mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.RingTheory.HahnSeries.Summable

Goal state before:
pp: case inr
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : CommRing R
x : R⟦Γ⟧
h : 0 < (x - 1).orderTop
h✝ : Nontrivial R
⊢ IsUnit x

Tactic:
·

Goal state after:
case inr
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : CommRing R
x : R⟦Γ⟧
h : 0 < (x - 1).orderTop
h✝ : Nontrivial R
⊢ IsUnit x
Module: Mathlib.RingTheory.HahnSeries.Summable

Goal state before:
pp: case inr.refine_1
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : CommRing R
x : R⟦Γ⟧
h : 0 < (x - 1).orderTop
h✝ : Nontrivial R
⊢ IsUnit x.leadingCoeff

pp: case inr.refine_2
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : CommRing R
x : R⟦Γ⟧
h : 0 < (x - 1).orderTop
h✝ : Nontrivial R
⊢ IsAddUnit x.order

Tactic:
·

Goal state after:
case inr.refine_1
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : CommRing R
x : R⟦Γ⟧
h : 0 < (x - 1).orderTop
h✝ : Nontrivial R
⊢ IsUnit x.leadingCoeff
Module: Mathlib.RingTheory.HahnSeries.Summable

Goal state before:
pp: case inr.refine_2
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : CommRing R
x : R⟦Γ⟧
h : 0 < (x - 1).orderTop
h✝ : Nontrivial R
⊢ IsAddUnit x.order

Tactic:
·

Goal state after:
case inr.refine_2
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommMonoid Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedCancelAddMonoid Γ
inst✝ : CommRing R
x : R⟦Γ⟧
h : 0 < (x - 1).orderTop
h✝ : Nontrivial R
⊢ IsAddUnit x.order
Module: Mathlib.RingTheory.HahnSeries.Summable

Goal state before:
pp: case mp
Γ : Type u_1
R : Type u_3
inst✝⁴ : AddCommGroup Γ
inst✝³ : LinearOrder Γ
inst✝² : IsOrderedAddMonoid Γ
inst✝¹ : CommRing R
inst✝ : IsDomain R
x : R⟦Γ⟧
⊢ IsUnit x → IsUnit x.leadingCoeff

pp: case mpr
Γ : Type u_1
R : Type u_3
inst✝⁴ : AddCommGroup Γ
inst✝³ : LinearOrder Γ
inst✝² : IsOrderedAddMonoid Γ
inst✝¹ : CommRing R
inst✝ : IsDomain R
x : R⟦Γ⟧
⊢ IsUnit x.leadingCoeff → IsUnit x

Tactic:
·

Goal state after:
case mp
Γ : Type u_1
R : Type u_3
inst✝⁴ : AddCommGroup Γ
inst✝³ : LinearOrder Γ
inst✝² : IsOrderedAddMonoid Γ
inst✝¹ : CommRing R
inst✝ : IsDomain R
x : R⟦Γ⟧
⊢ IsUnit x → IsUnit x.leadingCoeff
Module: Mathlib.RingTheory.HahnSeries.Summable

Goal state before:
pp: case mpr
Γ : Type u_1
R : Type u_3
inst✝⁴ : AddCommGroup Γ
inst✝³ : LinearOrder Γ
inst✝² : IsOrderedAddMonoid Γ
inst✝¹ : CommRing R
inst✝ : IsDomain R
x : R⟦Γ⟧
⊢ IsUnit x.leadingCoeff → IsUnit x

Tactic:
·

Goal state after:
case mpr
Γ : Type u_1
R : Type u_3
inst✝⁴ : AddCommGroup Γ
inst✝³ : LinearOrder Γ
inst✝² : IsOrderedAddMonoid Γ
inst✝¹ : CommRing R
inst✝ : IsDomain R
x : R⟦Γ⟧
⊢ IsUnit x.leadingCoeff → IsUnit x
Module: Mathlib.RingTheory.HahnSeries.Summable

Goal state before:
pp: case inl
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommGroup Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedAddMonoid Γ
inst✝ : Field R
a : Γ
⊢ ((single a) 0)⁻¹ = (single (-a)) 0⁻¹

pp: case inr
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommGroup Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedAddMonoid Γ
inst✝ : Field R
a : Γ
r : R
hr : r ≠ 0
⊢ ((single a) r)⁻¹ = (single (-a)) r⁻¹

Tactic:
·

Goal state after:
case inl
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommGroup Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedAddMonoid Γ
inst✝ : Field R
a : Γ
⊢ ((single a) 0)⁻¹ = (single (-a)) 0⁻¹
Module: Mathlib.RingTheory.HahnSeries.Summable

Goal state before:
pp: case inr
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommGroup Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedAddMonoid Γ
inst✝ : Field R
a : Γ
r : R
hr : r ≠ 0
⊢ ((single a) r)⁻¹ = (single (-a)) r⁻¹

Tactic:
·

Goal state after:
case inr
Γ : Type u_1
R : Type u_3
inst✝³ : AddCommGroup Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedAddMonoid Γ
inst✝ : Field R
a : Γ
r : R
hr : r ≠ 0
⊢ ((single a) r)⁻¹ = (single (-a)) r⁻¹
Module: Mathlib.RingTheory.Ideal.HasGoingUp

Goal state before:
pp: case inl
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : Algebra.HasGoingUp R S
p : Ideal R
P : Ideal S
inst✝² : P.IsPrime
inst✝¹ : P.LiesOver p
inst✝ : p.IsPrime
hle : p ≤ p
⊢ ∃ Q, P ≤ Q ∧ Q.IsPrime ∧ Q.LiesOver p

pp: case inr
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : Algebra.HasGoingUp R S
p q : Ideal R
inst✝² : q.IsPrime
P : Ideal S
inst✝¹ : P.IsPrime
inst✝ : P.LiesOver p
hle : p ≤ q
h : p ≠ q
⊢ ∃ Q, P ≤ Q ∧ Q.IsPrime ∧ Q.LiesOver q

Tactic:
·

Goal state after:
case inl
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : Algebra.HasGoingUp R S
p : Ideal R
P : Ideal S
inst✝² : P.IsPrime
inst✝¹ : P.LiesOver p
inst✝ : p.IsPrime
hle : p ≤ p
⊢ ∃ Q, P ≤ Q ∧ Q.IsPrime ∧ Q.LiesOver p
Module: Mathlib.RingTheory.Ideal.HasGoingUp

Goal state before:
pp: case inr
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : Algebra.HasGoingUp R S
p q : Ideal R
inst✝² : q.IsPrime
P : Ideal S
inst✝¹ : P.IsPrime
inst✝ : P.LiesOver p
hle : p ≤ q
h : p ≠ q
⊢ ∃ Q, P ≤ Q ∧ Q.IsPrime ∧ Q.LiesOver q

Tactic:
·

Goal state after:
case inr
R : Type u_1
S : Type u_2
inst✝⁶ : CommRing R
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
inst✝³ : Algebra.HasGoingUp R S
p q : Ideal R
inst✝² : q.IsPrime
P : Ideal S
inst✝¹ : P.IsPrime
inst✝ : P.LiesOver p
hle : p ≤ q
h : p ≠ q
⊢ ∃ Q, P ≤ Q ∧ Q.IsPrime ∧ Q.LiesOver q
Module: Mathlib.RingTheory.Henselian

Goal state before:
pp: case succ
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
n : ℕ
ih : Polynomial.eval (c n) f ∈ I ^ (n + 1)
⊢ Polynomial.eval (c n + -(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ) + -c n) ((taylor (c n)) f) ∈
    I ^ (n + 1 + 1)

Tactic:
add_neg_cancel_comm

Goal state after:
case succ
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
n : ℕ
ih : Polynomial.eval (c n) f ∈ I ^ (n + 1)
⊢ Polynomial.eval (-(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ)) ((taylor (c n)) f) ∈ I ^ (n + 1 + 1)
Module: Mathlib.RingTheory.HopfAlgebra.Quotient

Goal state before:
pp: case refine_1
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : HopfAlgebra R A
inst✝ : HopfAlgebraStruct R B
f : A →ₐc[R] B
hf : Function.Surjective ⇑f
hS : antipode R ∘ₗ f.toLinearMap = f.toLinearMap ∘ₗ antipode R
⊢ (toConv (antipode R) * toConv LinearMap.id).ofConv ∘ₗ f.toLinearMap = ofConv 1 ∘ₗ f.toLinearMap

pp: case refine_2
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : HopfAlgebra R A
inst✝ : HopfAlgebraStruct R B
f : A →ₐc[R] B
hf : Function.Surjective ⇑f
hS : antipode R ∘ₗ f.toLinearMap = f.toLinearMap ∘ₗ antipode R
⊢ (toConv LinearMap.id * toConv (antipode R)).ofConv ∘ₗ f.toLinearMap = ofConv 1 ∘ₗ f.toLinearMap

Tactic:
·

Goal state after:
case refine_1
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : HopfAlgebra R A
inst✝ : HopfAlgebraStruct R B
f : A →ₐc[R] B
hf : Function.Surjective ⇑f
hS : antipode R ∘ₗ f.toLinearMap = f.toLinearMap ∘ₗ antipode R
⊢ (toConv (antipode R) * toConv LinearMap.id).ofConv ∘ₗ f.toLinearMap = ofConv 1 ∘ₗ f.toLinearMap
Module: Mathlib.RingTheory.Henselian

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
n : ℕ
ih : Polynomial.eval (c n) f ∈ I ^ (n + 1)
⊢ ∀ (n_1 : ℕ), 0 * (-(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ)) ^ n_1 = 0

pp: case succ
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
n : ℕ
ih : Polynomial.eval (c n) f ∈ I ^ (n + 1)
⊢ ∑ k ∈ Finset.range 2, ((taylor (c n)) f).coeff k * (-(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ)) ^ k +
      ∑ k ∈ Finset.Ico 2 (((taylor (c n)) f).natDegree + 2),
        ((taylor (c n)) f).coeff k * (-(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ)) ^ k ∈
    I ^ (n + 1 + 1)

Tactic:
·

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
n : ℕ
ih : Polynomial.eval (c n) f ∈ I ^ (n + 1)
⊢ ∀ (n_1 : ℕ), 0 * (-(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ)) ^ n_1 = 0
Module: Mathlib.RingTheory.Henselian

Goal state before:
pp: case succ.refine_1
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
n : ℕ
ih : Polynomial.eval (c n) f ∈ I ^ (n + 1)
⊢ ∑ k ∈ Finset.range 2, ((taylor (c n)) f).coeff k * (-(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ)) ^ k ∈
    I ^ (n + 1 + 1)

pp: case succ.refine_2
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
n : ℕ
ih : Polynomial.eval (c n) f ∈ I ^ (n + 1)
⊢ ∑ k ∈ Finset.Ico 2 (((taylor (c n)) f).natDegree + 2),
      ((taylor (c n)) f).coeff k * (-(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ)) ^ k ∈
    I ^ (n + 1 + 1)

Tactic:
·

Goal state after:
case succ.refine_1
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
n : ℕ
ih : Polynomial.eval (c n) f ∈ I ^ (n + 1)
⊢ ∑ k ∈ Finset.range 2, ((taylor (c n)) f).coeff k * (-(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ)) ^ k ∈
    I ^ (n + 1 + 1)
Module: Mathlib.RingTheory.Ideal.HasGoingUp

Goal state before:
pp: case refine_1
R : Type u_3
S : Type u_4
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
⊢ HasGoingUp R S → SpecializingMap (PrimeSpectrum.comap (algebraMap R S))

pp: case refine_2
R : Type u_3
S : Type u_4
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
h : SpecializingMap (PrimeSpectrum.comap (algebraMap R S))
q : Ideal R
hq : q.IsPrime
P : Ideal S
hP : P.IsPrime
hlt : Ideal.under R P < q
⊢ ∃ Q, P ≤ Q ∧ Q.IsPrime ∧ Q.LiesOver q

Tactic:
·

Goal state after:
case refine_1
R : Type u_3
S : Type u_4
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
⊢ HasGoingUp R S → SpecializingMap (PrimeSpectrum.comap (algebraMap R S))
Module: Mathlib.RingTheory.HopfAlgebra.Quotient

Goal state before:
pp: case refine_2
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : HopfAlgebra R A
inst✝ : HopfAlgebraStruct R B
f : A →ₐc[R] B
hf : Function.Surjective ⇑f
hS : antipode R ∘ₗ f.toLinearMap = f.toLinearMap ∘ₗ antipode R
⊢ (toConv LinearMap.id * toConv (antipode R)).ofConv ∘ₗ f.toLinearMap = ofConv 1 ∘ₗ f.toLinearMap

Tactic:
·

Goal state after:
case refine_2
R : Type u_1
A : Type u_2
B : Type u_3
inst✝⁴ : CommSemiring R
inst✝³ : Semiring A
inst✝² : Semiring B
inst✝¹ : HopfAlgebra R A
inst✝ : HopfAlgebraStruct R B
f : A →ₐc[R] B
hf : Function.Surjective ⇑f
hS : antipode R ∘ₗ f.toLinearMap = f.toLinearMap ∘ₗ antipode R
⊢ (toConv LinearMap.id * toConv (antipode R)).ofConv ∘ₗ f.toLinearMap = ofConv 1 ∘ₗ f.toLinearMap
Module: Mathlib.RingTheory.Ideal.HasGoingUp

Goal state before:
pp: case refine_2
R : Type u_3
S : Type u_4
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
h : SpecializingMap (PrimeSpectrum.comap (algebraMap R S))
q : Ideal R
hq : q.IsPrime
P : Ideal S
hP : P.IsPrime
hlt : Ideal.under R P < q
⊢ ∃ Q, P ≤ Q ∧ Q.IsPrime ∧ Q.LiesOver q

Tactic:
·

Goal state after:
case refine_2
R : Type u_3
S : Type u_4
inst✝² : CommRing R
inst✝¹ : CommRing S
inst✝ : Algebra R S
h : SpecializingMap (PrimeSpectrum.comap (algebraMap R S))
q : Ideal R
hq : q.IsPrime
P : Ideal S
hP : P.IsPrime
hlt : Ideal.under R P < q
⊢ ∃ Q, P ≤ Q ∧ Q.IsPrime ∧ Q.LiesOver q
Module: Mathlib.RingTheory.Henselian

Goal state before:
pp: case succ.refine_2
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
n : ℕ
ih : Polynomial.eval (c n) f ∈ I ^ (n + 1)
⊢ ∑ k ∈ Finset.Ico 2 (((taylor (c n)) f).natDegree + 2),
      ((taylor (c n)) f).coeff k * (-(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ)) ^ k ∈
    I ^ (n + 1 + 1)

Tactic:
·

Goal state after:
case succ.refine_2
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
n : ℕ
ih : Polynomial.eval (c n) f ∈ I ^ (n + 1)
⊢ ∑ k ∈ Finset.Ico 2 (((taylor (c n)) f).natDegree + 2),
      ((taylor (c n)) f).coeff k * (-(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ)) ^ k ∈
    I ^ (n + 1 + 1)
Module: Mathlib.RingTheory.Ideal.HasGoingUp

Goal state before:
pp: case hf
R : Type u_3
S : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : HasGoingUp R S
inst✝ : HasGoingUp S T
⊢ SpecializingMap (PrimeSpectrum.comap (algebraMap S T))

pp: case hg
R : Type u_3
S : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : HasGoingUp R S
inst✝ : HasGoingUp S T
⊢ SpecializingMap (PrimeSpectrum.comap (algebraMap R S))

Tactic:
·

Goal state after:
case hf
R : Type u_3
S : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : HasGoingUp R S
inst✝ : HasGoingUp S T
⊢ SpecializingMap (PrimeSpectrum.comap (algebraMap S T))
Module: Mathlib.RingTheory.Ideal.HasGoingUp

Goal state before:
pp: case hg
R : Type u_3
S : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : HasGoingUp R S
inst✝ : HasGoingUp S T
⊢ SpecializingMap (PrimeSpectrum.comap (algebraMap R S))

Tactic:
·

Goal state after:
case hg
R : Type u_3
S : Type u_4
inst✝⁸ : CommRing R
inst✝⁷ : CommRing S
inst✝⁶ : Algebra R S
T : Type u_5
inst✝⁵ : CommRing T
inst✝⁴ : Algebra R T
inst✝³ : Algebra S T
inst✝² : IsScalarTower R S T
inst✝¹ : HasGoingUp R S
inst✝ : HasGoingUp S T
⊢ SpecializingMap (PrimeSpectrum.comap (algebraMap R S))
Module: Mathlib.RingTheory.Henselian

Goal state before:
pp: R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
⊢ ∃ a, f.IsRoot a ∧ a - a₀ ∈ I

Tactic:
have hfcI : ∀ n, f.eval (c n) ∈ I ^ (n + 1) := by
        intro n
        induction n with
        | zero => simpa only [Nat.rec_zero, zero_add, pow_one] using! h₁
        | succ n ih => ?_
        rw [← taylor_eval_sub (c n), hc, sub_eq_add_neg, sub_eq_add_neg,
          add_neg_cancel_comm]
        rw [eval_eq_sum, sum_over_range' _ _ _ (lt_add_of_pos_right _ zero_lt_two), ←
          Finset.sum_range_add_sum_Ico _ (Nat.le_add_left _ _)]
        swap
        · intro i
          rw [zero_mul]
        refine Ideal.add_mem _ ?_ ?_
        · rw [← one_add_one_eq_two, Finset.sum_range_succ, Finset.range_one, Finset.sum_singleton,
            taylor_coeff_zero, taylor_coeff_one, pow_zero, pow_one, mul_one, mul_neg,
            mul_left_comm, Ring.mul_inverse_cancel _ (hf'c n), mul_one, add_neg_cancel]
          exact Ideal.zero_mem _
        · refine Submodule.sum_mem _ ?_
          simp only [Finset.mem_Ico]
          rintro i ⟨h2i, _⟩
          have aux : n + 2 ≤ i * (n + 1) := by trans 2 * (n + 1) <;> nlinarith only [h2i]
          refine Ideal.mul_mem_left _ _ (Ideal.pow_le_pow_right aux ?_)
          rw [pow_mul']
          exact Ideal.pow_mem_pow ((Ideal.neg_mem_iff _).2 <| Ideal.mul_mem_right _ _ ih) _

Goal state after:
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
hfcI : ∀ (n : ℕ), Polynomial.eval (c n) f ∈ I ^ (n + 1)
⊢ ∃ a, f.IsRoot a ∧ a - a₀ ∈ I
Module: Mathlib.RingTheory.Henselian

Goal state before:
pp: case refine_1
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
hfcI : ∀ (n : ℕ), Polynomial.eval (c n) f ∈ I ^ (n + 1)
aux : ∀ (m n : ℕ), m ≤ n → c m ≡ c n [SMOD I ^ m • ⊤]
a : R
ha : ∀ (n : ℕ), c n ≡ a [SMOD I ^ n • ⊤]
⊢ f.IsRoot a

pp: case refine_2
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
hfcI : ∀ (n : ℕ), Polynomial.eval (c n) f ∈ I ^ (n + 1)
aux : ∀ (m n : ℕ), m ≤ n → c m ≡ c n [SMOD I ^ m • ⊤]
a : R
ha : ∀ (n : ℕ), c n ≡ a [SMOD I ^ n • ⊤]
⊢ a - a₀ ∈ I

Tactic:
·

Goal state after:
case refine_1
R : Type u_1
inst✝¹ : CommRing R
I : Ideal R
inst✝ : IsAdicComplete I R
f : R[X]
x✝ : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ I
h₂ : IsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (derivative f)))
f' : R[X] := derivative f
c : ℕ → R := fun n ↦ Nat.recOn n a₀ fun x b ↦ b - Polynomial.eval b f * (Polynomial.eval b f')⁻¹ʳ
hc : ∀ (n : ℕ), c (n + 1) = c n - Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ
hc_mod : ∀ (n : ℕ), c n ≡ a₀ [SMOD I]
hf'c : ∀ (n : ℕ), IsUnit (Polynomial.eval (c n) f')
hfcI : ∀ (n : ℕ), Polynomial.eval (c n) f ∈ I ^ (n + 1)
aux : ∀ (m n : ℕ), m ≤ n → c m ≡ c n [SMOD I ^ m • ⊤]
a : R
ha : ∀ (n : ℕ), c n ≡ a [SMOD I ^ n • ⊤]
⊢ f.IsRoot a