mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Jacobian R
P Q : Fin 3 → R
hPz : P z = 0
⊢ W'.addX P Q = (P x * Q z) ^ 2 * Q x

Tactic:
rw [addX, hPz]
  ring1

Goal state after:
Module: Mathlib.Topology.Algebra.WithZeroTopology

Goal state before:
pp: Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
s : Set Γ₀
⊢ IsOpen[topologicalSpace] s ↔ 0 ∉ s ∨ ∃ γ, γ ≠ 0 ∧ Iio γ ⊆ s

Tactic:
rw [isOpen_iff_mem_nhds, ← and_forall_ne (0 : Γ₀)]

Goal state after:
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
s : Set Γ₀
⊢ ((0 ∈ s → s ∈ 𝓝 0) ∧ ∀ (b : Γ₀), b ≠ 0 → b ∈ s → s ∈ 𝓝 b) ↔ 0 ∉ s ∨ ∃ γ, γ ≠ 0 ∧ Iio γ ⊆ s
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: R K : Type u
inst✝² : Ring R
inst✝¹ : DivisionRing K
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
this : UniformSpace R := IsTopologicalAddGroup.rightUniformSpace R
s : Set R
γ : (ofClass v).ValueGroup₀ˣ
⊢ True ∧ ↑(v.ltAddSubgroup ((Units.map ↑embedding) γ)) ⊆ s ↔ {x | v x < embedding ↑γ} ⊆ s

Tactic:
true_and

Goal state after:
R K : Type u
inst✝² : Ring R
inst✝¹ : DivisionRing K
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
this : UniformSpace R := IsTopologicalAddGroup.rightUniformSpace R
s : Set R
γ : (ofClass v).ValueGroup₀ˣ
⊢ ↑(v.ltAddSubgroup ((Units.map ↑embedding) γ)) ⊆ s ↔ {x | v x < embedding ↑γ} ⊆ s
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsTopologicalAddGroup R
H : ∀ {s : Set R}, s ∈ 𝓝 0 ↔ ∃ γ, {z | v.restrict z < ↑γ} ⊆ s
s : Set R
x : R
⊢ -x +ᵥ s ∈ 𝓝 (-x +ᵥ x) ↔ ∃ γ, {z | v.restrict (z - x) < ↑γ} ⊆ s

Tactic:
simp only [vadd_eq_add, neg_add_cancel, H, subset_vadd_set_iff, neg_neg]

Goal state after:
R : Type u_1
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
inst✝² : TopologicalSpace R
v : Valuation R Γ₀
inst✝¹ : v.Compatible
inst✝ : IsTopologicalAddGroup R
H : ∀ {s : Set R}, s ∈ 𝓝 0 ↔ ∃ γ, {z | v.restrict z < ↑γ} ⊆ s
s : Set R
x : R
⊢ (∃ γ, x +ᵥ {z | v.restrict z < ↑γ} ⊆ s) ↔ ∃ γ, {z | v.restrict (z - x) < ↑γ} ⊆ s
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
x : ↥(posSubmonoid R)
⊢ (valuation R) ↑x ≠ 0

Tactic:
by
  rw [ne_eq, valuation_eq_zero_iff]
  exact x.prop

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

Goal state before:
pp: R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
a : R
ha : a ≤ᵥ 1
n m : ℕ
hnm : n ≤ m
⊢ a ^ m ≤ᵥ a ^ n

Tactic:
obtain ⟨m, rfl⟩ := exists_add_of_le hnm
  simpa [pow_add] using mul_vle_mul_right (pow_vle_pow ha m) _

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

Goal state before:
pp: R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
a : R
ha : 1 ≤ᵥ a
n m : ℕ
hnm : n ≤ m
⊢ a ^ n ≤ᵥ a ^ m

Tactic:
obtain ⟨m, rfl⟩ := exists_add_of_le hnm
  simpa [pow_add] using mul_vle_mul_right (pow_vle_pow ha m) _

Goal state after:
Module: Mathlib.Topology.Algebra.GroupCompletion

Goal state before:
pp: α : Type u_3
β : Type u_4
inst✝⁵ : UniformSpace α
inst✝⁴ : AddGroup α
inst✝³ : IsUniformAddGroup α
inst✝² : UniformSpace β
inst✝¹ : AddGroup β
inst✝ : IsUniformAddGroup β
⊢ completion 0 ⋯ = 0

Tactic:
by
  ext x
  refine Completion.induction_on x ?_ ?_
  · apply isClosed_eq (AddMonoidHom.continuous_completion (0 : α →+ β) continuous_const)
    exact continuous_const
  · simp [(0 : α →+ β).completion_coe continuous_const, coe_zero]

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

Goal state before:
pp: case mk
R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
x✝ : R
y✝ : ↥(posSubmonoid R)
⊢ embedding ((embed (valuation R)) (ValueGroupWithZero.mk x✝ y✝)) = ValueGroupWithZero.mk x✝ y✝

Tactic:
simp [mk_eq_div]

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

Goal state before:
pp: R : Type u_2
Γ : Type u_3
inst✝³ : Ring R
inst✝² : ValuativeRel R
inst✝¹ : LinearOrderedCommGroupWithZero Γ
v : Valuation R Γ
inst✝ : v.Compatible
γ : ValueGroupWithZero R
x : R
⊢ v.restrict x < (orderMonoidIso v) γ ↔ (valuation R) x < γ

Tactic:
simpa using (valuation_lt_symm_orderMonoidIso v (orderMonoidIso v γ) x).symm

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

Goal state before:
pp: R : Type r
inst✝ : CommRing R
W' : Jacobian R
P Q : Fin 3 → R
hP : W'.Equation P
hPz : P z = 0
⊢ W'.addXYZ P Q = (P x * Q z) • Q

Tactic:
by
  rw [addXYZ, addX_of_Z_eq_zero_left hPz, addY_of_Z_eq_zero_left hP hPz, addZ_of_Z_eq_zero_left hPz,
    smul_fin3]

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

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P : Fin 3 → A
⊢ (W'⁄B).negY (⇑↑f ∘ P) = ((W'⁄A).map ↑f).negY (⇑↑f ∘ P)

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P : Fin 3 → A
⊢ (W'⁄B).negY (⇑↑f ∘ P) = (W'⁄B).negY (⇑↑f ∘ P)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P : Fin 3 → A
⊢ (W'⁄B).dblU (⇑↑f ∘ P) = ((W'⁄A).map ↑f).dblU (⇑↑f ∘ P)

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P : Fin 3 → A
⊢ (W'⁄B).dblU (⇑↑f ∘ P) = (W'⁄B).dblU (⇑↑f ∘ P)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P : Fin 3 → A
⊢ (W'⁄B).dblZ (⇑↑f ∘ P) = ((W'⁄A).map ↑f).dblZ (⇑↑f ∘ P)

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P : Fin 3 → A
⊢ (W'⁄B).dblZ (⇑↑f ∘ P) = (W'⁄B).dblZ (⇑↑f ∘ P)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P : Fin 3 → A
⊢ (W'⁄B).dblX (⇑↑f ∘ P) = ((W'⁄A).map ↑f).dblX (⇑↑f ∘ P)

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P : Fin 3 → A
⊢ (W'⁄B).dblX (⇑↑f ∘ P) = (W'⁄B).dblX (⇑↑f ∘ P)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P : Fin 3 → A
⊢ (W'⁄B).negDblY (⇑↑f ∘ P) = ((W'⁄A).map ↑f).negDblY (⇑↑f ∘ P)

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P : Fin 3 → A
⊢ (W'⁄B).negDblY (⇑↑f ∘ P) = (W'⁄B).negDblY (⇑↑f ∘ P)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P : Fin 3 → A
⊢ (W'⁄B).dblY (⇑↑f ∘ P) = ((W'⁄A).map ↑f).dblY (⇑↑f ∘ P)

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P : Fin 3 → A
⊢ (W'⁄B).dblY (⇑↑f ∘ P) = (W'⁄B).dblY (⇑↑f ∘ P)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P : Fin 3 → A
⊢ (W'⁄B).dblXYZ (⇑↑f ∘ P) = ((W'⁄A).map ↑f).dblXYZ (⇑↑f ∘ P)

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P : Fin 3 → A
⊢ (W'⁄B).dblXYZ (⇑↑f ∘ P) = (W'⁄B).dblXYZ (⇑↑f ∘ P)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P Q : Fin 3 → A
⊢ (W'⁄B).addX (⇑↑f ∘ P) (⇑↑f ∘ Q) = ((W'⁄A).map ↑f).addX (⇑↑f ∘ P) (⇑↑f ∘ Q)

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P Q : Fin 3 → A
⊢ (W'⁄B).addX (⇑↑f ∘ P) (⇑↑f ∘ Q) = (W'⁄B).addX (⇑↑f ∘ P) (⇑↑f ∘ Q)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P Q : Fin 3 → A
⊢ (W'⁄B).negAddY (⇑↑f ∘ P) (⇑↑f ∘ Q) = ((W'⁄A).map ↑f).negAddY (⇑↑f ∘ P) (⇑↑f ∘ Q)

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P Q : Fin 3 → A
⊢ (W'⁄B).negAddY (⇑↑f ∘ P) (⇑↑f ∘ Q) = (W'⁄B).negAddY (⇑↑f ∘ P) (⇑↑f ∘ Q)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P Q : Fin 3 → A
⊢ (W'⁄B).addY (⇑↑f ∘ P) (⇑↑f ∘ Q) = ((W'⁄A).map ↑f).addY (⇑↑f ∘ P) (⇑↑f ∘ Q)

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P Q : Fin 3 → A
⊢ (W'⁄B).addY (⇑↑f ∘ P) (⇑↑f ∘ Q) = (W'⁄B).addY (⇑↑f ∘ P) (⇑↑f ∘ Q)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P Q : Fin 3 → A
⊢ (W'⁄B).addXYZ (⇑↑f ∘ P) (⇑↑f ∘ Q) = ((W'⁄A).map ↑f).addXYZ (⇑↑f ∘ P) (⇑↑f ∘ Q)

Tactic:
map_baseChange

Goal state after:
R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P Q : Fin 3 → A
⊢ (W'⁄B).addXYZ (⇑↑f ∘ P) (⇑↑f ∘ Q) = (W'⁄B).addXYZ (⇑↑f ∘ P) (⇑↑f ∘ Q)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: R : Type r
S : Type s
A : Type u
B : Type v
inst✝¹⁰ : CommRing R
inst✝⁹ : CommRing S
inst✝⁸ : CommRing A
inst✝⁷ : CommRing B
W' : Jacobian R
inst✝⁶ : Algebra R S
inst✝⁵ : Algebra R A
inst✝⁴ : Algebra S A
inst✝³ : IsScalarTower R S A
inst✝² : Algebra R B
inst✝¹ : Algebra S B
inst✝ : IsScalarTower R S B
f : A →ₐ[S] B
P Q : Fin 3 → A
⊢ (W'⁄B).addXYZ (⇑f ∘ P) (⇑f ∘ Q) = ⇑f ∘ (W'⁄A).addXYZ P Q

Tactic:
rw [← RingHom.coe_coe, ← map_addXYZ, map_baseChange]

Goal state after:
Module: Mathlib.Order.Interval.Finset.DenselyOrdered

Goal state before:
pp: X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
H : DenselyOrdered X
⊢ ¬∃ x y, x < y

Tactic:
rintro ⟨a, b, hab⟩

Goal state after:
X : Type u_1
inst✝¹ : LinearOrder X
inst✝ : LocallyFiniteOrder X
H : DenselyOrdered X
a b : X
hab : a < b
⊢ False
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Reduction

Goal state before:
pp: case h.a₂
R : Type u_1
inst✝² : CommRing R
K : Type u_2
inst✝¹ : Field K
inst✝ : Algebra R K
W : WeierstrassCurve K
h₁ : ∃ r₁, (algebraMap R K) r₁ = W.a₁
h₂ : ∃ r₂, (algebraMap R K) r₂ = W.a₂
h₃ : ∃ r₃, (algebraMap R K) r₃ = W.a₃
h₄ : ∃ r₄, (algebraMap R K) r₄ = W.a₄
h₆ : ∃ r₆, (algebraMap R K) r₆ = W.a₆
⊢ W.a₂ = (algebraMap R K) h₂.choose

Tactic:
apply h₂.choose_spec.symm

Goal state after:
Module: Mathlib.GroupTheory.ArchimedeanDensely

Goal state before:
pp: case refine_2
G : Type u_2
inst✝² : LinearOrderedCommGroupWithZero G
inst✝¹ : Nontrivial Gˣ
inst✝ : MulArchimedean G
f : Gˣ ≃*o Multiplicative ℤ
⊢ G ≃*o WithZero (Multiplicative ℤ)

Tactic:
refine ⟨withZeroUnitsEquiv.symm.trans (MulEquiv.withZero f), ?_⟩

Goal state after:
case refine_2
G : Type u_2
inst✝² : LinearOrderedCommGroupWithZero G
inst✝¹ : Nontrivial Gˣ
inst✝ : MulArchimedean G
f : Gˣ ≃*o Multiplicative ℤ
⊢ ∀ {a b : G},
    (withZeroUnitsEquiv.symm.trans (MulEquiv.withZero ↑f)).toFun a ≤
        (withZeroUnitsEquiv.symm.trans (MulEquiv.withZero ↑f)).toFun b ↔
      a ≤ b
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: case refine_1
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
⊢ Function.Injective ⇑valueGroup₀_hom_extensionValuation

Tactic:
intro a b hab

Goal state after:
case refine_1
K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
hab : valueGroup₀_hom_extensionValuation a = valueGroup₀_hom_extensionValuation b
⊢ a = b
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : Field K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
hv : Valued K Γ₀
a b : (ofClass v).ValueGroup₀
hab : extensionValuation.restrict ↑⋯.choose = extensionValuation.restrict ↑⋯.choose
x : K := ⋯.choose
hx_def : x = ⋯.choose
y : K := ⋯.choose
hy_def : y = ⋯.choose
hx : v x = embedding a
hy : v y = embedding b
this : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)
ha0 : a = 0
⊢ failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)

Tactic:
simpa [ha0, extension_eq_zero_iff, map_eq_zero] using hx

Goal state after:
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
x : R
hx : x ≠ 0
⊢ -↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors) ≤ 0

Tactic:
Right.neg_nonpos_iff

Goal state after:
case inr
R : Type u_1
inst✝¹ : CommRing R
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
x : R
hx : x ≠ 0
⊢ 0 ≤ ↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x})).factors)
Module: Mathlib.RingTheory.DedekindDomain.AdicValuation

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
a b : R
hb : b ≠ 0
h : b ∈ v.asIdeal → a ∉ v.asIdeal
hv : (valuation K v) (↑a / ↑b) ≤ 1
⊢ b ∉ v.asIdeal

Tactic:
contrapose! hv

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
inst✝³ : IsDedekindDomain R
K : Type u_2
inst✝² : Field K
inst✝¹ : Algebra R K
inst✝ : IsFractionRing R K
v : HeightOneSpectrum R
a b : R
hb : b ≠ 0
h : b ∈ v.asIdeal → a ∉ v.asIdeal
hv : b ∈ v.asIdeal
⊢ 1 < (valuation K v) (↑a / ↑b)
Module: Mathlib.RingTheory.MvPowerSeries.Basic

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
f : MvPowerSeries σ R
m : ℕ
hf : constantCoeff f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : k ∈ (range n).finsuppAntidiag d
⊢ ∏ i ∈ range n, (coeff (k i)) f = 0

Tactic:
rw [mem_finsuppAntidiag] at hk

Goal state after:
σ : Type u_1
R : Type u_3
inst✝ : CommSemiring R
f : MvPowerSeries σ R
m : ℕ
hf : constantCoeff f ^ m = 0
d : σ →₀ ℕ
n : ℕ
hn : m + degree d ≤ n
k : ℕ →₀ σ →₀ ℕ
hk : (range n).sum ⇑k = d ∧ k.support ⊆ range n
⊢ ∏ i ∈ range n, (coeff (k i)) f = 0
Module: Mathlib.RingTheory.PowerSeries.Basic

Goal state before:
pp: R : Type u_1
inst✝ : Semiring R
φ : R[X]
⊢ ∑ i ∈ Finset.range (φ.natDegree + 1), PowerSeries.C (φ.coeff i) * PowerSeries.X ^ i =
    coeToPowerSeries.ringHom (∑ i ∈ Finset.range (φ.natDegree + 1), C (φ.coeff i) * X ^ i)

Tactic:
map_sum

Goal state after:
R : Type u_1
inst✝ : Semiring R
φ : R[X]
⊢ ∑ i ∈ Finset.range (φ.natDegree + 1), PowerSeries.C (φ.coeff i) * PowerSeries.X ^ i =
    ∑ x ∈ Finset.range (φ.natDegree + 1), coeToPowerSeries.ringHom (C (φ.coeff x) * X ^ x)
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f : MvPowerSeries σ R
h : ↑(weightedOrder w f).toNat = weightedOrder w f
⊢ ∃ d, (coeff d) f ≠ 0 ∧ (weight w) d = Nat.find ⋯

Tactic:
generalize_proofs h1

Goal state after:
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f : MvPowerSeries σ R
h : ↑(weightedOrder w f).toNat = weightedOrder w f
h1 : ∃ n d, (coeff d) f ≠ 0 ∧ (weight w) d = n
⊢ ∃ d, (coeff d) f ≠ 0 ∧ (weight w) d = Nat.find h1
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case insert
σ : Type u_1
w : σ → ℕ
R : Type u_4
ι : Type u_5
inst✝ : CommRing R
d : σ →₀ ℕ
f : MvPowerSeries σ R
g : ι → MvPowerSeries σ R
a : ι
s : Finset ι
ha : a ∉ s
ih : (∀ i ∈ s, ↑((weight w) d) < weightedOrder w (g i)) → (coeff d) (f * ∏ i ∈ s, (1 - g i)) = (coeff d) f
h : ↑((weight w) d) < weightedOrder w (g a) ∧ ∀ a ∈ s, ↑((weight w) d) < weightedOrder w (g a)
⊢ (coeff d) (f * ∏ x ∈ s, (1 - g x)) = (coeff d) f

Tactic:
ih h.2

Goal state after:
case insert
σ : Type u_1
w : σ → ℕ
R : Type u_4
ι : Type u_5
inst✝ : CommRing R
d : σ →₀ ℕ
f : MvPowerSeries σ R
g : ι → MvPowerSeries σ R
a : ι
s : Finset ι
ha : a ∉ s
ih : (∀ i ∈ s, ↑((weight w) d) < weightedOrder w (g i)) → (coeff d) (f * ∏ i ∈ s, (1 - g i)) = (coeff d) f
h : ↑((weight w) d) < weightedOrder w (g a) ∧ ∀ a ∈ s, ↑((weight w) d) < weightedOrder w (g a)
⊢ (coeff d) f = (coeff d) f
Module: Mathlib.RingTheory.MvPowerSeries.Order

Goal state before:
pp: case pos
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hd : (weight w) x.1 + (weight w) x.2 = p + q
hx : x.1 + x.2 = d
⊢ (coeff x.1) f * (coeff x.2) g =
    (if (weight w) x.1 = p then (coeff x.1) f else 0) * if (weight w) x.2 = q then (coeff x.2) g else 0

Tactic:
rcases trichotomy_of_add_eq_add hd with h | h | h

Goal state after:
case pos.inl
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hd : (weight w) x.1 + (weight w) x.2 = p + q
hx : x.1 + x.2 = d
h : (weight w) x.1 = p ∧ (weight w) x.2 = q
⊢ (coeff x.1) f * (coeff x.2) g =
    (if (weight w) x.1 = p then (coeff x.1) f else 0) * if (weight w) x.2 = q then (coeff x.2) g else 0

case pos.inr.inl
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hd : (weight w) x.1 + (weight w) x.2 = p + q
hx : x.1 + x.2 = d
h : (weight w) x.1 < p
⊢ (coeff x.1) f * (coeff x.2) g =
    (if (weight w) x.1 = p then (coeff x.1) f else 0) * if (weight w) x.2 = q then (coeff x.2) g else 0

case pos.inr.inr
σ : Type u_1
R : Type u_2
inst✝ : Semiring R
w : σ → ℕ
f g : MvPowerSeries σ R
p q : ℕ
hf : ↑p ≤ weightedOrder w f
hg : ↑q ≤ weightedOrder w g
d : σ →₀ ℕ
x : (σ →₀ ℕ) × (σ →₀ ℕ)
hd : (weight w) x.1 + (weight w) x.2 = p + q
hx : x.1 + x.2 = d
h : (weight w) x.2 < q
⊢ (coeff x.1) f * (coeff x.2) g =
    (if (weight w) x.1 = p then (coeff x.1) f else 0) * if (weight w) x.2 = q then (coeff x.2) g else 0
Module: Mathlib.RingTheory.PowerSeries.Trunc

Goal state before:
pp: case pos
R : Type u_1
inst✝ : Semiring R
n : ℕ
a : R
m : ℕ
H : ¬m < n + 1
h✝ : m = 0
⊢ 0 = a

Tactic:
try simp_all

Goal state after: