mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.Valuation.Quotient

Goal state before:
pp: R : Type u_1
Γ₀ : Type u_2
inst✝¹ : CommRing R
inst✝ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
⊢ (v.onQuot ⋯).supp = 0

Tactic:
by

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

Goal state before:
pp: R✝ Γ : Type
inst✝⁶ : Ring R✝
inst✝⁵ : DecidableEq R✝
inst✝⁴ : IsDomain R✝
inst✝³ : LinearOrderedCommGroupWithZero Γ
R : Type
inst✝² : Semiring R
inst✝¹ : DecidableEq R
inst✝ : IsDomain R
x✝¹ x✝ : R
⊢ (if x✝ = 0 then x✝¹ = 0 else True) ∨ if x✝¹ = 0 then x✝ = 0 else True

Tactic:
by

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

Goal state before:
pp: R✝ Γ : Type
inst✝⁶ : Ring R✝
inst✝⁵ : DecidableEq R✝
inst✝⁴ : IsDomain R✝
inst✝³ : LinearOrderedCommGroupWithZero Γ
R : Type
inst✝² : Semiring R
inst✝¹ : DecidableEq R
inst✝ : IsDomain R
z✝ y✝ x✝² : R
x✝¹ : if y✝ = 0 then x✝² = 0 else True
x✝ : if z✝ = 0 then y✝ = 0 else True
⊢ if z✝ = 0 then x✝² = 0 else True

Tactic:
by

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

Goal state before:
pp: R✝ Γ : Type
inst✝⁶ : Ring R✝
inst✝⁵ : DecidableEq R✝
inst✝⁴ : IsDomain R✝
inst✝³ : LinearOrderedCommGroupWithZero Γ
R : Type
inst✝² : Semiring R
inst✝¹ : DecidableEq R
inst✝ : IsDomain R
x✝² y✝ z✝ : R
x✝¹ : if z✝ = 0 then x✝² = 0 else True
x✝ : if z✝ = 0 then y✝ = 0 else True
⊢ if z✝ = 0 then x✝² + y✝ = 0 else True

Tactic:
by

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

Goal state before:
pp: R✝ Γ : Type
inst✝⁶ : Ring R✝
inst✝⁵ : DecidableEq R✝
inst✝⁴ : IsDomain R✝
inst✝³ : LinearOrderedCommGroupWithZero Γ
R : Type
inst✝² : Semiring R
inst✝¹ : DecidableEq R
inst✝ : IsDomain R
x✝² y✝ : R
x✝¹ : if y✝ = 0 then x✝² = 0 else True
x✝ : R
⊢ if y✝ * x✝ = 0 then x✝² * x✝ = 0 else True

Tactic:
by

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

Goal state before:
pp: R✝ Γ : Type
inst✝⁶ : Ring R✝
inst✝⁵ : DecidableEq R✝
inst✝⁴ : IsDomain R✝
inst✝³ : LinearOrderedCommGroupWithZero Γ
R : Type
inst✝² : Semiring R
inst✝¹ : DecidableEq R
inst✝ : IsDomain R
x✝¹ y✝ z✝ : R
x✝ : ¬if 0 = 0 then z✝ = 0 else True
⊢ (if y✝ * z✝ = 0 then x✝¹ * z✝ = 0 else True) → if y✝ = 0 then x✝¹ = 0 else True

Tactic:
by

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

Goal state before:
pp: R✝ Γ : Type
inst✝⁶ : Ring R✝
inst✝⁵ : DecidableEq R✝
inst✝⁴ : IsDomain R✝
inst✝³ : LinearOrderedCommGroupWithZero Γ
R : Type
inst✝² : Semiring R
inst✝¹ : DecidableEq R
inst✝ : IsDomain R
⊢ ¬if 0 = 0 then 1 = 0 else True

Tactic:
by

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

Goal state before:
pp: R✝ Γ : Type
inst✝⁶ : Ring R✝
inst✝⁵ : DecidableEq R✝
inst✝⁴ : IsDomain R✝
inst✝³ : LinearOrderedCommGroupWithZero Γ
R : Type
inst✝² : Semiring R
inst✝¹ : DecidableEq R
inst✝ : IsDomain R
x✝¹ x✝ : R
⊢ if x✝ * x✝¹ = 0 then x✝¹ * x✝ = 0 else True

Tactic:
by

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

Goal state before:
pp: R Γ : Type
inst✝³ : Ring R
inst✝² : DecidableEq R
inst✝¹ : IsDomain R
inst✝ : LinearOrderedCommGroupWithZero Γ
h : ValuativeRel R
hv : Valuation.Compatible 1
⊢ h = trivialRel

Tactic:
by

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

Goal state before:
pp: R Γ : Type
inst✝³ : Ring R
inst✝² : DecidableEq R
inst✝¹ : IsDomain R
inst✝ : LinearOrderedCommGroupWithZero Γ
⊢ trivialRel = ofValuation 1

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝⁵ : CommRing R
M : Type u_2
N : Type u_3
inst✝⁴ : AddCommGroup M
inst✝³ : AddCommGroup N
inst✝² : Module R M
inst✝¹ : Module R N
I : Ideal R
f : M →ₗ[R] N
surj : Function.Surjective ⇑f
inst✝ : Module.Flat R N
⊢ f.ker ⊓ I • ⊤ = I • f.ker

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝¹⁰ : CommRing R
S : Type u_2
inst✝⁹ : CommRing S
R' : Type u_3
S' : Type u_4
inst✝⁸ : CommRing R'
inst✝⁷ : CommRing S'
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R R'
inst✝⁴ : Algebra R' S'
inst✝³ : Algebra S S'
inst✝² : Algebra R S'
inst✝¹ : IsScalarTower R S S'
inst✝ : IsScalarTower R R' S'
surjRS : Function.Surjective ⇑(algebraMap R S)
eqmap : RingHom.ker (algebraMap S S') = Ideal.map (algebraMap R S) (RingHom.ker (algebraMap R R'))
⊢ Ideal.comap (algebraMap R R') (RingHom.ker (algebraMap R' S')) =
    RingHom.ker (algebraMap R R') ⊔ RingHom.ker (algebraMap R S)

Tactic:
by

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

Goal state before:
pp: R Γ : Type
inst✝⁵ : Ring R
inst✝⁴ : DecidableEq R
inst✝³ : IsDomain R
inst✝² : LinearOrderedCommGroupWithZero Γ
inst✝¹ : ValuativeRel R
inst✝ : Valuation.Compatible 1
⊢ Subsingleton (ValueGroupWithZero R)ˣ

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
J I : Ideal R
sq : I * I = ⊥
f : J.Cotangent →ₗ[R] J.Cotangent
le : f.range ≤ Submodule.map J.toCotangent (Submodule.comap (Submodule.subtype J) (I * J))
x : R
h : x ∈ I * J
⊢ f (J.toCotangent ⟨x, ⋯⟩) = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Extension

Goal state before:
pp: R : Type u_1
A : Type u_2
ΓR : Type u_3
ΓA : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : Ring A
inst✝³ : LinearOrderedCommMonoidWithZero ΓR
inst✝² : LinearOrderedCommMonoidWithZero ΓA
inst✝¹ : Algebra R A
vR : Valuation R ΓR
vA : Valuation A ΓA
inst✝ : vR.HasExtension vA
x y : R
⊢ vA ((algebraMap R A) x) < vA ((algebraMap R A) y) ↔ vR x < vR y

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Extension

Goal state before:
pp: R : Type u_1
A : Type u_2
ΓR : Type u_3
ΓA : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : Ring A
inst✝³ : LinearOrderedCommMonoidWithZero ΓR
inst✝² : LinearOrderedCommMonoidWithZero ΓA
inst✝¹ : Algebra R A
vR : Valuation R ΓR
vA : Valuation A ΓA
inst✝ : vR.HasExtension vA
x : R
⊢ vA ((algebraMap R A) x) ≤ 1 ↔ vR x ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Extension

Goal state before:
pp: R : Type u_1
A : Type u_2
ΓR : Type u_3
ΓA : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : Ring A
inst✝³ : LinearOrderedCommMonoidWithZero ΓR
inst✝² : LinearOrderedCommMonoidWithZero ΓA
inst✝¹ : Algebra R A
vR : Valuation R ΓR
vA : Valuation A ΓA
inst✝ : vR.HasExtension vA
x : R
⊢ vA ((algebraMap R A) x) < 1 ↔ vR x < 1

Tactic:
simpa only [map_one, not_le] using (val_map_le_iff vR vA 1 x).not

Goal state after:
Module: Mathlib.RingTheory.Valuation.Extension

Goal state before:
pp: R : Type u_1
A : Type u_2
ΓR : Type u_3
ΓA : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : Ring A
inst✝³ : LinearOrderedCommMonoidWithZero ΓR
inst✝² : LinearOrderedCommMonoidWithZero ΓA
inst✝¹ : Algebra R A
vR : Valuation R ΓR
vA : Valuation A ΓA
inst✝ : vR.HasExtension vA
x : R
⊢ vA ((algebraMap R A) x) < 1 ↔ vR x < 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Extension

Goal state before:
pp: R : Type u_1
A : Type u_2
ΓR : Type u_3
ΓA : Type u_4
inst✝⁵ : CommRing R
inst✝⁴ : Ring A
inst✝³ : LinearOrderedCommMonoidWithZero ΓR
inst✝² : LinearOrderedCommMonoidWithZero ΓA
inst✝¹ : Algebra R A
vR : Valuation R ΓR
vA : Valuation A ΓA
inst✝ : vR.HasExtension vA
x : R
⊢ vA ((algebraMap R A) x) = 1 ↔ vR x = 1

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
J I : Ideal R
sq : I ^ 2 = ⊥
f : J.Cotangent →ₗ[R] J.Cotangent
le : f.range ≤ Submodule.map J.toCotangent (Submodule.comap (Submodule.subtype J) (I * J))
⊢ Submodule.map J.toCotangent (Submodule.comap (Submodule.subtype J) (I * J)) ≤ f.ker

Tactic:
by

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

Goal state before:
pp: R Γ : Type
inst✝⁵ : Ring R
inst✝⁴ : DecidableEq R
inst✝³ : IsDomain R
inst✝² : LinearOrderedCommGroupWithZero Γ
inst✝¹ : ValuativeRel R
inst✝ : Valuation.Compatible 1
⊢ ¬IsNontrivial R

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Extension

Goal state before:
pp: R : Type u_1
A : Type u_2
ΓR : Type u_3
ΓA : Type u_4
inst✝⁴ : CommRing R
inst✝³ : Ring A
inst✝² : LinearOrderedCommMonoidWithZero ΓR
inst✝¹ : LinearOrderedCommMonoidWithZero ΓA
inst✝ : Algebra R A
vR : Valuation R ΓR
vA : Valuation A ΓA
⊢ vR.IsEquiv (comap (algebraMap R R) vR)

Tactic:
by

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

Goal state before:
pp: R Γ : Type
inst✝⁵ : Ring R
inst✝⁴ : DecidableEq R
inst✝³ : IsDomain R
inst✝² : LinearOrderedCommGroupWithZero Γ
inst✝¹ : ValuativeRel R
inst✝ : Valuation.Compatible 1
⊢ IsDiscrete R

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝¹¹ : CommRing R
S : Type u_2
inst✝¹⁰ : CommRing S
R' : Type u_3
S' : Type u_4
inst✝⁹ : CommRing R'
inst✝⁸ : CommRing S'
inst✝⁷ : Algebra R S
inst✝⁶ : Algebra R R'
inst✝⁵ : Algebra R' S'
inst✝⁴ : Algebra S S'
inst✝³ : Algebra R S'
inst✝² : IsScalarTower R S S'
inst✝¹ : IsScalarTower R R' S'
inst✝ : Module.Flat R S
surj : Function.Surjective ⇑(algebraMap R R')
surjS : Function.Surjective ⇑(algebraMap S S')
eqmap : RingHom.ker (algebraMap S S') = Ideal.map (algebraMap R S) (RingHom.ker (algebraMap R R'))
sq0 : RingHom.ker (algebraMap R R') ^ 2 = ⊥
smoothq : FormallySmooth R' S'
P : Extension R S := (Generators.self R S).toExtension
this : FormallySmooth R P.Ring := instFormallySmoothMvPolynomial S
IP : Ideal P.Ring := Ideal.map (algebraMap R P.Ring) (RingHom.ker (algebraMap R R'))
s' : S'
⊢ (MvPolynomial.aeval ⇑(algebraMap S S')) (MvPolynomial.X (Classical.choose ⋯)) = s'

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Extension

Goal state before:
pp: A : Type u_2
inst✝⁴ : Ring A
K : Type u_5
inst✝³ : Field K
inst✝² : Algebra K A
ΓA : Type u_7
ΓK : Type u_8
inst✝¹ : LinearOrderedCommGroupWithZero ΓK
inst✝ : LinearOrderedCommGroupWithZero ΓA
vK : Valuation K ΓK
vA : Valuation A ΓA
h : Subring.comap (algebraMap K A) vA.integer = vK.integer
⊢ vK.IsEquiv (comap (algebraMap K A) vA)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Extension

Goal state before:
pp: R : Type u_1
A : Type u_2
ΓR✝ : Type u_3
ΓA✝ : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : Ring A
inst✝⁸ : LinearOrderedCommMonoidWithZero ΓR✝
inst✝⁷ : LinearOrderedCommMonoidWithZero ΓA✝
inst✝⁶ : Algebra R A
vR✝ : Valuation R ΓR✝
vA✝ : Valuation A ΓA✝
K : Type u_5
inst✝⁵ : Field K
inst✝⁴ : Algebra K A
ΓR : Type u_6
ΓA : Type u_7
ΓK : Type u_8
inst✝³ : LinearOrderedCommGroupWithZero ΓR
inst✝² : LinearOrderedCommGroupWithZero ΓK
inst✝¹ : LinearOrderedCommGroupWithZero ΓA
vR : Valuation R ΓR
vK : Valuation K ΓK
vA : Valuation A ΓA
inst✝ : vR.HasExtension vA
⊢ ∀ x ∈ vR.integer, (algebraMap R A) x ∈ vA.integer

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Extension

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Ring A
inst✝³ : Algebra R A
ΓR : Type u_6
ΓA : Type u_7
inst✝² : LinearOrderedCommGroupWithZero ΓR
inst✝¹ : LinearOrderedCommGroupWithZero ΓA
vR : Valuation R ΓR
vA : Valuation A ΓA
inst✝ : vR.HasExtension vA
r : ↥vR.integer
a : ↥vA.integer
⊢ ↑(r • a) = ↑r • ↑a

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Extension

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝⁵ : CommRing R
inst✝⁴ : Ring A
inst✝³ : Algebra R A
ΓR : Type u_6
ΓA : Type u_7
inst✝² : LinearOrderedCommGroupWithZero ΓR
inst✝¹ : LinearOrderedCommGroupWithZero ΓA
vR : Valuation R ΓR
vA : Valuation A ΓA
inst✝ : vR.HasExtension vA
r : ↥vR.integer
⊢ ↑((algebraMap ↥vR.integer ↥vA.integer) r) = (algebraMap R A) ↑r

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Extension

Goal state before:
pp: R : Type u_1
A : Type u_2
ΓR✝ : Type u_3
ΓA✝ : Type u_4
inst✝¹⁰ : CommRing R
inst✝⁹ : Ring A
inst✝⁸ : LinearOrderedCommMonoidWithZero ΓR✝
inst✝⁷ : LinearOrderedCommMonoidWithZero ΓA✝
inst✝⁶ : Algebra R A
vR✝ : Valuation R ΓR✝
vA✝ : Valuation A ΓA✝
K : Type u_5
inst✝⁵ : Field K
inst✝⁴ : Algebra K A
ΓR : Type u_6
ΓA : Type u_7
ΓK : Type u_8
inst✝³ : LinearOrderedCommGroupWithZero ΓR
inst✝² : LinearOrderedCommGroupWithZero ΓK
inst✝¹ : LinearOrderedCommGroupWithZero ΓA
vR : Valuation R ΓR
vK : Valuation K ΓK
vA : Valuation A ΓA
inst✝ : vR.HasExtension vA
x : ↥vR.integer
y : ↥vA.integer
z : A
⊢ (x • y) • z = x • y • z

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Extension

Goal state before:
pp: R : Type u_1
A : Type u_2
ΓR✝ : Type u_3
ΓA✝ : Type u_4
inst✝¹² : CommRing R
inst✝¹¹ : Ring A
inst✝¹⁰ : LinearOrderedCommMonoidWithZero ΓR✝
inst✝⁹ : LinearOrderedCommMonoidWithZero ΓA✝
inst✝⁸ : Algebra R A
vR✝ : Valuation R ΓR✝
vA✝ : Valuation A ΓA✝
K : Type u_5
inst✝⁷ : Field K
inst✝⁶ : Algebra K A
ΓR : Type u_6
ΓA : Type u_7
ΓK : Type u_8
inst✝⁵ : LinearOrderedCommGroupWithZero ΓR
inst✝⁴ : LinearOrderedCommGroupWithZero ΓK
inst✝³ : LinearOrderedCommGroupWithZero ΓA
vR : Valuation R ΓR
vK : Valuation K ΓK
vA : Valuation A ΓA
inst✝² : vR.HasExtension vA
inst✝¹ : IsDomain R
inst✝ : IsTorsionFree R A
⊢ ∀ (r : ↥vR.integer) (m : ↥vA.integer), r • m = 0 → r = 0 ∨ m = 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Extension

Goal state before:
pp: R : Type u_1
inst✝⁶ : CommRing R
ΓR : Type u_6
inst✝⁵ : LinearOrderedCommGroupWithZero ΓR
vR : Valuation R ΓR
S : Type u_9
ΓS : Type u_10
inst✝⁴ : CommRing S
inst✝³ : LinearOrderedCommGroupWithZero ΓS
inst✝² : Algebra R S
inst✝¹ : IsLocalHom (algebraMap R S)
vS : Valuation S ΓS
inst✝ : vR.HasExtension vS
r : ↥vR.integer
hr : IsUnit ((algebraMap ↥vR.integer ↥vS.integer) r)
⊢ IsUnit r

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Extension

Goal state before:
pp: K : Type u_5
L : Type u_6
Γ₀ : Type u_7
Γ₁ : Type u_8
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : LinearOrderedCommGroupWithZero Γ₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₁
vK : Valuation K Γ₀
vL : Valuation L Γ₁
inst✝ : vK.HasExtension vL
x : ↥K₀
⊢ (algebraMap K L) ↑x ∈ L₀

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Extension

Goal state before:
pp: K : Type u_5
L : Type u_6
Γ₀ : Type u_7
Γ₁ : Type u_8
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : LinearOrderedCommGroupWithZero Γ₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₁
vK : Valuation K Γ₀
vL : Valuation L Γ₁
inst✝ : vK.HasExtension vL
x : ↥K₀
⊢ (algebraMap ↥K₀ ↥L₀) x ∈ maximalIdeal ↥L₀ ↔ x ∈ maximalIdeal ↥K₀

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Extension

Goal state before:
pp: K : Type u_5
L : Type u_6
Γ₀ : Type u_7
Γ₁ : Type u_8
inst✝⁵ : Field K
inst✝⁴ : Field L
inst✝³ : Algebra K L
inst✝² : LinearOrderedCommGroupWithZero Γ₀
inst✝¹ : LinearOrderedCommGroupWithZero Γ₁
vK : Valuation K Γ₀
vL : Valuation L Γ₁
inst✝ : vK.HasExtension vL
x✝ : ↥K₀
⊢ x✝ ∈ Ideal.comap (algebraMap ↥K₀ ↥L₀) (maximalIdeal ↥L₀) ↔ x✝ ∈ maximalIdeal ↥K₀

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝¹¹ : CommRing R
S : Type u_2
inst✝¹⁰ : CommRing S
R' : Type u_3
S' : Type u_4
inst✝⁹ : CommRing R'
inst✝⁸ : CommRing S'
inst✝⁷ : Algebra R S
inst✝⁶ : Algebra R R'
inst✝⁵ : Algebra R' S'
inst✝⁴ : Algebra S S'
inst✝³ : Algebra R S'
inst✝² : IsScalarTower R S S'
inst✝¹ : IsScalarTower R R' S'
inst✝ : Module.Flat R S
surj : Function.Surjective ⇑(algebraMap R R')
surjS : Function.Surjective ⇑(algebraMap S S')
eqmap : RingHom.ker (algebraMap S S') = Ideal.map (algebraMap R S) (RingHom.ker (algebraMap R R'))
sq0 : RingHom.ker (algebraMap R R') ^ 2 = ⊥
smoothq : FormallySmooth R' S'
P : Extension R S := (Generators.self R S).toExtension
this✝² : FormallySmooth R P.Ring := instFormallySmoothMvPolynomial S
IP : Ideal P.Ring := Ideal.map (algebraMap R P.Ring) (RingHom.ker (algebraMap R R'))
Gen : Generators R' S' S :=
  { val := ⇑(algebraMap S S'), σ' := fun s' ↦ MvPolynomial.X (Classical.choose ⋯), aeval_val_σ' := ⋯,
    algebraMap_eq := ⋯ }
P' : Extension R' S' := Gen.toExtension
this✝¹ : FormallySmooth R' P'.Ring := instFormallySmoothMvPolynomial S
this✝ : Algebra P.Ring P'.Ring := MvPolynomial.algebraMvPolynomial
this : IsScalarTower R P.Ring P'.Ring :=
  IsScalarTower.of_algebraMap_eq fun x ↦ Eq.symm (MvPolynomial.map_C (algebraMap R R') x)
algInst✝ : Algebra P.Ring S' := ((algebraMap S S').comp (algebraMap P.Ring S)).toAlgebra
scalarTowerInst✝ : IsScalarTower P.Ring S S' := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap P.Ring S'))
⊢ IsScalarTower P.Ring P'.Ring S'

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.Compare

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharP R p
i : ℕ
hin : i < n
⊢ ↑p ^ i = (WittVector.truncate n) (↑p ^ i)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.Compare

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
R : Type u_1
inst✝¹ : CommRing R
inst✝ : CharP R p
i : ℕ
hin : i ≤ n
hpi : ↑p ^ i = 0
⊢ i = n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.Compare

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
⊢ Fintype.card (TruncatedWittVector p n (ZMod p)) = p ^ n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.Compare

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n : ℕ
R : Type u_1
inst✝ : CommRing R
x : ZMod (p ^ n)
⊢ p ^ n ∣ p ^ n

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.Compare

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n m : ℕ
hm : n ≤ m
x : TruncatedWittVector p m (ZMod p)
⊢ (zmodEquivTrunc p n).symm ((truncate hm) x) = (ZMod.castHom ⋯ (ZMod (p ^ n))) ((zmodEquivTrunc p m).symm x)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.DiscreteValuationRing

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
a : kˣ
A : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
hA : A.coeff 0 = ↑a
n : ℕ
a✝ : (A * { coeff := inverseCoeff a A }).coeff n = coeff 1 n
H_coeff : k :=
  A.coeff (n + 1) * ↑(a⁻¹ ^ p ^ (n + 1)) + nthRemainder p n (truncateFun (n + 1) A) fun i ↦ inverseCoeff a A ↑i
H : ↑(a ^ p ^ (n + 1)) * ↑(a ^ p ^ (n + 1))⁻¹ = 1
⊢ ↑a ^ p ^ (n + 1) = ↑(a ^ p ^ (n + 1))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.Compare

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
n m : ℕ
hm : n ≤ m
⊢ (zmodEquivTrunc p n).symm.toRingHom.comp (truncate hm) =
    (ZMod.castHom ⋯ (ZMod (p ^ n))).comp (zmodEquivTrunc p m).symm.toRingHom

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.DiscreteValuationRing

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
a : kˣ
A : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
hA : A.coeff 0 = ↑a
n : ℕ
a✝ : (A * { coeff := inverseCoeff a A }).coeff n = coeff 1 n
H_coeff : k :=
  A.coeff (n + 1) * ↑(a⁻¹ ^ p ^ (n + 1)) + nthRemainder p n (truncateFun (n + 1) A) fun i ↦ inverseCoeff a A ↑i
H : ↑(a ^ p ^ (n + 1)) * ↑(a ^ p ^ (n + 1))⁻¹ = 1
ha : ↑a ^ p ^ (n + 1) = ↑(a ^ p ^ (n + 1))
⊢ ↑a⁻¹ ^ p ^ (n + 1) = ↑(a ^ p ^ (n + 1))⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.Compare

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
m n : ℕ
h : m ≤ n
⊢ (ZMod.castHom ⋯ (ZMod (p ^ m))).comp ((zmodEquivTrunc p n).symm.toRingHom.comp (truncate n)) =
    ((zmodEquivTrunc p m).symm.toRingHom.comp (TruncatedWittVector.truncate h)).comp (truncate n)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.Compare

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
m n : ℕ
h : m ≤ n
⊢ ((zmodEquivTrunc p m).symm.toRingHom.comp (TruncatedWittVector.truncate h)).comp (truncate n) =
    (zmodEquivTrunc p m).symm.toRingHom.comp (truncate m)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.DiscreteValuationRing

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : CommRing k
inst✝ : CharP k p
a : kˣ
A : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
hA : A.coeff 0 = ↑a
⊢ A * { coeff := inverseCoeff a A } = 1

Tactic:
by

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

Goal state before:
pp: R : Type u_1
inst✝¹¹ : CommRing R
S : Type u_2
inst✝¹⁰ : CommRing S
R' : Type u_3
S' : Type u_4
inst✝⁹ : CommRing R'
inst✝⁸ : CommRing S'
inst✝⁷ : Algebra R S
inst✝⁶ : Algebra R R'
inst✝⁵ : Algebra R' S'
inst✝⁴ : Algebra S S'
inst✝³ : Algebra R S'
inst✝² : IsScalarTower R S S'
inst✝¹ : IsScalarTower R R' S'
inst✝ : Module.Flat R S
surj : Function.Surjective ⇑(algebraMap R R')
surjS : Function.Surjective ⇑(algebraMap S S')
eqmap : RingHom.ker (algebraMap S S') = Ideal.map (algebraMap R S) (RingHom.ker (algebraMap R R'))
sq0 : RingHom.ker (algebraMap R R') ^ 2 = ⊥
smoothq : FormallySmooth R' S'
P : Extension R S := (Generators.self R S).toExtension
this✝³ : FormallySmooth R P.Ring := instFormallySmoothMvPolynomial S
IP✝ : Ideal P.Ring := Ideal.map (algebraMap R P.Ring) (RingHom.ker (algebraMap R R'))
Gen : Generators R' S' S :=
  { val := ⇑(algebraMap S S'), σ' := fun s' ↦ MvPolynomial.X (Classical.choose ⋯), aeval_val_σ' := ⋯,
    algebraMap_eq := ⋯ }
P' : Extension R' S' := Gen.toExtension
this✝² : FormallySmooth R' P'.Ring := instFormallySmoothMvPolynomial S
this✝¹ : Algebra P.Ring P'.Ring := MvPolynomial.algebraMvPolynomial
this✝ : IsScalarTower R P.Ring P'.Ring :=
  IsScalarTower.of_algebraMap_eq fun x ↦ Eq.symm (MvPolynomial.map_C (algebraMap R R') x)
algInst✝ : Algebra P.Ring S' := ((algebraMap S S').comp (algebraMap P.Ring S)).toAlgebra
scalarTowerInst✝ : IsScalarTower P.Ring S S' := IsScalarTower.of_algebraMap_eq' (Eq.refl (algebraMap P.Ring S'))
this : IsScalarTower P.Ring P'.Ring S'
J : Ideal P.Ring := RingHom.ker (algebraMap P.Ring S)
J' : Ideal P'.Ring := RingHom.ker (algebraMap P'.Ring S')
I : Ideal R := RingHom.ker (algebraMap R R')
IS : Ideal S := Ideal.map (algebraMap R S) I
kerS : RingHom.ker (algebraMap S S') = IS
IP : Ideal P.Ring := Ideal.map (algebraMap R P.Ring) I
kerP : RingHom.ker (algebraMap P.Ring P'.Ring) = IP
surjPP' : Function.Surjective ⇑(algebraMap P.Ring P'.Ring)
⊢ IS = Ideal.map (algebraMap P.Ring S) IP

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.Compare

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k₁ k₂ : ℕ
hk : k₁ ≤ k₂
⊢ (TruncatedWittVector.truncate hk).comp ((zmodEquivTrunc p k₂).toRingHom.comp (PadicInt.toZModPow k₂)) =
    (zmodEquivTrunc p k₁).toRingHom.comp (PadicInt.toZModPow k₁)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.DiscreteValuationRing

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : Field k
inst✝ : CharP k p
x : 𝕎 k
hx : x.coeff 0 ≠ 0
⊢ IsUnit x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.DiscreteValuationRing

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : Field k
inst✝ : CharP k p
⊢ ¬IsUnit ↑p

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.DiscreteValuationRing

Goal state before:
pp: p : ℕ
hp✝ : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : Field k
inst✝ : CharP k p
hp : ¬IsUnit ↑p
a b : 𝕎 k
hab : ↑p = a * b
⊢ a ≠ 0 ∧ b ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.Compare

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
⊢ (toPadicInt p).comp (fromPadicInt p) = RingHom.id ℤ_[p]

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.Compare

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
x : ℤ_[p]
⊢ ((toPadicInt p).comp (fromPadicInt p)) x = (RingHom.id ℤ_[p]) x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.Compare

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
⊢ (fromPadicInt p).comp (toPadicInt p) = RingHom.id (𝕎 (ZMod p))

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.Compare

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
x : 𝕎 (ZMod p)
⊢ ((fromPadicInt p).comp (toPadicInt p)) x = (RingHom.id (𝕎 (ZMod p))) x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.WittVector.DiscreteValuationRing

Goal state before:
pp: p : ℕ
hp : Fact (Nat.Prime p)
k : Type u_1
inst✝¹ : Field k
inst✝ : CharP k p
⊢ Irreducible ↑p

Tactic:
by

Goal state after: