mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Extension.Presentation.Submersive

Goal state before:
pp: R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
h : Function.Bijective ⇑(algebraMap R S)
⊢ (algebraMap (ofBijectiveAlgebraMap h).Ring S).mapMatrix (ofBijectiveAlgebraMap h).jacobiMatrix = 1

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
h : Function.Bijective ⇑(algebraMap R S)
⊢ (ofBijectiveAlgebraMap h).jacobian = 1

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
r : R
inst✝ : IsLocalization.Away r S
⊢ (pderiv ()) (C r * X () - 1) = C r

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
r : R
inst✝ : IsLocalization.Away r S
⊢ (localizationAway S r).jacobiMatrix = Matrix.diagonal fun x ↦ C r

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
r : R
inst✝ : IsLocalization.Away r S
⊢ (localizationAway S r).jacobian = (algebraMap R S) r

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
P✝ : PreSubmersivePresentation R S ι σ
ι' : Type u_1
σ' : Type u_2
T : Type u_3
inst✝³ : CommRing T
inst✝² : Algebra R T
inst✝¹ : Algebra S T
inst✝ : IsScalarTower R S T
Q : PreSubmersivePresentation S T ι' σ'
P : PreSubmersivePresentation R S ι σ
⊢ ∀ (a : σ') (b : σ), Sum.inl (Q.map a) ≠ Sum.inr (P.map b)

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : Algebra R S
ι' : Type u_1
σ' : Type u_2
T : Type u_3
inst✝⁷ : CommRing T
inst✝⁶ : Algebra R T
inst✝⁵ : Algebra S T
inst✝⁴ : IsScalarTower R S T
Q : PreSubmersivePresentation S T ι' σ'
P : PreSubmersivePresentation R S ι σ
inst✝³ : Finite ι
inst✝² : Finite ι'
inst✝¹ : Finite σ
inst✝ : Finite σ'
⊢ (Q.comp P).dimension = Q.dimension + P.dimension

Tactic:
by

Goal state after:
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'.negDblY P * P z ^ 2 =
    -(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

Tactic:
by

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

Goal state before:
pp: 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 : σ'
j : σ
⊢ (Q.comp P).jacobiMatrix (Sum.inl i) (Sum.inr j) = 0

Tactic:
by

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

Goal state before:
pp: 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'
⊢ (Q.comp P).jacobiMatrix.toBlocks₁₂ = 0

Tactic:
by

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

Goal state before:
pp: 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'
⊢ (aeval (Sum.elim X (⇑C ∘ P.val))) ((Q.comp P).jacobiMatrix (Sum.inl j) (Sum.inl i)) = Q.jacobiMatrix j i

Tactic:
by

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

Goal state before:
pp: 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'
⊢ (aeval (Q.comp P).val) (Q.comp P).jacobiMatrix.toBlocks₁₁.det = Q.jacobian

Tactic:
by

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

Goal state before:
pp: F : Type u
inst✝ : Field F
W : Projective F
P : Fin 3 → F
hP : W.Equation P
hPz : P z ≠ 0
⊢ W.negDblY P =
    (-(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) /
      P z ^ 2

Tactic:
by

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

Goal state before:
pp: 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
⊢ (Q.comp P).jacobiMatrix (Sum.inr i) (Sum.inr j) = (rename Sum.inr) (P.jacobiMatrix i j)

Tactic:
by

Goal state after:
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
s : M →ₗ[R] K
hs : s ∘ₗ f = LinearMap.id
hfg : Function.Exact ⇑f ⇑g
ι : Type u_5
κ : Type u_6
v : ι → M
a : κ → ι
hainj : Function.Injective a
hsa : ∀ (i : κ), s (v (a i)) = 0
hli : LinearIndependent R v
y : K
hy : f y ∈ Submodule.span R (Set.range (v ∘ a))
⊢ s (f y) = 0

Tactic:
by

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

Goal state before:
pp: 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'
⊢ (aeval (Q.comp P).val) (Q.comp P).jacobiMatrix.toBlocks₂₂.det = (algebraMap S T) P.jacobian

Tactic:
by

Goal state after:
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
s : M →ₗ[R] K
hs : s ∘ₗ f = LinearMap.id
hfg : Function.Exact ⇑f ⇑g
ι : Type u_5
κ : Type u_6
v : ι → M
a : κ → ι
hainj : Function.Injective a
hsa : ∀ (i : κ), s (v (a i)) = 0
hli : LinearIndependent R v
⊢ LinearIndependent R (⇑g ∘ v ∘ a)

Tactic:
by

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

Goal state before:
pp: 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'
⊢ (Q.comp P).jacobian = P.jacobian • Q.jacobian

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
T : Type u_1
inst✝² : CommRing T
inst✝¹ : Algebra R T
P : PreSubmersivePresentation R S ι σ
inst✝ : Finite σ
val✝ : Fintype σ
⊢ (baseChange T P).jacobiMatrix = (MvPolynomial.map (algebraMap R T)).mapMatrix P.jacobiMatrix

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
T : Type u_1
inst✝² : CommRing T
inst✝¹ : Algebra R T
P : PreSubmersivePresentation R S ι σ
inst✝ : Finite σ
⊢ (baseChange T P).jacobian = 1 ⊗ₜ[R] P.jacobian

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
P✝ P : PreSubmersivePresentation R S ι σ
ι' : Type u_1
σ' : Type u_2
e : ι' ≃ ι
f : σ' ≃ σ
⊢ Function.Injective (⇑e.symm ∘ P.map ∘ ⇑f)

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
P : PreSubmersivePresentation R S ι σ
ι' : Type u_1
σ' : Type u_2
e : ι' ≃ ι
f : σ' ≃ σ
inst✝³ : Fintype σ'
inst✝² : DecidableEq σ'
inst✝¹ : Fintype σ
inst✝ : DecidableEq σ
⊢ (P.reindex e f).jacobiMatrix = ((Matrix.reindex f.symm f.symm) P.jacobiMatrix).map ⇑(rename ⇑e.symm)

Tactic:
by

Goal state after:
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
s : M →ₗ[R] K
hs : s ∘ₗ f = LinearMap.id
hfg : Function.Exact ⇑f ⇑g
κ : Type u_6
σ : Type u_7
v : κ ⊕ σ → M
hg : Function.Surjective ⇑g
hslzero : ∀ (i : κ), s (v (Sum.inl i)) = 0
hli : LinearIndependent R (⇑s ∘ v ∘ Sum.inr)
hsp : ⊤ ≤ Submodule.span R (Set.range v)
⊢ ⊤ ≤ Submodule.span R (Set.range (⇑g ∘ v ∘ Sum.inl))

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
P : PreSubmersivePresentation R S ι σ
ι' : Type u_1
σ' : Type u_2
e : ι' ≃ ι
f : σ' ≃ σ
inst✝¹ : Finite σ
inst✝ : Finite σ'
⊢ (P.reindex e f).jacobian = P.jacobian

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
inst✝² : Finite σ
P : SubmersivePresentation R S ι σ
T : Type u_1
inst✝¹ : CommRing T
inst✝ : Algebra R T
e : S ≃ₐ[R] T
⊢ IsUnit (P.ofAlgEquiv e).jacobian

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Finite σ
h : Function.Bijective ⇑(algebraMap R S)
⊢ IsUnit (PreSubmersivePresentation.ofBijectiveAlgebraMap h).jacobian

Tactic:
by

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

Goal state before:
pp: 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
inst✝⁵ : Finite σ
T : Type u_1
ι' : Type u_2
σ' : Type u_3
inst✝⁴ : CommRing T
inst✝³ : Algebra R T
inst✝² : Algebra S T
inst✝¹ : IsScalarTower R S T
inst✝ : Finite σ'
Q : SubmersivePresentation S T ι' σ'
P : SubmersivePresentation R S ι σ
⊢ IsUnit (Q.comp P.toPreSubmersivePresentation).jacobian

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Finite σ
r : R
inst✝ : IsLocalization.Away r S
⊢ (fun x ↦ r ^ x) 1 = r

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Finite σ
r : R
inst✝ : IsLocalization.Away r S
⊢ IsUnit (PreSubmersivePresentation.localizationAway S r).jacobian

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Finite σ
P : SubmersivePresentation R S ι σ
ι' : Type u_1
σ' : Type u_2
inst✝ : Finite σ'
e : ι' ≃ ι
f : σ' ≃ σ
⊢ IsUnit (P.reindex e f).jacobian

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Finite σ
inst✝ : Subsingleton S
⊢ Ideal.span (Set.range fun x ↦ 1) = { val := fun x ↦ 1, σ' := fun x ↦ 1, aeval_val_σ' := ⋯, algebraMap_eq := ⋯ }.ker

Tactic:
by

Goal state after:
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
s : M →ₗ[R] K
hs : s ∘ₗ f = LinearMap.id
hfg : Function.Exact ⇑f ⇑g
ι : Type u_5
κ : Type u_6
σ : Type u_7
v : ι → M
a : κ → ι
b : σ → ι
hg : Function.Surjective ⇑g
hsa : ∀ (i : κ), s (v (a i)) = 0
hlib : LinearIndependent R (⇑s ∘ v ∘ b)
hab : Codisjoint (Set.range a) (Set.range b)
hsp : ⊤ ≤ span R (Set.range v)
⊢ ⊤ ≤ span R (Set.range (⇑g ∘ v ∘ a))

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Finite σ
P : SubmersivePresentation R S ι σ
this : Fintype σ
⊢ IsUnit ((LinearMap.toMatrix (Pi.basisFun S σ) (Pi.basisFun S σ)) P.aevalDifferential).det

Tactic:
by

Goal state after:
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
s : M →ₗ[R] K
hs : s ∘ₗ f = LinearMap.id
hfg : Function.Exact ⇑f ⇑g
ι : Type u_5
κ : Type u_6
σ : Type u_7
v✝ : ι → M
a : κ → ι
b : σ → ι
hg : Function.Surjective ⇑g
v : Basis ι R M
hainj : Function.Injective a
hsa : ∀ (i : κ), s (v (a i)) = 0
hlib : LinearIndependent R (⇑s ∘ ⇑v ∘ b)
hab : Codisjoint (Set.range a) (Set.range b)
⊢ ⊤ ≤ Submodule.span R (Set.range ⇑v)

Tactic:
by

Goal state after:
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
s : M →ₗ[R] K
hs : s ∘ₗ f = LinearMap.id
hfg : Function.Exact ⇑f ⇑g
ι : Type u_5
κ : Type u_6
σ : Type u_7
a : κ → ι
b : σ → ι
hg : Function.Surjective ⇑g
v : Basis ι R M
hainj : Function.Injective a
hsa : ∀ (i : κ), s (v (a i)) = 0
hlib : LinearIndependent R (⇑s ∘ ⇑v ∘ b)
hab : Codisjoint (Set.range a) (Set.range b)
k : κ
⊢ (ofSplitExact hs hfg hg v hainj hsa hlib hab) k = g (v (a k))

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Finite σ
P : SubmersivePresentation R S ι σ
i j : σ
⊢ P.basisDeriv i j = (aeval P.val) ((pderiv (P.map j)) (P.relation i))

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝³ : CommRing R
inst✝² : CommRing S
inst✝¹ : Algebra R S
inst✝ : Finite σ
P : SubmersivePresentation R S ι σ
⊢ LinearIndependent S fun i j ↦ (aeval P.val) ((pderiv (P.map j)) (P.relation i))

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Finite σ
inst✝ : Finite ι
P : SubmersivePresentation R S ι σ
⊢ IsStandardSmooth R S

Tactic:
by

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

Goal state before:
pp: n : ℕ
R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Finite σ
inst✝ : Finite ι
P : SubmersivePresentation R S ι σ
hP : P.dimension = n
⊢ IsStandardSmoothOfRelativeDimension n R S

Tactic:
by

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

Goal state before:
pp: n m : ℕ
R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Finite σ
inst✝ : IsStandardSmooth R S
⊢ FinitePresentation R S

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
T : Type u_1
inst✝² : CommRing T
inst✝¹ : Algebra R T
e : S ≃ₐ[R] T
inst✝ : IsStandardSmooth R S
⊢ IsStandardSmooth R T

Tactic:
by

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

Goal state before:
pp: n : ℕ
R : Type u
S : Type v
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
T : Type u_1
inst✝² : CommRing T
inst✝¹ : Algebra R T
e : S ≃ₐ[R] T
inst✝ : IsStandardSmoothOfRelativeDimension n R S
w✝³ w✝² : Type
w✝¹ : Finite w✝²
w✝ : Finite w✝³
P : SubmersivePresentation R S w✝³ w✝²
hP : P.dimension = n
⊢ (P.ofAlgEquiv e).dimension = n

Tactic:
by

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

Goal state before:
pp: n : ℕ
R : Type u
S : Type v
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : Algebra R S
T : Type u_1
inst✝² : CommRing T
inst✝¹ : Algebra R T
e : S ≃ₐ[R] T
inst✝ : IsStandardSmoothOfRelativeDimension n R S
⊢ IsStandardSmoothOfRelativeDimension n R T

Tactic:
by

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

Goal state before:
pp: R : Type u
S : Type v
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
T : Type u_1
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsStandardSmooth R S
inst✝ : IsStandardSmooth S T
⊢ ∃ ι σ, ∃ (x : Finite σ), Finite ι ∧ Nonempty (SubmersivePresentation R T ι σ)

Tactic:
by

Goal state after:
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
p q : Submodule R M
hpq : IsCompl p q
hmap : map g q = ⊤
x : ↥p
triv : x ∈ ⊤
⊢ g ↑x ∈ map g q

Tactic:
by

Goal state after:
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
p q : Submodule R M
hpq : IsCompl p q
hmap : map g q = ⊤
x : ↥p
triv : x ∈ ⊤
a : M
haq : a ∈ ↑q
ha : g a = g ↑x
⊢ ↑x - a ∈ ↑g.ker

Tactic:
by

Goal state after:
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
p q : Submodule R M
hpq : IsCompl p q
hmap : map g q = ⊤
x : ↥p
triv : x ∈ ⊤
a : M
haq : a ∈ ↑q
ha : g a = g ↑x
⊢ (p.projectionOnto q hpq) (↑x - a) = x

Tactic:
by

Goal state after:
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
p q : Submodule R M
hpq : IsCompl p q
hmap : map g q = ⊤
⊢ Function.Surjective ⇑(p.projectionOnto q hpq ∘ₗ f)

Tactic:
by

Goal state after:
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 = ⊤
⊢ Function.Bijective ⇑(p.projectionOnto q hpq ∘ₗ f)

Tactic:
by

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

Goal state before:
pp: n m : ℕ
R : Type u
S : Type v
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
T : Type u_1
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : IsStandardSmoothOfRelativeDimension n R S
inst✝ : IsStandardSmoothOfRelativeDimension m S T
⊢ ∃ ι σ, ∃ (x : Finite σ) (_ : Finite ι), ∃ P, P.dimension = m + n

Tactic:
by

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

Goal state before:
pp: n m : ℕ
R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : Finite σ
T : Type u_1
inst✝² : CommRing T
inst✝¹ : Algebra R T
inst✝ : IsStandardSmooth R S
⊢ ∃ ι σ, ∃ (x : Finite σ), Finite ι ∧ Nonempty (SubmersivePresentation T (T ⊗[R] S) ι σ)

Tactic:
by

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

Goal state before:
pp: n m : ℕ
R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : Finite σ
T : Type u_1
inst✝² : CommRing T
inst✝¹ : Algebra R T
inst✝ : IsStandardSmoothOfRelativeDimension n R S
⊢ ∃ ι σ, ∃ (x : Finite σ) (_ : Finite ι), ∃ P, P.dimension = n

Tactic:
by

Goal state after:
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✝⁸ : Ring R
inst✝⁷ : AddCommGroup M
inst✝⁶ : AddCommGroup K
inst✝⁵ : AddCommGroup P
inst✝⁴ : Module R M
inst✝³ : Module R K
inst✝² : Module R P
f : K →ₗ[R] M
g : M →ₗ[R] P
hfg : Function.Exact ⇑f ⇑g
hf : Function.Injective ⇑f
q : Submodule R M
E : Type u_8
inst✝¹ : AddCommGroup E
inst✝ : Module R E
i : E →ₗ[R] M
hi : Function.Injective ⇑i
h : IsCompl i.range q
hker : Disjoint g.ker q
hmap : Submodule.map g q = ⊤
⊢ Function.Bijective ⇑(linearProjOfIsCompl q i hi h ∘ₗ f)

Tactic:
by

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

Goal state before:
pp: n m : ℕ
R : Type u
S : Type v
ι : Type w
σ : Type t
inst✝⁴ : CommRing R
inst✝³ : CommRing S
inst✝² : Algebra R S
inst✝¹ : Finite σ
inst✝ : Subsingleton S
⊢ (SubmersivePresentation.ofSubsingleton R S).dimension = 0

Tactic:
by

Goal state after: