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

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
t : α
ht : ¬∃ n, t = ↑n
⊢ ContinuousAt (uncurry f ∘ Prod.map id fract) (s, t)

Tactic:
replace ht : t ≠ ⌊t⌋ := fun ht' => ht ⟨_, ht'⟩

Goal state after:
case inr
α : Type u_1
β : Type u_2
γ : Type u_3
inst✝⁷ : Ring α
inst✝⁶ : LinearOrder α
inst✝⁵ : FloorRing α
inst✝⁴ : TopologicalSpace α
inst✝³ : IsStrictOrderedRing α
inst✝² : OrderTopology α
inst✝¹ : TopologicalSpace β
inst✝ : TopologicalSpace γ
f : β → α → γ
h : ContinuousOn (uncurry f) (univ ×ˢ Icc 0 1)
hf : ∀ (s : β), f s 0 = f s 1
s : β
t : α
ht : t ≠ ↑⌊t⌋
⊢ ContinuousAt (uncurry f ∘ Prod.map id fract) (s, t)
Module: Mathlib.MeasureTheory.Measure.Haar.Quotient

Goal state before:
pp: case cpos.a
G : Type u_1
inst✝¹⁴ : Group G
inst✝¹³ : MeasurableSpace G
inst✝¹² : TopologicalSpace G
inst✝¹¹ : IsTopologicalGroup G
inst✝¹⁰ : BorelSpace G
inst✝⁹ : PolishSpace G
Γ : Subgroup G
inst✝⁸ : Γ.Normal
inst✝⁷ : T2Space (G ⧸ Γ)
inst✝⁶ : SecondCountableTopology (G ⧸ Γ)
μ : Measure (G ⧸ Γ)
inst✝⁵ : Countable ↥Γ
ν : Measure G
inst✝⁴ : ν.IsHaarMeasure
inst✝³ : ν.IsMulRightInvariant
inst✝² : LocallyCompactSpace G
inst✝¹ : QuotientMeasureEqMeasurePreimage ν μ
inst✝ : IsFiniteMeasure μ
K : PositiveCompacts G
K' : PositiveCompacts (G ⧸ Γ) := ⋯
this : μ.IsMulLeftInvariant
s : Set G
finiteCovol : ν s ≠ ∞
fund_dom_s : IsFundamentalDomain (↥Γ.op) s ν
h : ν (Quotient.mk (MulAction.orbitRel (↥Γ.op) G) ⁻¹' ↑K' ∩ s) = 0
i' : ν.IsOpenPosMeasure
⊢ IsOpen (interior ↑K)

Tactic:
exact isOpen_interior

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝² : TopologicalSpace ε
inst✝¹ : ContinuousENorm ε
μ : Measure α
p : ℝ≥0∞
f : α → ε
hf : MemLp f p μ
ν : Measure β
inst✝ : IsFiniteMeasure ν
hf' : MemLp f p (ν Set.univ • μ)
⊢ AEStronglyMeasurable f (Measure.map Prod.fst (μ.prod ν))

Tactic:
simpa using hf'.1

Goal state after:
Module: Mathlib.MeasureTheory.Function.LpSeminorm.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
ε : Type u_3
mα : MeasurableSpace α
mβ : MeasurableSpace β
inst✝³ : TopologicalSpace ε
inst✝² : ContinuousENorm ε
ν : Measure β
p : ℝ≥0∞
f : β → ε
hf : MemLp f p ν
μ : Measure α
inst✝¹ : IsFiniteMeasure μ
inst✝ : SFinite ν
hf' : MemLp f p (μ Set.univ • ν)
⊢ AEStronglyMeasurable f (Measure.map Prod.snd (μ.prod ν))

Tactic:
simpa using hf'.1

Goal state after:
Module: Mathlib.MeasureTheory.Integral.CircleIntegral

Goal state before:
pp: E : Type u_1
inst✝¹ : NormedAddCommGroup E
inst✝ : NormedSpace ℂ E
f : ℂ → E
c : ℂ
R C : ℝ
hR : 0 ≤ R
hf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C
this : |R| = R
⊢ 2 * π * |R| * C = 2 * π * R * C

Tactic:
by rw [this]

Goal state after:
Module: Mathlib.Analysis.BoxIntegral.DivergenceTheorem

Goal state before:
pp: E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J ↦ integral J GP (fun x ↦ f (i.insertNth y x)) BoxAdditiveMap.volume
⊢ HasIntegral I GP (fun x ↦ (f' x) (Pi.single i 1)) BoxAdditiveMap.volume
    (integral (I.face i) GP (fun x ↦ f (i.insertNth (I.upper i) x)) BoxAdditiveMap.volume -
      integral (I.face i) GP (fun x ↦ f (i.insertNth (I.lower i) x)) BoxAdditiveMap.volume)

Tactic:
set fb : Icc (I.lower i) (I.upper i) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x =>
    (integrable_of_continuousOn GP (Box.continuousOn_face_Icc Hc x.2) volume).toBoxAdditive

Goal state after:
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
n : ℕ
inst✝ : CompleteSpace E
I : Box (Fin (n + 1))
f : (Fin (n + 1) → ℝ) → E
f' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E
s : Set (Fin (n + 1) → ℝ)
hs : s.Countable
Hs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) x
Hd : ∀ x ∈ Box.Icc I \ s, HasFDerivWithinAt f (f' x) (Box.Icc I) x
i : Fin (n + 1)
Hc : ContinuousOn f (Box.Icc I)
fI : ℝ → Box (Fin n) → E := fun y J ↦ integral J GP (fun x ↦ f (i.insertNth y x)) BoxAdditiveMap.volume
fb : ↑(Set.Icc (I.lower i) (I.upper i)) → Fin n →ᵇᵃ[↑(I.face i)] E := fun x ↦ ⋯.toBoxAdditive
⊢ HasIntegral I GP (fun x ↦ (f' x) (Pi.single i 1)) BoxAdditiveMap.volume
    (integral (I.face i) GP (fun x ↦ f (i.insertNth (I.upper i) x)) BoxAdditiveMap.volume -
      integral (I.face i) GP (fun x ↦ f (i.insertNth (I.lower i) x)) BoxAdditiveMap.volume)
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: σ : Type u_5
R : Type u_6
inst✝² : CommSemiring R
inst✝¹ : Fintype σ
inst✝ : DecidableEq σ
⊢ (fun x ↦ x ∈ Set.univ) = fun x ↦ Nat.Partition.ofSym x = Nat.Partition.indiscrete 1

Tactic:
simp_rw [Set.mem_univ, Nat.Partition.ofSym_one]

Goal state after:
Module: Mathlib.RingTheory.MvPolynomial.Symmetric.Defs

Goal state before:
pp: τ : Type u_2
σ : Type u_5
R : Type u_6
inst✝⁴ : CommSemiring R
inst✝³ : Fintype σ
inst✝² : Fintype τ
inst✝¹ : DecidableEq σ
inst✝ : DecidableEq τ
n : ℕ
μ : n.Partition
e : σ ≃ τ
⊢ (rename ⇑e) (∑ s, (Multiset.map X ↑↑s).prod) = msymm τ R μ

Tactic:
map_sum

Goal state after:
τ : Type u_2
σ : Type u_5
R : Type u_6
inst✝⁴ : CommSemiring R
inst✝³ : Fintype σ
inst✝² : Fintype τ
inst✝¹ : DecidableEq σ
inst✝ : DecidableEq τ
n : ℕ
μ : n.Partition
e : σ ≃ τ
⊢ ∑ x, (rename ⇑e) (Multiset.map X ↑↑x).prod = msymm τ R μ
Module: Mathlib.Topology.Algebra.Polynomial

Goal state before:
pp: case refine_1
R : Type u_1
S : Type u_2
k : Type u_3
α : Type u_4
inst✝⁵ : Semiring R
inst✝⁴ : Ring S
inst✝³ : Field k
inst✝² : LinearOrder k
inst✝¹ : IsStrictOrderedRing k
f : R →+* S
abv : S → k
inst✝ : IsAbsoluteValue abv
l : Filter α
z : α → S
hz : Tendsto (abv ∘ z) l atTop
a✝ : R
hc : f (C a✝ * X).leadingCoeff ≠ 0
⊢ Tendsto (fun x ↦ abv (eval₂ f (z x) (C a✝ * X))) l atTop

Tactic:
rw [leadingCoeff_mul_X, leadingCoeff_C] at hc

Goal state after:
case refine_1
R : Type u_1
S : Type u_2
k : Type u_3
α : Type u_4
inst✝⁵ : Semiring R
inst✝⁴ : Ring S
inst✝³ : Field k
inst✝² : LinearOrder k
inst✝¹ : IsStrictOrderedRing k
f : R →+* S
abv : S → k
inst✝ : IsAbsoluteValue abv
l : Filter α
z : α → S
hz : Tendsto (abv ∘ z) l atTop
a✝ : R
hc : f a✝ ≠ 0
⊢ Tendsto (fun x ↦ abv (eval₂ f (z x) (C a✝ * X))) l atTop
Module: Mathlib.Analysis.Complex.CauchyIntegral

Goal state before:
pp: case ha
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
R : ℝ≥0
hw : w ∈ ball c ↑R
hc : ContinuousOn f (closedBall c ↑R)
hd : ∀ x ∈ ball c ↑R \ s, DifferentiableAt ℂ f x
hR : 0 < ↑R
this : w ∈ closure (ball c ↑R \ s)
A : ContinuousAt (fun w ↦ (2 * ↑π * I)⁻¹ • ∮ (z : ℂ) in C(c, ↑R), (z - w)⁻¹ • f z) w
B : ContinuousAt f w
z : ℂ
hz : z ∈ ball c ↑R \ s
⊢ 2 * ↑π * I ≠ 0

Tactic:
simp [Real.pi_ne_zero, I_ne_zero]

Goal state after:
Module: Mathlib.Analysis.Complex.CauchyIntegral

Goal state before:
pp: case ha
E : Type u
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℂ E
inst✝ : CompleteSpace E
R : ℝ
c w : ℂ
f : ℂ → E
s : Set ℂ
hs : s.Countable
hw : w ∈ ball c R
hc : ContinuousOn f (closedBall c R)
hd : ∀ x ∈ ball c R \ s, DifferentiableAt ℂ f x
⊢ 2 * ↑π * I ≠ 0

Tactic:
simp [Real.pi_ne_zero, I_ne_zero]

Goal state after:
Module: Mathlib.RingTheory.Polynomial.Vieta

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
s : Multiset R
k : ℕ
h : k ≤ s.card
⊢ (map (fun t ↦ X - C t) s).prod.coeff k = (-1) ^ (s.card - k) * s.esymm (s.card - k)

Tactic:
by
  conv_lhs =>
    congr
    congr
    congr
    ext x
    rw [sub_eq_add_neg]
    rw [← map_neg C x]
  convert! prod_X_add_C_coeff (map (fun t => -t) s) _ using 1
  · rw [map_map]; rfl
  · rw [esymm_neg, card_map]
  · rwa [card_map]

Goal state after:
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, v.restrict x ≠ 0 ∧ v.restrict x ≠ 1

Tactic:
by
    obtain ⟨x, ⟨hx0, hx1⟩⟩ := IsNontrivial.exists_val_nontrivial (v := v)
    exact ⟨x, by simp [hx0], by simpa⟩

Goal state after:
Module: Mathlib.Analysis.Complex.Polynomial.Basic

Goal state before:
pp: p : ℝ[X]
z : ℂ
h0 : (aeval z) p = 0
hz : z.im ≠ 0
⊢ (X - C ((starRingEnd ℂ) z)) * (X - C z) ∣ map (algebraMap ℝ ℂ) p

Tactic:
apply IsCoprime.mul_dvd

Goal state after:
case H
p : ℝ[X]
z : ℂ
h0 : (aeval z) p = 0
hz : z.im ≠ 0
⊢ IsCoprime (X - C ((starRingEnd ℂ) z)) (X - C z)

case H1
p : ℝ[X]
z : ℂ
h0 : (aeval z) p = 0
hz : z.im ≠ 0
⊢ X - C ((starRingEnd ℂ) z) ∣ map (algebraMap ℝ ℂ) p

case H2
p : ℝ[X]
z : ℂ
h0 : (aeval z) p = 0
hz : z.im ≠ 0
⊢ X - C z ∣ map (algebraMap ℝ ℂ) p
Module: Mathlib.RingTheory.MvPowerSeries.LexOrder

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
h : ¬φ = 0
ne : (⇑toLex '' Function.support φ).Nonempty
⊢ Lex (σ →₀ ℕ) → Lex (σ →₀ ℕ) → Prop

pp: σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
h : ¬φ = 0
ne : (⇑toLex '' Function.support φ).Nonempty
⊢ WellFounded ?m.40

Tactic:
· exact Finsupp.instLTLex.lt

Goal state after:
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
h : ¬φ = 0
ne : (⇑toLex '' Function.support φ).Nonempty
⊢ WellFounded LT.lt
Module: Mathlib.RingTheory.MvPowerSeries.LexOrder

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
w : WithTop (Lex (σ →₀ ℕ))
h : ∀ (d : σ →₀ ℕ), ↑(toLex d) < w → (coeff d) φ = 0
h' : φ.lexOrder < w
⊢ φ ≠ 0

Tactic:
rw [ne_eq, ← lexOrder_eq_top_iff_eq_zero]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ : MvPowerSeries σ R
w : WithTop (Lex (σ →₀ ℕ))
h : ∀ (d : σ →₀ ℕ), ↑(toLex d) < w → (coeff d) φ = 0
h' : φ.lexOrder < w
⊢ ¬φ.lexOrder = ⊤
Module: Mathlib.Topology.Algebra.Valued.NormedValued

Goal state before:
pp: case refine_1
K : Type u_1
hK : NormedField K
inst✝ : IsUltrametricDist K
U : Set K
⊢ (∃ ε > 0, Metric.ball 0 ε ⊆ U) → ∃ γ, {x | valuation.restrict x < ↑γ} ⊆ U

pp: case refine_2
K : Type u_1
hK : NormedField K
inst✝ : IsUltrametricDist K
U : Set K
⊢ (∃ γ, {x | valuation.restrict x < ↑γ} ⊆ U) → ∃ ε > 0, Metric.ball 0 ε ⊆ U

Tactic:
· rintro ⟨ε, hε, h⟩
        rcases RankLeOne.exists_val_lt (valuation (K := K)) with H | H
        · use Units.mk0 (valuation.restrict 1) (by simp)
          intro x hx
          simp only [Units.val_mk0, mem_ofPred_eq, map_one] at hx
          by_cases hx0 : x = 0
          · exact h (hx0 ▸ Metric.mem_ball_self hε)
          · exfalso
            rw [← (valuation (K := K)).restrict.zero_iff, ← ne_eq, ← isUnit_iff_ne_zero] at hx0
            apply not_le.mpr hx
            apply le_of_eq
            rw [eq_comm]
            simpa only [Units.ext_iff, hx0.unit_spec, Units.val_one,
              Submonoid.mk_eq_one] using! H.elim hx0.unit 1
        · obtain ⟨x, hx, hxy⟩ := H (γ := ⟨ε, le_of_lt hε⟩) (pos_iff_ne_zero.mp hε)
          use Units.mk0 (valuation.restrict x) (by simp [Valuation.restrict_def, hx])
          intro y hy
          apply h
          simp only [Metric.mem_ball, dist_zero_right]
          simp only [Units.val_mk0, mem_ofPred_eq, restrict_lt_iff, ← NNReal.coe_lt_coe] at hy
          apply lt_trans hy
          simpa [RankLeOne.hom', valuation.restrict_def] using! hxy

Goal state after:
case refine_2
K : Type u_1
hK : NormedField K
inst✝ : IsUltrametricDist K
U : Set K
⊢ (∃ γ, {x | valuation.restrict x < ↑γ} ⊆ U) → ∃ ε > 0, Metric.ball 0 ε ⊆ U
Module: Mathlib.RingTheory.MvPowerSeries.LexOrder

Goal state before:
pp: case inr.inl
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
p q : σ →₀ ℕ
hp : φ.lexOrder = ↑(toLex p)
hq : ψ.lexOrder = ↑(toLex q)
u v : σ →₀ ℕ
h' : (u, v) ≠ (p, q)
h : u + v = p + q
h'' : u < p
⊢ (coeff (u, v).1) φ * (coeff (u, v).2) ψ = 0

Tactic:
coeff_eq_zero_of_lt_lexOrder (d := u),

Goal state after:
case inr.inl
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
p q : σ →₀ ℕ
hp : φ.lexOrder = ↑(toLex p)
hq : ψ.lexOrder = ↑(toLex q)
u v : σ →₀ ℕ
h' : (u, v) ≠ (p, q)
h : u + v = p + q
h'' : u < p
⊢ 0 * (coeff (u, v).2) ψ = 0

case inr.inl
σ : Type u_1
R : Type u_2
inst✝² : Semiring R
inst✝¹ : LinearOrder σ
inst✝ : WellFoundedGT σ
φ ψ : MvPowerSeries σ R
p q : σ →₀ ℕ
hp : φ.lexOrder = ↑(toLex p)
hq : ψ.lexOrder = ↑(toLex q)
u v : σ →₀ ℕ
h' : (u, v) ≠ (p, q)
h : u + v = p + q
h'' : u < p
⊢ ↑(toLex u) < φ.lexOrder
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
v : Valuation L Γ₀
hv : v.RankOne
x : L
⊢ 0 < v.restrict x ↔ 0 < v x

Tactic:
v.restrict_pos_iff

Goal state after:
L : Type u_1
inst✝¹ : Field L
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation L Γ₀
hv : v.RankOne
x : L
⊢ 0 < v x ↔ 0 < v x
Module: Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors

Goal state before:
pp: case mpr
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
n : σ →₀ ℕ
r : R
⊢ r ∈ nonZeroDivisorsRight R → (monomial n) r ∈ nonZeroDivisorsRight (MvPowerSeries σ R)

Tactic:
intro H p hrp
    ext i
    have := congr(coeff (i + n) $hrp)
    rw [coeff_mul_monomial, if_pos le_add_self, add_tsub_cancel_right] at this
    simpa using H _ this

Goal state after:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.ModelsWithJ

Goal state before:
pp: case pos
F : Type u_2
inst✝¹ : Field F
j : F
inst✝ : DecidableEq F
h0 : j = 0
h3 : 3 = 0
this : Fact (IsUnit 2)
⊢ (ofJ1728 F).j = 0

Tactic:
ofJ1728_j

Goal state after:
case pos
F : Type u_2
inst✝¹ : Field F
j : F
inst✝ : DecidableEq F
h0 : j = 0
h3 : 3 = 0
this : Fact (IsUnit 2)
⊢ 1728 = 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.ModelsWithJ

Goal state before:
pp: case pos
F : Type u_2
inst✝¹ : Field F
j : F
inst✝ : DecidableEq F
h0 : ¬j = 0
h1728 : j = 1728
h2 : 2 ≠ 0
this : Fact (IsUnit 2)
⊢ (ofJ1728 F).j = 1728

Tactic:
ofJ1728_j

Goal state after:
Module: Mathlib.AlgebraicGeometry.Geometrically.Connected

Goal state before:
pp: X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
inst✝³ : GeometricallyConnected f
inst✝² : GeometricallyConnected g
inst✝¹ : UniversallyOpen f
inst✝ : UniversallyOpen g
K : Type u_1
x✝ : Field K
x : Spec (CommRingCat.of K) ⟶ Z
⊢ ConnectedSpace ↥(pullback (f ≫ g) x)

Tactic:
← (pullbackRightPullbackFstIso g x f).hom.homeomorph.connectedSpace_iff

Goal state after:
X Y Z : Scheme
f : X ⟶ Y
g : Y ⟶ Z
inst✝³ : GeometricallyConnected f
inst✝² : GeometricallyConnected g
inst✝¹ : UniversallyOpen f
inst✝ : UniversallyOpen g
K : Type u_1
x✝ : Field K
x : Spec (CommRingCat.of K) ⟶ Z
⊢ ConnectedSpace ↥(pullback f (pullback.fst g x))
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.CategoryTheory.Sites.Preserves

Goal state before:
pp: C : Type u
inst✝³ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
α : Type u_1
inst✝² : Small.{w, u_1} α
X : α → C
c : Cofan X
hc : IsColimit c
inst✝¹ : (ofArrows X c.inj).HasPairwisePullbacks
inst✝ : PreservesLimit (Discrete.functor fun x ↦ op (X x)) F
this : HasCoproduct X
hi : IsIso (piComparison F fun x ↦ op (X x))
⊢ ∀ (y : Equalizer.Presieve.Arrows.FirstObj F X),
    (ConcreteCategory.hom (Equalizer.Presieve.Arrows.firstMap F X c.inj)) y =
        (ConcreteCategory.hom (Equalizer.Presieve.Arrows.secondMap F X c.inj)) y →
      ∃! x, (ConcreteCategory.hom (Equalizer.Presieve.Arrows.forkMap F X c.inj)) x = y

Tactic:
rw [piComparison_fac (hc := hc), isIso_iff_bijective, Function.bijective_iff_existsUnique] at hi

Goal state after:
C : Type u
inst✝³ : Category.{v, u} C
F : Cᵒᵖ ⥤ Type w
α : Type u_1
inst✝² : Small.{w, u_1} α
X : α → C
c : Cofan X
hc : IsColimit c
inst✝¹ : (ofArrows X c.inj).HasPairwisePullbacks
inst✝ : PreservesLimit (Discrete.functor fun x ↦ op (X x)) F
this : HasCoproduct X
hi :
  ∀ (b : (fun X ↦ X) (∏ᶜ fun b ↦ F.obj (op (X b)))),
    ∃! a,
      (ConcreteCategory.hom
            (F.map (opCoproductIsoProduct' hc (productIsProduct fun x ↦ op (X x))).inv ≫
              Equalizer.Presieve.Arrows.forkMap F X c.inj))
          a =
        b
⊢ ∀ (y : Equalizer.Presieve.Arrows.FirstObj F X),
    (ConcreteCategory.hom (Equalizer.Presieve.Arrows.firstMap F X c.inj)) y =
        (ConcreteCategory.hom (Equalizer.Presieve.Arrows.secondMap F X c.inj)) y →
      ∃! x, (ConcreteCategory.hom (Equalizer.Presieve.Arrows.forkMap F X c.inj)) x = y
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point

Goal state before:
pp: case neg
F : Type u
inst✝ : Field F
W : Projective F
P : Fin 3 → F
u : F
hu : IsUnit u
hP : ¬W.Nonsingular P
⊢ 0 = toAffine W P

Tactic:
toAffine_of_singular hP

Goal state after:
case neg
F : Type u
inst✝ : Field F
W : Projective F
P : Fin 3 → F
u : F
hu : IsUnit u
hP : ¬W.Nonsingular P
⊢ 0 = 0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point

Goal state before:
pp: case pos
F : Type u
inst✝ : Field F
W : Projective F
P : Fin 3 → F
hP : W.Nonsingular P
hPz : P z = 0
⊢ toAffine W (W.neg P) = -toAffine W P

Tactic:
rw [neg_of_Z_eq_zero hP.left hPz, toAffine_smul _ (isUnit_Y_of_Z_eq_zero hP hPz).neg,
      toAffine_zero, toAffine_of_Z_eq_zero hPz, Affine.Point.neg_zero]

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

Goal state before:
pp: R : Type r
inst✝¹⁰ : CommRing R
W' : Projective R
S : Type s
inst✝⁹ : CommRing S
A : Type u
inst✝⁸ : CommRing A
B : Type v
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
⊢ (W'⁄B).polynomial = ((W'⁄A).map ↑f).polynomial

Tactic:
map_baseChange

Goal state after:
R : Type r
inst✝¹⁰ : CommRing R
W' : Projective R
S : Type s
inst✝⁹ : CommRing S
A : Type u
inst✝⁸ : CommRing A
B : Type v
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
⊢ (W'⁄B).polynomial = (W'⁄B).polynomial
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic

Goal state before:
pp: case e'_3
R : Type r
inst✝¹⁰ : CommRing R
W' : Projective R
S : Type s
inst✝⁹ : CommRing S
A : Type u
inst✝⁸ : CommRing A
B : Type v
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P : Fin 3 → A
h : (W'⁄A).Equation P
⊢ W'⁄B = (W'⁄A).map ↑f

Tactic:
map_baseChange

Goal state after:
case e'_3
R : Type r
inst✝¹⁰ : CommRing R
W' : Projective R
S : Type s
inst✝⁹ : CommRing S
A : Type u
inst✝⁸ : CommRing A
B : Type v
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P : Fin 3 → A
h : (W'⁄A).Equation P
⊢ W'⁄B = W'⁄B
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic

Goal state before:
pp: R : Type r
inst✝¹⁰ : CommRing R
W' : Projective R
S : Type s
inst✝⁹ : CommRing S
A : Type u
inst✝⁸ : CommRing A
B : Type v
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
hf : Function.Injective ⇑f
P : Fin 3 → A
⊢ (W'⁄B).Equation (⇑↑f ∘ P) ↔ ((W'⁄A).map ↑f).Equation (⇑↑f ∘ P)

Tactic:
map_baseChange

Goal state after:
R : Type r
inst✝¹⁰ : CommRing R
W' : Projective R
S : Type s
inst✝⁹ : CommRing S
A : Type u
inst✝⁸ : CommRing A
B : Type v
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
hf : Function.Injective ⇑f
P : Fin 3 → A
⊢ (W'⁄B).Equation (⇑↑f ∘ P) ↔ (W'⁄B).Equation (⇑↑f ∘ P)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic

Goal state before:
pp: R : Type r
inst✝¹⁰ : CommRing R
W' : Projective R
S : Type s
inst✝⁹ : CommRing S
A : Type u
inst✝⁸ : CommRing A
B : Type v
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
⊢ (W'⁄B).polynomialX = ((W'⁄A).map ↑f).polynomialX

Tactic:
map_baseChange

Goal state after:
R : Type r
inst✝¹⁰ : CommRing R
W' : Projective R
S : Type s
inst✝⁹ : CommRing S
A : Type u
inst✝⁸ : CommRing A
B : Type v
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
⊢ (W'⁄B).polynomialX = (W'⁄B).polynomialX
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic

Goal state before:
pp: R : Type r
inst✝¹⁰ : CommRing R
W' : Projective R
S : Type s
inst✝⁹ : CommRing S
A : Type u
inst✝⁸ : CommRing A
B : Type v
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
⊢ (W'⁄B).polynomialY = ((W'⁄A).map ↑f).polynomialY

Tactic:
map_baseChange

Goal state after:
R : Type r
inst✝¹⁰ : CommRing R
W' : Projective R
S : Type s
inst✝⁹ : CommRing S
A : Type u
inst✝⁸ : CommRing A
B : Type v
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
⊢ (W'⁄B).polynomialY = (W'⁄B).polynomialY
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic

Goal state before:
pp: R : Type r
inst✝¹⁰ : CommRing R
W' : Projective R
S : Type s
inst✝⁹ : CommRing S
A : Type u
inst✝⁸ : CommRing A
B : Type v
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
⊢ (W'⁄B).polynomialZ = ((W'⁄A).map ↑f).polynomialZ

Tactic:
map_baseChange

Goal state after:
R : Type r
inst✝¹⁰ : CommRing R
W' : Projective R
S : Type s
inst✝⁹ : CommRing S
A : Type u
inst✝⁸ : CommRing A
B : Type v
inst✝⁷ : CommRing B
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
⊢ (W'⁄B).polynomialZ = (W'⁄B).polynomialZ