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

Goal state before:
pp: R : Type u_1
inst✝¹⁰ : CommRing R
S : Type u_2
inst✝⁹ : CommRing S
φ : R →+* S
a : S
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformSpace S
inst✝⁶ : IsUniformAddGroup R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : T2Space S
inst✝² : CompleteSpace S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous ⇑φ
ha : HasEval a
f : R⟦X⟧
this :
  HasSum ((fun d ↦ φ ((MvPowerSeries.coeff d) f) * d.prod fun s e ↦ a ^ e) ∘ Finsupp.single ())
    (MvPowerSeries.eval₂ φ (fun x ↦ a) f)
⊢ HasSum (fun d ↦ φ ((coeff d) f) * a ^ d) (MvPowerSeries.eval₂ φ (fun x ↦ a) f)

pp: R : Type u_1
inst✝¹⁰ : CommRing R
S : Type u_2
inst✝⁹ : CommRing S
φ : R →+* S
a : S
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformSpace S
inst✝⁶ : IsUniformAddGroup R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : T2Space S
inst✝² : CompleteSpace S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous ⇑φ
ha : HasEval a
f : R⟦X⟧
this : HasSum (fun d ↦ φ ((MvPowerSeries.coeff d) f) * d.prod fun s e ↦ a ^ e) (MvPowerSeries.eval₂ φ (fun x ↦ a) f)
⊢ ∀ x ∉ Set.range (Finsupp.single ()), (φ ((MvPowerSeries.coeff x) f) * x.prod fun s e ↦ a ^ e) = 0

Tactic:
]

Goal state after:
R : Type u_1
inst✝¹⁰ : CommRing R
S : Type u_2
inst✝⁹ : CommRing S
φ : R →+* S
a : S
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformSpace S
inst✝⁶ : IsUniformAddGroup R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : T2Space S
inst✝² : CompleteSpace S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous ⇑φ
ha : HasEval a
f : R⟦X⟧
this :
  HasSum ((fun d ↦ φ ((MvPowerSeries.coeff d) f) * d.prod fun s e ↦ a ^ e) ∘ Finsupp.single ())
    (MvPowerSeries.eval₂ φ (fun x ↦ a) f)
⊢ HasSum (fun d ↦ φ ((coeff d) f) * a ^ d) (MvPowerSeries.eval₂ φ (fun x ↦ a) f)

R : Type u_1
inst✝¹⁰ : CommRing R
S : Type u_2
inst✝⁹ : CommRing S
φ : R →+* S
a : S
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformSpace S
inst✝⁶ : IsUniformAddGroup R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : T2Space S
inst✝² : CompleteSpace S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous ⇑φ
ha : HasEval a
f : R⟦X⟧
this : HasSum (fun d ↦ φ ((MvPowerSeries.coeff d) f) * d.prod fun s e ↦ a ^ e) (MvPowerSeries.eval₂ φ (fun x ↦ a) f)
⊢ ∀ x ∉ Set.range (Finsupp.single ()), (φ ((MvPowerSeries.coeff x) f) * x.prod fun s e ↦ a ^ e) = 0
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : UniformSpace R
S : Type u_3
inst✝⁸ : CommRing S
inst✝⁷ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : IsUniformAddGroup R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : CompleteSpace S
inst✝² : T2Space S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁹.toTopologicalSpace, inst✝⁷.toTopologicalSpace] ⇑φ
ha : HasEval a
f : MvPowerSeries σ R
⊢ HasSum (fun d ↦ φ ((coeff d) f) * d.prod fun s e ↦ a s ^ e) (⋯.extend (MvPolynomial.eval₂ φ a) f)

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : UniformSpace R
S : Type u_3
inst✝⁸ : CommRing S
inst✝⁷ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : IsUniformAddGroup R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : CompleteSpace S
inst✝² : T2Space S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁹.toTopologicalSpace, inst✝⁷.toTopologicalSpace] ⇑φ
ha : HasEval a
f : MvPowerSeries σ R
⊢ HasSum (fun d ↦ φ ((coeff d) f) * d.prod fun s e ↦ a s ^ e) (⋯.extend (MvPolynomial.eval₂ φ a) f)
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : UniformSpace R
S : Type u_3
inst✝⁸ : CommRing S
inst✝⁷ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : IsUniformAddGroup R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : CompleteSpace S
inst✝² : T2Space S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁹.toTopologicalSpace, inst✝⁷.toTopologicalSpace] ⇑φ
ha : HasEval a
f : MvPowerSeries σ R
⊢ Continuous[instTopologicalSpace R, inst✝⁷.toTopologicalSpace] (eval₂ φ a)

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : UniformSpace R
S : Type u_3
inst✝⁸ : CommRing S
inst✝⁷ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : IsUniformAddGroup R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : CompleteSpace S
inst✝² : T2Space S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁹.toTopologicalSpace, inst✝⁷.toTopologicalSpace] ⇑φ
ha : HasEval a
f : MvPowerSeries σ R
⊢ Continuous[instTopologicalSpace R, inst✝⁷.toTopologicalSpace] (eval₂ φ a)
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : UniformSpace R
S : Type u_3
inst✝⁸ : CommRing S
inst✝⁷ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : IsUniformAddGroup R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : CompleteSpace S
inst✝² : T2Space S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁹.toTopologicalSpace, inst✝⁷.toTopologicalSpace] ⇑φ
ha : HasEval a
ε : MvPowerSeries σ R → S
hε : Continuous[instTopologicalSpace R, inst✝⁷.toTopologicalSpace] ε
h : ∀ (p : MvPolynomial σ R), ε ↑p = MvPolynomial.eval₂ φ a p
⊢ ε = ⇑(eval₂Hom hφ ha)

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝¹⁰ : CommRing R
inst✝⁹ : UniformSpace R
S : Type u_3
inst✝⁸ : CommRing S
inst✝⁷ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝⁶ : IsTopologicalSemiring R
inst✝⁵ : IsUniformAddGroup R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : CompleteSpace S
inst✝² : T2Space S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous[inst✝⁹.toTopologicalSpace, inst✝⁷.toTopologicalSpace] ⇑φ
ha : HasEval a
ε : MvPowerSeries σ R → S
hε : Continuous[instTopologicalSpace R, inst✝⁷.toTopologicalSpace] ε
h : ∀ (p : MvPolynomial σ R), ε ↑p = MvPolynomial.eval₂ φ a p
⊢ ε = ⇑(eval₂Hom hφ ha)
Module: Mathlib.RingTheory.PowerSeries.Evaluation

Goal state before:
pp: R : Type u_1
inst✝¹⁰ : CommRing R
S : Type u_2
inst✝⁹ : CommRing S
φ : R →+* S
a : S
inst✝⁸ : UniformSpace R
inst✝⁷ : UniformSpace S
inst✝⁶ : IsUniformAddGroup R
inst✝⁵ : IsTopologicalSemiring R
inst✝⁴ : IsUniformAddGroup S
inst✝³ : T2Space S
inst✝² : CompleteSpace S
inst✝¹ : IsTopologicalRing S
inst✝ : IsLinearTopology S S
hφ : Continuous ⇑φ
ha : HasEval a
ε : R⟦X⟧ → S
hε : Continuous ε
h : ∀ (p : Polynomial R), ε ↑p = Polynomial.eval₂ φ a p
p : MvPolynomial Unit R
⊢ Polynomial.eval₂ φ a ((MvPolynomial.uniqueAlgEquiv R PUnit.{1}) p) =
    Polynomial.eval₂ φ a ((MvPolynomial.uniqueAlgEquiv R Unit) p)

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.PowerSeries.Evaluation

Goal state before:
pp: R : Type u_1
inst✝¹² : CommRing R
S : Type u_2
inst✝¹¹ : CommRing S
a : S
inst✝¹⁰ : UniformSpace R
inst✝⁹ : UniformSpace S
inst✝⁸ : IsUniformAddGroup R
inst✝⁷ : IsTopologicalSemiring R
inst✝⁶ : IsUniformAddGroup S
inst✝⁵ : T2Space S
inst✝⁴ : CompleteSpace S
inst✝³ : IsTopologicalRing S
inst✝² : IsLinearTopology S S
inst✝¹ : Algebra R S
inst✝ : ContinuousSMul R S
ha : HasEval a
p : Polynomial R
⊢ Polynomial.eval₂ (algebraMap R S) a p = Polynomial.eval₂ (algebraMap R S) a p

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: case h
σ : Type u_1
R : Type u_2
inst✝¹⁷ : CommRing R
inst✝¹⁶ : UniformSpace R
S : Type u_3
inst✝¹⁵ : CommRing S
inst✝¹⁴ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝¹³ : IsTopologicalSemiring R
inst✝¹² : IsUniformAddGroup R
inst✝¹¹ : IsUniformAddGroup S
inst✝¹⁰ : CompleteSpace S
inst✝⁹ : T2Space S
inst✝⁸ : IsTopologicalRing S
inst✝⁷ : IsLinearTopology S S
hφ : Continuous[inst✝¹⁶.toTopologicalSpace, inst✝¹⁴.toTopologicalSpace] ⇑φ
ha : HasEval a
T : Type u_4
inst✝⁶ : UniformSpace T
inst✝⁵ : CompleteSpace T
inst✝⁴ : T2Space T
inst✝³ : CommRing T
inst✝² : IsTopologicalRing T
inst✝¹ : IsLinearTopology T T
inst✝ : IsUniformAddGroup T
ε : S →+* T
hε : Continuous[inst✝¹⁴.toTopologicalSpace, inst✝⁶.toTopologicalSpace] ⇑ε
p : MvPolynomial σ R
⊢ MvPolynomial.eval₂ (ε.comp φ) (fun i ↦ ε (a i)) p = MvPolynomial.eval₂ (ε.comp φ) (fun s ↦ ε (a s)) p

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹² : CommRing R
inst✝¹¹ : UniformSpace R
S : Type u_3
inst✝¹⁰ : CommRing S
inst✝⁹ : UniformSpace S
φ : R →+* S
a : σ → S
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : IsUniformAddGroup R
inst✝⁶ : IsUniformAddGroup S
inst✝⁵ : CompleteSpace S
inst✝⁴ : T2Space S
inst✝³ : IsTopologicalRing S
inst✝² : IsLinearTopology S S
inst✝¹ : Algebra R S
inst✝ : ContinuousSMul R S
ha : HasEval a
r : R
⊢ (algebraMap R S) r = (algebraMap R S) r

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹² : CommRing R
inst✝¹¹ : UniformSpace R
S : Type u_3
inst✝¹⁰ : CommRing S
inst✝⁹ : UniformSpace S
a : σ → S
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : IsUniformAddGroup R
inst✝⁶ : IsUniformAddGroup S
inst✝⁵ : CompleteSpace S
inst✝⁴ : T2Space S
inst✝³ : IsTopologicalRing S
inst✝² : IsLinearTopology S S
inst✝¹ : Algebra R S
inst✝ : ContinuousSMul R S
ha : HasEval a
⊢ Continuous[instTopologicalSpace R, inst✝⁹.toTopologicalSpace] (eval₂ (algebraMap R S) a)

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝¹² : CommRing R
inst✝¹¹ : UniformSpace R
S : Type u_3
inst✝¹⁰ : CommRing S
inst✝⁹ : UniformSpace S
a : σ → S
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : IsUniformAddGroup R
inst✝⁶ : IsUniformAddGroup S
inst✝⁵ : CompleteSpace S
inst✝⁴ : T2Space S
inst✝³ : IsTopologicalRing S
inst✝² : IsLinearTopology S S
inst✝¹ : Algebra R S
inst✝ : ContinuousSMul R S
ha : HasEval a
⊢ Continuous[instTopologicalSpace R, inst✝⁹.toTopologicalSpace] (eval₂ (algebraMap R S) a)
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹² : CommRing R
inst✝¹¹ : UniformSpace R
S : Type u_3
inst✝¹⁰ : CommRing S
inst✝⁹ : UniformSpace S
a : σ → S
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : IsUniformAddGroup R
inst✝⁶ : IsUniformAddGroup S
inst✝⁵ : CompleteSpace S
inst✝⁴ : T2Space S
inst✝³ : IsTopologicalRing S
inst✝² : IsLinearTopology S S
inst✝¹ : Algebra R S
inst✝ : ContinuousSMul R S
ha : HasEval a
p : MvPolynomial σ R
⊢ MvPolynomial.eval₂ (algebraMap R S) a p = MvPolynomial.eval₂ (algebraMap R S) a p

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝¹² : CommRing R
inst✝¹¹ : UniformSpace R
S : Type u_3
inst✝¹⁰ : CommRing S
inst✝⁹ : UniformSpace S
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : IsUniformAddGroup R
inst✝⁶ : IsUniformAddGroup S
inst✝⁵ : CompleteSpace S
inst✝⁴ : T2Space S
inst✝³ : IsTopologicalRing S
inst✝² : IsLinearTopology S S
inst✝¹ : Algebra R S
inst✝ : ContinuousSMul R S
ε : MvPowerSeries σ R →ₐ[R] S
hε : Continuous[instTopologicalSpace R, inst✝⁹.toTopologicalSpace] ⇑ε
⊢ (eval₂ (algebraMap R S) fun s ↦ ε.toRingHom (X s)) = ⇑ε

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝¹² : CommRing R
inst✝¹¹ : UniformSpace R
S : Type u_3
inst✝¹⁰ : CommRing S
inst✝⁹ : UniformSpace S
inst✝⁸ : IsTopologicalSemiring R
inst✝⁷ : IsUniformAddGroup R
inst✝⁶ : IsUniformAddGroup S
inst✝⁵ : CompleteSpace S
inst✝⁴ : T2Space S
inst✝³ : IsTopologicalRing S
inst✝² : IsLinearTopology S S
inst✝¹ : Algebra R S
inst✝ : ContinuousSMul R S
ε : MvPowerSeries σ R →ₐ[R] S
hε : Continuous[instTopologicalSpace R, inst✝⁹.toTopologicalSpace] ⇑ε
⊢ (eval₂ (algebraMap R S) fun s ↦ ε.toRingHom (X s)) = ⇑ε
Module: Mathlib.RingTheory.MvPowerSeries.Evaluation

Goal state before:
pp: σ : Type u_1
R : Type u_2
inst✝²¹ : CommRing R
inst✝²⁰ : UniformSpace R
S : Type u_3
inst✝¹⁹ : CommRing S
inst✝¹⁸ : UniformSpace S
a : σ → S
inst✝¹⁷ : IsTopologicalSemiring R
inst✝¹⁶ : IsUniformAddGroup R
inst✝¹⁵ : IsUniformAddGroup S
inst✝¹⁴ : CompleteSpace S
inst✝¹³ : T2Space S
inst✝¹² : IsTopologicalRing S
inst✝¹¹ : IsLinearTopology S S
inst✝¹⁰ : Algebra R S
inst✝⁹ : ContinuousSMul R S
ha : HasEval a
T : Type u_4
inst✝⁸ : CommRing T
inst✝⁷ : UniformSpace T
inst✝⁶ : IsUniformAddGroup T
inst✝⁵ : IsTopologicalRing T
inst✝⁴ : IsLinearTopology T T
inst✝³ : T2Space T
inst✝² : Algebra R T
inst✝¹ : ContinuousSMul R T
inst✝ : CompleteSpace T
ε : S →ₐ[R] T
hε : Continuous[inst✝¹⁸.toTopologicalSpace, inst✝⁷.toTopologicalSpace] ⇑ε
⊢ eval₂ ((↑ε).comp (algebraMap R S)) (⇑↑ε ∘ a) = eval₂ (algebraMap R T) fun s ↦ ε.toRingHom (a s)

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_2
inst✝²¹ : CommRing R
inst✝²⁰ : UniformSpace R
S : Type u_3
inst✝¹⁹ : CommRing S
inst✝¹⁸ : UniformSpace S
a : σ → S
inst✝¹⁷ : IsTopologicalSemiring R
inst✝¹⁶ : IsUniformAddGroup R
inst✝¹⁵ : IsUniformAddGroup S
inst✝¹⁴ : CompleteSpace S
inst✝¹³ : T2Space S
inst✝¹² : IsTopologicalRing S
inst✝¹¹ : IsLinearTopology S S
inst✝¹⁰ : Algebra R S
inst✝⁹ : ContinuousSMul R S
ha : HasEval a
T : Type u_4
inst✝⁸ : CommRing T
inst✝⁷ : UniformSpace T
inst✝⁶ : IsUniformAddGroup T
inst✝⁵ : IsTopologicalRing T
inst✝⁴ : IsLinearTopology T T
inst✝³ : T2Space T
inst✝² : Algebra R T
inst✝¹ : ContinuousSMul R T
inst✝ : CompleteSpace T
ε : S →ₐ[R] T
hε : Continuous[inst✝¹⁸.toTopologicalSpace, inst✝⁷.toTopologicalSpace] ⇑ε
⊢ eval₂ ((↑ε).comp (algebraMap R S)) (⇑↑ε ∘ a) = eval₂ (algebraMap R T) fun s ↦ ε.toRingHom (a s)
Module: Mathlib.SetTheory.Cardinal.Continuum

Goal state before:
pp: ⊢ 𝔠 = 𝔠

Tactic:
]

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Continuum

Goal state before:
pp: c : Cardinal.{u}
⊢ 𝔠 ≤ c ↔ 𝔠 ≤ c

Tactic:
]

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Continuum

Goal state before:
pp: c : Cardinal.{u}
⊢ c ≤ 𝔠 ↔ c ≤ 𝔠

Tactic:
]

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Continuum

Goal state before:
pp: c : Cardinal.{u}
⊢ 𝔠 < c ↔ 𝔠 < c

Tactic:
]

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Continuum

Goal state before:
pp: c : Cardinal.{u}
⊢ c < 𝔠 ↔ c < 𝔠

Tactic:
]

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Continuum

Goal state before:
pp: ⊢ Order.succ ℵ₀ ≤ 𝔠

Tactic:
]

Goal state after:
⊢ Order.succ ℵ₀ ≤ 𝔠
Module: Mathlib.SetTheory.Cardinal.Continuum

Goal state before:
pp: ⊢ 2 ^ ℵ₀ = 2 ^ ℵ₀

Tactic:
]

Goal state after:
Module: Mathlib.SetTheory.Cardinal.Continuum

Goal state before:
pp: case a
x : Cardinal.{u_1}
h₁ : 2 ≤ x
h₂ : x ≤ 𝔠
⊢ x ^ ℵ₀ ≤ 𝔠 ^ ℵ₀

Tactic:
]

Goal state after:
case a
x : Cardinal.{u_1}
h₁ : 2 ≤ x
h₂ : x ≤ 𝔠
⊢ x ^ ℵ₀ ≤ 𝔠 ^ ℵ₀
Module: Mathlib.SetTheory.Cardinal.Continuum

Goal state before:
pp: case a
x : Cardinal.{u_1}
h₁ : 2 ≤ x
h₂ : x ≤ 𝔠
⊢ 2 ^ ℵ₀ ≤ x ^ ℵ₀

Tactic:
]

Goal state after:
case a
x : Cardinal.{u_1}
h₁ : 2 ≤ x
h₂ : x ≤ 𝔠
⊢ 2 ^ ℵ₀ ≤ x ^ ℵ₀
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
τ : Type u_4
S : Type u_5
inst✝ : CommRing S
a b : σ → MvPowerSeries τ S
this : UniformSpace S := ⊥
hb : HasEval b
ha : HasEval a
⊢ HasEval (a + b)

Tactic:
]

Goal state after:
σ : Type u_1
τ : Type u_4
S : Type u_5
inst✝ : CommRing S
a b : σ → MvPowerSeries τ S
this : UniformSpace S := ⊥
hb : HasEval b
ha : HasEval a
⊢ HasEval (a + b)
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
τ : Type u_4
S : Type u_5
inst✝ : CommRing S
a b : σ → MvPowerSeries τ S
ha : HasSubst a
hb : HasSubst b
⊢ HasSubst (a + b)

Tactic:
by
  let : UniformSpace S := ⊥
  rw [hasSubst_iff_hasEval_of_discreteTopology] at ha hb ⊢
  exact ha.add hb

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
τ : Type u_4
S : Type u_5
inst✝ : CommRing S
b a : σ → MvPowerSeries τ S
this : UniformSpace S := ⊥
ha : HasEval a
⊢ HasEval (b * a)

Tactic:
]

Goal state after:
σ : Type u_1
τ : Type u_4
S : Type u_5
inst✝ : CommRing S
b a : σ → MvPowerSeries τ S
this : UniformSpace S := ⊥
ha : HasEval a
⊢ HasEval (b * a)
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
a : σ → MvPowerSeries τ S
x : σ → ℕ
inst✝ : Finite τ
ha : HasSubst a
i : σ
⊢ IsNilpotent (if 0 < x i then constantCoeff (a i) else 0)

Tactic:
]

Goal state after:
σ : Type u_1
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
a : σ → MvPowerSeries τ S
x : σ → ℕ
inst✝ : Finite τ
ha : HasSubst a
i : σ
⊢ IsNilpotent (if 0 < x i then constantCoeff (a i) else 0)
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
p : MvPolynomial σ R
this✝ : UniformSpace R := ⊥
this : UniformSpace S := ⊥
⊢ MvPolynomial.eval₂ (algebraMap R (MvPowerSeries τ S)) a p = MvPolynomial.eval₂ (algebraMap R (MvPowerSeries τ S)) a p

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
ha : HasSubst a
this✝ : UniformSpace R := ⊥
this : UniformSpace S := ⊥
⊢ eval₂ (algebraMap R (MvPowerSeries τ S)) a = eval₂ (algebraMap R (MvPowerSeries τ S)) a

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : CommRing R
⊢ ⇑(substAlgHom ⋯) = id

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_3
inst✝ : CommRing R
⊢ ⇑(substAlgHom ⋯) = id
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: c : ℝ
f : ℕ → Bool
h1 : 0 ≤ c
h2 : c < 1
⊢ cantorFunctionAux c f 0 + ∑' (b : ℕ), cantorFunctionAux c f (b + 1) =
    (bif f 0 then 1 else 0) + c * cantorFunction c fun n ↦ f (n + 1)

Tactic:
]

Goal state after:
c : ℝ
f : ℕ → Bool
h1 : 0 ≤ c
h2 : c < 1
⊢ cantorFunctionAux c f 0 + ∑' (b : ℕ), cantorFunctionAux c f (b + 1) =
    (bif f 0 then 1 else 0) + c * cantorFunction c fun n ↦ f (n + 1)
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝ : CommRing R
this✝ : UniformSpace R := ⊥
f : MvPowerSeries σ R
this : ∀ (x : MvPowerSeries σ R), (aeval ⋯) x = (AlgHom.id R (MvPowerSeries σ R)) x
⊢ (aeval ⋯) f = id f

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_3
inst✝ : CommRing R
this✝ : UniformSpace R := ⊥
f : MvPowerSeries σ R
this : ∀ (x : MvPowerSeries σ R), (aeval ⋯) x = (AlgHom.id R (MvPowerSeries σ R)) x
⊢ (aeval ⋯) f = id f
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: c : ℝ
f : ℕ → Bool
h1 : 0 ≤ c
h2 : c < 1
⊢ (bif f 0 then 1 else 0) + c * ∑' (x : ℕ), cantorFunctionAux c (fun n ↦ f (n + 1)) x =
    (bif f 0 then 1 else 0) + c * ∑' (n : ℕ), cantorFunctionAux c (fun n ↦ f (n + 1)) n

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
ha : HasSubst a
f : MvPowerSeries σ R
⊢ subst a f = subst a f

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case zero
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn✝ : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n ↦ rec false (fun x x_1 ↦ true) n
hn : false = true
⊢ f_max 0 = true

Tactic:
]

Goal state after:
case zero
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn✝ : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n ↦ rec false (fun x x_1 ↦ true) n
hn : false = true
⊢ f_max 0 = true
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: case zero
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn✝ : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n ↦ rec false (fun x x_1 ↦ true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n ↦ rec true (fun x x_1 ↦ false) n
hn : g_min 0 = true
⊢ true = true

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
ha : HasSubst a
s : σ
⊢ a s = a s

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
ha : HasSubst a
e : σ →₀ ℕ
r : R
⊢ ((algebraMap R (MvPowerSeries τ S)) r * e.prod fun i k ↦ a i ^ k) =
    (algebraMap R (MvPowerSeries τ S)) r * e.prod fun s n ↦ a s ^ n

Tactic:
]

Goal state after:
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n ↦ rec false (fun x x_1 ↦ true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n ↦ rec true (fun x x_1 ↦ false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
⊢ c + c < 1

pp: c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n ↦ rec false (fun x x_1 ↦ true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n ↦ rec true (fun x x_1 ↦ false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
⊢ 0 < 1 - c

Tactic:
]

Goal state after:
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n ↦ rec false (fun x x_1 ↦ true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n ↦ rec true (fun x x_1 ↦ false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
⊢ c + c < 1

c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n ↦ rec false (fun x x_1 ↦ true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n ↦ rec true (fun x x_1 ↦ false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
⊢ 0 < 1 - c
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝² : CommRing R
τ : Type u_4
S : Type u_5
inst✝¹ : CommRing S
inst✝ : Algebra R S
a : σ → MvPowerSeries τ S
ha : HasSubst a
e : σ →₀ ℕ
r : R
⊢ ((algebraMap R (MvPowerSeries τ S)) r * e.prod fun s n ↦ a s ^ n) =
    (algebraMap R (MvPowerSeries τ S)) r * e.prod fun s n ↦ a s ^ n

Tactic:
]

Goal state after:
Module: Mathlib.RingTheory.MvPowerSeries.Substitution

Goal state before:
pp: σ : Type u_1
R : Type u_3
inst✝⁶ : CommRing R
τ : Type u_4
S : Type u_5
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
a : σ → MvPowerSeries τ S
ha : HasSubst a
inst✝³ : UniformSpace R
inst✝² : DiscreteUniformity R
inst✝¹ : UniformSpace S
inst✝ : DiscreteUniformity S
⊢ Continuous (eval₂ (algebraMap R (MvPowerSeries τ S)) a)

Tactic:
]

Goal state after:
σ : Type u_1
R : Type u_3
inst✝⁶ : CommRing R
τ : Type u_4
S : Type u_5
inst✝⁵ : CommRing S
inst✝⁴ : Algebra R S
a : σ → MvPowerSeries τ S
ha : HasSubst a
inst✝³ : UniformSpace R
inst✝² : DiscreteUniformity R
inst✝¹ : UniformSpace S
inst✝ : DiscreteUniformity S
⊢ Continuous (eval₂ (algebraMap R (MvPowerSeries τ S)) a)
Module: Mathlib.Analysis.Real.Cardinality

Goal state before:
pp: c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n ↦ rec false (fun x x_1 ↦ true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n ↦ rec true (fun x x_1 ↦ false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
⊢ c < 1

Tactic:
]

Goal state after:
c : ℝ
h1 : 0 < c
h2 : c < 1 / 2
h3 : c < 1
f g : ℕ → Bool
hn : ∀ k < 0, f k = g k
fn : f 0 = false
gn : g 0 = true
f_max : ℕ → Bool := fun n ↦ rec false (fun x x_1 ↦ true) n
hf_max : ∀ (n : ℕ), f n = true → f_max n = true
g_min : ℕ → Bool := fun n ↦ rec true (fun x x_1 ↦ false) n
hg_min : ∀ (n : ℕ), g_min n = true → g n = true
⊢ c < 1