mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: R✝ : Type u_1
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
R : Type u_3
inst✝ : Ring R
v : Valuation R Γ₀
⊢ ∀ (x y : ↥(ofClass Valued.v).valueGroup),
    (Equiv.setCongr ⋯).toFun (x * y) = (Equiv.setCongr ⋯).toFun x * (Equiv.setCongr ⋯).toFun y

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
R : Type u_3
inst✝ : Ring R
v : Valuation R Γ₀
x✝² x✝¹ : ↥(ofClass Valued.v).valueGroup
x✝ : x✝² < x✝¹
⊢ (valueGroupEquiv v) x✝² < (valueGroupEquiv v) x✝¹

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
R : Type u_3
inst✝ : Ring R
v : Valuation R Γ₀
x✝² x✝¹ : ↥(ofClass v).valueGroup
x✝ : x✝² < x✝¹
⊢ (valueGroupEquiv v).symm x✝² < (valueGroupEquiv v).symm x✝¹

Tactic:
by

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
hQ : W'.Equation Q
hQz : Q z = 0
⊢ W'.addXYZ P Q = -(Q x * P z) • P

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: R✝ : Type u_1
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
R : Type u_3
inst✝ : Ring R
v : Valuation R Γ₀
x : (ofClass Valued.v).ValueGroup₀
⊢ (WithZero.map' ↑(valueGroupEquiv v).symm) ((WithZero.map' ↑(valueGroupEquiv v)) x) = x

Tactic:
by

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

Goal state before:
pp: F : Type u
inst✝ : Field F
W : Jacobian F
P Q : Fin 3 → F
hP : W.Equation P
hQ : W.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z ^ 2 = Q x * P z ^ 2
⊢ W.addXYZ P Q = addU P Q • ![1, 1, 0]

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: R✝ : Type u_1
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
R : Type u_3
inst✝ : Ring R
v : Valuation R Γ₀
y : (ofClass v).ValueGroup₀
⊢ (WithZero.map' ↑(valueGroupEquiv v)) ((WithZero.map' ↑(valueGroupEquiv v).symm) y) = y

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: R✝ : Type u_1
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
R : Type u_3
inst✝ : Ring R
v : Valuation R Γ₀
⊢ ∀ (x y : (ofClass Valued.v).ValueGroup₀),
    (WithZero.map' ↑(valueGroupEquiv v)) (x * y) =
      (WithZero.map' ↑(valueGroupEquiv v)) x * (WithZero.map' ↑(valueGroupEquiv v)) y

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: R✝ : Type u_1
Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
R : Type u_3
inst✝ : Ring R
v : Valuation R Γ₀
a b : (ofClass Valued.v).ValueGroup₀
⊢ (WithZero.map' ↑(valueGroupEquiv v)) a ≤ (WithZero.map' ↑(valueGroupEquiv v)) b ↔ a ≤ b

Tactic:
by

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

Goal state before:
pp: Γ : Type u_1
inst✝¹ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝ : Field K
v : Valuation K Γ
hv : v.IsRankOneDiscrete
r : ↥v.valuationSubring
hr : r ≠ 0
π : v.Uniformizer
⊢ ∃ n u, ↑r = ↑(π.val ^ n) * ↑↑u

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
R : Type u_3
inst✝ : Ring R
v : Valuation R Γ₀
b : WithVal v
⊢ (valueGroupOrderIso₀ v) ((valuation v).restrict b) = v.restrict b.ofVal

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: Γ₀ : Type u_2
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
R : Type u_3
inst✝ : Ring R
v : Valuation R Γ₀
b : R
⊢ (valueGroupOrderIso₀ v).symm (v.restrict b) = Valued.v.restrict (toVal v b)

Tactic:
by

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

Goal state before:
pp: Γ : Type u_1
inst✝¹ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝ : Field K
v : Valuation K Γ
hv : v.IsRankOneDiscrete
π : v.Uniformizer
x : ↥v.valuationSubring
hx : x ∈ maximalIdeal ↥v.valuationSubring
hx₀ : ¬x = 0
n : ℕ
u : (↥v.valuationSubring)ˣ
hu : x = π.val ^ n * ↑u
hn : ¬IsUnit x
h : n = 0
⊢ False

Tactic:
by

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

Goal state before:
pp: Γ : Type u_1
inst✝¹ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝ : Field K
v : Valuation K Γ
hv : v.IsRankOneDiscrete
π : v.Uniformizer
⊢ maximalIdeal ↥v.valuationSubring = span {π.val}

Tactic:
by

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

Goal state before:
pp: Γ : Type u_1
inst✝¹ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝ : Field K
v : Valuation K Γ
hv : v.IsRankOneDiscrete
π : v.Uniformizer
n : ℕ
⊢ maximalIdeal ↥v.valuationSubring ^ n = span {π.val ^ n}

Tactic:
by

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

Goal state before:
pp: Γ : Type u_1
inst✝³ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝² : Field K
v : Valuation K Γ
inst✝¹ : Nontrivial ↥(MonoidWithZeroHom.ofClass v).valueGroup
inst✝ : IsCyclic ↥(MonoidWithZeroHom.ofClass v).valueGroup
⊢ ¬IsField ↥v.valuationSubring

Tactic:
by

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

Goal state before:
pp: Γ : Type u_1
inst✝² : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝¹ : Field K
v : Valuation K Γ
inst✝ : v.IsRankOneDiscrete
r : ↥v.valuationSubring
hr : maximalIdeal ↥v.valuationSubring = span {r}
⊢ r ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: R : Type u_4
Γ₀ : Type u_5
Γ₀' : Type u_6
inst✝² : Ring R
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ₀'
v : Valuation R Γ₀
w : Valuation R Γ₀'
hval : Valued R Γ₀'
hv : Valued.v = w
h : v.IsEquiv w
γ : (ofClass Valued.v).ValueGroup₀ˣ
r s : R
hr₀ : 0 < Valued.v r
hs₀ : 0 < Valued.v s
hr : Valued.v.restrict r / Valued.v.restrict s = ↑γ
⊢ Valued.v.restrict ((equiv v).symm r) / Valued.v.restrict ((equiv v).symm s) ≠ 0

Tactic:
by

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

Goal state before:
pp: Γ : Type u_1
inst✝² : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝¹ : Field K
v : Valuation K Γ
inst✝ : v.IsRankOneDiscrete
r : ↥v.valuationSubring
hr : maximalIdeal ↥v.valuationSubring = span {r}
⊢ v.IsUniformizer ↑r

Tactic:
by

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

Goal state before:
pp: Γ : Type u_1
inst✝³ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝² : Field K
v : Valuation K Γ
inst✝¹ : IsCyclic ↥(MonoidWithZeroHom.ofClass v).valueGroup
inst✝ : Nontrivial ↥(MonoidWithZeroHom.ofClass v).valueGroup
I : Ideal ↥v.valuationSubring
this : ∀ (P : Ideal ↥v.valuationSubring), P.IsPrime → Submodule.IsPrincipal P
⊢ Submodule.IsPrincipal I

Tactic:
by

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

Goal state before:
pp: F : Type u
inst✝¹ : Field F
W : Jacobian F
inst✝ : DecidableEq F
P Q : Fin 3 → F
hP : W.Equation P
hQ : W.Equation Q
hPz : P z ≠ 0
hQz : Q z ≠ 0
hx : P x * Q z ^ 2 ≠ Q x * P z ^ 2
⊢ W.addXYZ P Q =
    addZ P Q •
      ![W.toAffine.addX (P x / P z ^ 2) (Q x / Q z ^ 2)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        W.toAffine.addY (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3)
          (W.toAffine.slope (P x / P z ^ 2) (Q x / Q z ^ 2) (P y / P z ^ 3) (Q y / Q z ^ 3)),
        1]

Tactic:
by

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

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

Tactic:
by

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

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

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: Γ : Type u_1
inst✝³ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝² : Field K
v : Valuation K Γ
inst✝¹ : IsCyclic ↥(MonoidWithZeroHom.ofClass v).valueGroup
inst✝ : Nontrivial ↥(MonoidWithZeroHom.ofClass v).valueGroup
I P : Ideal ↥v.valuationSubring
hP : P.IsPrime
h_ne_bot : ¬P = ⊥
⊢ Nonempty v.Uniformizer

