mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Smooth.Locus

Goal state before:
pp: case e'_2
R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : EssFiniteType R A
p : PrimeSpectrum A
⊢ Module.Projective (Localization.AtPrime p.asIdeal) Ω[Localization.AtPrime p.asIdeal⁄R] ↔
    Module.Free (Localization.AtPrime p.asIdeal) (LocalizedModule p.asIdeal.primeCompl Ω[A⁄R])

Tactic:
trans Module.Free (Localization.AtPrime p.asIdeal) Ω[Localization.AtPrime p.asIdeal⁄R]

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✝ : EssFiniteType R A
p : PrimeSpectrum A
⊢ Module.Projective (Localization.AtPrime p.asIdeal) Ω[Localization.AtPrime p.asIdeal⁄R] ↔
    Module.Free (Localization.AtPrime p.asIdeal) Ω[Localization.AtPrime p.asIdeal⁄R]

R : Type u_1
A : Type u_2
inst✝³ : CommRing R
inst✝² : CommRing A
inst✝¹ : Algebra R A
inst✝ : EssFiniteType R A
p : PrimeSpectrum A
⊢ Module.Free (Localization.AtPrime p.asIdeal) Ω[Localization.AtPrime p.asIdeal⁄R] ↔
    Module.Free (Localization.AtPrime p.asIdeal) (LocalizedModule p.asIdeal.primeCompl Ω[A⁄R])
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Projective R
P : Fin 3 → R
hP : W'.Equation P
⊢ W'.dblX P * P z =
    ((eval P) W'.polynomialX ^ 2 - W'.a₁ * (eval P) W'.polynomialX * P z * (P y - W'.negY P) -
          W'.a₂ * P z ^ 2 * (P y - W'.negY P) ^ 2 -
        2 * P x * P z * (P y - W'.negY P) ^ 2) *
      (P y - W'.negY P)

Tactic:
linear_combination (norm := (rw [dblX, eval_polynomialX, negY]; ring1))
    9 * (W'.a₁ * P x ^ 2 + 2 * P x * P y) * (equation_iff _).mp hP

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

Goal state before:
pp: R : Type u_1
S : Type u_2
P : Type u_3
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
inst✝¹¹ : Module.Flat R S
inst✝¹⁰ : CommRing P
inst✝⁹ : Algebra R P
inst✝⁸ : Algebra P S
inst✝⁷ : IsScalarTower R P S
inst✝⁶ : IsLocalRing R
inst✝⁵ : IsLocalRing S
inst✝⁴ : IsLocalHom (algebraMap R S)
inst✝³ : FormallySmooth 𝓀[R] (𝓀[R] ⊗[R] S)
inst✝² : FormallySmooth R P
inst✝¹ : Module.Free P Ω[P⁄R]
inst✝ : Module.Finite P Ω[P⁄R]
h₁ : Function.Surjective ⇑(algebraMap P S)
h₂ : (RingHom.ker (algebraMap P S)).FG
Sp : Type (max u_1 u_2) := 𝓀[R] ⊗[R] S
Pp : Type (max u_1 u_3) := 𝓀[R] ⊗[R] P
φ : Pp →ₐ[𝓀[R]] Sp := TensorProduct.map (AlgHom.id 𝓀[R] 𝓀[R]) (IsScalarTower.toAlgHom R P S)
ψ : Sp →ₐ[R] 𝓀[S] := TensorProduct.lift (IsScalarTower.toAlgHom R 𝓀[R] 𝓀[S]) (IsScalarTower.toAlgHom R S 𝓀[S]) ⋯
algInst✝³ : Algebra Pp Sp := φ.toAlgebra
algInst✝² : Algebra P Sp := (φ.comp (algebraMap P Pp)).toAlgebra
algInst✝¹ : Algebra Sp 𝓀[S] := ψ.toAlgebra
algInst✝ : Algebra Pp 𝓀[S] := (ψ.comp φ.toRingHom).toAlgebra
scalarTowerInst✝¹ : IsScalarTower P Pp Sp := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap P Sp))
scalarTowerInst✝ : IsScalarTower Pp Sp 𝓀[S] := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap Pp 𝓀[S]))
this✝¹ : IsScalarTower 𝓀[R] Pp Sp
this✝ : IsScalarTower R Sp 𝓀[S]
this : IsScalarTower P S Sp
⊢ FormallySmooth R S

