mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: case pos
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hp : p ∈ Icc 0 1
a b : A
hab : a ≤ b
ha : 0 ≤ a
⊢ (fun a ↦ a ^ p) a ≤ (fun a ↦ a ^ p) b

pp: case neg
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hp : p ∈ Icc 0 1
a b : A
hab : a ≤ b
ha : ¬0 ≤ a
⊢ (fun a ↦ a ^ p) a ≤ (fun a ↦ a ^ p) b

Tactic:
·

Goal state after:
case pos
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hp : p ∈ Icc 0 1
a b : A
hab : a ≤ b
ha : 0 ≤ a
⊢ (fun a ↦ a ^ p) a ≤ (fun a ↦ a ^ p) b
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: case inl.inl
p : ℤ[X]
a : ℤ
hp : p.natDegree ≤ 0
⊢ ∃ z, eval₂ (Int.castRingHom ℝ) (↑a / ↑0) p * ↑0 ^ 0 = ↑z

pp: case inl.inr
p : ℤ[X]
a : ℤ
k : ℕ
hp : p.natDegree ≤ k
hk : k > 0
⊢ ∃ z, eval₂ (Int.castRingHom ℝ) (↑a / ↑0) p * ↑0 ^ k = ↑z

Tactic:
·

Goal state after:
case inl.inl
p : ℤ[X]
a : ℤ
hp : p.natDegree ≤ 0
⊢ ∃ z, eval₂ (Int.castRingHom ℝ) (↑a / ↑0) p * ↑0 ^ 0 = ↑z
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: case pos.inl.inl
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
a b : A
hab : a ≤ b
ha : 0 ≤ a
hb : 0 ≤ b
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
hp : p ∈ Ioo 0 1
⊢ (fun a ↦ a ^ p) a ≤ (fun a ↦ a ^ p) b

pp: case pos.inl.inr
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
a b : A
hab : a ≤ b
ha : 0 ≤ a
hb : 0 ≤ b
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
hp : p ∈ {0}
⊢ (fun a ↦ a ^ p) a ≤ (fun a ↦ a ^ p) b

pp: case pos.inr
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
a b : A
hab : a ≤ b
ha : 0 ≤ a
hb : 0 ≤ b
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
hp : p ∈ {1}
⊢ (fun a ↦ a ^ p) a ≤ (fun a ↦ a ^ p) b

Tactic:
·

Goal state after:
case pos.inl.inl
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
a b : A
hab : a ≤ b
ha : 0 ≤ a
hb : 0 ≤ b
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
hp : p ∈ Ioo 0 1
⊢ (fun a ↦ a ^ p) a ≤ (fun a ↦ a ^ p) b
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: case pos.inl.inr
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
a b : A
hab : a ≤ b
ha : 0 ≤ a
hb : 0 ≤ b
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
hp : p ∈ {0}
⊢ (fun a ↦ a ^ p) a ≤ (fun a ↦ a ^ p) b

pp: case pos.inr
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
a b : A
hab : a ≤ b
ha : 0 ≤ a
hb : 0 ≤ b
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
hp : p ∈ {1}
⊢ (fun a ↦ a ^ p) a ≤ (fun a ↦ a ^ p) b

Tactic:
·

Goal state after:
case pos.inl.inr
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
a b : A
hab : a ≤ b
ha : 0 ≤ a
hb : 0 ≤ b
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
hp : p ∈ {0}
⊢ (fun a ↦ a ^ p) a ≤ (fun a ↦ a ^ p) b
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: case pos.inr
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
a b : A
hab : a ≤ b
ha : 0 ≤ a
hb : 0 ≤ b
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
hp : p ∈ {1}
⊢ (fun a ↦ a ^ p) a ≤ (fun a ↦ a ^ p) b

Tactic:
·

Goal state after:
case pos.inr
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
a b : A
hab : a ≤ b
ha : 0 ≤ a
hb : 0 ≤ b
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
hp : p ∈ {1}
⊢ (fun a ↦ a ^ p) a ≤ (fun a ↦ a ^ p) b
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: case neg
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hp : p ∈ Icc 0 1
a b : A
hab : a ≤ b
ha : ¬0 ≤ a
⊢ (fun a ↦ a ^ p) a ≤ (fun a ↦ a ^ p) b

Tactic:
·

Goal state after:
case neg
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hp : p ∈ Icc 0 1
a b : A
hab : a ≤ b
ha : ¬0 ≤ a
⊢ (fun a ↦ a ^ p) a ≤ (fun a ↦ a ^ p) b
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: case refine_1
n : ℕ
x : ℝ
hx : x ∈ Ioc (-1) 1
⊢ 0 ≤ 1 - x ^ 2

pp: case refine_2
n : ℕ
x : ℝ
hx : x ∈ Ioc (-1) 1
⊢ 0 ≤ cos (x * (π / 2))

Tactic:
·

Goal state after:
case refine_1
n : ℕ
x : ℝ
hx : x ∈ Ioc (-1) 1
⊢ 0 ≤ 1 - x ^ 2
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: a : ℝ
⊢ Tendsto (fun n ↦ a ^ (2 * n + 1) / ↑n !) atTop (nhds 0)

Tactic:
rw [← mul_zero a]
  refine ((FloorSemiring.tendsto_pow_div_factorial_atTop (a ^ 2)).const_mul a).congr (fun x => ?_)
  rw [← pow_mul, mul_div_assoc', _root_.pow_succ']

Goal state after:
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: ⊢ Irrational (π / ↑2)

Tactic:
Nat.cast_two

Goal state after:
⊢ Irrational (π / 2)
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: case inl.inl
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
hp : p ∈ Ioo 0 1
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ p

pp: case inl.inr
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
hp : p ∈ {0}
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ p

pp: case inr
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
hp : p ∈ {1}
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ p

Tactic:
·

Goal state after:
case inl.inl
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
hp : p ∈ Ioo 0 1
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ p
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: case inl.inr
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
hp : p ∈ {0}
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ p

pp: case inr
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
hp : p ∈ {1}
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ p

Tactic:
·

Goal state after:
case inl.inr
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
hp : p ∈ {0}
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ p
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: case inr
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
hp : p ∈ {1}
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ p

Tactic:
·

Goal state after:
case inr
A : Type u_1
inst✝² : NonUnitalCStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ≥0
hIcc : Icc 0 1 = Ioo 0 1 ∪ {0} ∪ {1}
hp : p ∈ {1}
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ p
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: case inl
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : q = 0
⊢ Monotone fun a ↦ a ^ ↑q

pp: case inr
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : 0 < q
⊢ Monotone fun a ↦ a ^ ↑q

Tactic:
·

Goal state after:
case inl
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : q = 0
⊢ Monotone fun a ↦ a ^ ↑q
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: case pos
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : q = 0
a b : A
hab : a ≤ b
ha : 0 ≤ a
⊢ (fun a ↦ a ^ ↑0) a ≤ (fun a ↦ a ^ ↑0) b

pp: case neg
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : q = 0
a b : A
hab : a ≤ b
ha : ¬0 ≤ a
⊢ (fun a ↦ a ^ ↑0) a ≤ (fun a ↦ a ^ ↑0) b

Tactic:
·

Goal state after:
case pos
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : q = 0
a b : A
hab : a ≤ b
ha : 0 ≤ a
⊢ (fun a ↦ a ^ ↑0) a ≤ (fun a ↦ a ^ ↑0) b
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: case neg
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : q = 0
a b : A
hab : a ≤ b
ha : ¬0 ≤ a
⊢ (fun a ↦ a ^ ↑0) a ≤ (fun a ↦ a ^ ↑0) b

Tactic:
·

Goal state after:
case neg
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : q = 0
a b : A
hab : a ≤ b
ha : ¬0 ≤ a
⊢ (fun a ↦ a ^ ↑0) a ≤ (fun a ↦ a ^ ↑0) b
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: case inr
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : 0 < q
⊢ Monotone fun a ↦ a ^ ↑q

Tactic:
·

Goal state after:
case inr
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : 0 < q
⊢ Monotone fun a ↦ a ^ ↑q
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: case inl
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : q = 0
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ ↑q

pp: case inr
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : 0 < q
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ ↑q

Tactic:
·

Goal state after:
case inl
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : q = 0
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ ↑q
Module: Mathlib.Analysis.Real.Pi.Irrational

Goal state before:
pp: h' : ¬Irrational (π / 2)
a : ℤ
b : ℕ
hb : 0 < b
h : π / 2 = ↑a / ↑b
ha : 0 < ↑a
k : ∀ (n : ℕ), 0 < ↑a ^ (2 * n + 1) / ↑n !
j : ∀ᶠ (n : ℕ) in atTop, ↑a ^ (2 * n + 1) / ↑n ! * I n (π / 2) < 1
n : ℕ
hn : ↑a ^ (2 * n + 1) / ↑n ! * I n (π / 2) < 1
hn' : 0 < ↑a ^ (2 * n + 1) / ↑n ! * I n (π / 2)
⊢ n ≤ 2 * n + 1

Tactic:
by lia

Goal state after:
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Order

Goal state before:
pp: case inr
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : 0 < q
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ ↑q

Tactic:
·

Goal state after:
case inr
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
p : ℝ
hp : p ∈ Icc 0 1
q : ℝ≥0 := ⟨p, ⋯⟩
hq : 0 < q
⊢ ConcaveOn ℝ (Ici 0) fun a ↦ a ^ ↑q
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Order

Goal state before:
pp: A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
⊢ Tendsto (fun i ↦ if a ∈ {b | IsStrictlyPositive b} then cfc (fun x ↦ i⁻¹ * (x ^ i - 1)) a else 0) (𝓝[>] 0)
    (𝓝 (if a ∈ {b | IsStrictlyPositive b} then log a else 0))

Tactic:
by_cases ha : IsStrictlyPositive a

Goal state after:
case pos
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : IsStrictlyPositive a
⊢ Tendsto (fun i ↦ if a ∈ {b | IsStrictlyPositive b} then cfc (fun x ↦ i⁻¹ * (x ^ i - 1)) a else 0) (𝓝[>] 0)
    (𝓝 (if a ∈ {b | IsStrictlyPositive b} then log a else 0))

case neg
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : ¬IsStrictlyPositive a
⊢ Tendsto (fun i ↦ if a ∈ {b | IsStrictlyPositive b} then cfc (fun x ↦ i⁻¹ * (x ^ i - 1)) a else 0) (𝓝[>] 0)
    (𝓝 (if a ∈ {b | IsStrictlyPositive b} then log a else 0))
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Order

Goal state before:
pp: case pos
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : IsStrictlyPositive a
⊢ Tendsto (fun i ↦ if a ∈ {b | IsStrictlyPositive b} then cfc (fun x ↦ i⁻¹ * (x ^ i - 1)) a else 0) (𝓝[>] 0)
    (𝓝 (if a ∈ {b | IsStrictlyPositive b} then log a else 0))

pp: case neg
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : ¬IsStrictlyPositive a
⊢ Tendsto (fun i ↦ if a ∈ {b | IsStrictlyPositive b} then cfc (fun x ↦ i⁻¹ * (x ^ i - 1)) a else 0) (𝓝[>] 0)
    (𝓝 (if a ∈ {b | IsStrictlyPositive b} then log a else 0))

Tactic:
·

Goal state after:
case pos
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : IsStrictlyPositive a
⊢ Tendsto (fun i ↦ if a ∈ {b | IsStrictlyPositive b} then cfc (fun x ↦ i⁻¹ * (x ^ i - 1)) a else 0) (𝓝[>] 0)
    (𝓝 (if a ∈ {b | IsStrictlyPositive b} then log a else 0))
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Order

Goal state before:
pp: case neg
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : ¬IsStrictlyPositive a
⊢ Tendsto (fun i ↦ if a ∈ {b | IsStrictlyPositive b} then cfc (fun x ↦ i⁻¹ * (x ^ i - 1)) a else 0) (𝓝[>] 0)
    (𝓝 (if a ∈ {b | IsStrictlyPositive b} then log a else 0))

Tactic:
·

Goal state after:
case neg
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
a : A
ha : ¬IsStrictlyPositive a
⊢ Tendsto (fun i ↦ if a ∈ {b | IsStrictlyPositive b} then cfc (fun x ↦ i⁻¹ * (x ^ i - 1)) a else 0) (𝓝[>] 0)
    (𝓝 (if a ∈ {b | IsStrictlyPositive b} then log a else 0))
Module: Mathlib.Analysis.SpecialFunctions.FrullaniIntegral

Goal state before:
pp: case h
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
inst✝ : CompleteSpace E
hf : LocallyIntegrableOn f (Ioi 0) volume
ha : 0 < a
hb : 0 < b
V : E
δ : ℝ
hδ : 0 ≤ δ
h : ∀ x ∈ uIoc a b, ‖f x - V‖ ≤ δ
hsub : uIcc a b ⊆ Ioi 0
hint_f : IntervalIntegrable (fun x ↦ x⁻¹ • f x) volume a b
hint_V : IntervalIntegrable (fun x ↦ x⁻¹ • V) volume a b
hint_inv : IntervalIntegrable (fun x ↦ x⁻¹ * δ) volume a b
⊢ ∀ᵐ (t : ℝ) ∂volume.restrict (uIoc a b), ‖t⁻¹ • (f t - V)‖ ≤ t⁻¹ * δ

pp: case hbound
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
inst✝ : CompleteSpace E
hf : LocallyIntegrableOn f (Ioi 0) volume
ha : 0 < a
hb : 0 < b
V : E
δ : ℝ
hδ : 0 ≤ δ
h : ∀ x ∈ uIoc a b, ‖f x - V‖ ≤ δ
hsub : uIcc a b ⊆ Ioi 0
hint_f : IntervalIntegrable (fun x ↦ x⁻¹ • f x) volume a b
hint_V : IntervalIntegrable (fun x ↦ x⁻¹ • V) volume a b
hint_inv : IntervalIntegrable (fun x ↦ x⁻¹ * δ) volume a b
⊢ IntervalIntegrable (fun t ↦ t⁻¹ * δ) volume a b

Tactic:
·

Goal state after:
case h
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
inst✝ : CompleteSpace E
hf : LocallyIntegrableOn f (Ioi 0) volume
ha : 0 < a
hb : 0 < b
V : E
δ : ℝ
hδ : 0 ≤ δ
h : ∀ x ∈ uIoc a b, ‖f x - V‖ ≤ δ
hsub : uIcc a b ⊆ Ioi 0
hint_f : IntervalIntegrable (fun x ↦ x⁻¹ • f x) volume a b
hint_V : IntervalIntegrable (fun x ↦ x⁻¹ • V) volume a b
hint_inv : IntervalIntegrable (fun x ↦ x⁻¹ * δ) volume a b
⊢ ∀ᵐ (t : ℝ) ∂volume.restrict (uIoc a b), ‖t⁻¹ • (f t - V)‖ ≤ t⁻¹ * δ
Module: Mathlib.Analysis.SpecialFunctions.FrullaniIntegral

Goal state before:
pp: case hbound
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
inst✝ : CompleteSpace E
hf : LocallyIntegrableOn f (Ioi 0) volume
ha : 0 < a
hb : 0 < b
V : E
δ : ℝ
hδ : 0 ≤ δ
h : ∀ x ∈ uIoc a b, ‖f x - V‖ ≤ δ
hsub : uIcc a b ⊆ Ioi 0
hint_f : IntervalIntegrable (fun x ↦ x⁻¹ • f x) volume a b
hint_V : IntervalIntegrable (fun x ↦ x⁻¹ • V) volume a b
hint_inv : IntervalIntegrable (fun x ↦ x⁻¹ * δ) volume a b
⊢ IntervalIntegrable (fun t ↦ t⁻¹ * δ) volume a b

Tactic:
·

Goal state after:
case hbound
E : Type u_1
inst✝² : NormedAddCommGroup E
inst✝¹ : NormedSpace ℝ E
f : ℝ → E
a b : ℝ
inst✝ : CompleteSpace E
hf : LocallyIntegrableOn f (Ioi 0) volume
ha : 0 < a
hb : 0 < b
V : E
δ : ℝ
hδ : 0 ≤ δ
h : ∀ x ∈ uIoc a b, ‖f x - V‖ ≤ δ
hsub : uIcc a b ⊆ Ioi 0
hint_f : IntervalIntegrable (fun x ↦ x⁻¹ • f x) volume a b
hint_V : IntervalIntegrable (fun x ↦ x⁻¹ • V) volume a b
hint_inv : IntervalIntegrable (fun x ↦ x⁻¹ * δ) volume a b
⊢ IntervalIntegrable (fun t ↦ t⁻¹ * δ) volume a b
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.RingInverseOrder

Goal state before:
pp: case pos
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x : A
xpos : IsStrictlyPositive x
y : A
ypos : IsStrictlyPositive y
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
z : A := (conjSqrt x⁻¹ʳ) y
zpos : IsStrictlyPositive z
xinvpos : IsStrictlyPositive x⁻¹ʳ
hsp : IsStrictlyPositive (a • 1 + b • z)
r : ℝ
hr : r ∈ spectrum ℝ z
ha' : a = 0
⊢ 0 < a + b * r

pp: case neg
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x : A
xpos : IsStrictlyPositive x
y : A
ypos : IsStrictlyPositive y
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
z : A := (conjSqrt x⁻¹ʳ) y
zpos : IsStrictlyPositive z
xinvpos : IsStrictlyPositive x⁻¹ʳ
hsp : IsStrictlyPositive (a • 1 + b • z)
r : ℝ
hr : r ∈ spectrum ℝ z
ha' : ¬a = 0
⊢ 0 < a + b * r

Tactic:
·

Goal state after:
case pos
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x : A
xpos : IsStrictlyPositive x
y : A
ypos : IsStrictlyPositive y
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
z : A := (conjSqrt x⁻¹ʳ) y
zpos : IsStrictlyPositive z
xinvpos : IsStrictlyPositive x⁻¹ʳ
hsp : IsStrictlyPositive (a • 1 + b • z)
r : ℝ
hr : r ∈ spectrum ℝ z
ha' : a = 0
⊢ 0 < a + b * r
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.RingInverseOrder

Goal state before:
pp: case neg
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x : A
xpos : IsStrictlyPositive x
y : A
ypos : IsStrictlyPositive y
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
z : A := (conjSqrt x⁻¹ʳ) y
zpos : IsStrictlyPositive z
xinvpos : IsStrictlyPositive x⁻¹ʳ
hsp : IsStrictlyPositive (a • 1 + b • z)
r : ℝ
hr : r ∈ spectrum ℝ z
ha' : ¬a = 0
⊢ 0 < a + b * r

Tactic:
·

Goal state after:
case neg
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x : A
xpos : IsStrictlyPositive x
y : A
ypos : IsStrictlyPositive y
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
z : A := (conjSqrt x⁻¹ʳ) y
zpos : IsStrictlyPositive z
xinvpos : IsStrictlyPositive x⁻¹ʳ
hsp : IsStrictlyPositive (a • 1 + b • z)
r : ℝ
hr : r ∈ spectrum ℝ z
ha' : ¬a = 0
⊢ 0 < a + b * r
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.RingInverseOrder

Goal state before:
pp: case h.refine_1
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x : A
xpos : IsStrictlyPositive x
y : A
ypos : IsStrictlyPositive y
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
z : A := (conjSqrt x⁻¹ʳ) y
zpos : IsStrictlyPositive z
xinvpos : IsStrictlyPositive x⁻¹ʳ
hsp : IsStrictlyPositive (a • 1 + b • z)
h₁ : (a • 1 + b • z) ^ (-1) = cfc (fun r ↦ (a + b * r) ^ (-1)) z
h₂ : a • 1 + b • z ^ (-1) = cfc (fun r ↦ a + b * r ^ (-1)) z
⊢ ContinuousOn (fun r ↦ (a + b * r) ^ (-1)) (spectrum ℝ z)

pp: case h.refine_2
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x : A
xpos : IsStrictlyPositive x
y : A
ypos : IsStrictlyPositive y
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
z : A := (conjSqrt x⁻¹ʳ) y
zpos : IsStrictlyPositive z
xinvpos : IsStrictlyPositive x⁻¹ʳ
hsp : IsStrictlyPositive (a • 1 + b • z)
h₁ : (a • 1 + b • z) ^ (-1) = cfc (fun r ↦ (a + b * r) ^ (-1)) z
h₂ : a • 1 + b • z ^ (-1) = cfc (fun r ↦ a + b * r ^ (-1)) z
⊢ ContinuousOn (fun r ↦ a + b * r ^ (-1)) (spectrum ℝ z)

pp: case h.refine_3
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x : A
xpos : IsStrictlyPositive x
y : A
ypos : IsStrictlyPositive y
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
z : A := (conjSqrt x⁻¹ʳ) y
zpos : IsStrictlyPositive z
xinvpos : IsStrictlyPositive x⁻¹ʳ
hsp : IsStrictlyPositive (a • 1 + b • z)
h₁ : (a • 1 + b • z) ^ (-1) = cfc (fun r ↦ (a + b * r) ^ (-1)) z
h₂ : a • 1 + b • z ^ (-1) = cfc (fun r ↦ a + b * r ^ (-1)) z
⊢ ∀ x ∈ spectrum ℝ z, (a + b * x) ^ (-1) ≤ a + b * x ^ (-1)

Tactic:
·

Goal state after:
case h.refine_1
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x : A
xpos : IsStrictlyPositive x
y : A
ypos : IsStrictlyPositive y
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
z : A := (conjSqrt x⁻¹ʳ) y
zpos : IsStrictlyPositive z
xinvpos : IsStrictlyPositive x⁻¹ʳ
hsp : IsStrictlyPositive (a • 1 + b • z)
h₁ : (a • 1 + b • z) ^ (-1) = cfc (fun r ↦ (a + b * r) ^ (-1)) z
h₂ : a • 1 + b • z ^ (-1) = cfc (fun r ↦ a + b * r ^ (-1)) z
⊢ ContinuousOn (fun r ↦ (a + b * r) ^ (-1)) (spectrum ℝ z)
Module: Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.RingInverseOrder

Goal state before:
pp: case h.refine_2
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x : A
xpos : IsStrictlyPositive x
y : A
ypos : IsStrictlyPositive y
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
z : A := (conjSqrt x⁻¹ʳ) y
zpos : IsStrictlyPositive z
xinvpos : IsStrictlyPositive x⁻¹ʳ
hsp : IsStrictlyPositive (a • 1 + b • z)
h₁ : (a • 1 + b • z) ^ (-1) = cfc (fun r ↦ (a + b * r) ^ (-1)) z
h₂ : a • 1 + b • z ^ (-1) = cfc (fun r ↦ a + b * r ^ (-1)) z
⊢ ContinuousOn (fun r ↦ a + b * r ^ (-1)) (spectrum ℝ z)

pp: case h.refine_3
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x : A
xpos : IsStrictlyPositive x
y : A
ypos : IsStrictlyPositive y
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
z : A := (conjSqrt x⁻¹ʳ) y
zpos : IsStrictlyPositive z
xinvpos : IsStrictlyPositive x⁻¹ʳ
hsp : IsStrictlyPositive (a • 1 + b • z)
h₁ : (a • 1 + b • z) ^ (-1) = cfc (fun r ↦ (a + b * r) ^ (-1)) z
h₂ : a • 1 + b • z ^ (-1) = cfc (fun r ↦ a + b * r ^ (-1)) z
⊢ ∀ x ∈ spectrum ℝ z, (a + b * x) ^ (-1) ≤ a + b * x ^ (-1)

Tactic:
·

Goal state after:
case h.refine_2
A : Type u_1
inst✝² : CStarAlgebra A
inst✝¹ : PartialOrder A
inst✝ : StarOrderedRing A
x : A
xpos : IsStrictlyPositive x
y : A
ypos : IsStrictlyPositive y
a b : ℝ
ha : 0 ≤ a
hb : 0 ≤ b
hab : a + b = 1
z : A := (conjSqrt x⁻¹ʳ) y
zpos : IsStrictlyPositive z
xinvpos : IsStrictlyPositive x⁻¹ʳ
hsp : IsStrictlyPositive (a • 1 + b • z)
h₁ : (a • 1 + b • z) ^ (-1) = cfc (fun r ↦ (a + b * r) ^ (-1)) z
h₂ : a • 1 + b • z ^ (-1) = cfc (fun r ↦ a + b * r ^ (-1)) z
⊢ ContinuousOn (fun r ↦ a + b * r ^ (-1)) (spectrum ℝ z)