mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.Complex.Polynomial.Basic

Goal state before:
pp: p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a ↦ ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
z : ℂ
hz1 : (aeval z) p = 0
hz2 : z.im = 0
key : (IsScalarTower.toAlgHom ℚ ℝ ℂ) z.re = z
⊢ (aeval z) p = (IsScalarTower.toAlgHom ℚ ℝ ℂ) 0

Tactic:
map_zero

Goal state after:
p : ℚ[X]
hp : ¬p = 0
inj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)
a : Finset ℂ := (p.rootSet ℂ).toFinset
b : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset
c : Finset ℂ :=
  Finset.image (fun a ↦ ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe))).support
ha : ∀ (z : ℂ), z ∈ a ↔ (aeval z) p = 0
z : ℂ
hz1 : (aeval z) p = 0
hz2 : z.im = 0
key : (IsScalarTower.toAlgHom ℚ ℝ ℂ) z.re = z
⊢ (aeval z) p = 0
Module: Mathlib.RingTheory.Valuation.RankOne

Goal state before:
pp: R : Type u_1
Γ₀ : Type u_2
inst✝² : Ring R
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.IsNontrivial
a✝ : MulArchimedean (ofClass v).ValueGroup₀
f : Additive (ofClass v).ValueGroup₀ˣ →+o ℝ
hf : Injective ⇑f
e : (ofClass v).ValueGroup₀ˣ →* Multiplicative ℝ := AddMonoidHom.toMultiplicativeRight ↑f
he : StrictMono ⇑e
x : Multiplicative ℝ
⊢ NNReal.mk (2 ^ (↑x).log) ⋯ ≠ 0

Tactic:
simp only [ne_eq, NNReal.eq_iff, NNReal.coe_mk, NNReal.coe_zero]
        positivity

Goal state after:
Module: Mathlib.Data.Real.Embedding

Goal state before:
pp: case mp.refine_2
M : Type u_1
inst✝⁶ : AddCommGroup M
inst✝⁵ : LinearOrder M
inst✝⁴ : IsOrderedAddMonoid M
inst✝³ : One M
inst✝² : ZeroLEOneClass M
inst✝¹ : NeZero 1
inst✝ : Archimedean M
x y : M
a : ℚ
h : a.num • 1 < a.den • (x + y)
k : ℕ
hk : 1 + 1 ≤ k • (a.den • (x + y) - a.num • 1)
hk0 : k ≠ 0
hka0 : k * a.den ≠ 0
m : ℤ
right✝ : ∀ (y : ℤ), (fun m ↦ 0 + m • 1 ∈ Set.Ico (k • a.den • x - 1) (k • a.den • x - 1 + 1)) y → y = m
hm1 : k • a.den • x - 1 ≤ 0 + m • 1
hm2 : 0 + m • 1 < k • a.den • x - 1 + 1
⊢ mkRat (↑k * a.num - m) (k * a.den) ∈ ratLt y

Tactic:
have hk' : 1 + (k • a.num • 1 - k • a.den • y) ≤ k • a.den • x - 1 := by
        rw [smul_add, smul_sub, smul_add, le_sub_iff_add_le, ← sub_le_iff_le_add] at hk
        rw [le_sub_iff_add_le]
        convert! hk using 1
        abel

Goal state after:
case mp.refine_2
M : Type u_1
inst✝⁶ : AddCommGroup M
inst✝⁵ : LinearOrder M
inst✝⁴ : IsOrderedAddMonoid M
inst✝³ : One M
inst✝² : ZeroLEOneClass M
inst✝¹ : NeZero 1
inst✝ : Archimedean M
x y : M
a : ℚ
h : a.num • 1 < a.den • (x + y)
k : ℕ
hk : 1 + 1 ≤ k • (a.den • (x + y) - a.num • 1)
hk0 : k ≠ 0
hka0 : k * a.den ≠ 0
m : ℤ
right✝ : ∀ (y : ℤ), (fun m ↦ 0 + m • 1 ∈ Set.Ico (k • a.den • x - 1) (k • a.den • x - 1 + 1)) y → y = m
hm1 : k • a.den • x - 1 ≤ 0 + m • 1
hm2 : 0 + m • 1 < k • a.den • x - 1 + 1
hk' : 1 + (k • a.num • 1 - k • a.den • y) ≤ k • a.den • x - 1
⊢ mkRat (↑k * a.num - m) (k * a.den) ∈ ratLt y
Module: Mathlib.RingTheory.Valuation.RankOne

Goal state before:
pp: R : Type u_1
Γ₀ : Type u_2
inst✝¹ : Ring R
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
hv : v.RankOne
x : (ofClass v).ValueGroup₀
hx : (hom v) x = 0
h_lt : 0 < x
hs : 0 < (hom v) x
⊢ False

Tactic:
hx

Goal state after:
R : Type u_1
Γ₀ : Type u_2
inst✝¹ : Ring R
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
hv : v.RankOne
x : (ofClass v).ValueGroup₀
hx : (hom v) x = 0
h_lt : 0 < x
hs : 0 < 0
⊢ False
Module: Mathlib.RingTheory.Valuation.RankOne

Goal state before:
pp: R : Type u_1
Γ₀ : Type u_2
inst✝¹ : Ring R
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
hv : v.RankOne
x : (ofClass v).ValueGroup₀
h : x = 0
⊢ (hom v) 0 = 0

Tactic:
map_zero

Goal state after:
R : Type u_1
Γ₀ : Type u_2
inst✝¹ : Ring R
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
hv : v.RankOne
x : (ofClass v).ValueGroup₀
h : x = 0
⊢ 0 = 0
Module: Mathlib.RingTheory.Valuation.RankOne

Goal state before:
pp: case neg.inr.inr
R : Type u_3
inst✝¹ : Ring R
inst✝ : ValuativeRel R
h : MulArchimedean (ValueGroupWithZero R)
a b : ValueGroupWithZero R
H : a < b
H' : 1 b ≤ 1 a
ha : a ≠ 0
ha' : a ≠ 1
⊢ IsNontrivial R

Tactic:
exact ⟨⟨a, ha, ha'⟩⟩

Goal state after:
Module: Mathlib.Data.Real.Embedding

Goal state before:
pp: case a
M : Type u_1
inst✝⁶ : AddCommGroup M
inst✝⁵ : LinearOrder M
inst✝⁴ : IsOrderedAddMonoid M
inst✝³ : One M
inst✝² : ZeroLEOneClass M
inst✝¹ : NeZero 1
inst✝ : Archimedean M
x : ℚ
⊢ x.num • 1 < ↑x.den • 1 → ↑x ≤ 1

Tactic:
hx

Goal state after:
case a
M : Type u_1
inst✝⁶ : AddCommGroup M
inst✝⁵ : LinearOrder M
inst✝⁴ : IsOrderedAddMonoid M
inst✝³ : One M
inst✝² : ZeroLEOneClass M
inst✝¹ : NeZero 1
inst✝ : Archimedean M
x : ℚ
hx : x.num • 1 < ↑x.den • 1
⊢ ↑x ≤ 1
Module: Mathlib.Data.Real.Embedding

Goal state before:
pp: case a
M : Type u_1
inst✝⁶ : AddCommGroup M
inst✝⁵ : LinearOrder M
inst✝⁴ : IsOrderedAddMonoid M
inst✝³ : One M
inst✝² : ZeroLEOneClass M
inst✝¹ : NeZero 1
inst✝ : Archimedean M
⊢ 1 ≤ embedRealFun 1

Tactic:
rw [le_csSup_iff (ratLt'_bddAbove (1 : M)) (ratLt'_nonempty 1)]

Goal state after:
case a
M : Type u_1
inst✝⁶ : AddCommGroup M
inst✝⁵ : LinearOrder M
inst✝⁴ : IsOrderedAddMonoid M
inst✝³ : One M
inst✝² : ZeroLEOneClass M
inst✝¹ : NeZero 1
inst✝ : Archimedean M
⊢ ∀ b ∈ upperBounds (ratLt' 1), 1 ≤ b
Module: Mathlib.Topology.MetricSpace.Ultra.Basic

Goal state before:
pp: case inl
X : Type u_1
inst✝¹ : PseudoMetricSpace X
inst✝ : IsUltrametricDist X
x : X
r : ℝ
hr : r ≠ 0
h : r < 0
⊢ IsOpen (closedBall x r)

Tactic:
simp [closedBall_eq_empty.mpr h]

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

Goal state before:
pp: case h
K : Type u_1
hK : NormedField K
inst✝ : IsUltrametricDist K
U : Set K
ε : ℝ
hε : ε > 0
h : Metric.ball 0 ε ⊆ U
H : Subsingleton (ofClass valuation).ValueGroup₀ˣ
x : K
⊢ x ∈ {x | valuation.restrict x < ↑(Units.mk0 (valuation.restrict 1) ⋯)} → x ∈ U

Tactic:
hx

Goal state after:
case h
K : Type u_1
hK : NormedField K
inst✝ : IsUltrametricDist K
U : Set K
ε : ℝ
hε : ε > 0
h : Metric.ball 0 ε ⊆ U
H : Subsingleton (ofClass valuation).ValueGroup₀ˣ
x : K
hx : x ∈ {x | valuation.restrict x < ↑(Units.mk0 (valuation.restrict 1) ⋯)}
⊢ x ∈ U
Module: Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
φ : MvPowerSeries σ R
hφ : constantCoeff φ ∈ nonZeroDivisorsRight R
x : MvPowerSeries σ R
⊢ x * φ = 0 → x = 0

Tactic:
hx

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
φ : MvPowerSeries σ R
hφ : constantCoeff φ ∈ nonZeroDivisorsRight R
x : MvPowerSeries σ R
hx : x * φ = 0
⊢ x = 0
Module: Mathlib.Topology.Algebra.Valued.NormedValued

Goal state before:
pp: L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
this : Nonempty { ε // ε > 0 }
U : Set (L × L)
⊢ (∃ i, True ∧ {p | v.restrict (p.2 - p.1) < ↑i} ⊆ U) ↔ U ∈ ⨅ x, 𝓟 {p | v.norm (p.1 - p.2) < ↑x}

Tactic:
mem_iInf_of_directed

Goal state after:
L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
this : Nonempty { ε // ε > 0 }
U : Set (L × L)
⊢ (∃ i, True ∧ {p | v.restrict (p.2 - p.1) < ↑i} ⊆ U) ↔ ∃ i, U ∈ 𝓟 {p | v.norm (p.1 - p.2) < ↑i}

case h
L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
this : Nonempty { ε // ε > 0 }
U : Set (L × L)
⊢ Directed (fun x1 x2 ↦ x1 ≥ x2) fun x ↦ 𝓟 {p | v.norm (p.1 - p.2) < ↑x}
Module: Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
φ : MvPowerSeries σ R
hφ : constantCoeff φ ∈ nonZeroDivisorsLeft R
x : MvPowerSeries σ R
⊢ φ * x = 0 → x = 0

Tactic:
hx

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
φ : MvPowerSeries σ R
hφ : constantCoeff φ ∈ nonZeroDivisorsLeft R
x : MvPowerSeries σ R
hx : φ * x = 0
⊢ x = 0
Module: Mathlib.Topology.Algebra.Valued.NormedValued

Goal state before:
pp: L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
this : Nonempty { ε // ε > 0 }
U : Set (L × L)
x✝ : ∃ i, {p | v.restrict (p.2 - p.1) < ↑i} ⊆ U
ε : (ofClass v).ValueGroup₀ˣ
hε : {p | v.restrict (p.2 - p.1) < ↑ε} ⊆ U
δ : ℝ≥0 := (RankOne.hom v) ↑ε
hδ : δ = (RankOne.hom v) ↑ε
hδ_pos : 0 < δ
x : L × L
⊢ x ∈ {p | v.norm (p.1 - p.2) < ↑δ} → x ∈ {p | v.restrict (p.2 - p.1) < ↑ε}

Tactic:
hx

Goal state after:
L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
this : Nonempty { ε // ε > 0 }
U : Set (L × L)
x✝ : ∃ i, {p | v.restrict (p.2 - p.1) < ↑i} ⊆ U
ε : (ofClass v).ValueGroup₀ˣ
hε : {p | v.restrict (p.2 - p.1) < ↑ε} ⊆ U
δ : ℝ≥0 := (RankOne.hom v) ↑ε
hδ : δ = (RankOne.hom v) ↑ε
hδ_pos : 0 < δ
x : L × L
hx : x ∈ {p | v.norm (p.1 - p.2) < ↑δ}
⊢ x ∈ {p | v.restrict (p.2 - p.1) < ↑ε}
Module: Mathlib.Topology.Algebra.Valued.NormedValued

Goal state before:
pp: L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
this✝ : Nonempty { ε // ε > 0 }
U : Set (L × L)
x✝ : ∃ a, 0 < a ∧ {p | v.norm (p.1 - p.2) < a} ⊆ U
r : ℝ
hr_pos : 0 < r
hr : {p | v.norm (p.1 - p.2) < r} ⊆ U
this : Nontrivial Γ₀ˣ
u : (ofClass v).ValueGroup₀ˣ
hu : ↑((RankOne.hom v) ↑u) < r
x : L × L
⊢ x ∈ {p | v.restrict (p.2 - p.1) < ↑u} → x ∈ {p | v.norm (p.1 - p.2) < r}

Tactic:
hx

Goal state after:
L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
val : Valued L Γ₀
hv : v.RankOne
this✝ : Nonempty { ε // ε > 0 }
U : Set (L × L)
x✝ : ∃ a, 0 < a ∧ {p | v.norm (p.1 - p.2) < a} ⊆ U
r : ℝ
hr_pos : 0 < r
hr : {p | v.norm (p.1 - p.2) < r} ⊆ U
this : Nontrivial Γ₀ˣ
u : (ofClass v).ValueGroup₀ˣ
hu : ↑((RankOne.hom v) ↑u) < r
x : L × L
hx : x ∈ {p | v.restrict (p.2 - p.1) < ↑u}
⊢ x ∈ {p | v.norm (p.1 - p.2) < r}
Module: Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors

Goal state before:
pp: case h₀
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
φ : MvPowerSeries σ R
hφ : constantCoeff φ ∈ nonZeroDivisorsLeft R
x : MvPowerSeries σ R
hx : φ * x = 0
d e : σ →₀ ℕ
he : ∀ y < e, (coeff y) x = (coeff y) 0
⊢ ∀ b ∈ antidiagonal e, b ≠ (0, e) → (coeff b.1) φ * (coeff b.2) x = 0

pp: case h₁
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
φ : MvPowerSeries σ R
hφ : constantCoeff φ ∈ nonZeroDivisorsLeft R
x : MvPowerSeries σ R
hx : φ * x = 0
d e : σ →₀ ℕ
he : ∀ y < e, (coeff y) x = (coeff y) 0
⊢ (0, e) ∉ antidiagonal e → (coeff (0, e).1) φ * (coeff (0, e).2) x = 0

Tactic:
· rintro ⟨_, u⟩ huv _
    suffices u < e by simp only [he u this, mul_zero, map_zero]
    apply lt_of_le_of_ne
    · simp only [← mem_antidiagonal.mp huv, le_add_iff_nonneg_left, zero_le]
    · rintro rfl
      simp_all

Goal state after:
case h₁
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
φ : MvPowerSeries σ R
hφ : constantCoeff φ ∈ nonZeroDivisorsLeft R
x : MvPowerSeries σ R
hx : φ * x = 0
d e : σ →₀ ℕ
he : ∀ y < e, (coeff y) x = (coeff y) 0
⊢ (0, e) ∉ antidiagonal e → (coeff (0, e).1) φ * (coeff (0, e).2) x = 0
Module: Mathlib.RingTheory.PowerSeries.Inverse

Goal state before:
pp: case neg.e_a.h.inr.hc.refine_2
R : Type u_1
inst✝ : Ring R
n : ℕ
a : R
φ : R⟦X⟧
h✝ : ¬n = 0
i j : ℕ
_hij : (i, j) ∈ antidiagonal n
H : j < n
hh :
  single () n ≤
    (match (i, j) with
      | (a, b) => (single () a, single () b)).2
⊢ n ≤ j

Tactic:
simpa [Finsupp.single_eq_same] using hh ()

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

Goal state before:
pp: R : Type u_1
k : Type u_2
inst✝ : Field k
f : k⟦X⟧
hf : f ≠ 0
⊢ Invertible (constantCoeff f.divXPowOrder)

Tactic:
apply invertibleOfNonzero
    simpa [constantCoeff_divXPowOrder_eq_zero_iff.not]

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

Goal state before:
pp: k : Type u_2
inst✝ : Field k
P : k[X]
hP : P ≠ 0
⊢ ∀ (c : ℕ), ↑c ≤ emultiplicity X ↑P ↔ ↑c ≤ emultiplicity Polynomial.X P

Tactic:
simp only [← pow_dvd_iff_le_emultiplicity, Polynomial.X_pow_dvd_iff,
    PowerSeries.X_pow_dvd_iff, Polynomial.coeff_coe P, implies_true, ne_eq, coe_eq_zero_iff, hP,
    not_false_eq_true]

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

Goal state before:
pp: k : Type u_2
inst✝ : Field k
P : k[X]
hP : P ≠ 0
⊢ ↑P ≠ 0

Tactic:
simp only [← pow_dvd_iff_le_emultiplicity, Polynomial.X_pow_dvd_iff,
    PowerSeries.X_pow_dvd_iff, Polynomial.coeff_coe P, implies_true, ne_eq, coe_eq_zero_iff, hP,
    not_false_eq_true]

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Projective R
P : Fin 3 → R
⊢ W'.addX P (W'.neg P) = 0

Tactic:
rw [addX, neg_X, neg_Y, neg_Z, negY]
  ring1

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic

Goal state before:
pp: case a
F : Type u
inst✝ : Field F
W : Projective F
P : Fin 3 → F
hPz : P z ≠ 0
⊢ (eval P) W.polynomial / P z ^ 3 +
        (P y ^ 2 / P z ^ 2 * 1 + W.a₁ * P x * P y / P z ^ 2 * 1 + W.a₃ * P y / P z * 1 +
              W.a₂ * P x ^ 2 / P z ^ 2 * (P z / P z) +
            W.a₄ * P x / P z * (P z ^ 2 / P z ^ 2) +
          W.a₆ * (P z ^ 3 / P z ^ 3)) -
      (Polynomial.evalEval (P x / P z) (P y / P z) W.toAffine.polynomial +
        (P y ^ 2 / P z ^ 2 * (P z / P z) + W.a₁ * P x * P y / P z ^ 2 * (P z / P z) +
                W.a₃ * P y / P z * (P z ^ 2 / P z ^ 2) +
              W.a₂ * P x ^ 2 / P z ^ 2 * 1 +
            W.a₄ * P x / P z * 1 +
          W.a₆ * 1)) =
    0

Tactic:
rw [eval_polynomial, Affine.evalEval_polynomial]; ring1

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Projective R
P : Fin 3 → R
⊢ 3 * (eval P) W'.polynomial =
    P x * (eval P) W'.polynomialX + P y * (eval P) W'.polynomialY + P z * (eval P) W'.polynomialZ

Tactic:
rw [eval_polynomial, eval_polynomialX, eval_polynomialY, eval_polynomialZ]

Goal state after:
R : Type r
inst✝ : CommRing R
W' : Projective R
P : Fin 3 → R
⊢ 3 *
      (P y ^ 2 * P z + W'.a₁ * P x * P y * P z + W'.a₃ * P y * P z ^ 2 -
        (P x ^ 3 + W'.a₂ * P x ^ 2 * P z + W'.a₄ * P x * P z ^ 2 + W'.a₆ * P z ^ 3)) =
    P x * (W'.a₁ * P y * P z - (3 * P x ^ 2 + 2 * W'.a₂ * P x * P z + W'.a₄ * P z ^ 2)) +
        P y * (2 * P y * P z + W'.a₁ * P x * P z + W'.a₃ * P z ^ 2) +
      P z *
        (P y ^ 2 + W'.a₁ * P x * P y + 2 * W'.a₃ * P y * P z -
          (W'.a₂ * P x ^ 2 + 2 * W'.a₄ * P x * P z + 3 * W'.a₆ * P z ^ 2))
Module: Mathlib.CategoryTheory.Sites.CoproductSheafCondition

Goal state before:
pp: case convert_2
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
S : C
ι : Type u_3
X : ι → C
f : (i : ι) → X i ⟶ S
inst✝⁴ : (ofArrows X f).HasPairwisePullbacks
c : Cofan X
hc : IsColimit c
hc' : IsUniversalColimit c
inst✝³ : HasPullback (Cofan.IsColimit.desc hc f) (Cofan.IsColimit.desc hc f)
inst✝² : ∀ (i : ι), HasPullback (f i) (Cofan.IsColimit.desc hc f)
F : Cᵒᵖ ⥤ Type u_4
inst✝¹ : PreservesLimit (Discrete.functor fun i ↦ op (X i)) F
inst✝ : PreservesLimit (Discrete.functor fun ij ↦ op (Limits.pullback (f ij.1) (f ij.2))) F
E : PreZeroHypercover S := ⋯
this✝¹ : (E.sigmaOfIsColimit hc).HasPullbacks
this✝ : ∀ (i : E.I₀), HasPullback (E.f i) ((E.sigmaOfIsColimit hc).f PUnit.unit)
this : PreservesLimit (Discrete.functor fun i ↦ op (E.toPreOneHypercover.X i)) F
⊢ (Discrete.equivalence (Equiv.sigmaPUnit (E.toPreOneHypercover.I₀ × E.toPreOneHypercover.I₀))).inverse.Initial

pp: case convert_9
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
S : C
ι : Type u_3
X : ι → C
f : (i : ι) → X i ⟶ S
inst✝⁴ : (ofArrows X f).HasPairwisePullbacks
c : Cofan X
hc : IsColimit c
hc' : IsUniversalColimit c
inst✝³ : HasPullback (Cofan.IsColimit.desc hc f) (Cofan.IsColimit.desc hc f)
inst✝² : ∀ (i : ι), HasPullback (f i) (Cofan.IsColimit.desc hc f)
F : Cᵒᵖ ⥤ Type u_4
inst✝¹ : PreservesLimit (Discrete.functor fun i ↦ op (X i)) F
inst✝ : PreservesLimit (Discrete.functor fun ij ↦ op (Limits.pullback (f ij.1) (f ij.2))) F
E : PreZeroHypercover S := ⋯
this✝¹ : (E.sigmaOfIsColimit hc).HasPullbacks
this✝ : ∀ (i : E.I₀), HasPullback (E.f i) ((E.sigmaOfIsColimit hc).f PUnit.unit)
this : PreservesLimit (Discrete.functor fun i ↦ op (E.toPreOneHypercover.X i)) F
⊢ PreservesLimit
    ((Discrete.equivalence (Equiv.sigmaPUnit (E.toPreOneHypercover.I₀ × E.toPreOneHypercover.I₀))).inverse ⋙
      Discrete.functor fun i ↦ op (E.toPreOneHypercover.Y' i))
    F

Tactic:
· infer_instance

Goal state after:
case convert_9
C : Type u_1
inst✝⁵ : Category.{v_1, u_1} C
S : C
ι : Type u_3
X : ι → C
f : (i : ι) → X i ⟶ S
inst✝⁴ : (ofArrows X f).HasPairwisePullbacks
c : Cofan X
hc : IsColimit c
hc' : IsUniversalColimit c
inst✝³ : HasPullback (Cofan.IsColimit.desc hc f) (Cofan.IsColimit.desc hc f)
inst✝² : ∀ (i : ι), HasPullback (f i) (Cofan.IsColimit.desc hc f)
F : Cᵒᵖ ⥤ Type u_4
inst✝¹ : PreservesLimit (Discrete.functor fun i ↦ op (X i)) F
inst✝ : PreservesLimit (Discrete.functor fun ij ↦ op (Limits.pullback (f ij.1) (f ij.2))) F
E : PreZeroHypercover S := ⋯
this✝¹ : (E.sigmaOfIsColimit hc).HasPullbacks
this✝ : ∀ (i : E.I₀), HasPullback (E.f i) ((E.sigmaOfIsColimit hc).f PUnit.unit)
this : PreservesLimit (Discrete.functor fun i ↦ op (E.toPreOneHypercover.X i)) F
⊢ PreservesLimit
    ((Discrete.equivalence (Equiv.sigmaPUnit (E.toPreOneHypercover.I₀ × E.toPreOneHypercover.I₀))).inverse ⋙
      Discrete.functor fun i ↦ op (E.toPreOneHypercover.Y' i))
    F
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W' : Projective R
S : Type s
inst✝ : CommRing S
f : R →+* S
P : Fin 3 → R
h : W'.Equation P
⊢ f 0 = 0

Tactic:
map_zero

Goal state after:
R : Type r
inst✝¹ : CommRing R
W' : Projective R
S : Type s
inst✝ : CommRing S
f : R →+* S
P : Fin 3 → R
h : W'.Equation P
⊢ 0 = 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point

Goal state before:
pp: R : Type r
S : Type s
A F : Type u
B K : Type v
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Field F
inst✝ : Field K
W' : Projective R
W : Projective F
P : W.Point
⊢ -P + P = 0

Tactic:
apply (toAffineAddEquiv W).injective
    simp only [map_add, toAffineAddEquiv_apply, toAffineLift_neg, neg_add_cancel, toAffineLift_zero]

Goal state after:
Module: Mathlib.RingTheory.RingHom.Smooth

Goal state before:
pp: R S : Type u_6
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set S
hs : Ideal.span s = ⊤
hf : ∀ (r : ↑s), (fun {R S} [CommRing R] [CommRing S] ↦ Smooth) ((algebraMap S (Localization.Away ↑r)).comp f)
this : f.FinitePresentation
algInst✝ : Algebra R S := f.toAlgebra
algebraizeInst✝ : Algebra.FinitePresentation R S
⊢ Set.univ ⊆ Algebra.smoothLocus R S

Tactic:
← TopologicalSpace.Opens.coe_top,

Goal state after:
R S : Type u_6
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
s : Set S
hs : Ideal.span s = ⊤
hf : ∀ (r : ↑s), (fun {R S} [CommRing R] [CommRing S] ↦ Smooth) ((algebraMap S (Localization.Away ↑r)).comp f)
this : f.FinitePresentation
algInst✝ : Algebra R S := f.toAlgebra
algebraizeInst✝ : Algebra.FinitePresentation R S
⊢ ↑⊤ ⊆ Algebra.smoothLocus R S
Module: Mathlib.RingTheory.Smooth.Locus

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
x : PrimeSpectrum A
⊢ x ∈ smoothLocus R A → ∃ t ⊆ smoothLocus R A, IsOpen t ∧ x ∈ t

Tactic:
hx

Goal state after:
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : FinitePresentation R A
x : PrimeSpectrum A
hx : x ∈ smoothLocus R A
⊢ ∃ t ⊆ smoothLocus R A, IsOpen t ∧ x ∈ t
Module: Mathlib.RingTheory.Smooth.Pi

Goal state before:
pp: case map_mul
R : Type u_1
I : Type u_2
A : I → Type u_3
inst✝³ : CommRing R
inst✝² : (i : I) → CommRing (A i)
inst✝¹ : (i : I) → Algebra R (A i)
inst✝ : Finite I
val✝ : Fintype I
H : ∀ (i : I), FormallySmooth R (A i)
B : Type (max u_1 u_2 u_3)
x✝¹ : CommRing B
x✝ : Algebra R B
J : Ideal B
hJ : J ^ 2 = ⊥
g : ((i : I) → A i) →ₐ[R] B ⧸ J
hJ' : ∀ x ∈ RingHom.ker (Ideal.Quotient.mk J), IsNilpotent x
e : I → B
he : CompleteOrthogonalIdempotents e
he' : ∀ (i : I), (Ideal.Quotient.mk J) (e i) = g (Pi.single i 1)
iso : B ≃ₐ[R] (i : I) → B ⧸ Ideal.span {1 - e i} := ⋯
J' : (i : I) → Ideal (B ⧸ Ideal.span {1 - e i}) := ⋯
ι : (i : I) → B ⧸ J →ₐ[R] (B ⧸ Ideal.span {1 - e i}) ⧸ J' i := ⋯
hι : ∀ (i : I) (x : B ⧸ J), (ι i) x = 0 → (Ideal.Quotient.mk J) (e i) * x = 0
i : I
⊢ ∀ (x y : A i),
    (((ι i).comp g).toLinearMap ∘ₗ LinearMap.single R A i) (x * y) =
      (((ι i).comp g).toLinearMap ∘ₗ LinearMap.single R A i) x *
        (((ι i).comp g).toLinearMap ∘ₗ LinearMap.single R A i) y

Tactic:
· intro x y; simp [Pi.single_mul]

Goal state after:
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : EssFiniteType K L
this✝ : FormallyUnramified K L
this : Module.Finite K L
B : Type (max u_1 u_2)
x✝¹ : CommRing B
x✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
pb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L
x : B
hx : (Ideal.Quotient.mk I) x = f pb.gen
helper : ∀ (x : B), (IsScalarTower.toAlgHom K B (B ⧸ I)) x = (Ideal.Quotient.mk I) x
⊢ f 0 = 0

Tactic:
map_zero

Goal state after:
K : Type u_1
L : Type u_2
inst✝⁴ : Field K
inst✝³ : Field L
inst✝² : Algebra K L
inst✝¹ : Algebra.IsSeparable K L
inst✝ : EssFiniteType K L
this✝ : FormallyUnramified K L
this : Module.Finite K L
B : Type (max u_1 u_2)
x✝¹ : CommRing B
x✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
pb : PowerBasis K L := Field.powerBasisOfFiniteOfSeparable K L
x : B
hx : (Ideal.Quotient.mk I) x = f pb.gen
helper : ∀ (x : B), (IsScalarTower.toAlgHom K B (B ⧸ I)) x = (Ideal.Quotient.mk I) x
⊢ 0 = 0
Module: Mathlib.RingTheory.Etale.Field

Goal state before:
pp: case refine_3
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsSeparable K L
B : Type (max u_1 u_2)
x✝¹ : CommRing B
x✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g ↦ (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g ↦ (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
H : ∀ (x y : L), ∃ α, x ∈ K⟮α⟯ ∧ y ∈ K⟮α⟯
⊢ (g 0) 0 = 0

Tactic:
map_zero

Goal state after:
case refine_3
K : Type u_1
L : Type u_2
inst✝³ : Field K
inst✝² : Field L
inst✝¹ : Algebra K L
inst✝ : Algebra.IsSeparable K L
B : Type (max u_1 u_2)
x✝¹ : CommRing B
x✝ : Algebra K B
I : Ideal B
h : I ^ 2 = ⊥
f : L →ₐ[K] B ⧸ I
g : (k : L) → ↥K⟮k⟯ →ₐ[K] B
hg₁ : ∀ (k : L), (fun g ↦ (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) (g k)
hg₂ :
  ∀ (k : L) (y : ↥K⟮k⟯ →ₐ[K] B),
    (fun g ↦ (Ideal.Quotient.mkₐ K I).comp g = f.comp (IsScalarTower.toAlgHom K (↥K⟮k⟯) L)) y → y = g k
hg₃ : ∀ (x y : L) (h : x ∈ K⟮y⟯), (g y) ⟨x, h⟩ = (g x) (IntermediateField.AdjoinSimple.gen K x)
H : ∀ (x y : L), ∃ α, x ∈ K⟮α⟯ ∧ y ∈ K⟮α⟯
⊢ 0 = 0