² : Semiring R
inst✝¹ : Semiring S
inst✝ : Semiring T
f : R →+* S
g : S →+* T
n : σ →₀ ℕ
⊢ f (if n = 0 then 1 else 0) = if n = 0 then 1 else 0
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: case insert
σ : Type u_1
R : Type u_3
inst✝² : CommSemiring R
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq σ
f : ι → MvPowerSeries σ R
a : ι
s : Finset ι
ha : a ∉ s
ih : ∀ (d : σ →₀ ℕ), (coeff d) (∏ j ∈ s, f j) = ∑ l ∈ s.finsuppAntidiag d, ∏ i ∈ s, (coeff (l i)) (f i)
u v : σ →₀ ℕ
⊢ (coeff u) (f a) * (coeff v) (∏ x ∈ s, f x) =
    ∑ x ∈ (s.finsuppAntidiag v).attach, ∏ x_1 ∈ insert a s, (coeff (Function.update (⇑↑x) a u x_1)) (f x_1)

Tactic:
rw [ih, Finset.mul_sum, ← Finset.sum_attach]

Goal state after:
case insert
σ : Type u_1
R : Type u_3
inst✝² : CommSemiring R
ι : Type u_4
inst✝¹ : DecidableEq ι
inst✝ : DecidableEq σ
f : ι → MvPowerSeries σ R
a : ι
s : Finset ι
ha : a ∉ s
ih : ∀ (d : σ →₀ ℕ), (coeff d) (∏ j ∈ s, f j) = ∑ l ∈ s.finsuppAntidiag d, ∏ i ∈ s, (coeff (l i)) (f i)
u v : σ →₀ ℕ
⊢ ∑ x ∈ (s.finsuppAntidiag v).attach, (coeff u) (f a) * ∏ i ∈ s, (coeff (↑x i)) (f i) =
    ∑ x ∈ (s.finsuppAntidiag v).attach, ∏ x_1 ∈ insert a s, (coeff (Function.update (⇑↑x) a u x_1)) (f x_1)
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
r : R
⊢ (valuation K v) ↑r ≤ 1

Tactic:
by
  rw [valuation_of_algebraMap]; exact v.intValuation_le_one r

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
f : MvPowerSeries σ R
m : ℕ
hf : constantCoeff f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : (range n).sum ⇑k = d ∧ k.support ⊆ range n
s : Finset ℕ := {i ∈ range n | k i = 0}
hs_def : s = {i ∈ range n | k i = 0}
hs : s ⊆ range n
i : ℕ
hi : i ∈ range n ∧ k i = 0
⊢ (coeff (k i)) f = constantCoeff f

Tactic:
hi.2,

Goal state after:
σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
f : MvPowerSeries σ R
m : ℕ
hf : constantCoeff f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : (range n).sum ⇑k = d ∧ k.support ⊆ range n
s : Finset ℕ := {i ∈ range n | k i = 0}
hs_def : s = {i ∈ range n | k i = 0}
hs : s ⊆ range n
i : ℕ
hi : i ∈ range n ∧ k i = 0
⊢ (coeff 0) f = constantCoeff f
Module: Mathlib.RingTheory.MvPowerSeries.PiTopology

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹ : TopologicalSpace R
inst✝ : Semiring R
ι : Type u_3
f : ι → MvPowerSeries σ R
u : Filter ι
g : MvPowerSeries σ R
⊢ Tendsto f u (pi fun i ↦ 𝓝 (g i)) ↔ ∀ (d : σ →₀ ℕ), Tendsto (fun i ↦ (coeff d) (f i)) u (𝓝 ((coeff d) g))

Tactic:
tendsto_pi

Goal state after:
σ : Type u_1
R : Type u_2
inst✝¹ : TopologicalSpace R
inst✝ : Semiring R
ι : Type u_3
f : ι → MvPowerSeries σ R
u : Filter ι
g : MvPowerSeries σ R
⊢ (∀ (i : σ →₀ ℕ), Tendsto (fun x ↦ f x i) u (𝓝 (g i))) ↔
    ∀ (d : σ →₀ ℕ), Tendsto (fun i ↦ (coeff d) (f i)) u (𝓝 ((coeff d) g))
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
inst✝⁵ : IsDedekindDomain R
K : Type u_2
S : Type u_3
inst✝⁴ : Field K
inst✝³ : CommSemiring S
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
v : HeightOneSpectrum R
inst✝ : Algebra S K
⊢ UniformContinuousConstSMul S (WithVal (valuation K v))

Tactic:
by
  refine ⟨fun l ↦ ?_⟩
  simp_rw [WithVal.smul_right_def, Algebra.smul_def]
  exact (Ring.uniformContinuousConstSMul (WithVal <| v.valuation K)).uniformContinuous_const_smul _

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: case neg.left
R : Type u_1
inst✝¹ : Semiring R
n : ℕ
a : R
inst✝ : Decidable (a = 0)
h : ¬a = 0
i : ℕ
hi : i = n
⊢ (coeff n) ((monomial n) a) ≠ 0

Tactic:
coeff_monomial_same

Goal state after:
case neg.left
R : Type u_1
inst✝¹ : Semiring R
n : ℕ
a : R
inst✝ : Decidable (a = 0)
h : ¬a = 0
i : ℕ
hi : i = n
⊢ a ≠ 0
Module: Mathlib.RingTheory.PowerSeries.Order

Goal state before:
pp: case insert
R : Type u_2
ι : Type u_3
inst✝ : CommRing R
k : ℕ
φ : R⟦X⟧
f : ι → R⟦X⟧
a : ι
s : Finset ι
ha : a ∉ s
ih : (∀ i ∈ s, ↑k < (f i).order) → (coeff k) (φ * ∏ i ∈ s, (1 - f i)) = (coeff k) φ
t : ∀ i ∈ insert a s, ↑k < (f i).order
⊢ (coeff k) (φ * ∏ i ∈ insert a s, (1 - f i)) = (coeff k) φ

Tactic:
simp only [Finset.mem_insert, forall_eq_or_imp] at t

Goal state after:
case insert
R : Type u_2
ι : Type u_3
inst✝ : CommRing R
k : ℕ
φ : R⟦X⟧
f : ι → R⟦X⟧
a : ι
s : Finset ι
ha : a ∉ s
ih : (∀ i ∈ s, ↑k < (f i).order) → (coeff k) (φ * ∏ i ∈ s, (1 - f i)) = (coeff k) φ
t : ↑k < (f a).order ∧ ∀ a ∈ s, ↑k < (f a).order
⊢ (coeff k) (φ * ∏ i ∈ insert a s, (1 - f i)) = (coeff k) φ
Module: Mathlib.RingTheory.PowerSeries.Evaluation

Goal state before:
pp: R : Type u_1
inst✝¹⁷ : CommRing R
S : Type u_2
inst✝¹⁶ : CommRing S
φ : R →+* S
a : S
inst✝¹⁵ : UniformSpace R
inst✝¹⁴ : UniformSpace S
inst✝¹³ : IsUniformAddGroup R
inst✝¹² : IsTopologicalSemiring R
inst✝¹¹ : IsUniformAddGroup S
inst✝¹⁰ : T2Space S
inst✝⁹ : CompleteSpace S
inst✝⁸ : IsTopologicalRing S
inst✝⁷ : IsLinearTopology S S
hφ : Continuous ⇑φ
ha : HasEval a
T : Type u_3
inst✝⁶ : UniformSpace T
inst✝⁵ : CompleteSpace T
inst✝⁴ : T2Space T
inst✝³ : CommRing T
inst✝² : IsTopologicalRing T
inst✝¹ : IsLinearTopology T T
inst✝ : IsUniformAddGroup T
ε : S →+* T
hε : Continuous ⇑ε
⊢ Continuous ⇑(ε.comp φ)

Tactic:
by simp only [RingHom.coe_comp, hε.comp hφ]

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

Goal state before:
pp: case a
x : Cardinal.{u_1}
h₁ : 2 ≤ x
h₂ : x ≤ 𝔠
⊢ 𝔠 ≤ x ^ ℵ₀

Tactic:
rw [← two_power_aleph0]

Goal state after:
case a
x : Cardinal.{u_1}
h₁ : 2 ≤ x
h₂ : x ≤ 𝔠
⊢ 2 ^ ℵ₀ ≤ x ^ ℵ₀