Tactic:
by

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

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

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: R : Type u_4
Γ₀ : Type u_5
Γ₀' : Type u_6
inst✝² : Ring R
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ₀'
v : Valuation R Γ₀
w : Valuation R Γ₀'
hval : Valued R Γ₀'
hv : Valued.v = w
h : v.IsEquiv w
γ : (ofClass Valued.v).ValueGroup₀ˣ
r s : R
hr₀ : 0 < Valued.v r
hs₀ : 0 < Valued.v s
hr : Valued.v.restrict r / Valued.v.restrict s = ↑γ
x : WithVal v
hx :
  x ∈
    {x |
      Valued.v.restrict x <
        ↑(Units.mk0 (Valued.v.restrict ((equiv v).symm r) / Valued.v.restrict ((equiv v).symm s)) ⋯)}
y : R := (equiv v) x
hy : toVal v y = x
⊢ 0 < Valued.v.restrict (toVal v s)

Tactic:
by

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

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

Tactic:
by

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

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

Tactic:
by

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

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

Tactic:
by

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

Goal state before:
pp: F : Type u
K : Type v
inst✝¹ : Field F
inst✝ : Field K
f : F →+* K
P Q : Fin 3 → F
⊢ addU (⇑f ∘ P) (⇑f ∘ Q) = f (addU P Q)

Tactic:
by

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

Goal state before:
pp: R : Type r
S : Type s
inst✝¹ : CommRing R
inst✝ : CommRing S
f : R →+* S
P Q : Fin 3 → R
⊢ addZ (⇑f ∘ P) (⇑f ∘ Q) = f (addZ P Q)

Tactic:
by

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

Goal state before:
pp: R : Type r
S : Type s
inst✝¹ : CommRing R
inst✝ : CommRing S
W' : Jacobian R
f : R →+* S
P Q : Fin 3 → R
⊢ (W'.map f).addX (⇑f ∘ P) (⇑f ∘ Q) = f (W'.addX P Q)

Tactic:
by

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

Goal state before:
pp: R : Type r
S : Type s
inst✝¹ : CommRing R
inst✝ : CommRing S
W' : Jacobian R
f : R →+* S
P Q : Fin 3 → R
⊢ (W'.map f).negAddY (⇑f ∘ P) (⇑f ∘ Q) = f (W'.negAddY P Q)

Tactic:
by

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

Goal state before:
pp: R : Type r
S : Type s
inst✝¹ : CommRing R
inst✝ : CommRing S
W' : Jacobian R
f : R →+* S
P Q : Fin 3 → R
⊢ (W'.map f).addY (⇑f ∘ P) (⇑f ∘ Q) = f (W'.addY P Q)

Tactic:
by

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

Goal state before:
pp: R : Type r
S : Type s
inst✝¹ : CommRing R
inst✝ : CommRing S
W' : Jacobian R
f : R →+* S
P Q : Fin 3 → R
⊢ (W'.map f).addXYZ (⇑f ∘ P) (⇑f ∘ Q) = ⇑f ∘ W'.addXYZ P Q

Tactic:
by

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) = f ((W'⁄A).negY P)

Tactic:
by

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).dblU (⇑f ∘ P) = f ((W'⁄A).dblU P)

Tactic:
by

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).dblZ (⇑f ∘ P) = f ((W'⁄A).dblZ P)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: R : Type u_4
Γ₀ : Type u_5
Γ₀' : Type u_6
inst✝² : Ring R
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ₀'
v : Valuation R Γ₀
w : Valuation R Γ₀'
hval : Valued R Γ₀'
hv : Valued.v = w
h : v.IsEquiv w
⊢ UniformContinuous ⇑(equiv v)

Tactic:
by

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).dblX (⇑f ∘ P) = f ((W'⁄A).dblX P)

Tactic:
by

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).negDblY (⇑f ∘ P) = f ((W'⁄A).negDblY P)

Tactic:
by

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

Goal state before:
pp: Γ : Type u_1
inst✝³ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝² : Field K
v : Valuation K Γ
inst✝¹ : IsCyclic ↥(MonoidWithZeroHom.ofClass v).valueGroup
inst✝ : Nontrivial ↥(MonoidWithZeroHom.ofClass v).valueGroup
I : Ideal ↥v.valuationSubring
⊢ Submodule.IsPrincipal I

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: R : Type u_4
Γ₀ : Type u_5
Γ₀' : Type u_6
inst✝² : Ring R
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ₀'
v : Valuation R Γ₀
w : Valuation R Γ₀'
hval : Valued R Γ₀'
hv : Valued.v = w
h : w.IsEquiv v
γ : (ofClass Valued.v).ValueGroup₀ˣ
r s : WithVal v
hr₀ : 0 < Valued.v r
hs₀ : 0 < Valued.v s
hr : Valued.v.restrict r / Valued.v.restrict s = ↑γ
h' : w.restrict.IsEquiv v.restrict
⊢ Valued.v.restrict ((equiv v) r) / Valued.v.restrict ((equiv v) s) ≠ 0

Tactic:
by

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).dblY (⇑f ∘ P) = f ((W'⁄A).dblY P)

Tactic:
by

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

Goal state before:
pp: Γ : Type u_1
inst✝³ : LinearOrderedCommGroupWithZero Γ
K : Type u_2
inst✝² : Field K
v : Valuation K Γ
inst✝¹ : IsCyclic ↥(MonoidWithZeroHom.ofClass v).valueGroup
inst✝ : Nontrivial ↥(MonoidWithZeroHom.ofClass v).valueGroup
⊢ maximalIdeal ↥v.valuationSubring ≠ ⊥

Tactic:
by

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).dblXYZ (⇑f ∘ P) = ⇑f ∘ (W'⁄A).dblXYZ P

Tactic:
by

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 Q : Fin 3 → A
⊢ (W'⁄B).addX (⇑f ∘ P) (⇑f ∘ Q) = f ((W'⁄A).addX P Q)

Tactic:
by

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 Q : Fin 3 → A
⊢ (W'⁄B).negAddY (⇑f ∘ P) (⇑f ∘ Q) = f ((W'⁄A).negAddY P Q)

Tactic:
by

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 Q : Fin 3 → A
⊢ (W'⁄B).addY (⇑f ∘ P) (⇑f ∘ Q) = f ((W'⁄A).addY P Q)

Tactic:
by

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 Q : Fin 3 → A
⊢ (W'⁄B).addXYZ (⇑f ∘ P) (⇑f ∘ Q) = ⇑f ∘ (W'⁄A).addXYZ P Q

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing

Goal state before:
pp: A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ IsLocalRing.maximalIdeal A ≠ ⊥

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing

Goal state before:
pp: A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
v : Valuation K (WithZero (Multiplicative ℤ)) := valuation K (maximalIdeal A)
π : K := ⋯.choose
hπ : v π = ↑(ofAdd (-1))
⊢ v π ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing

Goal state before:
pp: A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ Nontrivial ↥(MonoidWithZeroHom.ofClass (valuation K (maximalIdeal A))).valueGroup

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing

Goal state before:
pp: A : Type u_1
K : Type u_2
inst✝⁵ : CommRing A
inst✝⁴ : IsDomain A
inst✝³ : IsDiscreteValuationRing A
inst✝² : Field K
inst✝¹ : Algebra A K
inst✝ : IsFractionRing A K
⊢ (valuation K (maximalIdeal A)).IsRankOneDiscrete

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation K Γ₀
x y : K
γ : Γ₀ˣ
y_ne : y ≠ 0
h : v (x - y) < min (↑γ * (v y * v y)) (v y)
hyp1 : v (x - y) < ↑γ * (v y * v y)
hyp1' : v (x - y) * (v y * v y)⁻¹ < ↑γ
hyp2 : v (x - y) < v y
key : v x = v y
⊢ x ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: R : Type u_4
Γ₀ : Type u_5
Γ₀' : Type u_6
inst✝² : Ring R
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ₀'
v : Valuation R Γ₀
w : Valuation R Γ₀'
hval : Valued R Γ₀'
hv : Valued.v = w
h : w.IsEquiv v
⊢ UniformContinuous ⇑(equiv v).symm

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation K Γ₀
x y : K
γ : Γ₀ˣ
y_ne : y ≠ 0
h : v (x - y) < min (↑γ * (v y * v y)) (v y)
hyp1 : v (x - y) < ↑γ * (v y * v y)
hyp1' : v (x - y) * (v y * v y)⁻¹ < ↑γ
hyp2 : v (x - y) < v y
key : v x = v y
x_ne : x ≠ 0
⊢ x⁻¹ - y⁻¹ = x⁻¹ * (y - x) * y⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation K Γ₀
x y : K
γ : Γ₀ˣ
y_ne : y ≠ 0
h : v (x - y) < min (↑γ * (v y * v y)) (v y)
hyp1 : v (x - y) < ↑γ * (v y * v y)
hyp1' : v (x - y) * (v y * v y)⁻¹ < ↑γ
hyp2 : v (x - y) < v y
key : v x = v y
x_ne : x ≠ 0
decomp : x⁻¹ - y⁻¹ = x⁻¹ * (y - x) * y⁻¹
⊢ v (x⁻¹ - y⁻¹) = v (x⁻¹ * (y - x) * y⁻¹)

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.WithVal

Goal state before:
pp: R : Type u_4
Γ₀ : Type u_5
Γ₀' : Type u_6
inst✝² : Ring R
inst✝¹ : LinearOrderedCommGroupWithZero Γ₀
inst✝ : LinearOrderedCommGroupWithZero Γ₀'
v : Valuation R Γ₀
w : Valuation R Γ₀'
h : v.IsEquiv w
h_val : Valued.v.IsEquiv Valued.v
h_res : v.restrict.IsEquiv w.restrict
x : (ofClass Valued.v).ValueGroup₀ˣ
u : ↥(ofClass Valued.v).valueGroup := WithZero.unzero ⋯
a : R
ha : (ofClass Valued.v) a ≠ 0
y : R
hu : (ofClass Valued.v) a * ↑↑u = (ofClass Valued.v) y
y₀ : (ofClass Valued.v).ValueGroup₀ := h_val.orderMonoidIso ↑x
hy₀_def : y₀ = h_val.orderMonoidIso ↑x
⊢ y₀ ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation K Γ₀
x y : K
γ : Γ₀ˣ
y_ne : y ≠ 0
h : v (x - y) < min (↑γ * (v y * v y)) (v y)
hyp1 : v (x - y) < ↑γ * (v y * v y)
hyp1' : v (x - y) * (v y * v y)⁻¹ < ↑γ
hyp2 : v (x - y) < v y
key : v x = v y
x_ne : x ≠ 0
decomp : x⁻¹ - y⁻¹ = x⁻¹ * (y - x) * y⁻¹
⊢ v (x⁻¹ * (y - x) * y⁻¹) = v x⁻¹ * v (y - x) * v y⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.Topology.Algebra.Valued.ValuedField

Goal state before:
pp: K : Type u_1
inst✝¹ : DivisionRing K
Γ₀ : Type u_2
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation K Γ₀
x y : K
γ : Γ₀ˣ
y_ne : y ≠ 0
h : v (x - y) < min (↑γ * (v y * v y)) (v y)
hyp1 : v (x - y) < ↑γ * (v y * v y)
hyp1' : v (x - y) * (v y * v y)⁻¹ < ↑γ
hyp2 : v (x - y) < v y
key : v x = v y
x_ne : x ≠ 0
decomp : x⁻¹ - y⁻¹ = x⁻¹ * (y - x) * y⁻¹
⊢ v x⁻¹ * v (y - x) * v y⁻¹ = (v x)⁻¹ * v (y - x) * (v y)⁻¹

Tactic:
by

Goal state after: