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

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_3
P : Type u_4
inst✝³ : AddCommGroup N
inst✝² : AddCommGroup P
inst✝¹ : Module R N
inst✝ : Module R P
f : N →ₗ[R] M
g : M →ₗ[R] P
hf : Function.Injective ⇑f
hg : Function.Surjective ⇑g
H : Function.Exact ⇑f ⇑g
hP : IsFiniteLength R P
hN : IsFiniteLength R N
s : CompositionSeries (Submodule R P)
hs₁ : RelSeries.head s = ⊥
hs₂ : RelSeries.last s = ⊤
t : CompositionSeries (Submodule R N)
ht₁ : RelSeries.head t = ⊥
ht₂ : RelSeries.last t = ⊤
s' : CompositionSeries (Submodule R M) := RelSeries.map s { toFun := Submodule.comap g, map_rel' := ⋯ }
t' : CompositionSeries (Submodule R M) := RelSeries.map t { toFun := Submodule.map f, map_rel' := ⋯ }
⊢ Submodule.map f ⊤ = Submodule.comap g ⊥

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_3
P : Type u_4
inst✝³ : AddCommGroup N
inst✝² : AddCommGroup P
inst✝¹ : Module R N
inst✝ : Module R P
f : N →ₗ[R] M
g : M →ₗ[R] P
hf : Function.Injective ⇑f
hg : Function.Surjective ⇑g
H : Function.Exact ⇑f ⇑g
hP : IsFiniteLength R P
hN : IsFiniteLength R N
s : CompositionSeries (Submodule R P)
hs₁ : RelSeries.head s = ⊥
hs₂ : RelSeries.last s = ⊤
t : CompositionSeries (Submodule R N)
ht₁ : RelSeries.head t = ⊥
ht₂ : RelSeries.last t = ⊤
s' : CompositionSeries (Submodule R M) := RelSeries.map s { toFun := Submodule.comap g, map_rel' := ⋯ }
t' : CompositionSeries (Submodule R M) := RelSeries.map t { toFun := Submodule.map f, map_rel' := ⋯ }
hfg : Submodule.map f ⊤ = Submodule.comap g ⊥
⊢ RelSeries.last t' = RelSeries.head s'

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_3
P : Type u_4
inst✝³ : AddCommGroup N
inst✝² : AddCommGroup P
inst✝¹ : Module R N
inst✝ : Module R P
f : N →ₗ[R] M
g : M →ₗ[R] P
hf : Function.Injective ⇑f
hg : Function.Surjective ⇑g
H : Function.Exact ⇑f ⇑g
hP : IsFiniteLength R P
hN : IsFiniteLength R N
s : CompositionSeries (Submodule R P)
hs₁ : RelSeries.head s = ⊥
hs₂ : RelSeries.last s = ⊤
t : CompositionSeries (Submodule R N)
ht₁ : RelSeries.head t = ⊥
ht₂ : RelSeries.last t = ⊤
s' : CompositionSeries (Submodule R M) := RelSeries.map s { toFun := Submodule.comap g, map_rel' := ⋯ }
t' : CompositionSeries (Submodule R M) := RelSeries.map t { toFun := Submodule.map f, map_rel' := ⋯ }
hfg : Submodule.map f ⊤ = Submodule.comap g ⊥
r : RelSeries {(x, y) | JordanHolderLattice.IsMaximal x y} := RelSeries.smash t' s' ⋯
⊢ r.head = ⊥

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_3
P : Type u_4
inst✝³ : AddCommGroup N
inst✝² : AddCommGroup P
inst✝¹ : Module R N
inst✝ : Module R P
f : N →ₗ[R] M
g : M →ₗ[R] P
hf : Function.Injective ⇑f
hg : Function.Surjective ⇑g
H : Function.Exact ⇑f ⇑g
hP : IsFiniteLength R P
hN : IsFiniteLength R N
s : CompositionSeries (Submodule R P)
hs₁ : RelSeries.head s = ⊥
hs₂ : RelSeries.last s = ⊤
t : CompositionSeries (Submodule R N)
ht₁ : RelSeries.head t = ⊥
ht₂ : RelSeries.last t = ⊤
s' : CompositionSeries (Submodule R M) := RelSeries.map s { toFun := Submodule.comap g, map_rel' := ⋯ }
t' : CompositionSeries (Submodule R M) := RelSeries.map t { toFun := Submodule.map f, map_rel' := ⋯ }
hfg : Submodule.map f ⊤ = Submodule.comap g ⊥
r : RelSeries {(x, y) | JordanHolderLattice.IsMaximal x y} := RelSeries.smash t' s' ⋯
⊢ r.last = ⊤

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Bezout

Goal state before:
pp: R : Type u
inst✝ : CommRing R
⊢ IsBezout R ↔ ∀ (x y : R), Submodule.IsPrincipal (Ideal.span {x, y})

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Bezout

Goal state before:
pp: R : Type u
inst✝² : CommRing R
S : Type v
inst✝¹ : CommRing S
f : R →+* S
hf : Function.Surjective ⇑f
inst✝ : IsBezout R
⊢ IsBezout S

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Bezout

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : IsBezout R
inst✝ : IsDomain R
tfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R
tfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R
tfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R
h : WellFounded DvdNotUnit
f : { J // J.FG } → R
hf : ∀ (I : { J // J.FG }), ↑I = Ideal.span {f I}
x y : { N // N.FG }
e : f x = f y
⊢ x = y

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
N : Type u_3
P : Type u_4
inst✝³ : AddCommGroup N
inst✝² : AddCommGroup P
inst✝¹ : Module R N
inst✝ : Module R P
f : N →ₗ[R] M
g : M →ₗ[R] P
hf : Function.Injective ⇑f
hg : Function.Surjective ⇑g
H : Function.Exact ⇑f ⇑g
⊢ length R M = length R N + length R P

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
N : Type u_3
inst✝¹ : AddCommGroup N
inst✝ : Module R N
f : N →ₗ[R] M
hf : Function.Injective ⇑f
⊢ length R N ≤ length R M

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
P : Type u_4
inst✝¹ : AddCommGroup P
inst✝ : Module R P
g : M →ₗ[R] P
hg : Function.Surjective ⇑g
⊢ length R P ≤ length R M

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Bezout

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : IsBezout R
inst✝ : IsDomain R
tfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R
tfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R
tfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R
h : WellFounded DvdNotUnit
f : { J // J.FG } → R
hf : ∀ (I : { J // J.FG }), ↑I = Ideal.span {f I}
⊢ ∀ {a b : { N // N.FG }}, DvdNotUnit ({ toFun := f, inj' := ⋯ } a) ({ toFun := f, inj' := ⋯ } b) ↔ b < a

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Bezout

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : IsBezout R
inst✝ : IsDomain R
tfae_1_to_2 : IsNoetherianRing R → IsPrincipalIdealRing R
tfae_2_to_3 : IsPrincipalIdealRing R → UniqueFactorizationMonoid R
tfae_3_to_4 : UniqueFactorizationMonoid R → WfDvdMonoid R
h : WellFounded DvdNotUnit
⊢ IsNoetherianRing R

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Bezout

Goal state before:
pp: R : Type u
inst✝² : CommRing R
inst✝¹ : IsBezout R
inst✝ : IsDomain R
⊢ [IsNoetherianRing R, IsPrincipalIdealRing R, UniqueFactorizationMonoid R, WfDvdMonoid R].TFAE

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: case h_empty
R : Type u_1
inst✝² : Ring R
M : PEmpty.{u_5 + 1} → Type u_6
inst✝¹ : (i : PEmpty.{u_5 + 1}) → AddCommGroup (M i)
inst✝ : (i : PEmpty.{u_5 + 1}) → Module R (M i)
⊢ length R ((i : PEmpty.{u_5 + 1}) → M i) = ∑ i, length R (M i)

Tactic:
simp [Module.length_eq_zero]

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: R : Type u_1
inst✝ : Ring R
⊢ ∀ {ι : Type u_5} [inst : Fintype ι] (M : ι → Type u_6) [inst_1 : (i : ι) → AddCommGroup (M i)]
    [inst_2 : (i : ι) → Module R (M i)], length R ((i : ι) → M i) = ∑ i, length R (M i)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
ι : Type u_5
h✝ : Infinite ι
a✝ : Nontrivial M
m : ℕ
s : Finset ι
hs : s.card = m
⊢ length R (↥s →₀ M) = ↑m * length R M

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
ι : Type u_5
⊢ length R (ι →₀ M) = ENat.card ι * length R M

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
ι : Type u_5
⊢ length R (ι → M) = ENat.card ι * length R M

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝³ : Ring R
inst✝² : AddCommGroup M
inst✝¹ : Module R M
inst✝ : Free R M
⊢ length R M = Cardinal.toENat (Module.rank R M) * length R R

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁵ : Ring R
inst✝⁴ : AddCommGroup M
inst✝³ : Module R M
inst✝² : StrongRankCondition R
inst✝¹ : Free R M
inst✝ : Module.Finite R M
⊢ length R M = ↑(finrank R M) * length R R

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝² : Ring R
inst✝¹ : AddCommGroup M
inst✝ : Module R M
⊢ length R M = 1 ↔ IsSimpleModule R M

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: K : Type u_5
M : Type u_6
inst✝² : DivisionRing K
inst✝¹ : AddCommGroup M
inst✝ : Module K M
⊢ length K M = Cardinal.toENat (Module.rank K M)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: K : Type u_5
M : Type u_6
inst✝³ : DivisionRing K
inst✝² : AddCommGroup M
inst✝¹ : Module K M
inst✝ : Module.Finite K M
⊢ length K M = ↑(finrank K M)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁶ : Ring R
inst✝⁵ : AddCommGroup M
inst✝⁴ : Module R M
A : Type u_5
inst✝³ : Ring A
inst✝² : SMul A R
inst✝¹ : Module A M
inst✝ : IsScalarTower A R M
p : Submodule R M
⊢ Module.length R ↥p ≤ Module.length A ↥(restrictScalars A p)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Length

Goal state before:
pp: R : Type u_1
M : Type u_2
inst✝⁴ : Ring R
inst✝³ : AddCommGroup M
inst✝² : Module R M
inst✝¹ : IsArtinian R M
inst✝ : IsNoetherian R M
p : Submodule R M
h : p ≠ ⊥
⊢ Module.length R (M ⧸ p) < Module.length R M

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: case h
A : Type u
inst✝² : CommRing A
K : Type v
inst✝¹ : Field K
inst✝ : Algebra A K
x y : ValueGroup A K
a b : K
c d : Aˣ
e : A
he : e • b = a
⊢ (↑c * e * ↑d⁻¹) • d • b = c • a

Tactic:
simp_rw [Units.smul_def, ← he, mul_smul]

Goal state after:
case h
A : Type u
inst✝² : CommRing A
K : Type v
inst✝¹ : Field K
inst✝ : Algebra A K
x y : ValueGroup A K
a b : K
c d : Aˣ
e : A
he : e • b = a
⊢ ↑c • e • ↑d⁻¹ • ↑d • b = ↑c • e • b
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: A : Type u
inst✝² : CommRing A
K : Type v
inst✝¹ : Field K
inst✝ : Algebra A K
x y : ValueGroup A K
⊢ ∀ (a₁ a₂ b₁ b₂ : K),
    (MulAction.orbitRel Aˣ K) a₁ b₁ → (MulAction.orbitRel Aˣ K) a₂ b₂ → (∃ c, c • a₂ = a₁) = ∃ c, c • b₂ = b₁

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: A : Type u
inst✝² : CommRing A
K : Type v
inst✝¹ : Field K
inst✝ : Algebra A K
x y : ValueGroup A K
⊢ ∀ (a₁ a₂ b₁ b₂ : K),
    (MulAction.orbitRel Aˣ K) a₁ b₁ →
      (MulAction.orbitRel Aˣ K) a₂ b₂ → Quotient.mk'' (a₁ * a₂) = Quotient.mk'' (b₁ * b₂)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: A : Type u
inst✝² : CommRing A
K : Type v
inst✝¹ : Field K
inst✝ : Algebra A K
x y : ValueGroup A K
⊢ ∀ (a₁ a₂ b₁ b₂ : K),
    (MulAction.orbitRel Aˣ K) a₁ b₁ →
      (MulAction.orbitRel Aˣ K) a₂ b₂ → Quotient.mk'' (a₁ * a₂) = Quotient.mk'' (b₁ * b₂)

Tactic:
by
        rintro _ _ a b ⟨c, rfl⟩ ⟨d, rfl⟩
        apply Quotient.sound'
        dsimp
        use c * d
        simp only [mul_smul, Algebra.smul_def, Units.smul_def]
        ring

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: A : Type u
inst✝² : CommRing A
K : Type v
inst✝¹ : Field K
inst✝ : Algebra A K
x : ValueGroup A K
⊢ ∀ (a b : K), (MulAction.orbitRel Aˣ K) a b → Quotient.mk'' a⁻¹ = Quotient.mk'' b⁻¹

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: A : Type u
inst✝² : CommRing A
K : Type v
inst✝¹ : Field K
inst✝ : Algebra A K
c : 0 = 1
⊢ False

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Integers

Goal state before:
pp: R : Type u
Γ₀ : Type v
inst✝¹ : Ring R
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
x y : R
hx : x ∈ {x | v x ≤ 1}
hy : y ∈ {x | v x ≤ 1}
⊢ x * y ∈ {x | v x ≤ 1}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Integers

Goal state before:
pp: R : Type u
Γ₀ : Type v
inst✝¹ : Ring R
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
⊢ 0 ∈ {x | v x ≤ 1}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Integers

Goal state before:
pp: R : Type u
Γ₀ : Type v
inst✝¹ : Ring R
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
x : R
hx : x ∈ {x | v x ≤ 1}
⊢ -x ∈ {x | v x ≤ 1}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Integers

Goal state before:
pp: R : Type u
Γ₀ : Type v
inst✝¹ : Ring R
inst✝ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
r : R
⊢ r ∈ v.integer ↔ v r ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Integers

Goal state before:
pp: R : Type u
Γ₀ : Type v
inst✝³ : CommRing R
inst✝² : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
O : Type w
inst✝¹ : CommRing O
inst✝ : Algebra O R
x : O
hx : IsUnit x
H : ∀ (x : O), v ((algebraMap O R) x) ≤ 1
u : Oˣ
hu : ↑u = x
⊢ 1 ≤ v ((algebraMap O R) x)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Integers

Goal state before:
pp: R : Type u
Γ₀ : Type v
inst✝³ : CommRing R
inst✝² : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
O : Type w
inst✝¹ : CommRing O
inst✝ : Algebra O R
hv : v.Integers O
x : O
hx : IsUnit ((algebraMap O R) x)
hvx : v ((algebraMap O R) x) = 1
u : Rˣ
hu : ↑u = (algebraMap O R) x
h1 : v ↑u ≤ 1
⊢ v ↑u⁻¹ ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Integers

Goal state before:
pp: R : Type u
Γ₀ : Type v
inst✝³ : CommRing R
inst✝² : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
O : Type w
inst✝¹ : CommRing O
inst✝ : Algebra O R
hv : v.Integers O
x : O
hx : IsUnit ((algebraMap O R) x)
hvx : v ((algebraMap O R) x) = 1
u : Rˣ
hu : ↑u = (algebraMap O R) x
h1 : v ↑u ≤ 1
h2 : v ↑u⁻¹ ≤ 1
r1 : O
hr1 : (algebraMap O R) r1 = ↑u
r2 : O
hr2 : (algebraMap O R) r2 = ↑u⁻¹
⊢ (algebraMap O R) (r1 * r2) = (algebraMap O R) 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: case mk.mk.inl
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
a b : ValueGroup A K
xa ya : A
hya : ya ∈ nonZeroDivisors A
xb yb : A
hyb : yb ∈ nonZeroDivisors A
this✝ : (algebraMap A K) ya ≠ 0
this : (algebraMap A K) yb ≠ 0
c : A
h : xa * yb * c = xb * ya
⊢ Quot.mk (⇑(MulAction.orbitRel Aˣ K)) ((algebraMap A K) xa / (algebraMap A K) ya) ≤
      Quot.mk (⇑(MulAction.orbitRel Aˣ K)) ((algebraMap A K) xb / (algebraMap A K) yb) ∨
    Quot.mk (⇑(MulAction.orbitRel Aˣ K)) ((algebraMap A K) xb / (algebraMap A K) yb) ≤
      Quot.mk (⇑(MulAction.orbitRel Aˣ K)) ((algebraMap A K) xa / (algebraMap A K) ya)

pp: case mk.mk.inr
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
a b : ValueGroup A K
xa ya : A
hya : ya ∈ nonZeroDivisors A
xb yb : A
hyb : yb ∈ nonZeroDivisors A
this✝ : (algebraMap A K) ya ≠ 0
this : (algebraMap A K) yb ≠ 0
c : A
h : xb * ya * c = xa * yb
⊢ Quot.mk (⇑(MulAction.orbitRel Aˣ K)) ((algebraMap A K) xa / (algebraMap A K) ya) ≤
      Quot.mk (⇑(MulAction.orbitRel Aˣ K)) ((algebraMap A K) xb / (algebraMap A K) yb) ∨
    Quot.mk (⇑(MulAction.orbitRel Aˣ K)) ((algebraMap A K) xb / (algebraMap A K) yb) ≤
      Quot.mk (⇑(MulAction.orbitRel Aˣ K)) ((algebraMap A K) xa / (algebraMap A K) ya)

