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

Goal state before:
pp: case of_j_ne_zero.of_j_ne_zero
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
C : VariableChange F
inst✝¹ : (C • E).IsCharThreeJNeZeroNF
C' : VariableChange F
heq : (C • E).j = (C' • E').j
inst✝ : (C' • E').IsCharThreeJNeZeroNF
⊢ ∃ C, C • E = E'

Tactic:
obtain ⟨C'', hC⟩ := exists_variableChange_of_char_three_of_j_ne_zero _ _ heq

Goal state after:
case of_j_ne_zero.of_j_ne_zero
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
C : VariableChange F
inst✝¹ : (C • E).IsCharThreeJNeZeroNF
C' : VariableChange F
heq : (C • E).j = (C' • E').j
inst✝ : (C' • E').IsCharThreeJNeZeroNF
C'' : VariableChange F
hC : C'' • C • E = C' • E'
⊢ ∃ C, C • E = E'
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: case of_j_eq_zero.of_j_ne_zero
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
heq : E.j = E'.j
C : VariableChange F
inst✝¹ : (C • E).IsShortNF
C' : VariableChange F
inst✝ : (C' • E').IsCharThreeJNeZeroNF
h : E.j ≠ 0
⊢ ∃ C, C • E = E'

Tactic:
← variableChange_j E C,

Goal state after:
case of_j_eq_zero.of_j_ne_zero
F : Type u_1
inst✝⁶ : Field F
inst✝⁵ : IsSepClosed F
E E' : WeierstrassCurve F
inst✝⁴ : E.IsElliptic
inst✝³ : E'.IsElliptic
inst✝² : CharP F 3
heq : E.j = E'.j
C : VariableChange F
inst✝¹ : (C • E).IsShortNF
C' : VariableChange F
inst✝ : (C' • E').IsCharThreeJNeZeroNF
h : (C • E).j ≠ 0
⊢ ∃ C, C • E = E'
Module: Mathlib.AlgebraicGeometry.EllipticCurve.IsomOfJ

Goal state before:
pp: 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 := invertibleOfNonzero hchar2
this : Invertible 3 := invertibleOfNonzero hchar3
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)
⊢ u ^ 4 = E.a₄ / E'.a₄

Tactic:
rw [pow_mul u 2 2, hu]

Goal state after:
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 := invertibleOfNonzero hchar2
this : Invertible 3 := invertibleOfNonzero hchar3
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)
⊢ (E.a₆ / E'.a₆ / (E.a₄ / E'.a₄)) ^ 2 = E.a₄ / E'.a₄
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Basic

Goal state before:
pp: case h
F : Type u
inst✝ : Field F
W : Jacobian F
P Q : Fin 3 → F
hPz : P z = 0
hQz : Q z = 0
hPx : IsUnit (P x)
hPy : IsUnit (P y)
hQx : IsUnit (Q x)
hQy : IsUnit (Q y)
hP : P y ^ 2 = P x ^ 3 ∧ (3 * P x ^ 2 ≠ 0 ∨ 2 * P y ≠ 0 ∨ W.a₁ * P x * P y ≠ 0)
hQ : Q y ^ 2 = Q x ^ 3 ∧ (3 * Q x ^ 2 ≠ 0 ∨ 2 * Q y ≠ 0 ∨ W.a₁ * Q x * Q y ≠ 0)
⊢ (fun m ↦ m • Q) (hPy.unit / hPx.unit * (hQx.unit / hQy.unit)) = P

Tactic:
simp only [Units.smul_def, smul_fin3, Units.val_mul, Units.val_div_eq_div_val, IsUnit.unit_spec,
    mul_pow, div_pow, hQz, mul_zero]

Goal state after:
case h
F : Type u
inst✝ : Field F
W : Jacobian F
P Q : Fin 3 → F
hPz : P z = 0
hQz : Q z = 0
hPx : IsUnit (P x)
hPy : IsUnit (P y)
hQx : IsUnit (Q x)
hQy : IsUnit (Q y)
hP : P y ^ 2 = P x ^ 3 ∧ (3 * P x ^ 2 ≠ 0 ∨ 2 * P y ≠ 0 ∨ W.a₁ * P x * P y ≠ 0)
hQ : Q y ^ 2 = Q x ^ 3 ∧ (3 * Q x ^ 2 ≠ 0 ∨ 2 * Q y ≠ 0 ∨ W.a₁ * Q x * Q y ≠ 0)
⊢ ![P y ^ 2 / P x ^ 2 * (Q x ^ 2 / Q y ^ 2) * Q x, P y ^ 3 / P x ^ 3 * (Q x ^ 3 / Q y ^ 3) * Q y, 0] = P
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
⊢ 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

Tactic:
neg_X,

Goal state after:
case a
R : Type r
inst✝ : CommRing R
W' : Jacobian R
P : Fin 3 → R
hP : W'.Equation P
⊢ P x * P x ^ 2 * P z ^ 2 - 2 * P y * W'.neg P y * P z * W'.neg P z + P x ^ 2 * 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 * P x * P z * W'.neg P z ^ 2 +
                    2 * W'.a₂ * P x * 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 * 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.RingTheory.Valuation.ExtendToLocalization

Goal state before:
pp: A : Type u_1
inst✝⁴ : CommRing A
Γ : Type u_2
inst✝³ : LinearOrderedCommGroupWithZero Γ
v : Valuation A Γ
S : Submonoid A
hS : S ≤ v.supp.primeCompl
B : Type u_3
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : IsLocalization S B
f : S.LocalizationMap B := IsLocalization.toLocalizationMap S B
h : ∀ (s : ↥S), IsUnit (↑v.toMonoidWithZeroHom ↑s) := fun s ↦ isUnit_iff_ne_zero.mpr (hS s.property)
x y : B
⊢ (↑(f.lift h)).toFun (x + y) ≤ max ((↑(f.lift h)).toFun x) ((↑(f.lift h)).toFun y)

Tactic:
obtain ⟨a, b, s, rfl, rfl⟩ : ∃ (a b : A) (s : S), f.mk' a s = x ∧ f.mk' b s = y := by
        obtain ⟨a, s, rfl⟩ := f.mk'_surjective x
        obtain ⟨b, t, rfl⟩ := f.mk'_surjective y
        use a * t, b * s, s * t
        constructor <;>
          · rw [f.mk'_eq_iff_eq, Submonoid.coe_mul]
            ring_nf

Goal state after:
A : Type u_1
inst✝⁴ : CommRing A
Γ : Type u_2
inst✝³ : LinearOrderedCommGroupWithZero Γ
v : Valuation A Γ
S : Submonoid A
hS : S ≤ v.supp.primeCompl
B : Type u_3
inst✝² : CommRing B
inst✝¹ : Algebra A B
inst✝ : IsLocalization S B
f : S.LocalizationMap B := IsLocalization.toLocalizationMap S B
h : ∀ (s : ↥S), IsUnit (↑v.toMonoidWithZeroHom ↑s) := fun s ↦ isUnit_iff_ne_zero.mpr (hS s.property)
a b : A
s : ↥S
⊢ (↑(f.lift h)).toFun (f.mk' a s + f.mk' b s) ≤ max ((↑(f.lift h)).toFun (f.mk' a s)) ((↑(f.lift h)).toFun (f.mk' b s))
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'.negY ![W'.dblX P, W'.negDblY P, W'.dblZ P] = (P x ^ 2) ^ 3

Tactic:
negY_eq,

Goal state after:
R : Type r
inst✝ : CommRing R
W' : Jacobian R
P : Fin 3 → R
hP : W'.Equation P
hPz : P z = 0
⊢ -W'.negDblY P - W'.a₁ * W'.dblX P * W'.dblZ P - W'.a₃ * W'.dblZ P ^ 3 = (P x ^ 2) ^ 3
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula

Goal state before:
pp: R : Type r
inst✝¹ : CommRing R
W' : Jacobian R
inst✝ : NoZeroDivisors R
P Q : Fin 3 → R
hQz : Q z ≠ 0
hx : P x * Q z ^ 2 = Q x * P z ^ 2
hy : P y * Q z ^ 3 = Q y * P z ^ 3
hy' : P y * Q z ^ 3 = W'.negY Q * P z ^ 3
⊢ W'.negY ![W'.dblX P, W'.negDblY P, W'.dblZ P] = W'.dblU P ^ 3

Tactic:
negY_eq,

Goal state after:
R : Type r
inst✝¹ : CommRing R
W' : Jacobian R
inst✝ : NoZeroDivisors R
P Q : Fin 3 → R
hQz : Q z ≠ 0
hx : P x * Q z ^ 2 = Q x * P z ^ 2
hy : P y * Q z ^ 3 = Q y * P z ^ 3
hy' : P y * Q z ^ 3 = W'.negY Q * P z ^ 3
⊢ -W'.negDblY P - W'.a₁ * W'.dblX P * W'.dblZ P - W'.a₃ * W'.dblZ P ^ 3 = W'.dblU P ^ 3
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
a : A
a_in : a ∈ (B i).carrier
⊢ (B i).carrier ∈ 𝓝 a

Tactic:
rw [(hB.hasBasis_nhds a).mem_iff]

Goal state after:
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
a : A
a_in : a ∈ (B i).carrier
⊢ ∃ i_1, True ∧ {b | b - a ∈ B i_1} ⊆ (B i).carrier
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x y : R
u : ↥(posSubmonoid R)
⊢ x * ↑u ≤ᵥ y * ↑u ↔ x ≤ᵥ y

Tactic:
by simp

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

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x y : R
u : ↥(posSubmonoid R)
⊢ ↑u * x ≤ᵥ ↑u * y ↔ x ≤ᵥ y

Tactic:
by simp

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

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x : R
y : ↥(posSubmonoid R)
h : x ≤ᵥ 0
⊢ 0 * ↑y ≤ᵥ x * ↑1

Tactic:
by simp

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

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x : ↥(posSubmonoid R)
⊢ ↑x * ↑1 ≤ᵥ 1 * ↑x

Tactic:
by simp

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

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x : ↥(posSubmonoid R)
⊢ 1 * ↑x ≤ᵥ ↑x * ↑1

Tactic:
by simp

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

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
⊢ 1 * ↑1 ≤ᵥ 1 * ↑1

Tactic:
by simp

Goal state after:
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✝ x y z w : R
⊢ ∀ (t s u v : ↥(posSubmonoid R)),
    x * ↑t ≤ᵥ y * ↑s →
      y * ↑s ≤ᵥ x * ↑t →
        z * ↑u ≤ᵥ w * ↑v →
          w * ↑v ≤ᵥ z * ↑u → ValueGroupWithZero.mk (x * z) (s * v) = ValueGroupWithZero.mk (y * w) (t * u)

Tactic:
t

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z✝ x y z w : R
t : ↥(posSubmonoid R)
⊢ ∀ (s u v : ↥(posSubmonoid R)),
    x * ↑t ≤ᵥ y * ↑s →
      y * ↑s ≤ᵥ x * ↑t →
        z * ↑u ≤ᵥ w * ↑v →
          w * ↑v ≤ᵥ z * ↑u → ValueGroupWithZero.mk (x * z) (s * v) = ValueGroupWithZero.mk (y * w) (t * u)
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 y : R
⊢ ∀ (t s : ↥(posSubmonoid R)),
    x * ↑t ≤ᵥ y * ↑s → y * ↑s ≤ᵥ x * ↑t → ValueGroupWithZero.mk (x ^ n) (s ^ n) = ValueGroupWithZero.mk (y ^ n) (t ^ n)

Tactic:
t

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z : R
n : ℕ
x y : R
t : ↥(posSubmonoid R)
⊢ ∀ (s : ↥(posSubmonoid R)),
    x * ↑t ≤ᵥ y * ↑s → y * ↑s ≤ᵥ x * ↑t → ValueGroupWithZero.mk (x ^ n) (s ^ n) = ValueGroupWithZero.mk (y ^ n) (t ^ n)
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✝ x y z w : R
⊢ ∀ (t s u v : ↥(posSubmonoid R)),
    x * ↑t ≤ᵥ y * ↑s → y * ↑s ≤ᵥ x * ↑t → z * ↑u ≤ᵥ w * ↑v → w * ↑v ≤ᵥ z * ↑u → (x * ↑v ≤ᵥ z * ↑s) = (y * ↑u ≤ᵥ w * ↑t)

Tactic:
t

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z✝ x y z w : R
t : ↥(posSubmonoid R)
⊢ ∀ (s u v : ↥(posSubmonoid R)),
    x * ↑t ≤ᵥ y * ↑s → y * ↑s ≤ᵥ x * ↑t → z * ↑u ≤ᵥ w * ↑v → w * ↑v ≤ᵥ z * ↑u → (x * ↑v ≤ᵥ z * ↑s) = (y * ↑u ≤ᵥ w * ↑t)
Module: Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Point

Goal state before:
pp: R : Type r
S : Type s
A F : Type u
B K : Type v
inst✝⁵ : CommRing R
inst✝⁴ : CommRing S
inst✝³ : CommRing A
inst✝² : CommRing B
inst✝¹ : Field F
inst✝ : Field K
W' : Jacobian R
W : Jacobian F
x✝¹ x✝ : W.Point
⊢ x✝¹ + x✝ = x✝ + x✝¹

Tactic:
apply (toAffineAddEquiv W).injective
    simp only [map_add, add_comm]

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

Goal state before:
pp: Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
γ : Γ₀ˣ
⊢ {↑γ} ∈ 𝓝 ↑γ

Tactic:
by simp

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

Goal state before:
pp: α : Type u_1
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
γ γ₁ γ₂ : Γ₀
l : Filter α
f : α → Γ₀
⊢ IsClosed[instTopologicalSpaceProd] {p | p.1 ≤ p.2}

Tactic:
by
    simp only [← isOpen_compl_iff, compl_ofPred, not_le, isOpen_iff_mem_nhds]
    rintro ⟨a, b⟩ (hab : b < a)
    rw [nhds_prod_eq, nhds_of_ne_zero hab.ne_zero, pure_prod]
    exact Iio_mem_nhds hab

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

Goal state before:
pp: case mk.mk.mk
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z a₁ : R
a₂ : ↥(posSubmonoid R)
b₁ : R
b₂ : ↥(posSubmonoid R)
hab : ValueGroupWithZero.mk a₁ a₂ ≤ ValueGroupWithZero.mk b₁ b₂
c₁ : R
c₂ : ↥(posSubmonoid R)
hbc : ValueGroupWithZero.mk b₁ b₂ ≤ ValueGroupWithZero.mk c₁ c₂
⊢ ValueGroupWithZero.mk a₁ a₂ ≤ ValueGroupWithZero.mk c₁ c₂

Tactic:
rw [ValueGroupWithZero.mk_le_mk] at hab hbc ⊢

Goal state after:
case mk.mk.mk
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x x' y y' z a₁ : R
a₂ : ↥(posSubmonoid R)
b₁ : R
b₂ : ↥(posSubmonoid R)
hab : a₁ * ↑b₂ ≤ᵥ b₁ * ↑a₂
c₁ : R
c₂ : ↥(posSubmonoid R)
hbc : b₁ * ↑c₂ ≤ᵥ c₁ * ↑b₂
⊢ a₁ * ↑c₂ ≤ᵥ c₁ * ↑a₂
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
inst✝¹ : TopologicalSpace R
v : Valuation R Γ₀
inst✝ : v.Compatible
H : ∀ {s : Set R} {x : R}, s ∈ 𝓝 x ↔ ∃ γ, {z | v.restrict (z - x) < ↑γ} ⊆ s
s : Set R
x : R
x✝ : ∃ γ, (fun x_1 ↦ x + x_1) '' {z | (valuation R) z < ↑γ} ⊆ s
γ : (ValueGroupWithZero R)ˣ
hγ : (fun x_1 ↦ x + x_1) '' {z | (valuation R) z < ↑γ} ⊆ s
⊢ (orderMonoidIso v) ↑γ ≠ 0

Tactic:
by simp

Goal state after:
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝⁴ : Ring R
inst✝³ : ValuativeRel R
Γ₀ : Type u_3
inst✝² : LinearOrderedCommGroupWithZero Γ₀
inst✝¹ : TopologicalSpace R
v : Valuation R Γ₀
inst✝ : v.Compatible
H : ∀ {s : Set R} {x : R}, s ∈ 𝓝 x ↔ ∃ γ, {z | v.restrict (z - x) < ↑γ} ⊆ s
s : Set R
x : R
h_mem : s ∈ 𝓝 x
γ : (ofClass v).ValueGroup₀ˣ
hγ : {z | v.restrict (z - x) < ↑γ} ⊆ s
⊢ (orderMonoidIso v).symm ↑γ ≠ 0

Tactic:
by simp

Goal state after:
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 x y : R
⊢ ∀ (t s : ↥(posSubmonoid R)),
    x * ↑t ≤ᵥ y * ↑s →
      y * ↑s ≤ᵥ x * ↑t →
        (if h : x ≤ᵥ 0 then 0 else ValueGroupWithZero.mk ↑s ⟨x, h⟩) =
          if h : y ≤ᵥ 0 then 0 else ValueGroupWithZero.mk ↑t ⟨y, h⟩

Tactic:
t

Goal state after:
R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
x✝ x' y✝ y' z x y : R
t : ↥(posSubmonoid R)
⊢ ∀ (s : ↥(posSubmonoid R)),
    x * ↑t ≤ᵥ y * ↑s →
      y * ↑s ≤ᵥ x * ↑t →
        (if h : x ≤ᵥ 0 then 0 else ValueGroupWithZero.mk ↑s ⟨x, h⟩) =
          if h : y ≤ᵥ 0 then 0 else ValueGroupWithZero.mk ↑t ⟨y, h⟩
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝³ : Ring R
inst✝² : ValuativeRel R
inst✝¹ : TopologicalSpace R
inst✝ : IsValuativeTopology R
x : R
γ : (ValueGroupWithZero R)ˣ
x✝ : True
⊢ ↑γ ≠ 0 ∧ {y | (valuation R) (y - x) < ↑γ} ⊆ {z | (valuation R) (z - x) < ↑γ}

Tactic:
by simp

Goal state after:
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝³ : Ring R
inst✝² : ValuativeRel R
inst✝¹ : TopologicalSpace R
inst✝ : IsValuativeTopology R
x : R
γ : ValueGroupWithZero R
hγ : γ ≠ 0
⊢ True ∧ {z | (valuation R) (z - x) < ↑(Units.mk0 γ hγ)} ⊆ {y | (valuation R) (y - x) < γ}

Tactic:
by simp

Goal state after:
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝³ : Ring R
inst✝² : ValuativeRel R
inst✝¹ : TopologicalSpace R
inst✝ : IsValuativeTopology R
γ : (ValueGroupWithZero R)ˣ
x✝ : True
⊢ ↑γ ≠ 0 ∧ {x | (valuation R) x < ↑γ} ⊆ {x | (valuation R) x < ↑γ}

Tactic:
by simp

Goal state after:
Module: Mathlib.Topology.Algebra.ValuativeRel.ValuativeTopology

Goal state before:
pp: R : Type u_1
inst✝³ : Ring R
inst✝² : ValuativeRel R
inst✝¹ : TopologicalSpace R
inst✝ : IsValuativeTopology R
γ : ValueGroupWithZero R
hγ : γ ≠ 0
⊢ True ∧ {x | (valuation R) x < ↑(Units.mk0 γ hγ)} ⊆ {x | (valuation R) x < γ}

Tactic:
by simp

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

Goal state before:
pp: R✝ : Type u_1
inst✝³ : Semiring R✝
inst✝² : ValuativeRel R✝
x✝² x' y y' z : R✝
R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
x x✝¹ x✝ : R
⊢ ValueGroupWithZero.mk (x✝¹ * x✝) 1 = ValueGroupWithZero.mk x✝¹ 1 * ValueGroupWithZero.mk x✝ 1

Tactic:
by simp

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

Goal state before:
pp: R✝ : Type u_1
inst✝³ : Semiring R✝
inst✝² : ValuativeRel R✝
x✝² x' y y' z : R✝
R : Type u_2
inst✝¹ : Ring R
inst✝ : ValuativeRel R
x x✝¹ x✝ : R
⊢ x✝¹ ≤ᵥ x✝ ↔ (valuation R) x✝¹ ≤ (valuation R) x✝

Tactic:
by simp [valuation]

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

Goal state before:
pp: R✝ : Type u_1
inst✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
x✝¹ x' y y' z : R✝
R : Type u_2
inst✝³ : Ring R
inst✝² : ValuativeRel R
x : R
S : Type u_3
Γ : Type u_4
inst✝¹ : Ring S
inst✝ : LinearOrderedCommGroupWithZero Γ
v : Valuation S Γ
x✝ y✝ z✝ : S
h0 : ¬v z✝ ≤ v 0
h : v (x✝ * z✝) ≤ v (y✝ * z✝)
⊢ v x✝ ≤ v y✝

Tactic:
simp only [map_mul] at h
    apply le_of_mul_le_mul_right h
    simpa [pos_iff_ne_zero] using h0

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

Goal state before:
pp: R✝ : Type u_1
inst✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
x✝ x' y y' z : R✝
R : Type u_2
inst✝³ : Ring R
inst✝² : ValuativeRel R
x : R
S : Type u_3
Γ : Type u_4
inst✝¹ : Ring S
inst✝ : LinearOrderedCommGroupWithZero Γ
v : Valuation S Γ
⊢ ¬v 1 ≤ v 0

Tactic:
by simp

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

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
s : (ValueGroupWithZero R)ˣ
h : 1 ≠ s
⊢ ↑s ≠ 0

Tactic:
by simp

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

Goal state before:
pp: R : Type u_1
inst✝¹ : Semiring R
inst✝ : ValuativeRel R
r s : (ValueGroupWithZero R)ˣ
h : r ≠ s
hr : r ≠ 1
⊢ ↑r ≠ 0

Tactic:
by simp

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

Goal state before:
pp: R : Type u_2
inst✝³ : Ring R
inst✝² : ValuativeRel R
Γ₀ : Type u_3
inst✝¹ : LinearOrderedCommMonoidWithZero Γ₀
v : Valuation R Γ₀
inst✝ : v.Compatible
r : R
s : ↥(posSubmonoid R)
hr✝ : ValueGroupWithZero.mk r s ≠ 0
hr' : ValueGroupWithZero.mk r s ≠ 1
hγ : v r ≠ 0
hγ' : v r ≤ v ↑s → v r < v ↑s
hr : v r = 1
⊢ v ↑s ≠ 0

Tactic:
by simp

Goal state after:
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 * c + b * c)

Tactic:
add_mul

Goal state after:
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 * c + b * c) = ↑(a * c + b * c)
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: case hb
R✝ : Type u_1
inst✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
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
x y : R
r s : ↥(posSubmonoid R)
⊢ (restrict₀ (ofClass v)) ↑s ≠ 0

Tactic:
simp [ValueGroup₀.restrict₀]

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

Goal state before:
pp: case hd
R✝ : Type u_1
inst✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
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
x y : R
r s : ↥(posSubmonoid R)
⊢ (restrict₀ (ofClass v)) ↑r ≠ 0

Tactic:
simp [ValueGroup₀.restrict₀]

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

Goal state before:
pp: R✝ : Type u_1
inst✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
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
⊢ ValueGroupWithZero.lift (fun r s ↦ (restrict₀ (ofClass v)) r / (restrict₀ (ofClass v)) ↑s) ⋯ 1 = 1

Tactic:
simp [ValueGroup₀.restrict₀]

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

Goal state before:
pp: K : Type u_1
inst✝² : Field K
inst✝¹ : UniformSpace K
inst✝ : CompletableTopField K
x : hat K
h : x ≠ 0
y : hat K
y_ne : y ≠ 0
this : (fun x ↦ ↑x⁻¹) = (fun y ↦ ↑y) ∘ fun x ↦ x⁻¹
⊢ Cauchy (map (fun x ↦ x⁻¹) (Filter.comap coe' (𝓝 y)))

Tactic:
apply CompletableTopField.nice

Goal state after:
case a
K : Type u_1
inst✝² : Field K
inst✝¹ : UniformSpace K
inst✝ : CompletableTopField K
x : hat K
h : x ≠ 0
y : hat K
y_ne : y ≠ 0
this : (fun x ↦ ↑x⁻¹) = (fun y ↦ ↑y) ∘ fun x ↦ x⁻¹
⊢ Cauchy (Filter.comap coe' (𝓝 y))

case a
K : Type u_1
inst✝² : Field K
inst✝¹ : UniformSpace K
inst✝ : CompletableTopField K
x : hat K
h : x ≠ 0
y : hat K
y_ne : y ≠ 0
this : (fun x ↦ ↑x⁻¹) = (fun y ↦ ↑y) ∘ fun x ↦ x⁻¹
⊢ 𝓝 0 ⊓ Filter.comap coe' (𝓝 y) = ⊥
Module: Mathlib.Topology.Algebra.UniformField

Goal state before:
pp: case a
K : Type u_1
inst✝² : Field K
inst✝¹ : UniformSpace K
inst✝ : CompletableTopField K
x : hat K
h : x ≠ 0
y : hat K
y_ne : y ≠ 0
this✝ : (fun x ↦ ↑x⁻¹) = (fun y ↦ ↑y) ∘ fun x ↦ x⁻¹
this : (Filter.comap coe' (𝓝 y)).NeBot
⊢ Filter.comap (fun p ↦ (↑p.1, ↑p.2)) (uniformity (hat K)) ≤ uniformity K

Tactic:
Completion.comap_coe_eq_uniformity

Goal state after:
case a
K : Type u_1
inst✝² : Field K
inst✝¹ : UniformSpace K
inst✝ : CompletableTopField K
x : hat K
h : x ≠ 0
y : hat K
y_ne : y ≠ 0
this✝ : (fun x ↦ ↑x⁻¹) = (fun y ↦ ↑y) ∘ fun x ↦ x⁻¹
this : (Filter.comap coe' (𝓝 y)).NeBot
⊢ uniformity K ≤ uniformity K
Module: Mathlib.RingTheory.Valuation.ValuativeRel.Basic

Goal state before:
pp: R✝ : Type u_1
inst✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
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
x : ValueGroupWithZero R
⊢ (fun x ↦ embedding (⋯.orderMonoidIso x)) ((↑(embed v)).toFun x) = x

Tactic:
by simp

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

Goal state before:
pp: R✝ : Type u_1
inst✝⁵ : Semiring R✝
inst✝⁴ : ValuativeRel R✝
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
x : ValueGroupWithZero R
⊢ embedding (⋯.orderMonoidIso ((↑(embed v)).toFun x)) = x

Tactic:
by simp

Goal state after: