mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: case coe.inr
a x✝ : ℝ≥0
h : a < x✝
ha : a ≠ 0
⊢ x✝⁻¹ < a⁻¹

Tactic:
]

Goal state after:
case coe.inr
a x✝ : ℝ≥0
h : a < x✝
ha : a ≠ 0
⊢ x✝⁻¹ < a⁻¹
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a : ℝ≥0∞
⊢ 1 ≤ a ↔ 1 ≤ a

Tactic:
]

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a : ℝ≥0∞
⊢ a ≤ 1 ↔ a ≤ 1

Tactic:
]

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a : ℝ≥0∞
⊢ 1 < a ↔ 1 < a

Tactic:
]

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a : ℝ≥0∞
⊢ a < 1 ↔ a < 1

Tactic:
]

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a : ℝ≥0∞
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: case inr.inr
a b : ℝ≥0∞
h₀ : a = 0 → b = 0
h₁ : a = ∞ → b = 0
ha : a ≠ 0
ha' : a ≠ ∞
⊢ b = b

Tactic:
]

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
⊢ -count K v I = -count K v I

Tactic:
]

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: case top
a : ℝ≥0∞
c : ℝ≥0
h0 : ∞ ≠ 0 ∨ ↑c ≠ 0
ht : ∞ ≠ ∞ ∨ ↑c ≠ ∞
⊢ a = 0 ↔ a * ∞ ≤ ↑c

Tactic:
]

Goal state after:
case top
a : ℝ≥0∞
c : ℝ≥0
h0 : ∞ ≠ 0 ∨ ↑c ≠ 0
ht : ∞ ≠ ∞ ∨ ↑c ≠ ∞
⊢ a = 0 ↔ a * ∞ ≤ ↑c
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: case coe.inr
a c : ℝ≥0∞
b : ℝ≥0
h0 : ↑b ≠ 0 ∨ c ≠ 0
ht : ↑b ≠ ∞ ∨ c ≠ ∞
hb : ↑b ≠ 0
⊢ a ≤ c / ↑b ↔ a * ↑b ≤ c

Tactic:
]

Goal state after:
case coe.inr
a c : ℝ≥0∞
b : ℝ≥0
h0 : ↑b ≠ 0 ∨ c ≠ 0
ht : ↑b ≠ ∞ ∨ c ≠ ∞
hb : ↑b ≠ 0
⊢ a ≤ c / ↑b ↔ a * ↑b ≤ c
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: case coe.inr
a c : ℝ≥0∞
b : ℝ≥0
h0 : ↑b ≠ 0 ∨ c ≠ 0
ht : ↑b ≠ ∞ ∨ c ≠ ∞
hb : ↑b ≠ 0
⊢ a * ↑b ≤ c ↔ a * ↑b ≤ c

Tactic:
]

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

Goal state before:
pp: case ofNat
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
n : ℕ
⊢ count K v (I ^ n) = ↑n * count K v I

Tactic:
]

Goal state after:
case ofNat
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
n : ℕ
⊢ count K v (I ^ n) = ↑n * count K v I
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a : ℝ≥0∞
⊢ a ≤ a

Tactic:
]

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b c : ℝ≥0∞
h : a ≤ b / c
⊢ a * c⁻¹⁻¹ ≤ b

Tactic:
]

Goal state after:
a b c : ℝ≥0∞
h : a ≤ b / c
⊢ a * c⁻¹⁻¹ ≤ b
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b c : ℝ≥0∞
h : a ≤ b / c
⊢ a * c ≤ b

Tactic:
rw [← inv_inv c]
  exact div_le_of_le_mul h

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b c : ℝ≥0∞
h : a < b * c
⊢ a < b * c

Tactic:
]

Goal state after:
a b c : ℝ≥0∞
h : a < b * c
⊢ a < b * c
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: case negSucc
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
n : ℕ
⊢ -(↑n.succ * count K v I) = -↑n.succ * count K v I

Tactic:
]

Goal state after:
case negSucc
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
I : FractionalIdeal R⁰ K
n : ℕ
⊢ -(↑n.succ * count K v I) = -↑n.succ * count K v I
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
n : ℤ
⊢ n = n

Tactic:
]

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v w : HeightOneSpectrum R
hw : w ≠ v
⊢ ↑w.asIdeal = ↑w.asIdeal

Tactic:
]

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b : ℝ≥0∞
h₁ : b = ∞ → a ≠ 0
h₂ : a = ∞ → b ≠ 0
⊢ 1 ≤ a * b ↔ 1 ≤ a * b

pp: case hb0
a b : ℝ≥0∞
h₁ : b = ∞ → a ≠ 0
h₂ : a = ∞ → b ≠ 0
⊢ a ≠ 0 ∨ b ≠ ∞

pp: case hbt
a b : ℝ≥0∞
h₁ : b = ∞ → a ≠ 0
h₂ : a = ∞ → b ≠ 0
⊢ a ≠ ∞ ∨ b ≠ 0

Tactic:
]

Goal state after:
case hb0
a b : ℝ≥0∞
h₁ : b = ∞ → a ≠ 0
h₂ : a = ∞ → b ≠ 0
⊢ a ≠ 0 ∨ b ≠ ∞

case hbt
a b : ℝ≥0∞
h₁ : b = ∞ → a ≠ 0
h₂ : a = ∞ → b ≠ 0
⊢ a ≠ ∞ ∨ b ≠ 0
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b : ℝ≥0∞
⊢ a * b ≤ 1 ↔ a * b ≤ 1

pp: case h0
a b : ℝ≥0∞
⊢ b ≠ 0 ∨ 1 ≠ 0

pp: case ht
a b : ℝ≥0∞
⊢ b ≠ ∞ ∨ 1 ≠ ∞

Tactic:
]

Goal state after:
case h0
a b : ℝ≥0∞
⊢ b ≠ 0 ∨ 1 ≠ 0

case ht
a b : ℝ≥0∞
⊢ b ≠ ∞ ∨ 1 ≠ ∞
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b : ℝ≥0∞
h : a * b = 1
⊢ b⁻¹ = b⁻¹

pp: a b : ℝ≥0∞
h : a * b = 1
⊢ b ≠ ∞

Tactic:
]

Goal state after:
a b : ℝ≥0∞
h : a * b = 1
⊢ b ≠ ∞
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b r : ℝ≥0∞
hr₀ : r ≠ 0
hr₁ : r ≠ ∞
⊢ r * a ≤ b ↔ r * a ≤ b

Tactic:
]

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b c : ℝ≥0∞
ha : a ≠ 0
ha' : a ≠ ∞
h : b = c / a
⊢ c = c

Tactic:
]

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b c : ℝ≥0∞
ha : a ≠ 0
ha' : a ≠ ∞
h : a * b = c
⊢ b = b

Tactic:
]

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b c d : ℝ≥0∞
ha : a ≠ 0
ha' : a ≠ ∞
hb : b ≠ 0
hb' : b ≠ ∞
⊢ a * (c / b) = d ↔ a * c = b * d

Tactic:
]

Goal state after:
a b c d : ℝ≥0∞
ha : a ≠ 0
ha' : a ≠ ∞
hb : b ≠ 0
hb' : b ≠ ∞
⊢ a * (c / b) = d ↔ a * c = b * d
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v w : HeightOneSpectrum R
hw : w ≠ v
hw_fact : ↑w.asIdeal = spanSingleton R⁰ ((algebraMap R K) 1)⁻¹ * ↑w.asIdeal
hw_ne_zero : ↑w.asIdeal ≠ 0
hv : Irreducible (Associates.mk v.asIdeal)
hw' : Irreducible (Associates.mk w.asIdeal)
⊢ ¬⟨Associates.mk v.asIdeal, hv⟩ = ⟨Associates.mk w.asIdeal, hw'⟩

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v w : HeightOneSpectrum R
hw : w ≠ v
hw_fact : ↑w.asIdeal = spanSingleton R⁰ ((algebraMap R K) 1)⁻¹ * ↑w.asIdeal
hw_ne_zero : ↑w.asIdeal ≠ 0
hv : Irreducible (Associates.mk v.asIdeal)
hw' : Irreducible (Associates.mk w.asIdeal)
⊢ ¬⟨Associates.mk v.asIdeal, hv⟩ = ⟨Associates.mk w.asIdeal, hw'⟩
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b c d : ℝ≥0∞
ha : a ≠ 0
ha' : a ≠ ∞
hb : b ≠ 0
hb' : b ≠ ∞
⊢ a * (c / b) = d ↔ a * c = b * d

Tactic:
conv_rhs => rw [eq_comm]

Goal state after:
a b c d : ℝ≥0∞
ha : a ≠ 0
ha' : a ≠ ∞
hb : b ≠ 0
hb' : b ≠ ∞
⊢ a * (c / b) = d ↔ b * d = a * c
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b c d : ℝ≥0∞
ha : a ≠ 0
ha' : a ≠ ∞
hb : b ≠ 0
hb' : b ≠ ∞
⊢ d = a * (c / b) ↔ d = a * (c / b)

Tactic:
]

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b : ℝ≥0∞
hb₀ : b ≠ 0
hb₁ : b ≠ ∞
h : a / b = 1
⊢ b = b

Tactic:
]

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

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v w : HeightOneSpectrum R
hw : w ≠ v
hw_fact : ↑w.asIdeal = spanSingleton R⁰ ((algebraMap R K) 1)⁻¹ * ↑w.asIdeal
hw_ne_zero : ↑w.asIdeal ≠ 0
hv : Irreducible (Associates.mk v.asIdeal)
hw' : Irreducible (Associates.mk w.asIdeal)
⊢ ¬v = w

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v w : HeightOneSpectrum R
hw : w ≠ v
hw_fact : ↑w.asIdeal = spanSingleton R⁰ ((algebraMap R K) 1)⁻¹ * ↑w.asIdeal
hw_ne_zero : ↑w.asIdeal ≠ 0
hv : Irreducible (Associates.mk v.asIdeal)
hw' : Irreducible (Associates.mk w.asIdeal)
⊢ ¬v = w
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: ⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: ⊢ 3⁻¹ + 3⁻¹ + 3⁻¹ = 3 * 3⁻¹

Tactic:
]

Goal state after:
⊢ 3⁻¹ + 3⁻¹ + 3⁻¹ = 3 * 3⁻¹
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a : ℝ≥0∞
⊢ a = a

Tactic:
]

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

Goal state before:
pp: case pos
R : Type u_1
inst✝⁵ : CommRing R
K : Type u_2
inst✝⁴ : Field K
inst✝³ : Algebra R K
inst✝² : IsFractionRing R K
inst✝¹ : IsDedekindDomain R
v w : HeightOneSpectrum R
inst✝ : Decidable (w = v)
h : w = v
⊢ 1 = 1

Tactic:
]

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b : ℝ≥0∞
⊢ a ≠ 0 ∧ b ≠ ∞ ↔ a ≠ 0 ∧ b ≠ ∞

Tactic:
]

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: b c a : ℝ≥0∞
h0 : b ≠ 0 ∨ c ≠ 0
htop : b ≠ ∞ ∨ c ≠ ∞
⊢ a * b⁻¹ * c = a * (b⁻¹ * c)

pp: case ha
b c a : ℝ≥0∞
h0 : b ≠ 0 ∨ c ≠ 0
htop : b ≠ ∞ ∨ c ≠ ∞
⊢ b ≠ 0 ∨ c⁻¹ ≠ ∞

pp: case hb
b c a : ℝ≥0∞
h0 : b ≠ 0 ∨ c ≠ 0
htop : b ≠ ∞ ∨ c ≠ ∞
⊢ b ≠ ∞ ∨ c⁻¹ ≠ 0

Tactic:
]

Goal state after:
b c a : ℝ≥0∞
h0 : b ≠ 0 ∨ c ≠ 0
htop : b ≠ ∞ ∨ c ≠ ∞
⊢ a * b⁻¹ * c = a * (b⁻¹ * c)

case ha
b c a : ℝ≥0∞
h0 : b ≠ 0 ∨ c ≠ 0
htop : b ≠ ∞ ∨ c ≠ ∞
⊢ b ≠ 0 ∨ c⁻¹ ≠ ∞

case hb
b c a : ℝ≥0∞
h0 : b ≠ 0 ∨ c ≠ 0
htop : b ≠ ∞ ∨ c ≠ ∞
⊢ b ≠ ∞ ∨ c⁻¹ ≠ 0
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: case hp₁
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
exps : HeightOneSpectrum R → ℤ
I I' : FractionalIdeal R⁰ K
⊢ count K v I = 0 → count K v I' = 0 → count K v (I * I') = 0

Tactic:
hI

Goal state after:
case hp₁
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
exps : HeightOneSpectrum R → ℤ
I I' : FractionalIdeal R⁰ K
hI : count K v I = 0
⊢ count K v I' = 0 → count K v (I * I') = 0
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a : ℝ≥0
hz : a ≠ 0
⊢ ↑a / 2 < ↑a

Tactic:
]

Goal state after:
a : ℝ≥0
hz : a ≠ 0
⊢ ↑a / 2 < ↑a
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: case pos
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
exps : HeightOneSpectrum R → ℤ
I I' : FractionalIdeal R⁰ K
hI : count K v I = 0
hI' : count K v I' = 0
h : I ≠ 0 ∧ I' ≠ 0
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a : ℝ≥0
hz : a ≠ 0
⊢ a / 2 < a

pp: a : ℝ≥0
hz : a ≠ 0
⊢ 2 ≠ 0

Tactic:
]

Goal state after:
a : ℝ≥0
hz : a ≠ 0
⊢ a / 2 < a

a : ℝ≥0
hz : a ≠ 0
⊢ 2 ≠ 0
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: case neg
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
exps : HeightOneSpectrum R → ℤ
I I' : FractionalIdeal R⁰ K
hI : count K v I = 0
hI' : count K v I' = 0
h : ¬(I ≠ 0 ∧ I' ≠ 0)
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: ⊢ 1 / 2 = 1 / 2

Tactic:
]

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

Goal state before:
pp: case hp₂
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
exps : HeightOneSpectrum R → ℤ
w : HeightOneSpectrum R
hw : w ∈ fun i ↦ i = v → False
⊢ 0 = 0

Tactic:
]

Goal state after:
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b c : ℝ≥0∞
h₁ : a ≠ 0 ∨ b ≠ ∞
h₂ : a ≠ ∞ ∨ b ≠ 0
h : ∀ a' > a, ∀ b' > b, c ≤ a' * b'
⊢ a⁻¹ * b⁻¹ ≤ c⁻¹

Tactic:
]

Goal state after:
a b c : ℝ≥0∞
h₁ : a ≠ 0 ∨ b ≠ ∞
h₂ : a ≠ ∞ ∨ b ≠ 0
h : ∀ a' > a, ∀ b' > b, c ≤ a' * b'
⊢ a⁻¹ * b⁻¹ ≤ c⁻¹
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
exps : HeightOneSpectrum R →₀ ℤ
⊢ ∑ i ∈ exps.support, count K v (↑i.asIdeal ^ exps i) = exps v

pp: case hS
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
exps : HeightOneSpectrum R →₀ ℤ
⊢ ∀ i ∈ exps.support, ↑i.asIdeal ^ exps i ≠ 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
exps : HeightOneSpectrum R →₀ ℤ
⊢ ∑ i ∈ exps.support, count K v (↑i.asIdeal ^ exps i) = exps v

case hS
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
exps : HeightOneSpectrum R →₀ ℤ
⊢ ∀ i ∈ exps.support, ↑i.asIdeal ^ exps i ≠ 0
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a b : ℝ≥0∞
ha : a ≠ 0
hb : b ≠ ∞
n : ℕ
hn : b / a < ↑n
⊢ b / a < ↑n

Tactic:
]

Goal state after:
a b : ℝ≥0∞
ha : a ≠ 0
hb : b ≠ ∞
n : ℕ
hn : b / a < ↑n
⊢ b / a < ↑n
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: case right
a b : ℝ≥0∞
ha : a ≠ ∞
hb : b ≠ 0
n : ℕ
npos : n > 0
hn : a < ↑n * b
⊢ a / ↑n < b

Tactic:
]

Goal state after:
case right
a b : ℝ≥0∞
ha : a ≠ ∞
hb : b ≠ 0
n : ℕ
npos : n > 0
hn : a < ↑n * b
⊢ a / ↑n < b
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: a : ℝ≥0∞
ha : a ≠ 0
n : ℕ
hn : (↑n)⁻¹ < a
⊢ ↑n < 2 ^ n

Tactic:
]

Goal state after:
a : ℝ≥0∞
ha : a ≠ 0
n : ℕ
hn : (↑n)⁻¹ < a
⊢ ↑n < 2 ^ n
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
x y z : α
⊢ edist x y ≤ edist x z + edist z y

Tactic:
]

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
x y z : α
⊢ edist x y ≤ edist x z + edist z y
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: case a
α : Type u
inst✝ : PseudoEMetricSpace α
x y z : α
h : edist x y = 0
⊢ edist x z ≤ edist x y + edist y z

Tactic:
]

Goal state after:
case a
α : Type u
inst✝ : PseudoEMetricSpace α
x y z : α
h : edist x y = 0
⊢ edist x z ≤ edist x y + edist y z
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: case a
α : Type u
inst✝ : PseudoEMetricSpace α
x y z : α
h : edist y x = 0
⊢ edist y z ≤ edist x z

Tactic:
]

Goal state after:
case a
α : Type u
inst✝ : PseudoEMetricSpace α
x y z : α
h : edist y x = 0
⊢ edist y z ≤ edist x z
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: case a
α : Type u
inst✝ : PseudoEMetricSpace α
x y z : α
h : edist y x = 0
⊢ edist y z ≤ edist y x + edist x z

Tactic:
]

Goal state after:
case a
α : Type u
inst✝ : PseudoEMetricSpace α
x y z : α
h : edist y x = 0
⊢ edist y z ≤ edist y x + edist x z
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
x y z : α
h : edist x y = 0
⊢ edist x z = edist y z

Tactic:
by
  apply le_antisymm
  · rw [← zero_add (edist y z), ← h]
    apply edist_triangle
  · rw [edist_comm] at h
    rw [← zero_add (edist x z), ← h]
    apply edist_triangle

Goal state after:
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
x y z : α
h : edist x y = 0
⊢ edist x z = edist y z

Tactic:
]

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
x y z : α
h : edist x y = 0
⊢ edist x z = edist y z
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
x y z t : α
⊢ edist x ?y + edist ?y z + edist z t ≤ edist x y + edist y z + edist z t

pp: case y
α : Type u
inst✝ : PseudoEMetricSpace α
x y z t : α
⊢ α

Tactic:
]

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

Goal state before:
pp: case e'_2
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
exps : HeightOneSpectrum R → ℤ
h_exps : ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, exps v = 0
⊢ ∏ i ∈ Finite.toFinset h_exps, ↑i.asIdeal ^ exps i =
    ∏ a ∈ { support := Finite.toFinset h_exps, toFun := exps, mem_support_toFun := ⋯ }.support,
      ↑a.asIdeal ^ { support := Finite.toFinset h_exps, toFun := exps, mem_support_toFun := ⋯ } a

pp: case e'_2.h
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
exps : HeightOneSpectrum R → ℤ
h_exps : ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, exps v = 0
⊢ (mulSupport fun v ↦ ↑v.asIdeal ^ exps v) ⊆ ↑(Finite.toFinset h_exps)

Tactic:
]

Goal state after:
case e'_2
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
exps : HeightOneSpectrum R → ℤ
h_exps : ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, exps v = 0
⊢ ∏ i ∈ Finite.toFinset h_exps, ↑i.asIdeal ^ exps i =
    ∏ a ∈ { support := Finite.toFinset h_exps, toFun := exps, mem_support_toFun := ⋯ }.support,
      ↑a.asIdeal ^ { support := Finite.toFinset h_exps, toFun := exps, mem_support_toFun := ⋯ } a

case e'_2.h
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
exps : HeightOneSpectrum R → ℤ
h_exps : ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, exps v = 0
⊢ (mulSupport fun v ↦ ↑v.asIdeal ^ exps v) ⊆ ↑(Finite.toFinset h_exps)
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: case e'_2.h
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
exps : HeightOneSpectrum R → ℤ
h_exps : ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, exps v = 0
⊢ (mulSupport fun v ↦ ↑v.asIdeal ^ exps v) ⊆ {x | (fun v ↦ exps v = 0) x}ᶜ

Tactic:
]

Goal state after:
case e'_2.h
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
exps : HeightOneSpectrum R → ℤ
h_exps : ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, exps v = 0
⊢ (mulSupport fun v ↦ ↑v.asIdeal ^ exps v) ⊆ {x | (fun v ↦ exps v = 0) x}ᶜ
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: case e'_2.h
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v✝ : HeightOneSpectrum R
exps : HeightOneSpectrum R → ℤ
h_exps : ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, exps v = 0
v : HeightOneSpectrum R
hv : 1 ≠ 1
h : v ∈ {x | (fun v ↦ exps v = 0) x}
⊢ False

Tactic:
]

Goal state after:
case e'_2.h
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v✝ : HeightOneSpectrum R
exps : HeightOneSpectrum R → ℤ
h_exps : ∀ᶠ (v : HeightOneSpectrum R) in Filter.cofinite, exps v = 0
v : HeightOneSpectrum R
hv : 1 ≠ 1
h : v ∈ {x | (fun v ↦ exps v = 0) x}
⊢ False
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: case refine_1
x : ℝ≥0
hx : ↑x ≠ 0
y : ℝ≥0
hy : 1 < ↑y
A : y ≠ 0
n : ℤ
hn : y ^ n ≤ x
h'n : x < y ^ (n + 1)
⊢ y ^ n ≤ x

Tactic:
]

Goal state after:
case refine_1
x : ℝ≥0
hx : ↑x ≠ 0
y : ℝ≥0
hy : 1 < ↑y
A : y ≠ 0
n : ℤ
hn : y ^ n ≤ x
h'n : x < y ^ (n + 1)
⊢ y ^ n ≤ x
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: case refine_2
x : ℝ≥0
hx : ↑x ≠ 0
y : ℝ≥0
hy : 1 < ↑y
A : y ≠ 0
n : ℤ
hn : y ^ n ≤ x
h'n : x < y ^ (n + 1)
⊢ x < y ^ (n + 1)

Tactic:
]

Goal state after:
case refine_2
x : ℝ≥0
hx : ↑x ≠ 0
y : ℝ≥0
hy : 1 < ↑y
A : y ≠ 0
n : ℤ
hn : y ^ n ≤ x
h'n : x < y ^ (n + 1)
⊢ x < y ^ (n + 1)
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: α : Type u
β : Type v
X : Type u_1
inst✝¹ : PseudoEMetricSpace α
inst✝ : PseudoEMetricSpace β
x y : α × β
⊢ max (edist x.1 y.1) (edist x.2 y.2) = max (edist y.1 x.1) (edist y.2 x.2)

Tactic:
by simp [edist_comm]

Goal state after:
Module: Mathlib.Topology.MetricSpace.Pseudo.Defs

Goal state before:
pp: case inl
α✝ : Type u
β : Type v
X : Type u_1
ι : Type u_2
α : Type u_3
dist : α → α → ℝ
dist_comm : ∀ (x y : α), dist x y = dist y x
dist_triangle : ∀ (x y z : α), dist x z ≤ dist x y + dist y z
t : Set α
ht : t ∈ {s | ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C}
hs : ∅ ∈ {s | ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C}
⊢ t ∈ {s | ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C}

Tactic:
]

Goal state after:
case inl
α✝ : Type u
β : Type v
X : Type u_1
ι : Type u_2
α : Type u_3
dist : α → α → ℝ
dist_comm : ∀ (x y : α), dist x y = dist y x
dist_triangle : ∀ (x y z : α), dist x z ≤ dist x y + dist y z
t : Set α
ht : t ∈ {s | ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C}
hs : ∅ ∈ {s | ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C}
⊢ t ∈ {s | ∃ C, ∀ ⦃x : α⦄, x ∈ s → ∀ ⦃y : α⦄, y ∈ s → dist x y ≤ C}
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
x y : α
ε : ℝ≥0∞
⊢ edist y x < ε ↔ edist y x < ε

Tactic:
]

Goal state after:
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
x y : α
ε : ℝ≥0∞
⊢ edist y x ≤ ε ↔ edist y x ≤ ε

Tactic:
]

Goal state after:
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
x : α
ε : ℝ≥0∞
h : 0 < ε
⊢ 0 < ε

Tactic:
]

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
x : α
ε : ℝ≥0∞
h : 0 < ε
⊢ 0 < ε
Module: Mathlib.Topology.EMetricSpace.Defs

Goal state before:
pp: α : Type u
inst✝ : PseudoEMetricSpace α
x : α
ε : ℝ≥0∞
⊢ 0 ≤ ε

Tactic:
]

Goal state after:
α : Type u
inst✝ : PseudoEMetricSpace α
x : α
ε : ℝ≥0∞
⊢ 0 ≤ ε
Module: Mathlib.RingTheory.DedekindDomain.Factorization

Goal state before:
pp: R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
J : Ideal R
hJ : J ≠ 0
⊢ 0 = 0

pp: case hI
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
J : Ideal R
hJ : J ≠ 0
⊢ ↑J ≠ 0

pp: case h_aJ
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
J : Ideal R
hJ : J ≠ 0
⊢ ↑J = spanSingleton R⁰ ((algebraMap R K) 1)⁻¹ * ↑J

Tactic:
]

Goal state after:
case hI
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
J : Ideal R
hJ : J ≠ 0
⊢ ↑J ≠ 0

case h_aJ
R : Type u_1
inst✝⁴ : CommRing R
K : Type u_2
inst✝³ : Field K
inst✝² : Algebra R K
inst✝¹ : IsFractionRing R K
inst✝ : IsDedekindDomain R
v : HeightOneSpectrum R
J : Ideal R
hJ : J ≠ 0
⊢ ↑J = spanSingleton R⁰ ((algebraMap R K) 1)⁻¹ * ↑J
Module: Mathlib.Data.ENNReal.Inv

Goal state before:
pp: case refine_1
x : ℝ≥0
hx : ↑x ≠ 0
y : ℝ≥0
hy : 1 < ↑y
A : y ≠ 0
n : ℤ
hn : y ^ n < x
h'n : x ≤ y ^ (n + 1)
⊢ y ^ n < x

Tactic:
]

Goal state after:
case refine_1
x : ℝ≥0
hx : ↑x ≠ 0
y : ℝ≥0
hy : 1 < ↑y
A : y ≠ 0
n : ℤ
hn : y ^ n < x
h'n : x ≤ y ^ (n + 1)
⊢ y ^ n < x