Tactic:
have : IsScalarTower S Sp 𝓀[S] := .of_algebraMap_eq fun r ↦ by
    simp [RingHom.algebraMap_toAlgebra, ψ, Sp]

Goal state after:
R : Type u_1
S : Type u_2
P : Type u_3
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
inst✝¹¹ : Module.Flat R S
inst✝¹⁰ : CommRing P
inst✝⁹ : Algebra R P
inst✝⁸ : Algebra P S
inst✝⁷ : IsScalarTower R P S
inst✝⁶ : IsLocalRing R
inst✝⁵ : IsLocalRing S
inst✝⁴ : IsLocalHom (algebraMap R S)
inst✝³ : FormallySmooth 𝓀[R] (𝓀[R] ⊗[R] S)
inst✝² : FormallySmooth R P
inst✝¹ : Module.Free P Ω[P⁄R]
inst✝ : Module.Finite P Ω[P⁄R]
h₁ : Function.Surjective ⇑(algebraMap P S)
h₂ : (RingHom.ker (algebraMap P S)).FG
Sp : Type (max u_1 u_2) := 𝓀[R] ⊗[R] S
Pp : Type (max u_1 u_3) := 𝓀[R] ⊗[R] P
φ : Pp →ₐ[𝓀[R]] Sp := TensorProduct.map (AlgHom.id 𝓀[R] 𝓀[R]) (IsScalarTower.toAlgHom R P S)
ψ : Sp →ₐ[R] 𝓀[S] := TensorProduct.lift (IsScalarTower.toAlgHom R 𝓀[R] 𝓀[S]) (IsScalarTower.toAlgHom R S 𝓀[S]) ⋯
algInst✝³ : Algebra Pp Sp := φ.toAlgebra
algInst✝² : Algebra P Sp := (φ.comp (algebraMap P Pp)).toAlgebra
algInst✝¹ : Algebra Sp 𝓀[S] := ψ.toAlgebra
algInst✝ : Algebra Pp 𝓀[S] := (ψ.comp φ.toRingHom).toAlgebra
scalarTowerInst✝¹ : IsScalarTower P Pp Sp := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap P Sp))
scalarTowerInst✝ : IsScalarTower Pp Sp 𝓀[S] := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap Pp 𝓀[S]))
this✝² : IsScalarTower 𝓀[R] Pp Sp
this✝¹ : IsScalarTower R Sp 𝓀[S]
this✝ : IsScalarTower P S Sp
this : IsScalarTower S Sp 𝓀[S]
⊢ FormallySmooth R S
Module: Mathlib.RingTheory.Smooth.Fiber

Goal state before:
pp: case tmul
R : Type u_1
S : Type u_2
P : Type u_3
inst✝¹⁴ : CommRing R
inst✝¹³ : CommRing S
inst✝¹² : Algebra R S
inst✝¹¹ : Module.Flat R S
inst✝¹⁰ : CommRing P
inst✝⁹ : Algebra R P
inst✝⁸ : Algebra P S
inst✝⁷ : IsScalarTower R P S
inst✝⁶ : IsLocalRing R
inst✝⁵ : IsLocalRing S
inst✝⁴ : IsLocalHom (algebraMap R S)
inst✝³ : FormallySmooth 𝓀[R] (𝓀[R] ⊗[R] S)
inst✝² : FormallySmooth R P
inst✝¹ : Module.Free P Ω[P⁄R]
inst✝ : Module.Finite P Ω[P⁄R]
h₁ : Function.Surjective ⇑(algebraMap P S)
h₂ : (RingHom.ker (algebraMap P S)).FG
Sp : Type (max u_1 u_2) := 𝓀[R] ⊗[R] S
Pp : Type (max u_1 u_3) := 𝓀[R] ⊗[R] P
φ : Pp →ₐ[𝓀[R]] Sp := TensorProduct.map (AlgHom.id 𝓀[R] 𝓀[R]) (IsScalarTower.toAlgHom R P S)
ψ : Sp →ₐ[R] 𝓀[S] := TensorProduct.lift (IsScalarTower.toAlgHom R 𝓀[R] 𝓀[S]) (IsScalarTower.toAlgHom R S 𝓀[S]) ⋯
algInst✝³ : Algebra Pp Sp := φ.toAlgebra
algInst✝² : Algebra P Sp := (φ.comp (algebraMap P Pp)).toAlgebra
algInst✝¹ : Algebra Sp 𝓀[S] := ψ.toAlgebra
algInst✝ : Algebra Pp 𝓀[S] := (ψ.comp φ.toRingHom).toAlgebra
scalarTowerInst✝¹ : IsScalarTower P Pp Sp := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap P Sp))
scalarTowerInst✝ : IsScalarTower Pp Sp 𝓀[S] := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap Pp 𝓀[S]))
this✝⁷ : IsScalarTower 𝓀[R] Pp Sp
this✝⁶ : IsScalarTower R Sp 𝓀[S]
this✝⁵ : IsScalarTower P S Sp
this✝⁴ : IsScalarTower S Sp 𝓀[S]
this✝³ : IsScalarTower P Sp 𝓀[S]
this✝² : IsScalarTower P Pp 𝓀[S]
ePp : Pp ≃ₐ[P] P ⊗[R] 𝓀[R] :=
  let __spread.0 := TensorProduct.comm R 𝓀[R] P;
  { toEquiv := __spread.0.toEquiv, map_mul' := ⋯, map_add' := ⋯, commutes' := ⋯ }
e₀ : Ω[Pp⁄𝓀[R]] ≃ₗ[Pp] Pp ⊗[P] Ω[P⁄R] := (KaehlerDifferential.tensorKaehlerEquiv R 𝓀[R] P Pp).symm
this✝¹ : Module.Free Pp Ω[Pp⁄𝓀[R]]
this✝ : Module.Finite Pp Ω[Pp⁄𝓀[R]]
e₁ : ↥(RingHom.ker φ) ≃ₗ[Pp] Pp ⊗[P] ↥(RingHom.ker (algebraMap P S)) := kerTensorProductMapIdToAlgHomEquiv R P S 𝓀[R] h₁
h₁' : Function.Surjective ⇑φ
h₂' : (RingHom.ker φ).FG
h₃ : 𝓂[Sp] ≤ RingHom.ker ψ
this : Function.Injective ⇑(KaehlerDifferential.cotangentComplexBaseChange 𝓀[R] Sp Pp 𝓀[S])
eᵣ : 𝓀[S] ⊗[Pp] Ω[Pp⁄𝓀[R]] ≃ₗ[S] 𝓀[S] ⊗[P] Ω[P⁄R] :=
  LinearEquiv.restrictScalars S (AlgebraTensorModule.congr (LinearEquiv.refl 𝓀[S] 𝓀[S]) e₀) ≪≫ₗ
    LinearEquiv.restrictScalars S (AlgebraTensorModule.cancelBaseChange P Pp Sp 𝓀[S] Ω[P⁄R])
eₗ : 𝓀[S] ⊗[Pp] ↥(RingHom.ker φ) ≃ₗ[S] 𝓀[S] ⊗[P] ↥(RingHom.ker (algebraMap P S)) :=
  LinearEquiv.restrictScalars S (AlgebraTensorModule.congr (LinearEquiv.refl 𝓀[S] 𝓀[S]) e₁) ≪≫ₗ
    LinearEquiv.restrictScalars S (AlgebraTensorModule.cancelBaseChange P Pp Sp 𝓀[S] ↥(RingHom.ker (algebraMap P S)))
x : 𝓀[S]
y : ↥(RingHom.ker (algebraMap P S))
⊢ (x * 1) ⊗ₜ[P] (KaehlerDifferential.D R P) ↑y =
    (AlgebraTensorModule.cancelBaseChange P (𝓀[R] ⊗[R] P) Sp 𝓀[S] Ω[P⁄R])
      ((x * 1) ⊗ₜ[𝓀[R] ⊗[R] P] e₀ ((KaehlerDifferential.D 𝓀[R] (𝓀[R] ⊗[R] P)) (1 ⊗ₜ[R] (1 * ↑y))))

Tactic:
simp [e₀, Pp, ← TensorProduct.one_def]

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Functoriality

Goal state before:
pp: case isTrue
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
ι : Type u_6
M : ι → Type u_7
inst✝³ : (i : ι) → AddCommGroup (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
j : ι
x : AdicCauchySequence I (M j)
i : ι
n : ℕ
h✝ : j = i
⊢ Submodule.Quotient.mk (Eq.recOn h✝ (↑x n)) = ↑(Eq.recOn h✝ ((mk I (M j)) x)) n

pp: case isFalse
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
ι : Type u_6
M : ι → Type u_7
inst✝³ : (i : ι) → AddCommGroup (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
j : ι
x : AdicCauchySequence I (M j)
i : ι
n : ℕ
h✝ : ¬j = i
⊢ Submodule.Quotient.mk 0 = ↑0 n

Tactic:
· next h => subst h; simp

Goal state after:
case isFalse
R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
ι : Type u_6
M : ι → Type u_7
inst✝³ : (i : ι) → AddCommGroup (M i)
inst✝² : (i : ι) → Module R (M i)
inst✝¹ : Fintype ι
inst✝ : DecidableEq ι
j : ι
x : AdicCauchySequence I (M j)
i : ι
n : ℕ
h✝ : ¬j = i
⊢ Submodule.Quotient.mk 0 = ↑0 n
Module: Mathlib.RingTheory.AdicCompletion.Algebra

Goal state before:
pp: case hg
R : Type u_1
inst✝ : CommRing R
I : Ideal R
n : ℕ
⊢ Function.Surjective ⇑(Ideal.Quotient.factorₐ R ⋯)

pp: case hf
R : Type u_1
inst✝ : CommRing R
I : Ideal R
n : ℕ
⊢ Function.Surjective ⇑(AlgHom.ofLinearMap (eval I R n) ⋯ ⋯)

Tactic:
· exact factor_surjective Ideal.mul_le_left

Goal state after:
case hf
R : Type u_1
inst✝ : CommRing R
I : Ideal R
n : ℕ
⊢ Function.Surjective ⇑(AlgHom.ofLinearMap (eval I R n) ⋯ ⋯)
Module: Mathlib.RingTheory.AdicCompletion.Exactness

Goal state before:
pp: case e'_5
R : Type u
inst✝⁸ : CommRing R
I : Ideal R
M : Type u
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
N : Type u
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
P : Type u
inst✝³ : AddCommGroup P
inst✝² : Module R P
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R N
f : M →ₗ[R] N
g : N →ₗ[R] P
hf : Function.Injective ⇑f
hfg : Function.Exact ⇑f ⇑g
hg : Function.Surjective ⇑g
k : ℕ
hkn : ∀ n ≥ k, I ^ n • ⊤ ⊓ f.range = I ^ (n - k) • (I ^ k • ⊤ ⊓ f.range)
x : AdicCauchySequence I N
hker : ∀ (n : ℕ), g (↑x n) ∈ I ^ n • ⊤
n : ℕ
d : N
hdmem : d ∈ I ^ (k + n + 1) • ⊤
y yₙ : M
hd : f y = ↑x (k + n + 1) - d
hyₙ : f yₙ - ↑x (k + n) ∈ I ^ (k + n) • ⊤
⊢ ↑x (k + n + 1) - d - f yₙ = ↑x (k + n + 1) - ↑x (k + n) - d - (f yₙ - ↑x (k + n))

Tactic:
abel

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Exactness

Goal state before:
pp: case convert_6.refine_2
R : Type u
inst✝⁸ : CommRing R
I : Ideal R
M : Type u
inst✝⁷ : AddCommGroup M
inst✝⁶ : Module R M
N : Type u
inst✝⁵ : AddCommGroup N
inst✝⁴ : Module R N
P : Type u
inst✝³ : AddCommGroup P
inst✝² : Module R P
inst✝¹ : IsNoetherianRing R
inst✝ : Module.Finite R N
f : M →ₗ[R] N
g : N →ₗ[R] P
hf : Function.Injective ⇑f
hfg : Function.Exact ⇑f ⇑g
hg : Function.Surjective ⇑g
k : ℕ
hkn : ∀ n ≥ k, I ^ n • ⊤ ⊓ f.range = I ^ (n - k) • (I ^ k • ⊤ ⊓ f.range)
x : AdicCauchySequence I N
hker : ∀ (n : ℕ), g (↑x n) ∈ I ^ n • ⊤
n : ℕ
d : N
hdmem : d ∈ I ^ (k + n + 1) • ⊤
y yₙ : M
hd : f y = ↑x (k + n + 1) - d
hyₙ : f yₙ - ↑x (k + n) ∈ I ^ (k + n) • ⊤
⊢ d ∈ I ^ (k + n) • ⊤

Tactic:
exact (Submodule.smul_mono_left (Ideal.pow_le_pow_right (by lia))) hdmem

Goal state after:
Module: Mathlib.RingTheory.AdicCompletion.Functoriality

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
I : Ideal R
M : Type u_2
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : M →ₗ[R] N
h : Function.Surjective ⇑((I • ⊤).mkQ ∘ₗ f)
y : AdicCompletion I N
n : ℕ
⊢ { m // Submodule.Quotient.mk (f m) = (eval I N n) y }

Tactic:
induction n with
    | zero =>
      use 0
      apply_fun (Submodule.Quotient.equiv (I ^ 0 • ⊤) ⊤ (.refl R N) (by simp)).toEquiv
      exact Subsingleton.elim _ _
    | succ n xn =>
      choose z hz using exists_smodEq_pow_smul_top_and_mkQ_eq h
          (y' := eval _ _ (n + 1) y) (by simp) xn.2
      exact ⟨z, hz.2⟩

Goal state after:
Module: Mathlib.RingTheory.Extension.Presentation.Submersive

Goal state before:
pp: case mul_X
R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
ι' : Type u_1
σ' : Type u_2
T : Type u_3
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
Q : PreSubmersivePresentation S T ι' σ'
P : PreSubmersivePresentation R S ι σ
inst✝¹ : Fintype σ
inst✝ : Fintype σ'
i✝ j : σ
p : MvPolynomial ι R
i : ι
hp :
  { toRingHom := eval₂Hom (algebraMap R T) (Q.comp P).val, commutes' := ⋯ } ((rename Sum.inr) p) =
    eval₂ (algebraMap R T) (fun i ↦ (algebraMap S T) (P.val i)) p
⊢ { toRingHom := eval₂Hom (algebraMap R T) (Q.comp P).val, commutes' := ⋯ } ((rename Sum.inr) (p * X i)) =
    eval₂ (algebraMap R T) (fun i ↦ (algebraMap S T) (P.val i)) (p * X i)

Tactic:
simp only [map_mul, eval₂_mul, hp]

Goal state after:
case mul_X
R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
ι' : Type u_1
σ' : Type u_2
T : Type u_3
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
Q : PreSubmersivePresentation S T ι' σ'
P : PreSubmersivePresentation R S ι σ
inst✝¹ : Fintype σ
inst✝ : Fintype σ'
i✝ j : σ
p : MvPolynomial ι R
i : ι
hp :
  { toRingHom := eval₂Hom (algebraMap R T) (Q.comp P).val, commutes' := ⋯ } ((rename Sum.inr) p) =
    eval₂ (algebraMap R T) (fun i ↦ (algebraMap S T) (P.val i)) p
⊢ eval₂ (algebraMap R T) (fun i ↦ (algebraMap S T) (P.val i)) p *
      { toRingHom := eval₂Hom (algebraMap R T) (Q.comp P).val, commutes' := ⋯ } ((rename Sum.inr) (X i)) =
    eval₂ (algebraMap R T) (fun i ↦ (algebraMap S T) (P.val i)) p *
      eval₂ (algebraMap R T) (fun i ↦ (algebraMap S T) (P.val i)) (X i)
Module: Mathlib.RingTheory.Extension.Presentation.Submersive

Goal state before:
pp: case mul_X
R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
ι' : Type u_1
σ' : Type u_2
T : Type u_3
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
Q : PreSubmersivePresentation S T ι' σ'
P : PreSubmersivePresentation R S ι σ
inst✝¹ : Fintype σ
inst✝ : Fintype σ'
i✝ j : σ
p : MvPolynomial ι R
i : ι
hp :
  { toRingHom := eval₂Hom (algebraMap R T) (Q.comp P).val, commutes' := ⋯ } ((rename Sum.inr) p) =
    eval₂ (algebraMap R T) (fun i ↦ (algebraMap S T) (P.val i)) p
⊢ { toRingHom := eval₂Hom (algebraMap R T) (Q.comp P).val, commutes' := ⋯ } ((rename Sum.inr) (p * X i)) =
    eval₂ (algebraMap R T) (fun i ↦ (algebraMap S T) (P.val i)) (p * X i)

Tactic:
| mul_X p i hp =>

Goal state after:
Module: Mathlib.RingTheory.Extension.Presentation.Submersive

Goal state before:
pp: case intro.intro
R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
ι' : Type u_1
σ' : Type u_2
T : Type u_3
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
Q : PreSubmersivePresentation S T ι' σ'
P : PreSubmersivePresentation R S ι σ
inst✝¹ : Finite σ
inst✝ : Finite σ'
val✝¹ : Fintype σ'
val✝ : Fintype σ
⊢ (algebraMap (Q.comp P).Ring T)
      (Matrix.fromBlocks (Q.comp P).jacobiMatrix.toBlocks₁₁ (Q.comp P).jacobiMatrix.toBlocks₁₂
          (Q.comp P).jacobiMatrix.toBlocks₂₁ (Q.comp P).jacobiMatrix.toBlocks₂₂).det =
    P.jacobian • Q.jacobian

Tactic:
jacobiMatrix_comp_₁₂

Goal state after:
case intro.intro
R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
ι' : Type u_1
σ' : Type u_2
T : Type u_3
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
Q : PreSubmersivePresentation S T ι' σ'
P : PreSubmersivePresentation R S ι σ
inst✝¹ : Finite σ
inst✝ : Finite σ'
val✝¹ : Fintype σ'
val✝ : Fintype σ
⊢ (algebraMap (Q.comp P).Ring T)
      (Matrix.fromBlocks (Q.comp P).jacobiMatrix.toBlocks₁₁ 0 (Q.comp P).jacobiMatrix.toBlocks₂₁
          (Q.comp P).jacobiMatrix.toBlocks₂₂).det =
    P.jacobian • Q.jacobian
Module: Mathlib.LinearAlgebra.Basis.Exact

Goal state before:
pp: R : Type u_1
M : Type u_2
K : Type u_3
P : Type u_4
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : AddCommGroup K
inst✝³ : AddCommGroup P
inst✝² : Module R M
inst✝¹ : Module R K
inst✝ : Module R P
f : K →ₗ[R] M
g : M →ₗ[R] P
hfg : Function.Exact ⇑f ⇑g
hf : Function.Injective ⇑f
p q : Submodule R M
hpq : IsCompl p q
hker : Disjoint g.ker q
hmap : map g q = ⊤
⊢ Set.InjOn ⇑(p.projectionOnto q hpq) ↑f.range

Tactic:
simpa [← LinearMap.disjoint_ker_iff_injOn, ← hfg.linearMap_ker_eq]

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

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W' : Projective R
inst✝ : NoZeroDivisors R
P Q : Fin 3 → R
hP : W'.Equation P
hQz : Q z ≠ 0
hx : P x * Q z = Q x * P z
hy : P y * Q z = Q y * P z
hy' : P y * Q z = W'.negY Q * P z
⊢ W'.negDblY P * P z ^ 2 = -(eval P) W'.polynomialX ^ 3

Tactic:
negDblY_eq' hP,

Goal state after:
R : Type r
inst✝¹ : CommRing R
W' : Projective R
inst✝ : NoZeroDivisors R
P Q : Fin 3 → R
hP : W'.Equation P
hQz : Q z ≠ 0
hx : P x * Q z = Q x * P z
hy : P y * Q z = Q y * P z
hy' : P y * Q z = W'.negY Q * P z
⊢ -(eval P) W'.polynomialX *
        ((eval P) W'.polynomialX ^ 2 - W'.a₁ * (eval P) W'.polynomialX * P z * (P y - W'.negY P) -
              W'.a₂ * P z ^ 2 * (P y - W'.negY P) ^ 2 -
            2 * P x * P z * (P y - W'.negY P) ^ 2 -
          P x * P z * (P y - W'.negY P) ^ 2) +
      P y * P z ^ 2 * (P y - W'.negY P) ^ 3 =
    -(eval P) W'.polynomialX ^ 3
Module: Mathlib.RingTheory.Extension.Cotangent.LocalizationAway

Goal state before:
pp: R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
⊢ (algebraMap P.Ring (Localization.Away ((Ideal.Quotient.mk (P.ker ^ 2)) (P.σ g)))) (P.σ g) *
        IsLocalization.Away.invSelf ((Ideal.Quotient.mk (P.ker ^ 2)) (P.σ g)) -
      1 =
    0

Tactic:
IsScalarTower.algebraMap_apply P.Ring (P.Ring ⧸ P.ker ^ 2) (Localization.Away _)

Goal state after:
R : Type u_1
S : Type u_2
T : Type u_3
ι : Type u_4
inst✝⁷ : CommRing R
inst✝⁶ : CommRing S
inst✝⁵ : Algebra R S
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
g : S
inst✝ : IsLocalization.Away g T
P : Generators R S ι
⊢ (algebraMap (P.Ring ⧸ P.ker ^ 2) (Localization.Away ((Ideal.Quotient.mk (P.ker ^ 2)) (P.σ g))))
          ((algebraMap P.Ring (P.Ring ⧸ P.ker ^ 2)) (P.σ g)) *
        IsLocalization.Away.invSelf ((Ideal.Quotient.mk (P.ker ^ 2)) (P.σ g)) -
      1 =
    0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

Goal state before:
pp: case a
R : Type r
inst✝ : CommRing R
W' : Projective R
P Q : Fin 3 → R
hP : W'.Equation P
hQ : W'.Equation Q
⊢ W'.addZ P Q * (P z * Q z) +
        (Q z ^ 3 * 0 +
          P z ^ 3 *
            (Q y ^ 2 * Q z + W'.a₁ * Q x * Q y * Q z + W'.a₃ * Q y * Q z ^ 2 -
              (Q x ^ 3 + W'.a₂ * Q x ^ 2 * Q z + W'.a₄ * Q x * Q z ^ 2 + W'.a₆ * Q z ^ 3))) -
      ((P x * Q z - Q x * P z) ^ 3 +
        (Q z ^ 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 z ^ 3 * 0)) =
    0

Tactic:
rw [addZ]

Goal state after:
case a
R : Type r
inst✝ : CommRing R
W' : Projective R
P Q : Fin 3 → R
hP : W'.Equation P
hQ : W'.Equation Q
⊢ (-3 * P x ^ 2 * Q x * Q z + 3 * P x * Q x ^ 2 * P z + P y ^ 2 * Q z ^ 2 - Q y ^ 2 * P z ^ 2 +
                          W'.a₁ * P x * P y * Q z ^ 2 -
                        W'.a₁ * Q x * Q y * P z ^ 2 -
                      W'.a₂ * P x ^ 2 * Q z ^ 2 +
                    W'.a₂ * Q x ^ 2 * P z ^ 2 +
                  W'.a₃ * P y * P z * Q z ^ 2 -
                W'.a₃ * Q y * P z ^ 2 * Q z -
              W'.a₄ * P x * P z * Q z ^ 2 +
            W'.a₄ * Q x * P z ^ 2 * Q z) *
          (P z * Q z) +
        (Q z ^ 3 * 0 +
          P z ^ 3 *
            (Q y ^ 2 * Q z + W'.a₁ * Q x * Q y * Q z + W'.a₃ * Q y * Q z ^ 2 -
              (Q x ^ 3 + W'.a₂ * Q x ^ 2 * Q z + W'.a₄ * Q x * Q z ^ 2 + W'.a₆ * Q z ^ 3))) -
      ((P x * Q z - Q x * P z) ^ 3 +
        (Q z ^ 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 z ^ 3 * 0)) =
    0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula

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

Tactic:
rw [addZ]

Goal state after:
R : Type r
inst✝ : CommRing R
W' : Projective R
P : Fin 3 → R
⊢ -3 * P x ^ 2 * P x * P z + 3 * P x * P x ^ 2 * P z + P y ^ 2 * P z ^ 2 - P y ^ 2 * P z ^ 2 +
                    W'.a₁ * P x * P y * P z ^ 2 -
                  W'.a₁ * P x * P y * P z ^ 2 -
                W'.a₂ * P x ^ 2 * P z ^ 2 +
              W'.a₂ * P x ^ 2 * P z ^ 2 +
            W'.a₃ * P y * P z * P z ^ 2 -
          W'.a₃ * P y * P z ^ 2 * P z -
        W'.a₄ * P x * P z * P z ^ 2 +
      W'.a₄ * P x * P z ^ 2 * P z =
    0
Module: Mathlib.AlgebraicGeometry.Morphisms.Proper

Goal state before:
pp: X Y Z S : Scheme
f : X ⟶ Y
g : Y ⟶ Z
⊢ MorphismProperty.RespectsIso @IsProper

Tactic:
isProper_eq

Goal state after:
X Y Z S : Scheme
f : X ⟶ Y
g : Y ⟶ Z
⊢ (@IsSeparated ⊓ @UniversallyClosed ⊓ @LocallyOfFiniteType).RespectsIso
Module: Mathlib.AlgebraicGeometry.Morphisms.Proper

Goal state before:
pp: X Y Z S : Scheme
f : X ⟶ Y
g : Y ⟶ Z
⊢ MorphismProperty.IsStableUnderComposition @IsProper

Tactic:
isProper_eq

Goal state after:
X Y Z S : Scheme
f : X ⟶ Y
g : Y ⟶ Z
⊢ (@IsSeparated ⊓ @UniversallyClosed ⊓ @LocallyOfFiniteType).IsStableUnderComposition
Module: Mathlib.AlgebraicGeometry.Morphisms.Proper

Goal state before:
pp: X Y Z S : Scheme
f : X ⟶ Y
g : Y ⟶ Z
⊢ MorphismProperty.IsMultiplicative @IsProper

Tactic:
isProper_eq

Goal state after:
X Y Z S : Scheme
f : X ⟶ Y
g : Y ⟶ Z
⊢ (@IsSeparated ⊓ @UniversallyClosed ⊓ @LocallyOfFiniteType).IsMultiplicative
Module: Mathlib.AlgebraicGeometry.Morphisms.Proper

Goal state before:
pp: X Y Z S : Scheme
f : X ⟶ Y
g : Y ⟶ Z
⊢ MorphismProperty.IsStableUnderBaseChange @IsProper

Tactic:
isProper_eq

Goal state after:
X Y Z S : Scheme
f : X ⟶ Y
g : Y ⟶ Z
⊢ (@IsSeparated ⊓ @UniversallyClosed ⊓ @LocallyOfFiniteType).IsStableUnderBaseChange
Module: Mathlib.AlgebraicGeometry.Morphisms.Proper

Goal state before:
pp: X Y Z S : Scheme
f : X ⟶ Y
g : Y ⟶ Z
⊢ IsZariskiLocalAtTarget @IsProper

Tactic:
isProper_eq

Goal state after:
X Y Z S : Scheme
f : X ⟶ Y
g : Y ⟶ Z
⊢ IsZariskiLocalAtTarget (@IsSeparated ⊓ @UniversallyClosed ⊓ @LocallyOfFiniteType)