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

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
hR : HenselianLocalRing R
f : R[X]
hf : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ maximalIdeal R
h₂ : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a₀ (derivative f)) = 0
⊢ ¬IsUnit ((Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a₀ (derivative f)))

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
hR : HenselianLocalRing R
f : R[X]
hf : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ maximalIdeal R
h₂ : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a₀ (derivative f)) = 0
⊢ ¬IsUnit ((Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a₀ (derivative f)))
Module: Mathlib.RingTheory.Henselian

Goal state before:
pp: R : Type u_1
inst✝ : CommRing R
hR : HenselianLocalRing R
f : R[X]
hf : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ maximalIdeal R
h₂ : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a₀ (derivative f)) = 0
⊢ ¬IsUnit 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝ : CommRing R
hR : HenselianLocalRing R
f : R[X]
hf : f.Monic
a₀ : R
h₁ : Polynomial.eval a₀ f ∈ maximalIdeal R
h₂ : (Ideal.Quotient.mk (maximalIdeal R)) (Polynomial.eval a₀ (derivative f)) = 0
⊢ ¬IsUnit 0
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 1

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 1
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
this : x.order = 0
⊢ 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
this : x.order = 0
⊢ IsAddUnit x.order
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
this : x.order = 0
⊢ IsAddUnit 0

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
this : x.order = 0
⊢ IsAddUnit 0
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
u i : R⟦Γ⟧
ui : u * i = 1
iu : i * u = 1
⊢ 1 = 1

pp: case mp.hc
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
u i : R⟦Γ⟧
ui : u * i = 1
iu : i * u = 1
⊢ u.order + i.order = 0

Tactic:
]

Goal state after:
case mp.hc
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
u i : R⟦Γ⟧
ui : u * i = 1
iu : i * u = 1
⊢ u.order + i.order = 0
Module: Mathlib.RingTheory.HahnSeries.Summable

Goal state before:
pp: case mp.hc
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
u i : R⟦Γ⟧
ui : u * i = 1
iu : i * u = 1
⊢ 0 = 0

Tactic:
]

Goal state after:
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⟧
u i : R
ui : u * i = 1
iu : i * u = 1
hx : u = 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⟧
u i : R
ui : u * i = 1
iu : i * u = 1
hx : u = x.leadingCoeff
⊢ IsUnit x
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
n : ℕ
ih : c n ≡ a₀ [SMOD I]
⊢ c n + -(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ) ≡ a₀ + 0 [SMOD I]

Tactic:
]

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
n : ℕ
ih : c n ≡ a₀ [SMOD I]
⊢ c n + -(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ) ≡ a₀ + 0 [SMOD I]
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⟧
u i : R
ui : u * i = 1
iu : i * x.leadingCoeff = 1
hx : u = 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⟧
u i : R
ui : u * i = 1
iu : i * x.leadingCoeff = 1
hx : u = x.leadingCoeff
⊢ IsUnit x
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⟧
u i : R
ui : u * i = 1
iu : i * x.leadingCoeff = 1
hx : u = x.leadingCoeff
h : (single (-x.order)) i * x * (SummableFamily.powers (1 - (single (-x.order)) i * x)).hsum = 1
⊢ 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⟧
u i : R
ui : u * i = 1
iu : i * x.leadingCoeff = 1
hx : u = x.leadingCoeff
h : (single (-x.order)) i * x * (SummableFamily.powers (1 - (single (-x.order)) i * x)).hsum = 1
⊢ IsUnit x
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
n : ℕ
ih : c n ≡ a₀ [SMOD I]
⊢ Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ ∈ I

Tactic:
]

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
n : ℕ
ih : c n ≡ a₀ [SMOD I]
⊢ Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ ∈ I
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 ≡ 0 [SMOD 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
n : ℕ
ih : c n ≡ a₀ [SMOD I]
⊢ Polynomial.eval (c n) f ≡ 0 [SMOD I]

Tactic:
]

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 ≡ 0 [SMOD 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
n : ℕ
ih : c n ≡ a₀ [SMOD I]
⊢ Polynomial.eval (c n) f ≡ 0 [SMOD I]
Module: Mathlib.RingTheory.HahnSeries.Summable

Goal state before:
pp: Γ : Type u_1
R : Type u_3
inst✝³ : AddCommGroup Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedAddMonoid Γ
inst✝ : Field R
a b : Γ
r s : R
⊢ (single (a + -b)) (r * s⁻¹) = (single (a + -b)) (r * s⁻¹)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.HahnSeries.Summable

Goal state before:
pp: Γ : Type u_1
Γ' : Type u_2
R : Type u_3
V : Type u_4
α : Type u_5
β : Type u_6
inst✝³ : AddCommGroup Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedAddMonoid Γ
inst✝ : Field R
x : R⟦Γ⟧
x0 : x ≠ 0
h :
  (single (-x.order)) x.leadingCoeff⁻¹ * x *
      (SummableFamily.powers (1 - (single (-x.order)) x.leadingCoeff⁻¹ * x)).hsum =
    1
⊢ x * x⁻¹ = 1

Tactic:
]

Goal state after:
Γ : Type u_1
Γ' : Type u_2
R : Type u_3
V : Type u_4
α : Type u_5
β : Type u_6
inst✝³ : AddCommGroup Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedAddMonoid Γ
inst✝ : Field R
x : R⟦Γ⟧
x0 : x ≠ 0
h :
  (single (-x.order)) x.leadingCoeff⁻¹ * x *
      (SummableFamily.powers (1 - (single (-x.order)) x.leadingCoeff⁻¹ * x)).hsum =
    1
⊢ x * x⁻¹ = 1
Module: Mathlib.RingTheory.HahnSeries.Summable

Goal state before:
pp: Γ : Type u_1
Γ' : Type u_2
R : Type u_3
V : Type u_4
α : Type u_5
β : Type u_6
inst✝³ : AddCommGroup Γ
inst✝² : LinearOrder Γ
inst✝¹ : IsOrderedAddMonoid Γ
inst✝ : Field R
x : R⟦Γ⟧
x0 : x ≠ 0
h :
  (single (-x.order)) x.leadingCoeff⁻¹ * x *
      (SummableFamily.powers (1 - (single (-x.order)) x.leadingCoeff⁻¹ * x)).hsum =
    1
⊢ 1 = 1

Tactic:
]

Goal state after:
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 : under R P ≤ q
h : under R 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 : under R P ≤ q
h : under R P ≠ q
⊢ ∃ Q, P ≤ Q ∧ Q.IsPrime ∧ Q.LiesOver q
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

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.HopfAlgebra.MonoidAlgebra

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : HopfAlgebra R A
a : A
⊢ AddMonoidAlgebra.single (-0) ((antipode R) a) = C ((antipode R) a)

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : HopfAlgebra R A
a : A
⊢ AddMonoidAlgebra.single (-0) ((antipode R) a) = C ((antipode R) a)
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 (-(Polynomial.eval (c n) f * (Polynomial.eval (c n) f')⁻¹ʳ)) ((taylor (c n)) f) ∈ I ^ (n + 1 + 1)

Tactic:
]

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_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.HopfAlgebra.MonoidAlgebra

Goal state before:
pp: R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : HopfAlgebra R A
n : ℤ
⊢ AddMonoidAlgebra.single (-n) ((antipode R) 1) = AddMonoidAlgebra.single (-n) 1

Tactic:
]

Goal state after:
R : Type u_1
A : Type u_2
inst✝² : CommSemiring R
inst✝¹ : Semiring A
inst✝ : HopfAlgebra R A
n : ℤ
⊢ AddMonoidAlgebra.single (-n) ((antipode R) 1) = AddMonoidAlgebra.single (-n) 1
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)
⊢ ∑ 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)

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

Tactic:
]

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)
⊢ ∑ 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)

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.HopfAlgebra.Quotient

Goal state before:
pp: 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 (f.toLinearMap ∘ₗ antipode R) * toConv ((toConv LinearMap.id).ofConv ∘ₗ f.toLinearMap)).ofConv =
    (toConv (f.toLinearMap ∘ₗ antipode R) * toConv f.toLinearMap).ofConv

Tactic:
]

Goal state after:
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 (f.toLinearMap ∘ₗ antipode R) * toConv ((toConv LinearMap.id).ofConv ∘ₗ f.toLinearMap)).ofConv =
    (toConv (f.toLinearMap ∘ₗ antipode R) * toConv f.toLinearMap).ofConv
Module: Mathlib.RingTheory.HopfAlgebra.Quotient

Goal state before:
pp: 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 (f.toLinearMap ∘ₗ antipode R) * toConv f.toLinearMap).ofConv =
    (toConv ((↑f).toLinearMap ∘ₗ (toConv (antipode R)).ofConv) *
        toConv ((↑f).toLinearMap ∘ₗ (toConv LinearMap.id).ofConv)).ofConv

Tactic:
]

Goal state after:
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 (f.toLinearMap ∘ₗ antipode R) * toConv f.toLinearMap).ofConv =
    (toConv ((↑f).toLinearMap ∘ₗ (toConv (antipode R)).ofConv) *
        toConv ((↑f).toLinearMap ∘ₗ (toConv LinearMap.id).ofConv)).ofConv
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)
i : ℕ
⊢ 0 = 0

Tactic:
]

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

Goal state before:
pp: 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
⊢ ofConv 1 ∘ₗ f.toLinearMap = ofConv 1 ∘ₗ f.toLinearMap

Tactic:
]

Goal state after:
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
h : HasGoingUp R S
P : PrimeSpectrum S
q : PrimeSpectrum R
hq : PrimeSpectrum.comap (algebraMap R S) P ≤ q
⊢ ∃ a', flip (fun x1 x2 ↦ x1 ⤳ x2) a' P ∧ PrimeSpectrum.comap (algebraMap R S) a' = 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
h : HasGoingUp R S
P : PrimeSpectrum S
q : PrimeSpectrum R
hq : PrimeSpectrum.comap (algebraMap R S) P ≤ q
⊢ ∃ a', flip (fun x1 x2 ↦ x1 ⤳ x2) a' P ∧ PrimeSpectrum.comap (algebraMap R S) a' = q
Module: Mathlib.RingTheory.HopfAlgebra.Quotient

Goal state before:
pp: 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 ((toConv LinearMap.id).ofConv ∘ₗ f.toLinearMap) * toConv (f.toLinearMap ∘ₗ antipode R)).ofConv =
    (toConv f.toLinearMap * toConv (f.toLinearMap ∘ₗ antipode R)).ofConv

Tactic:
]

Goal state after:
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 ((toConv LinearMap.id).ofConv ∘ₗ f.toLinearMap) * toConv (f.toLinearMap ∘ₗ antipode R)).ofConv =
    (toConv f.toLinearMap * toConv (f.toLinearMap ∘ₗ antipode R)).ofConv