Tactic:
· right
    use c
    rw [Algebra.smul_def]
    field_simp
    simp only [← map_mul]; congr 1; linear_combination h

Goal state after:
case mk.mk.inr
A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
a b : ValueGroup A K
xa ya : A
hya : ya ∈ nonZeroDivisors A
xb yb : A
hyb : yb ∈ nonZeroDivisors A
this✝ : (algebraMap A K) ya ≠ 0
this : (algebraMap A K) yb ≠ 0
c : A
h : xb * ya * c = xa * yb
⊢ Quot.mk (⇑(MulAction.orbitRel Aˣ K)) ((algebraMap A K) xa / (algebraMap A K) ya) ≤
      Quot.mk (⇑(MulAction.orbitRel Aˣ K)) ((algebraMap A K) xb / (algebraMap A K) yb) ∨
    Quot.mk (⇑(MulAction.orbitRel Aˣ K)) ((algebraMap A K) xb / (algebraMap A K) yb) ≤
      Quot.mk (⇑(MulAction.orbitRel Aˣ K)) ((algebraMap A K) xa / (algebraMap A K) ya)
Module: Mathlib.RingTheory.Valuation.Integers

Goal state before:
pp: R : Type u
Γ₀ : Type v
inst✝³ : CommRing R
inst✝² : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
O : Type w
inst✝¹ : CommRing O
inst✝ : Algebra O R
hv : v.Integers O
x : O
hx : IsUnit ((algebraMap O R) x)
hvx : v ((algebraMap O R) x) = 1
u : Rˣ
hu : ↑u = (algebraMap O R) x
h1 : v ↑u ≤ 1
h2 : v ↑u⁻¹ ≤ 1
r1 : O
hr1 : (algebraMap O R) r1 = ↑u
r2 : O
hr2 : (algebraMap O R) r2 = ↑u⁻¹
⊢ (algebraMap O R) (r2 * r1) = (algebraMap O R) 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Integers

Goal state before:
pp: R : Type u
Γ₀ : Type v
inst✝³ : CommRing R
inst✝² : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
O : Type w
inst✝¹ : CommRing O
inst✝ : Algebra O R
hv : v.Integers O
x y : O
h : x ∣ y
⊢ v ((algebraMap O R) y) ≤ v ((algebraMap O R) x)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Integers

Goal state before:
pp: R : Type u
Γ₀ : Type v
inst✝³ : CommRing R
inst✝² : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
O : Type w
inst✝¹ : CommRing O
inst✝ : Algebra O R
hv : v.Integers O
h : Nontrivial R
⊢ v 0 ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Integers

Goal state before:
pp: R : Type u
Γ₀ : Type v
inst✝³ : CommRing R
inst✝² : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
O : Type w
inst✝¹ : CommRing O
inst✝ : Algebra O R
hv : v.Integers O
h : Nontrivial R
o0 : O
ho0 : (algebraMap O R) o0 = 0
⊢ v 1 ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Integers

Goal state before:
pp: R : Type u
Γ₀ : Type v
inst✝³ : CommRing R
inst✝² : LinearOrderedCommGroupWithZero Γ₀
v : Valuation R Γ₀
O : Type w
inst✝¹ : CommRing O
inst✝ : Algebra O R
hv : v.Integers O
⊢ Nontrivial O ↔ Nontrivial R

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Integers

Goal state before:
pp: F : Type u
Γ₀ : Type v
inst✝³ : Field F
inst✝² : LinearOrderedCommGroupWithZero Γ₀
v : Valuation F Γ₀
O : Type w
inst✝¹ : CommRing O
inst✝ : Algebra O F
hv : v.Integers O
x y : O
h : v ((algebraMap O F) x) ≤ v ((algebraMap O F) y)
hy : (algebraMap O F) y ≠ 0
⊢ v (((algebraMap O F) y)⁻¹ * (algebraMap O F) x) ≤ 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Integers

Goal state before:
pp: F : Type u
Γ₀ : Type v
inst✝³ : Field F
inst✝² : LinearOrderedCommGroupWithZero Γ₀
v : Valuation F Γ₀
O : Type w
inst✝¹ : CommRing O
inst✝ : Algebra O F
hv : v.Integers O
x : O
hvx : v ((algebraMap O F) x) = 1
⊢ IsUnit x

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Integers

Goal state before:
pp: F : Type u
Γ₀ : Type v
inst✝³ : Field F
inst✝² : LinearOrderedCommGroupWithZero Γ₀
v : Valuation F Γ₀
O : Type w
inst✝¹ : CommRing O
inst✝ : Algebra O F
hv : v.Integers O
x : Oˣ
⊢ v ((algebraMap O F) ↑x) = 1

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Integers

Goal state before:
pp: F : Type u
Γ₀ : Type v
inst✝³ : Field F
inst✝² : LinearOrderedCommGroupWithZero Γ₀
v : Valuation F Γ₀
O : Type w
inst✝¹ : CommRing O
inst✝ : Algebra O F
hv : v.Integers O
x : O
⊢ 0 < v ((algebraMap O F) x) ↔ x ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
a b : ValueGroup A K
⊢ a ≤ b ∨ b ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
⊢ ∀ (a : ValueGroup A K), a ≤ a

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Integers

Goal state before:
pp: F : Type u
Γ₀ : Type v
inst✝³ : Field F
inst✝² : LinearOrderedCommGroupWithZero Γ₀
v : Valuation F Γ₀
O : Type w
inst✝¹ : CommRing O
inst✝ : Algebra O F
hv : v.Integers O
x y : O
⊢ DvdNotUnit x y ↔ v ((algebraMap O F) y) < v ((algebraMap O F) x)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
⊢ ∀ (a b c : ValueGroup A K), a ≤ b → b ≤ c → a ≤ c

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Integers

Goal state before:
pp: F : Type u
Γ₀ : Type v
inst✝³ : Field F
inst✝² : LinearOrderedCommGroupWithZero Γ₀
v : Valuation F Γ₀
O : Type w
inst✝¹ : CommRing O
inst✝ : Algebra O F
hv : v.Integers O
x : F
⊢ ∃ a, x = (algebraMap O F) a ∨ x⁻¹ = (algebraMap O F) a

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
a✝ b✝ : ValueGroup A K
b : K
e f : A
hf : f • e • b = b
hb : ¬b = 0
⊢ IsUnit e

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
⊢ ∀ (a b : ValueGroup A K), a ≤ b → b ≤ a → a = b

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Integers

Goal state before:
pp: F : Type u
Γ₀ : Type v
inst✝³ : Field F
inst✝² : LinearOrderedCommGroupWithZero Γ₀
v : Valuation F Γ₀
O : Type w
inst✝¹ : CommRing O
inst✝ : Algebra O F
hv : v.Integers O
x : O
⊢ ↑(Ideal.span {x}) = {y | v ((algebraMap O F) y) ≤ v ((algebraMap O F) x)}

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
⊢ ∀ (a b c : ValueGroup A K), a * b * c = a * (b * c)

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
⊢ ∀ (a : ValueGroup A K), 1 * a = a

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
⊢ ∀ (a : ValueGroup A K), a * 1 = a

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.ValuationRing

Goal state before:
pp: A : Type u
inst✝⁵ : CommRing A
K : Type v
inst✝⁴ : Field K
inst✝³ : Algebra A K
inst✝² : IsDomain A
inst✝¹ : ValuationRing A
inst✝ : IsFractionRing A K
⊢ ∀ (a b : ValueGroup A K), a * b = b * a

Tactic:
by

Goal state after:
Module: Mathlib.RingTheory.Valuation.Integers

Goal state before:
pp: F : Type u
Γ₀ : Type v
inst✝⁴ : Field F
inst✝³ : LinearOrderedCommGroupWithZero Γ₀
v : Valuation F Γ₀
O : Type w
inst✝² : CommRing O
inst✝¹ : Algebra O F
hv : v.Integers O
inst✝ : Subsingleton (↥(MonoidHom.mrange v))ˣ
⊢ Function.Bijective ⇑(algebraMap O F)

Tactic:
by

Goal state after: