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

Goal state before:
pp: case h.a₄
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E✝ E'✝ : WeierstrassCurve F
inst✝⁴ : E✝.IsElliptic
inst✝³ : E'✝.IsElliptic
p : ℕ
inst✝² : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝³ : NeZero 2
this✝² : NeZero 4
this✝¹ : NeZero 6
this✝ : Invertible 2 := ⋯
this : Invertible 3 := ⋯
E : WeierstrassCurve F
inst✝¹ : E.IsElliptic
h✝¹ : E.IsShortNF
E' : WeierstrassCurve F
inst✝ : E'.IsElliptic
h✝ : E'.IsShortNF
heq : E.a₄ ^ 3 * E'.a₆ ^ 2 = E'.a₄ ^ 3 * E.a₆ ^ 2
ha₄ : ¬E.a₄ = 0
ha₆ : ¬E.a₆ = 0
ha₄' : E'.a₄ ≠ 0
ha₆' : E'.a₆ ≠ 0
u : F
hu : u ^ 2 = E.a₆ / E'.a₆ / (E.a₄ / E'.a₄)
hu4 : u ^ 4 = E.a₄ / E'.a₄
hu6 : u ^ 6 = E.a₆ / E'.a₆
hu0 : u ≠ 0
⊢ ({ u := Units.mk0 u hu0, r := 0, s := 0, t := 0 } • E).a₄ = E'.a₄

Tactic:
[

Goal state after:
case h.a₄
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E✝ E'✝ : WeierstrassCurve F
inst✝⁴ : E✝.IsElliptic
inst✝³ : E'✝.IsElliptic
p : ℕ
inst✝² : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝³ : NeZero 2
this✝² : NeZero 4
this✝¹ : NeZero 6
this✝ : Invertible 2 := ⋯
this : Invertible 3 := ⋯
E : WeierstrassCurve F
inst✝¹ : E.IsElliptic
h✝¹ : E.IsShortNF
E' : WeierstrassCurve F
inst✝ : E'.IsElliptic
h✝ : E'.IsShortNF
heq : E.a₄ ^ 3 * E'.a₆ ^ 2 = E'.a₄ ^ 3 * E.a₆ ^ 2
ha₄ : ¬E.a₄ = 0
ha₆ : ¬E.a₆ = 0
ha₄' : E'.a₄ ≠ 0
ha₆' : E'.a₆ ≠ 0
u : F
hu : u ^ 2 = E.a₆ / E'.a₆ / (E.a₄ / E'.a₄)
hu4 : u ^ 4 = E.a₄ / E'.a₄
hu6 : u ^ 6 = E.a₆ / E'.a₆
hu0 : u ≠ 0
⊢ ({ u := Units.mk0 u hu0, r := 0, s := 0, t := 0 } • E).a₄ = E'.a₄
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: case h.a₆
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E✝ E'✝ : WeierstrassCurve F
inst✝⁴ : E✝.IsElliptic
inst✝³ : E'✝.IsElliptic
p : ℕ
inst✝² : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝³ : NeZero 2
this✝² : NeZero 4
this✝¹ : NeZero 6
this✝ : Invertible 2 := ⋯
this : Invertible 3 := ⋯
E : WeierstrassCurve F
inst✝¹ : E.IsElliptic
h✝¹ : E.IsShortNF
E' : WeierstrassCurve F
inst✝ : E'.IsElliptic
h✝ : E'.IsShortNF
heq : E.a₄ ^ 3 * E'.a₆ ^ 2 = E'.a₄ ^ 3 * E.a₆ ^ 2
ha₄ : ¬E.a₄ = 0
ha₆ : ¬E.a₆ = 0
ha₄' : E'.a₄ ≠ 0
ha₆' : E'.a₆ ≠ 0
u : F
hu : u ^ 2 = E.a₆ / E'.a₆ / (E.a₄ / E'.a₄)
hu4 : u ^ 4 = E.a₄ / E'.a₄
hu6 : u ^ 6 = E.a₆ / E'.a₆
hu0 : u ≠ 0
⊢ ({ u := Units.mk0 u hu0, r := 0, s := 0, t := 0 } • E).a₆ = E'.a₆

Tactic:
[

Goal state after:
case h.a₆
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E✝ E'✝ : WeierstrassCurve F
inst✝⁴ : E✝.IsElliptic
inst✝³ : E'✝.IsElliptic
p : ℕ
inst✝² : CharP F p
hchar2 : 2 ≠ 0
hchar3 : 3 ≠ 0
this✝³ : NeZero 2
this✝² : NeZero 4
this✝¹ : NeZero 6
this✝ : Invertible 2 := ⋯
this : Invertible 3 := ⋯
E : WeierstrassCurve F
inst✝¹ : E.IsElliptic
h✝¹ : E.IsShortNF
E' : WeierstrassCurve F
inst✝ : E'.IsElliptic
h✝ : E'.IsShortNF
heq : E.a₄ ^ 3 * E'.a₆ ^ 2 = E'.a₄ ^ 3 * E.a₆ ^ 2
ha₄ : ¬E.a₄ = 0
ha₆ : ¬E.a₆ = 0
ha₄' : E'.a₄ ≠ 0
ha₆' : E'.a₆ ≠ 0
u : F
hu : u ^ 2 = E.a₆ / E'.a₆ / (E.a₄ / E'.a₄)
hu4 : u ^ 4 = E.a₄ / E'.a₄
hu6 : u ^ 6 = E.a₆ / E'.a₆
hu0 : u ≠ 0
⊢ ({ u := Units.mk0 u hu0, r := 0, s := 0, t := 0 } • E).a₆ = E'.a₆
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W' : Jacobian R
S : Type s
inst✝ : CommRing S
f : R →+* S
hf : Function.Injective ⇑f
P : Fin 3 → R
⊢ (W'.map f).Nonsingular (⇑f ∘ P) ↔ W'.Nonsingular P

Tactic:
by
  simp only [Nonsingular, W'.map_equation hf, map_polynomialX, map_polynomialY, map_polynomialZ,
    eval_map, ← eval₂_comp, map_ne_zero_iff f hf]

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

Goal state before:
pp: case pos
F : Type u
inst✝ : Field F
W : Jacobian F
P : Fin 3 → F
hP : W.Nonsingular P
hPz : P z = 0
⊢ W.Nonsingular (W.neg P)

Tactic:
simp only [neg_of_Z_eq_zero hP hPz, nonsingular_smul _
        ((isUnit_Y_of_Z_eq_zero hP hPz).div <| isUnit_X_of_Z_eq_zero hP hPz).neg, nonsingular_zero]

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

Goal state before:
pp: case a
R : Type r
inst✝ : CommRing R
W' : Jacobian R
P : Fin 3 → R
hP : W'.Equation P
⊢ W'.addX P (W'.neg P) + -2 * P z ^ 2 * 0 -
      (W'.dblZ P ^ 2 +
        -2 * P z ^ 2 *
          (P y ^ 2 + W'.a₁ * P x * P y * P z + W'.a₃ * P y * P z ^ 3 -
            (P x ^ 3 + W'.a₂ * P x ^ 2 * P z ^ 2 + W'.a₄ * P x * P z ^ 4 + W'.a₆ * P z ^ 6))) =
    0

Tactic:
addX,

Goal state after:
case a
R : Type r
inst✝ : CommRing R
W' : Jacobian R
P : Fin 3 → R
hP : W'.Equation P
⊢ P x * W'.neg P x ^ 2 * P z ^ 2 - 2 * P y * W'.neg P y * P z * W'.neg P z + P x ^ 2 * W'.neg P x * W'.neg P z ^ 2 -
                        W'.a₁ * P x * W'.neg P y * P z ^ 2 * W'.neg P z -
                      W'.a₁ * P y * W'.neg P x * P z * W'.neg P z ^ 2 +
                    2 * W'.a₂ * P x * W'.neg P x * P z ^ 2 * W'.neg P z ^ 2 -
                  W'.a₃ * W'.neg P y * P z ^ 4 * W'.neg P z -
                W'.a₃ * P y * P z * W'.neg P z ^ 4 +
              W'.a₄ * W'.neg P x * P z ^ 4 * W'.neg P z ^ 2 +
            W'.a₄ * P x * P z ^ 2 * W'.neg P z ^ 4 +
          2 * W'.a₆ * P z ^ 4 * W'.neg P z ^ 4 +
        -2 * P z ^ 2 * 0 -
      (W'.dblZ P ^ 2 +
        -2 * P z ^ 2 *
          (P y ^ 2 + W'.a₁ * P x * P y * P z + W'.a₃ * P y * P z ^ 3 -
            (P x ^ 3 + W'.a₂ * P x ^ 2 * P z ^ 2 + W'.a₄ * P x * P z ^ 4 + W'.a₆ * P z ^ 6))) =
    0
Module: Mathlib.AlgebraicGeometry.EllipticCurve.DivisionPolynomial.Degree

Goal state before:
pp: case neg
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
ih : ∀ (n : ℕ), (W.Φ ↑n).natDegree ≤ (↑n).natAbs ^ 2
n✝ : ℕ
⊢ (W.Φ (-↑n✝)).natDegree ≤ (-↑n✝).natAbs ^ 2

Tactic:
[

Goal state after:
case neg
R : Type u
inst✝ : CommRing R
W : WeierstrassCurve R
ih : ∀ (n : ℕ), (W.Φ ↑n).natDegree ≤ (↑n).natAbs ^ 2
n✝ : ℕ
⊢ (W.Φ (-↑n✝)).natDegree ≤ (-↑n✝).natAbs ^ 2
Module: Mathlib.Data.Int.WithZero

Goal state before:
pp: case pos
e : ℝ≥0
he : e ≠ 0
x y : WithZero (Multiplicative ℤ)
hxy : x * y = 0
⊢ (if hx : x * y = 0 then 0 else e ^ toAdd (unzero hx)) =
    (if hx : x = 0 then 0 else e ^ toAdd (unzero hx)) * if hx : y = 0 then 0 else e ^ toAdd (unzero hx)

Tactic:
rcases mul_eq_zero.mp hxy with hx | hy

Goal state after:
case pos.inl
e : ℝ≥0
he : e ≠ 0
x y : WithZero (Multiplicative ℤ)
hxy : x * y = 0
hx : x = 0
⊢ (if hx : x * y = 0 then 0 else e ^ toAdd (unzero hx)) =
    (if hx : x = 0 then 0 else e ^ toAdd (unzero hx)) * if hx : y = 0 then 0 else e ^ toAdd (unzero hx)

case pos.inr
e : ℝ≥0
he : e ≠ 0
x y : WithZero (Multiplicative ℤ)
hxy : x * y = 0
hy : y = 0
⊢ (if hx : x * y = 0 then 0 else e ^ toAdd (unzero hx)) =
    (if hx : x = 0 then 0 else e ^ toAdd (unzero hx)) * if hx : y = 0 then 0 else e ^ toAdd (unzero hx)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

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

Tactic:
by
  linear_combination (norm :=
      (rw [negDblY, dblU_of_Z_eq_zero hPz, dblX_of_Z_eq_zero hP hPz, negY_of_Z_eq_zero hPz]; ring1))
    (8 * P y ^ 2 - 4 * P x ^ 3) * (equation_of_Z_eq_zero hPz).mp hP

Goal state after:
Module: Mathlib.Topology.Algebra.Nonarchimedean.Bases

Goal state before:
pp: A : Type u_1
ι : Type u_2
inst✝¹ : Ring A
inst✝ : Nonempty ι
B : ι → AddSubgroup A
hB : RingSubgroupsBasis B
i : ι
x✝ : TopologicalSpace A := hB.topology
⊢ IsOpen[hB.topology] (B i).carrier

Tactic:
by
      rw [isOpen_iff_mem_nhds]
      intro a a_in
      rw [(hB.hasBasis_nhds a).mem_iff]
      use i, trivial
      rintro b b_in
      simpa using (B i).add_mem a_in b_in

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

Goal state before:
pp: case mpr
G : Type u_1
inst✝ : AddCommGroup G
B : AddGroupFilterBasis G
F : Filter G
this✝¹ : UniformSpace G := B.uniformSpace
this✝ : IsUniformAddGroup G
this : F ×ˢ F ≤ uniformity G ↔ ∀ U ∈ B, ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U
h' : F.NeBot
h : ∀ U ∈ B, ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U
⊢ F ×ˢ F ≤ uniformity G

Tactic:
this

Goal state after:
case mpr
G : Type u_1
inst✝ : AddCommGroup G
B : AddGroupFilterBasis G
F : Filter G
this✝¹ : UniformSpace G := B.uniformSpace
this✝ : IsUniformAddGroup G
this : F ×ˢ F ≤ uniformity G ↔ ∀ U ∈ B, ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U
h' : F.NeBot
h : ∀ U ∈ B, ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U
⊢ ∀ U ∈ B, ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: case a
R : Type r
inst✝ : CommRing R
W' : Jacobian R
P Q : Fin 3 → R
hP : W'.Equation P
hQ : W'.Equation Q
⊢ W'.addX P Q * (P z * Q z) ^ 2 +
        (-Q z ^ 6 * 0 +
          P z ^ 6 *
            (Q y ^ 2 + W'.a₁ * Q x * Q y * Q z + W'.a₃ * Q y * Q z ^ 3 -
              (Q x ^ 3 + W'.a₂ * Q x ^ 2 * Q z ^ 2 + W'.a₄ * Q x * Q z ^ 4 + W'.a₆ * Q z ^ 6))) -
      ((P y * Q z ^ 3 - Q y * P z ^ 3) ^ 2 + W'.a₁ * (P y * Q z ^ 3 - Q y * P z ^ 3) * P z * Q z * addZ P Q -
              W'.a₂ * P z ^ 2 * Q z ^ 2 * addZ P Q ^ 2 -
            P x * Q z ^ 2 * addZ P Q ^ 2 -
          Q x * P z ^ 2 * addZ P Q ^ 2 +
        (-Q z ^ 6 *
            (P y ^ 2 + W'.a₁ * P x * P y * P z + W'.a₃ * P y * P z ^ 3 -
              (P x ^ 3 + W'.a₂ * P x ^ 2 * P z ^ 2 + W'.a₄ * P x * P z ^ 4 + W'.a₆ * P z ^ 6)) +
          P z ^ 6 * 0)) =
    0

Tactic:
addX,

Goal state after:
case a
R : Type r
inst✝ : CommRing R
W' : Jacobian R
P Q : Fin 3 → R
hP : W'.Equation P
hQ : W'.Equation Q
⊢ (P x * Q x ^ 2 * P z ^ 2 - 2 * P y * Q y * P z * Q z + P x ^ 2 * Q x * Q z ^ 2 - W'.a₁ * P x * Q y * P z ^ 2 * Q z -
                        W'.a₁ * P y * Q x * P z * Q z ^ 2 +
                      2 * W'.a₂ * P x * Q x * P z ^ 2 * Q z ^ 2 -
                    W'.a₃ * Q y * P z ^ 4 * Q z -
                  W'.a₃ * P y * P z * Q z ^ 4 +
                W'.a₄ * Q x * P z ^ 4 * Q z ^ 2 +
              W'.a₄ * P x * P z ^ 2 * Q z ^ 4 +
            2 * W'.a₆ * P z ^ 4 * Q z ^ 4) *
          (P z * Q z) ^ 2 +
        (-Q z ^ 6 * 0 +
          P z ^ 6 *
            (Q y ^ 2 + W'.a₁ * Q x * Q y * Q z + W'.a₃ * Q y * Q z ^ 3 -
              (Q x ^ 3 + W'.a₂ * Q x ^ 2 * Q z ^ 2 + W'.a₄ * Q x * Q z ^ 4 + W'.a₆ * Q z ^ 6))) -
      ((P y * Q z ^ 3 - Q y * P z ^ 3) ^ 2 + W'.a₁ * (P y * Q z ^ 3 - Q y * P z ^ 3) * P z * Q z * addZ P Q -
              W'.a₂ * P z ^ 2 * Q z ^ 2 * addZ P Q ^ 2 -
            P x * Q z ^ 2 * addZ P Q ^ 2 -
          Q x * P z ^ 2 * addZ P Q ^ 2 +
        (-Q z ^ 6 *
            (P y ^ 2 + W'.a₁ * P x * P y * P z + W'.a₃ * P y * P z ^ 3 -
              (P x ^ 3 + W'.a₂ * P x ^ 2 * P z ^ 2 + W'.a₄ * P x * P z ^ 4 + W'.a₆ * P z ^ 6)) +
          P z ^ 6 * 0)) =
    0
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z : R
⊢ ∀ (x : R) (y : ↥(posSubmonoid R)), ValueGroupWithZero.mk x y ^ 0 = 1

Tactic:
[

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z : R
⊢ ∀ (x : R) (y : ↥(posSubmonoid R)), ValueGroupWithZero.mk x y ^ 0 = 1
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z : R
n : ℕ
⊢ ∀ (x : R) (y : ↥(posSubmonoid R)),
    ValueGroupWithZero.mk x y ^ (n + 1) = ValueGroupWithZero.mk x y ^ n * ValueGroupWithZero.mk x y

Tactic:
[

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z : R
n : ℕ
⊢ ∀ (x : R) (y : ↥(posSubmonoid R)),
    ValueGroupWithZero.mk x y ^ (n + 1) = ValueGroupWithZero.mk x y ^ n * ValueGroupWithZero.mk x y
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:
addX,

Goal state after:
R : Type r
inst✝ : CommRing R
W' : Jacobian R
P Q : Fin 3 → R
hPz : P z = 0
⊢ P x * Q x ^ 2 * P z ^ 2 - 2 * P y * Q y * P z * Q z + P x ^ 2 * Q x * Q z ^ 2 - W'.a₁ * P x * Q y * P z ^ 2 * Q z -
                  W'.a₁ * P y * Q x * P z * Q z ^ 2 +
                2 * W'.a₂ * P x * Q x * P z ^ 2 * Q z ^ 2 -
              W'.a₃ * Q y * P z ^ 4 * Q z -
            W'.a₃ * P y * P z * Q z ^ 4 +
          W'.a₄ * Q x * P z ^ 4 * Q z ^ 2 +
        W'.a₄ * P x * P z ^ 2 * Q z ^ 4 +
      2 * W'.a₆ * P z ^ 4 * Q z ^ 4 =
    (P x * Q z) ^ 2 * Q x
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

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

Tactic:
addX,

Goal state after:
R : Type r
inst✝ : CommRing R
W' : Jacobian R
P Q : Fin 3 → R
hQz : Q z = 0
⊢ P x * Q x ^ 2 * P z ^ 2 - 2 * P y * Q y * P z * Q z + P x ^ 2 * Q x * Q z ^ 2 - W'.a₁ * P x * Q y * P z ^ 2 * Q z -
                  W'.a₁ * P y * Q x * P z * Q z ^ 2 +
                2 * W'.a₂ * P x * Q x * P z ^ 2 * Q z ^ 2 -
              W'.a₃ * Q y * P z ^ 4 * Q z -
            W'.a₃ * P y * P z * Q z ^ 4 +
          W'.a₄ * Q x * P z ^ 4 * Q z ^ 2 +
        W'.a₄ * P x * P z ^ 2 * Q z ^ 4 +
      2 * W'.a₆ * P z ^ 4 * Q z ^ 4 =
    (-(Q x * P z)) ^ 2 * P x
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
⊢ (∃ i, True ∧ ↑(v.ltAddSubgroup ((Units.map ↑embedding) i)) ⊆ s) ↔ ∃ γ, {x | v.restrict x < ↑γ} ⊆ s

Tactic:
[

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
⊢ (∃ i, True ∧ ↑(v.ltAddSubgroup ((Units.map ↑embedding) i)) ⊆ s) ↔ ∃ γ, {x | v.restrict x < ↑γ} ⊆ s
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: case e'_5
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
x✝ : (ofClass v).ValueGroup₀ˣ
⊢ {p | v.restrict (p.2 - p.1) < ↑x✝} = (fun p ↦ p.2 + -p.1) ⁻¹' ↑(v.ltAddSubgroup ((Units.map ↑embedding) x✝))

Tactic:
[

Goal state after:
case e'_5
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
x✝ : (ofClass v).ValueGroup₀ˣ
⊢ {p | v.restrict (p.2 - p.1) < ↑x✝} = (fun p ↦ p.2 + -p.1) ⁻¹' ↑(v.ltAddSubgroup ((Units.map ↑embedding) x✝))
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
F : Filter R
⊢ (∀ U ∈ ⋯.toRingFilterBasis.toAddGroupFilterBasis, ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U) ↔
    ∀ (γ : (ofClass v).ValueGroup₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ

Tactic:
[

Goal state after:
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
F : Filter R
⊢ (∀ U ∈ ⋯.toRingFilterBasis.toAddGroupFilterBasis, ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U) ↔
    ∀ (γ : (ofClass v).ValueGroup₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: case mp
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
F : Filter R
h : ∀ (U : Set R), (∃ i, U = ↑(v.ltAddSubgroup ((Units.map ↑embedding) i))) → ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U
γ : (ofClass v).ValueGroup₀ˣ
⊢ ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ

Tactic:
[

Goal state after:
case mp
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
F : Filter R
h : ∀ (U : Set R), (∃ i, U = ↑(v.ltAddSubgroup ((Units.map ↑embedding) i))) → ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U
γ : (ofClass v).ValueGroup₀ˣ
⊢ ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: case mpr
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
F : Filter R
h : ∀ (γ : (ofClass v).ValueGroup₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ
γ : (ofClass v).ValueGroup₀ˣ
⊢ ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ ↑(v.ltAddSubgroup ((Units.map ↑embedding) γ))

Tactic:
[

Goal state after:
case mpr
R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
F : Filter R
h : ∀ (γ : (ofClass v).ValueGroup₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ
γ : (ofClass v).ValueGroup₀ˣ
⊢ ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ ↑(v.ltAddSubgroup ((Units.map ↑embedding) γ))
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₁ : Type u_3
Γ₂ : Type u_4
inst✝³ : LinearOrderedCommMonoidWithZero Γ₁
inst✝² : LinearOrderedCommMonoidWithZero Γ₂
v₁ : Valuation R Γ₁
v₂ : Valuation R Γ₂
inst✝¹ : v₁.Compatible
inst✝ : v₂.Compatible
x y : R
⊢ v₁ x ≤ v₁ y ↔ v₂ x ≤ v₂ y

Tactic:
[

Goal state after:
R : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₁ : Type u_3
Γ₂ : Type u_4
inst✝³ : LinearOrderedCommMonoidWithZero Γ₁
inst✝² : LinearOrderedCommMonoidWithZero Γ₂
v₁ : Valuation R Γ₁
v₂ : Valuation R Γ₂
inst✝¹ : v₁.Compatible
inst✝ : v₂.Compatible
x y : R
⊢ v₁ x ≤ v₁ y ↔ v₂ x ≤ v₂ y
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝³ : Ring R
inst✝² : ValuativeRel R
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.Compatible
x y : R
⊢ x =ᵥ y ↔ v x = v y

Tactic:
[

Goal state after:
R : Type u_1
inst✝³ : Ring R
inst✝² : ValuativeRel R
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.Compatible
x y : R
⊢ x =ᵥ y ↔ v x = v y
Module: Mathlib.Topology.Algebra.Valued.ValuationTopology

Goal state before:
pp: R : Type u
inst✝¹ : Ring R
Γ₀ : Type v
inst✝ : LinearOrderedCommGroupWithZero Γ₀
_i : Valued R Γ₀
r : (ofClass v).ValueGroup₀
hr : r ≠ 0
h : {x | v.restrict x = r} = {x | v.restrict x ≤ r} \ {x | v.restrict x < r}
⊢ IsClopen ({x | v.restrict x ≤ r} \ {x | v.restrict x < r})

Tactic:
exact IsClopen.diff (isClopen_closedBall _ hr) (isClopen_ball _ _)

Goal state after:
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
_u : UniformSpace R
inst✝² : IsUniformAddGroup R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
F : Filter R
⊢ (∀ U ∈ ⋯.toRingFilterBasis.toAddGroupFilterBasis, ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U) ↔
    ∀ (γ : (ofClass v).ValueGroup₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ

Tactic:
[

Goal state after:
R : Type u_1
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
_u : UniformSpace R
inst✝² : IsUniformAddGroup R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
F : Filter R
⊢ (∀ U ∈ ⋯.toRingFilterBasis.toAddGroupFilterBasis, ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U) ↔
    ∀ (γ : (ofClass v).ValueGroup₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: case mp
R : Type u_1
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
_u : UniformSpace R
inst✝² : IsUniformAddGroup R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
F : Filter R
h : ∀ (U : Set R), (∃ i, U = ↑(v.ltAddSubgroup ((Units.map ↑embedding) i))) → ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U
γ : (ofClass v).ValueGroup₀ˣ
⊢ ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ

Tactic:
[

Goal state after:
case mp
R : Type u_1
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
_u : UniformSpace R
inst✝² : IsUniformAddGroup R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
F : Filter R
h : ∀ (U : Set R), (∃ i, U = ↑(v.ltAddSubgroup ((Units.map ↑embedding) i))) → ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ U
γ : (ofClass v).ValueGroup₀ˣ
⊢ ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: case mpr
R : Type u_1
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
_u : UniformSpace R
inst✝² : IsUniformAddGroup R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
F : Filter R
h : ∀ (γ : (ofClass v).ValueGroup₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ
γ : (ofClass v).ValueGroup₀ˣ
⊢ ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ ↑(v.ltAddSubgroup ((Units.map ↑embedding) γ))

Tactic:
[

Goal state after:
case mpr
R : Type u_1
inst✝⁵ : Ring R
inst✝⁴ : ValuativeRel R
Γ₀ : Type u_3
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
_u : UniformSpace R
inst✝² : IsUniformAddGroup R
inst✝¹ : IsValuativeTopology R
v : Valuation R Γ₀
inst✝ : v.Compatible
F : Filter R
h : ∀ (γ : (ofClass v).ValueGroup₀ˣ), ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, v.restrict (y - x) < ↑γ
γ : (ofClass v).ValueGroup₀ˣ
⊢ ∃ M ∈ F, ∀ x ∈ M, ∀ y ∈ M, y - x ∈ ↑(v.ltAddSubgroup ((Units.map ↑embedding) γ))
Module: Mathlib.Topology.Algebra.UniformRing

Goal state before:
pp: α : Type u_1
inst✝³ : Ring α
inst✝² : UniformSpace α
inst✝¹ : IsTopologicalRing α
inst✝ : IsUniformAddGroup α
a✝ b✝ c✝ : Completion α
a b c : α
⊢ ↑a * ↑b * ↑c = ↑a * (↑b * ↑c)

Tactic:
rw [← coe_mul, ← coe_mul, ← coe_mul, ← coe_mul, mul_assoc]

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

Goal state before:
pp: α : Type u_1
inst✝⁹ : Ring α
inst✝⁸ : UniformSpace α
inst✝⁷ : IsTopologicalRing α
inst✝⁶ : IsUniformAddGroup α
β : Type u
inst✝⁵ : UniformSpace β
inst✝⁴ : Ring β
inst✝³ : IsUniformAddGroup β
inst✝² : IsTopologicalRing β
f : α →+* β
hf✝ : Continuous ⇑f
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
hf' : Continuous ⇑↑f
hf : UniformContinuous ⇑f
a✝ b✝ : Completion α
a b : α
⊢ Completion.extension (⇑f) (↑a * ↑b) = Completion.extension ⇑f ↑a * Completion.extension ⇑f ↑b

Tactic:
[

Goal state after:
α : Type u_1
inst✝⁹ : Ring α
inst✝⁸ : UniformSpace α
inst✝⁷ : IsTopologicalRing α
inst✝⁶ : IsUniformAddGroup α
β : Type u
inst✝⁵ : UniformSpace β
inst✝⁴ : Ring β
inst✝³ : IsUniformAddGroup β
inst✝² : IsTopologicalRing β
f : α →+* β
hf✝ : Continuous ⇑f
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
hf' : Continuous ⇑↑f
hf : UniformContinuous ⇑f
a✝ b✝ : Completion α
a b : α
⊢ Completion.extension (⇑f) (↑a * ↑b) = Completion.extension ⇑f ↑a * Completion.extension ⇑f ↑b
Module: Mathlib.Topology.Algebra.UniformRing

Goal state before:
pp: α : Type u_1
inst✝⁹ : Ring α
inst✝⁸ : UniformSpace α
inst✝⁷ : IsTopologicalRing α
inst✝⁶ : IsUniformAddGroup α
β : Type u
inst✝⁵ : UniformSpace β
inst✝⁴ : Ring β
inst✝³ : IsUniformAddGroup β
inst✝² : IsTopologicalRing β
f : α →+* β
hf✝ : Continuous ⇑f
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
hf' : Continuous ⇑↑f
hf : UniformContinuous ⇑f
⊢ Completion.extension (⇑f) 0 = 0

Tactic:
[

Goal state after:
α : Type u_1
inst✝⁹ : Ring α
inst✝⁸ : UniformSpace α
inst✝⁷ : IsTopologicalRing α
inst✝⁶ : IsUniformAddGroup α
β : Type u
inst✝⁵ : UniformSpace β
inst✝⁴ : Ring β
inst✝³ : IsUniformAddGroup β
inst✝² : IsTopologicalRing β
f : α →+* β
hf✝ : Continuous ⇑f
inst✝¹ : CompleteSpace β
inst✝ : T0Space β
hf' : Continuous ⇑↑f
hf : UniformContinuous ⇑f
⊢ Completion.extension (⇑f) 0 = 0