1. FormalConjectures
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  3. «34145»
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Mathoverflow 34145

Can the unit square be covered by $1/k$-by-$1/(k+1)$ rectangles (across $1 \le k$ natural)?

I am deliberately not requiring that the rotations can only be $0^\circ, 90^\circ, 180^\circ, \text{ or } 270^\circ$.

Because of indexing, since n : ℕ starts at 0, we change the side lengths to $1 / (n + 1)$ and $1 / (n + 2)$, so that the first rectangle is $1/1$ by $1/2$, the second is $1/2$ by $1/3$, etc.

Reference: mathoverflow/34145 asked by user Kaveh

open Real MeasureTheory Measure Modulenamespace Mathoverflow34145

A rectangle is specified by its width, height, starting point, and rotation. The rectangle is assumed to start in the lower left corner. For example, the unit square ${ (x, y) \mid 0 \le x \le 1, 0 \le y \le 1 }$ is specified as ⟨1, 1, (0, 0), 0⟩

structure Rectangle : Type where width : ℝ height : ℝ start : ℝ × ℝ rotation : Angle

A combination of a rotation and a translation to map the standard rectangle to the desired rectangle.

noncomputable def rigidMotion (start : ℝ × ℝ) (θ : Angle) (p : ℝ × ℝ) : (ℝ × ℝ) := (start.1 + p.1 * θ.cos - p.2 * θ.sin, start.2 + p.1 * θ.sin + p.2 * θ.cos) @[category test, AMS 51] lemma rigidMotion_test : rigidMotion (sqrt 2, sqrt 11) (2 * π / 3 : ℝ) (sqrt 5, sqrt 7) = (sqrt 2 - sqrt 5 / 2 - sqrt 21 / 2, -sqrt 7 / 2 + sqrt 11 + sqrt 15 / 2) := ⊢ rigidMotion (√2, √11) ↑(2 * π / 3) (√5, √7) = (√2 - √5 / 2 - √21 / 2, -√7 / 2 + √11 + √15 / 2) ⊢ √2 + √5 * (2 * (2 ^ 2)⁻¹ - 1) - √7 * (2 * (√3 / 2) * 2⁻¹) = √2 - √5 / 2 - √3 * (√7 / 2) ∧ √11 + √5 * (2 * (√3 / 2) * 2⁻¹) + √7 * (2 * (2 ^ 2)⁻¹ - 1) = -√7 / 2 + √11 + √3 * (√5 / 2) ⊢ True ∧ True; All goals completed! 🐙

A scaling to map the unit square to a standard rectangle.

noncomputable def scale (x y : ℝ) (p : ℝ × ℝ) : (ℝ × ℝ) := (x * p.1, y * p.2)

The unit square.

def unitSquare : Set (ℝ × ℝ) := { p | 0 ≤ p.1 ∧ p.1 ≤ 1 ∧ 0 ≤ p.2 ∧ p.2 ≤ 1 }

Converts a rectangle to a set in ℝ × ℝ.

def Rectangle.toSet (r : Rectangle) : Set (ℝ × ℝ) := rigidMotion r.start r.rotation '' (scale r.width r.height '' unitSquare)

The standard Lebesgue measure on ℝ².

noncomputable abbrev lbMeasure : Measure (ℝ × ℝ) := (Basis.finTwoProd ℝ).addHaar

lbMeasure is invariant under rigidMotion start θ.

@[category test, AMS 51] lemma lbMeasure_rigidMotion (start : ℝ × ℝ) (θ : Angle) (s : Set (ℝ × ℝ)) : lbMeasure (rigidMotion start θ '' s) = lbMeasure s := start:ℝ × ℝθ:Angles:Set (ℝ × ℝ)⊢ lbMeasure (rigidMotion start θ '' s) = lbMeasure s start:ℝ × ℝθ:Angles:Set (ℝ × ℝ)α:ℝ × ℝ → ℝ × ℝ := fun x => (x.1 * θ.cos - x.2 * θ.sin, x.1 * θ.sin + x.2 * θ.cos)⊢ lbMeasure (rigidMotion start θ '' s) = lbMeasure s start:ℝ × ℝθ:Angles:Set (ℝ × ℝ)α:ℝ × ℝ → ℝ × ℝ := fun x => (x.1 * θ.cos - x.2 * θ.sin, x.1 * θ.sin + x.2 * θ.cos)⊢ lbMeasure (rigidMotion start θ '' s) = (Basis.finTwoProd ℝ).addHaar (α '' s)start:ℝ × ℝθ:Angles:Set (ℝ × ℝ)α:ℝ × ℝ → ℝ × ℝ := fun x => (x.1 * θ.cos - x.2 * θ.sin, x.1 * θ.sin + x.2 * θ.cos)⊢ (Basis.finTwoProd ℝ).addHaar (α '' s) = lbMeasure s start:ℝ × ℝθ:Angles:Set (ℝ × ℝ)α:ℝ × ℝ → ℝ × ℝ := fun x => (x.1 * θ.cos - x.2 * θ.sin, x.1 * θ.sin + x.2 * θ.cos)⊢ lbMeasure (rigidMotion start θ '' s) = (Basis.finTwoProd ℝ).addHaar (α '' s) exact (measure_preimage_add_right _ start _).symm.trans (congr_arg _ (Set.ext <| start:ℝ × ℝθ:Angles:Set (ℝ × ℝ)α:ℝ × ℝ → ℝ × ℝ := fun x => (x.1 * θ.cos - x.2 * θ.sin, x.1 * θ.sin + x.2 * θ.cos)⊢ ∀ (x : ℝ × ℝ), x ∈ (fun h => h + start) ⁻¹' (rigidMotion start θ '' s) ↔ x ∈ α '' s All goals completed! 🐙)) start:ℝ × ℝθ:Angles:Set (ℝ × ℝ)α:ℝ × ℝ → ℝ × ℝ := fun x => (x.1 * θ.cos - x.2 * θ.sin, x.1 * θ.sin + x.2 * θ.cos)⊢ (Basis.finTwoProd ℝ).addHaar (α '' s) = lbMeasure s start:ℝ × ℝθ:Angles:Set (ℝ × ℝ)α:ℝ × ℝ → ℝ × ℝ := fun x => (x.1 * θ.cos - x.2 * θ.sin, x.1 * θ.sin + x.2 * θ.cos)β:Basis (Fin 2) ℝ (ℝ × ℝ) := Basis.finTwoProd ℝ⊢ (Basis.finTwoProd ℝ).addHaar (α '' s) = lbMeasure s start:ℝ × ℝθ:Angles:Set (ℝ × ℝ)α:ℝ × ℝ → ℝ × ℝ := fun x => (x.1 * θ.cos - x.2 * θ.sin, x.1 * θ.sin + x.2 * θ.cos)β:Basis (Fin 2) ℝ (ℝ × ℝ) := Basis.finTwoProd ℝ⊢ (Basis.finTwoProd ℝ).addHaar (α '' s) = β.addHaar (⇑((β.constr ℝ) ![α (β 0), α (β 1)]) '' s)start:ℝ × ℝθ:Angles:Set (ℝ × ℝ)α:ℝ × ℝ → ℝ × ℝ := fun x => (x.1 * θ.cos - x.2 * θ.sin, x.1 * θ.sin + x.2 * θ.cos)β:Basis (Fin 2) ℝ (ℝ × ℝ) := Basis.finTwoProd ℝ⊢ β.addHaar (⇑((β.constr ℝ) ![α (β 0), α (β 1)]) '' s) = lbMeasure s start:ℝ × ℝθ:Angles:Set (ℝ × ℝ)α:ℝ × ℝ → ℝ × ℝ := fun x => (x.1 * θ.cos - x.2 * θ.sin, x.1 * θ.sin + x.2 * θ.cos)β:Basis (Fin 2) ℝ (ℝ × ℝ) := Basis.finTwoProd ℝ⊢ (Basis.finTwoProd ℝ).addHaar (α '' s) = β.addHaar (⇑((β.constr ℝ) ![α (β 0), α (β 1)]) '' s) exact congr_arg _ <| Set.image_congr <| start:ℝ × ℝθ:Angles:Set (ℝ × ℝ)α:ℝ × ℝ → ℝ × ℝ := fun x => (x.1 * θ.cos - x.2 * θ.sin, x.1 * θ.sin + x.2 * θ.cos)β:Basis (Fin 2) ℝ (ℝ × ℝ) := Basis.finTwoProd ℝ⊢ ∀ a ∈ s, α a = ((β.constr ℝ) ![α (β 0), α (β 1)]) a All goals completed! 🐙 start:ℝ × ℝθ:Angles:Set (ℝ × ℝ)α:ℝ × ℝ → ℝ × ℝ := fun x => (x.1 * θ.cos - x.2 * θ.sin, x.1 * θ.sin + x.2 * θ.cos)β:Basis (Fin 2) ℝ (ℝ × ℝ) := Basis.finTwoProd ℝ⊢ β.addHaar (⇑((β.constr ℝ) ![α (β 0), α (β 1)]) '' s) = lbMeasure s start:ℝ × ℝθ:Angles:Set (ℝ × ℝ)α:ℝ × ℝ → ℝ × ℝ := fun x => (x.1 * θ.cos - x.2 * θ.sin, x.1 * θ.sin + x.2 * θ.cos)β:Basis (Fin 2) ℝ (ℝ × ℝ) := Basis.finTwoProd ℝ⊢ lbMeasure s = ENNReal.ofReal |(LinearMap.toMatrix β β) ((β.constr ℝ) ![α (β 0), α (β 1)]) 0 0 * (LinearMap.toMatrix β β) ((β.constr ℝ) ![α (β 0), α (β 1)]) 1 1 - (LinearMap.toMatrix β β) ((β.constr ℝ) ![α (β 0), α (β 1)]) 0 1 * (LinearMap.toMatrix β β) ((β.constr ℝ) ![α (β 0), α (β 1)]) 1 0| * β.addHaar s All goals completed! 🐙

lbMeasure is scaled by scale.

@[category test, AMS 51] lemma lbMeasure_scale (x y : ℝ) (s : Set (ℝ × ℝ)) : lbMeasure (scale x y '' s) = .ofReal |x * y| * lbMeasure s := x:ℝy:ℝs:Set (ℝ × ℝ)⊢ lbMeasure (scale x y '' s) = ENNReal.ofReal |x * y| * lbMeasure s let scaleLinear : (ℝ × ℝ) →ₗ[ℝ] (ℝ × ℝ) := { toFun p := (x * p.1, y * p.2) map_add' p q := x:ℝy:ℝs:Set (ℝ × ℝ)p:ℝ × ℝq:ℝ × ℝ⊢ (x * (p + q).1, y * (p + q).2) = (x * p.1, y * p.2) + (x * q.1, y * q.2) All goals completed! 🐙 map_smul' c p := x:ℝy:ℝs:Set (ℝ × ℝ)c:ℝp:ℝ × ℝ⊢ (x * (c • p).1, y * (c • p).2) = (RingHom.id ℝ) c • (x * p.1, y * p.2) All goals completed! 🐙 } x:ℝy:ℝs:Set (ℝ × ℝ)scaleLinear:ℝ × ℝ →ₗ[ℝ] ℝ × ℝ := { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ }h₁:scale x y = ⇑scaleLinear := rfl⊢ lbMeasure (scale x y '' s) = ENNReal.ofReal |x * y| * lbMeasure s have h₂ : scaleLinear.toMatrix (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ) = !![x, 0; 0, y] := x:ℝy:ℝs:Set (ℝ × ℝ)⊢ lbMeasure (scale x y '' s) = ENNReal.ofReal |x * y| * lbMeasure s x:ℝy:ℝs:Set (ℝ × ℝ)scaleLinear:ℝ × ℝ →ₗ[ℝ] ℝ × ℝ := { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ }h₁:scale x y = ⇑scaleLinear := rfli:Fin 2j:Fin 2⊢ (LinearMap.toMatrix (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)) scaleLinear i j = !![x, 0; 0, y] i j; x:ℝy:ℝs:Set (ℝ × ℝ)scaleLinear:ℝ × ℝ →ₗ[ℝ] ℝ × ℝ := { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ }h₁:scale x y = ⇑scaleLinear := rfli:Fin 2j:Fin 2⊢ (LinearMap.toMatrix (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)) { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ } i j = !![x, 0; 0, y] i j; x:ℝy:ℝs:Set (ℝ × ℝ)scaleLinear:ℝ × ℝ →ₗ[ℝ] ℝ × ℝ := { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ }h₁:scale x y = ⇑scaleLinear := rflj:Fin 2⊢ (LinearMap.toMatrix (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)) { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ } ((fun i => i) ⟨0, ⋯⟩) j = !![x, 0; 0, y] ((fun i => i) ⟨0, ⋯⟩) jx:ℝy:ℝs:Set (ℝ × ℝ)scaleLinear:ℝ × ℝ →ₗ[ℝ] ℝ × ℝ := { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ }h₁:scale x y = ⇑scaleLinear := rflj:Fin 2⊢ (LinearMap.toMatrix (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)) { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ } ((fun i => i) ⟨1, ⋯⟩) j = !![x, 0; 0, y] ((fun i => i) ⟨1, ⋯⟩) j x:ℝy:ℝs:Set (ℝ × ℝ)scaleLinear:ℝ × ℝ →ₗ[ℝ] ℝ × ℝ := { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ }h₁:scale x y = ⇑scaleLinear := rflj:Fin 2⊢ (LinearMap.toMatrix (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)) { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ } ((fun i => i) ⟨0, ⋯⟩) j = !![x, 0; 0, y] ((fun i => i) ⟨0, ⋯⟩) jx:ℝy:ℝs:Set (ℝ × ℝ)scaleLinear:ℝ × ℝ →ₗ[ℝ] ℝ × ℝ := { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ }h₁:scale x y = ⇑scaleLinear := rflj:Fin 2⊢ (LinearMap.toMatrix (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)) { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ } ((fun i => i) ⟨1, ⋯⟩) j = !![x, 0; 0, y] ((fun i => i) ⟨1, ⋯⟩) j x:ℝy:ℝs:Set (ℝ × ℝ)scaleLinear:ℝ × ℝ →ₗ[ℝ] ℝ × ℝ := { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ }h₁:scale x y = ⇑scaleLinear := rfl⊢ (LinearMap.toMatrix (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)) { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ } ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) = !![x, 0; 0, y] ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩)x:ℝy:ℝs:Set (ℝ × ℝ)scaleLinear:ℝ × ℝ →ₗ[ℝ] ℝ × ℝ := { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ }h₁:scale x y = ⇑scaleLinear := rfl⊢ (LinearMap.toMatrix (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)) { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ } ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) = !![x, 0; 0, y] ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) x:ℝy:ℝs:Set (ℝ × ℝ)scaleLinear:ℝ × ℝ →ₗ[ℝ] ℝ × ℝ := { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ }h₁:scale x y = ⇑scaleLinear := rfl⊢ (LinearMap.toMatrix (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)) { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ } ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) = !![x, 0; 0, y] ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩)x:ℝy:ℝs:Set (ℝ × ℝ)scaleLinear:ℝ × ℝ →ₗ[ℝ] ℝ × ℝ := { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ }h₁:scale x y = ⇑scaleLinear := rfl⊢ (LinearMap.toMatrix (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)) { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ } ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) = !![x, 0; 0, y] ((fun i => i) ⟨0, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩)x:ℝy:ℝs:Set (ℝ × ℝ)scaleLinear:ℝ × ℝ →ₗ[ℝ] ℝ × ℝ := { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ }h₁:scale x y = ⇑scaleLinear := rfl⊢ (LinearMap.toMatrix (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)) { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ } ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩) = !![x, 0; 0, y] ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨0, ⋯⟩)x:ℝy:ℝs:Set (ℝ × ℝ)scaleLinear:ℝ × ℝ →ₗ[ℝ] ℝ × ℝ := { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ }h₁:scale x y = ⇑scaleLinear := rfl⊢ (LinearMap.toMatrix (Basis.finTwoProd ℝ) (Basis.finTwoProd ℝ)) { toFun := fun p => (x * p.1, y * p.2), map_add' := ⋯, map_smul' := ⋯ } ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) = !![x, 0; 0, y] ((fun i => i) ⟨1, ⋯⟩) ((fun i => i) ⟨1, ⋯⟩) All goals completed! 🐙 All goals completed! 🐙

The Lebesgue measure of the unit square is 1.

@[category test, AMS 51] lemma lbMeasure_unitSquare : lbMeasure unitSquare = 1 := ⊢ lbMeasure unitSquare = 1 ⊢ unitSquare = ↑(Basis.finTwoProd ℝ).parallelepiped p:ℝ × ℝ⊢ p ∈ unitSquare ↔ p ∈ ↑(Basis.finTwoProd ℝ).parallelepiped p:ℝ × ℝ⊢ 0 ≤ p.1 ∧ p.1 ≤ 1 ∧ 0 ≤ p.2 ∧ p.2 ≤ 1 ↔ ∃ t, ((∀ (i : Fin 2), 0 i ≤ t i) ∧ ∀ (i : Fin 2), t i ≤ 1 i) ∧ p = (t 0, t 1) exact ⟨fun h ↦ ⟨![p.1, p.2], p:ℝ × ℝh:0 ≤ p.1 ∧ p.1 ≤ 1 ∧ 0 ≤ p.2 ∧ p.2 ≤ 1⊢ ((∀ (i : Fin 2), 0 i ≤ ![p.1, p.2] i) ∧ ∀ (i : Fin 2), ![p.1, p.2] i ≤ 1 i) ∧ p = (![p.1, p.2] 0, ![p.1, p.2] 1) All goals completed! 🐙⟩, fun ⟨t, ht⟩ ↦ ht.2 ▸ ⟨ht.1.1 0, ht.1.2 0, ht.1.1 1, ht.1.2 1⟩⟩

The Lebesgue measure of the a rectangle r is r.width * r.height

@[category test, AMS 51] lemma lbMeasure_rectangle_toSet (r : Rectangle) : lbMeasure r.toSet = .ofReal |r.width * r.height| := r:Rectangle⊢ lbMeasure r.toSet = ENNReal.ofReal |r.width * r.height| All goals completed! 🐙

The areas of the required rectangles sum to 1.

@[category test, AMS 51] lemma tsum_area_eq_one : ∑' (n : ℕ), ((1 / (n + 1)) * (1 / (n + 2)) : ℝ) = 1 := ⊢ ∑' (n : ℕ), 1 / (↑n + 1) * (1 / (↑n + 2)) = 1 have (n : ℕ) : ∑ i ∈ Finset.range n, (1 / (i + 1) * (1 / (i + 2)) : ℝ) = 1 - 1 / (n + 1) := ⊢ ∑' (n : ℕ), 1 / (↑n + 1) * (1 / (↑n + 2)) = 1 induction n with ⊢ ∑ i ∈ Finset.range 0, 1 / (↑i + 1) * (1 / (↑i + 2)) = 1 - 1 / (↑0 + 1) All goals completed! 🐙 n:ℕih:∑ i ∈ Finset.range n, 1 / (↑i + 1) * (1 / (↑i + 2)) = 1 - 1 / (↑n + 1)⊢ ∑ i ∈ Finset.range (n + 1), 1 / (↑i + 1) * (1 / (↑i + 2)) = 1 - 1 / (↑(n + 1) + 1) n:ℕih:∑ i ∈ Finset.range n, 1 / (↑i + 1) * (1 / (↑i + 2)) = 1 - 1 / (↑n + 1)⊢ 1 - 1 / (↑n + 1) + 1 / (↑n + 1) * (1 / (↑n + 2)) = 1 - 1 / (↑(n + 1) + 1); n:ℕih:∑ i ∈ Finset.range n, 1 / (↑i + 1) * (1 / (↑i + 2)) = 1 - 1 / (↑n + 1)⊢ ((↑n + 1 - 1) * (↑n + 2) + 1) * (↑(n + 1) + 1) = (↑n + 1) * (↑n + 2) * (↑(n + 1) + 1 - 1); n:ℕih:∑ i ∈ Finset.range n, 1 / (↑i + 1) * (1 / (↑i + 2)) = 1 - 1 / (↑n + 1)⊢ ((↑n + 1 - 1) * (↑n + 2) + 1) * (↑n + 1 + 1) = (↑n + 1) * (↑n + 2) * (↑n + 1 + 1 - 1); All goals completed! 🐙 this:∀ (n : ℕ), ∑ i ∈ Finset.range n, 1 / (↑i + 1) * (1 / (↑i + 2)) = 1 - 1 / (↑n + 1) := fun n => Nat.recAux (of_eq_true (Eq.trans (congr (congrArg Eq (Finset.sum_congr (Eq.refl ∅) fun x a => congr (congrArg HMul.hMul (one_div (↑x + 1))) (one_div (↑x + 2)))) (Eq.trans (congrArg (HSub.hSub 1) (Eq.trans (congrArg (HDiv.hDiv 1) (Eq.trans (congrArg (fun x => x + 1) (CharP.cast_eq_zero ℝ 0)) (zero_add 1))) (div_self (of_eq_true (Eq.trans (congrArg Not one_ne_zero._simp_1) not_false_eq_true))))) (_root_.sub_self 1))) (eq_self 0))) (fun n ih => Eq.mpr (id (congrArg (fun _a => _a = 1 - 1 / (↑(n + 1) + 1)) (Finset.sum_range_succ (fun i => 1 / (↑i + 1) * (1 / (↑i + 2))) n))) (Eq.mpr (id (congrArg (fun _a => _a + 1 / (↑n + 1) * (1 / (↑n + 2)) = 1 - 1 / (↑(n + 1) + 1)) ih)) (Eq.mpr (id (Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.subst_sub (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 1) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n + 1).2 (1, ↑n + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑n) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑n)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n).2 (1, ↑n).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑n))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 1)) (one_mul ((1, ↑n + 1) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑n + 1) ::ᵣ []))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 1 - 1)) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, ↑n + 1 - 1) (mul_one ((-1, ↑n + 1) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))))) (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 1 - 1) (Eq.symm (div_one 1)))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 2) ((1, ↑n + 1 - 1) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = ↑n + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 1 - 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n + 1 - 1).2 (1, ↑n + 1 - 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 1 - 1)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 2)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HMul.hMul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl)) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑n) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑n)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n).2 (1, ↑n).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑n))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 1)) (one_mul ((1, ↑n + 1) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑n + 1) ::ᵣ []))) (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑n) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑n)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n).2 (1, ↑n).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑n))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl 2) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval 2)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 2 (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 2 (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = 2) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, 2) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, 2).2 (1, 2).1)) (Mathlib.Tactic.FieldSimp.zpow'_one 2))))) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 2)) (one_mul ((1, ↑n + 2) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑n + 2) ::ᵣ []))) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (-1, ↑n + 2) (mul_one ((-1, ↑n + 1) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ((↑n + 1 - 1) * (↑n + 2) + 1)) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, (↑n + 1 - 1) * (↑n + 2) + 1) (mul_one ((-1, ↑n + 2) ::ᵣ ((-1, ↑n + 1) ::ᵣ [])).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (add_pos' (Nat.cast_pos'.mpr (Right.add_pos_of_nonneg_of_pos (zero_le n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 ((↑n + 1 - 1) * (↑n + 2) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1) + 1) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 ((↑n + 1 - 1) * (↑n + 2) + 1) (Eq.symm (div_one 1)))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1) + 1) ((1, (↑n + 1 - 1) * (↑n + 2) + 1) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = (↑n + 1 - 1) * (↑n + 2) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, (↑n + 1 - 1) * (↑n + 2) + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, (↑n + 1 - 1) * (↑n + 2) + 1).2 (1, (↑n + 1 - 1) * (↑n + 2) + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ((↑n + 1 - 1) * (↑n + 2) + 1)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑(n + 1) + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.subst_sub (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (add_pos' (Nat.cast_pos'.mpr (Right.add_pos_of_nonneg_of_pos (zero_le n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1) + 1) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑(n + 1) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1) + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑(n + 1) + 1).2 (1, ↑(n + 1) + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑(n + 1) + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑(n + 1)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑(n + 1))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑(n + 1)) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1)) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑(n + 1)) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1)) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑(n + 1)).2 (1, ↑(n + 1)).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑(n + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑(n + 1) + 1)) (one_mul ((1, ↑(n + 1) + 1) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑(n + 1) + 1) ::ᵣ []))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑(n + 1) + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑(n + 1) + 1 - 1)) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, ↑(n + 1) + 1 - 1) (mul_one ((-1, ↑(n + 1) + 1) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑(n + 1) + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑(n + 1) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))))) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1) + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 1) (Eq.symm (div_one 1))))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1) + 1 - 1) ((1, ↑n + 2) ::ᵣ ((1, ↑n + 1) ::ᵣ []))) (congr_arg₂ HMul.hMul (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 2) ((1, ↑n + 1) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = ↑n + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n + 1).2 (1, ↑n + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 1)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 2)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑(n + 1) + 1 - 1)))))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (add_pos' (Nat.cast_pos'.mpr (Right.add_pos_of_nonneg_of_pos (zero_le n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) one_ne_zero))))) (Eq.mpr (id (congr (congrArg (fun x => Eq (((↑n + 1 - 1) * (↑n + 2) + 1) * (x + 1))) (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))) (congrArg (fun x => (↑n + 1) * (↑n + 2) * (x + 1 - 1)) (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))))) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 2 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Eq.refl 4)))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 4) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 2) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 3))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 2 * Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 2 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 4) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 2) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 4) (Eq.refl 5)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 4 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))))))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 2 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Eq.refl 3)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 3))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 3))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 3)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 3 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 3) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 3 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 2) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 3))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 2 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 3) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 2) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 3) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 3) (Eq.refl 5)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 4 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))))))))))) ni:ℕ⊢ 0 ≤ 1 / (↑i + 1) * (1 / (↑i + 2))this:∀ (n : ℕ), ∑ i ∈ Finset.range n, 1 / (↑i + 1) * (1 / (↑i + 2)) = 1 - 1 / (↑n + 1) := fun n => Nat.recAux (of_eq_true (Eq.trans (congr (congrArg Eq (Finset.sum_congr (Eq.refl ∅) fun x a => congr (congrArg HMul.hMul (one_div (↑x + 1))) (one_div (↑x + 2)))) (Eq.trans (congrArg (HSub.hSub 1) (Eq.trans (congrArg (HDiv.hDiv 1) (Eq.trans (congrArg (fun x => x + 1) (CharP.cast_eq_zero ℝ 0)) (zero_add 1))) (div_self (of_eq_true (Eq.trans (congrArg Not one_ne_zero._simp_1) not_false_eq_true))))) (_root_.sub_self 1))) (eq_self 0))) (fun n ih => Eq.mpr (id (congrArg (fun _a => _a = 1 - 1 / (↑(n + 1) + 1)) (Finset.sum_range_succ (fun i => 1 / (↑i + 1) * (1 / (↑i + 2))) n))) (Eq.mpr (id (congrArg (fun _a => _a + 1 / (↑n + 1) * (1 / (↑n + 2)) = 1 - 1 / (↑(n + 1) + 1)) ih)) (Eq.mpr (id (Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.subst_sub (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 1) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n + 1).2 (1, ↑n + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑n) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑n)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n).2 (1, ↑n).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑n))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 1)) (one_mul ((1, ↑n + 1) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑n + 1) ::ᵣ []))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 1 - 1)) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, ↑n + 1 - 1) (mul_one ((-1, ↑n + 1) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))))) (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 1 - 1) (Eq.symm (div_one 1)))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 2) ((1, ↑n + 1 - 1) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = ↑n + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 1 - 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n + 1 - 1).2 (1, ↑n + 1 - 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 1 - 1)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 2)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HMul.hMul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl)) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑n) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑n)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n).2 (1, ↑n).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑n))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 1)) (one_mul ((1, ↑n + 1) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑n + 1) ::ᵣ []))) (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑n) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑n)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n).2 (1, ↑n).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑n))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl 2) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval 2)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 2 (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 2 (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = 2) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, 2) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, 2).2 (1, 2).1)) (Mathlib.Tactic.FieldSimp.zpow'_one 2))))) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 2)) (one_mul ((1, ↑n + 2) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑n + 2) ::ᵣ []))) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (-1, ↑n + 2) (mul_one ((-1, ↑n + 1) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ((↑n + 1 - 1) * (↑n + 2) + 1)) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, (↑n + 1 - 1) * (↑n + 2) + 1) (mul_one ((-1, ↑n + 2) ::ᵣ ((-1, ↑n + 1) ::ᵣ [])).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (add_pos' (Nat.cast_pos'.mpr (Right.add_pos_of_nonneg_of_pos (zero_le n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 ((↑n + 1 - 1) * (↑n + 2) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1) + 1) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 ((↑n + 1 - 1) * (↑n + 2) + 1) (Eq.symm (div_one 1)))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1) + 1) ((1, (↑n + 1 - 1) * (↑n + 2) + 1) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = (↑n + 1 - 1) * (↑n + 2) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, (↑n + 1 - 1) * (↑n + 2) + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, (↑n + 1 - 1) * (↑n + 2) + 1).2 (1, (↑n + 1 - 1) * (↑n + 2) + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ((↑n + 1 - 1) * (↑n + 2) + 1)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑(n + 1) + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.subst_sub (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (add_pos' (Nat.cast_pos'.mpr (Right.add_pos_of_nonneg_of_pos (zero_le n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1) + 1) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑(n + 1) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1) + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑(n + 1) + 1).2 (1, ↑(n + 1) + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑(n + 1) + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑(n + 1)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑(n + 1))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑(n + 1)) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1)) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑(n + 1)) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1)) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑(n + 1)).2 (1, ↑(n + 1)).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑(n + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑(n + 1) + 1)) (one_mul ((1, ↑(n + 1) + 1) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑(n + 1) + 1) ::ᵣ []))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑(n + 1) + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑(n + 1) + 1 - 1)) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, ↑(n + 1) + 1 - 1) (mul_one ((-1, ↑(n + 1) + 1) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑(n + 1) + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑(n + 1) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))))) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1) + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 1) (Eq.symm (div_one 1))))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1) + 1 - 1) ((1, ↑n + 2) ::ᵣ ((1, ↑n + 1) ::ᵣ []))) (congr_arg₂ HMul.hMul (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 2) ((1, ↑n + 1) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = ↑n + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n + 1).2 (1, ↑n + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 1)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 2)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑(n + 1) + 1 - 1)))))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (add_pos' (Nat.cast_pos'.mpr (Right.add_pos_of_nonneg_of_pos (zero_le n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) one_ne_zero))))) (Eq.mpr (id (congr (congrArg (fun x => Eq (((↑n + 1 - 1) * (↑n + 2) + 1) * (x + 1))) (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))) (congrArg (fun x => (↑n + 1) * (↑n + 2) * (x + 1 - 1)) (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))))) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 2 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Eq.refl 4)))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 4) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 2) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 3))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 2 * Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 2 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 4) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 2) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 4) (Eq.refl 5)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 4 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))))))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 2 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Eq.refl 3)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 3))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 3))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 3)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 3 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 3) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 3 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 2) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 3))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 2 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 3) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 2) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 3) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 3) (Eq.refl 5)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 4 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))))))))))) n⊢ Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, 1 / (↑i + 1) * (1 / (↑i + 2))) Filter.atTop (nhds 1) this:∀ (n : ℕ), ∑ i ∈ Finset.range n, 1 / (↑i + 1) * (1 / (↑i + 2)) = 1 - 1 / (↑n + 1) := fun n => Nat.recAux (of_eq_true (Eq.trans (congr (congrArg Eq (Finset.sum_congr (Eq.refl ∅) fun x a => congr (congrArg HMul.hMul (one_div (↑x + 1))) (one_div (↑x + 2)))) (Eq.trans (congrArg (HSub.hSub 1) (Eq.trans (congrArg (HDiv.hDiv 1) (Eq.trans (congrArg (fun x => x + 1) (CharP.cast_eq_zero ℝ 0)) (zero_add 1))) (div_self (of_eq_true (Eq.trans (congrArg Not one_ne_zero._simp_1) not_false_eq_true))))) (_root_.sub_self 1))) (eq_self 0))) (fun n ih => Eq.mpr (id (congrArg (fun _a => _a = 1 - 1 / (↑(n + 1) + 1)) (Finset.sum_range_succ (fun i => 1 / (↑i + 1) * (1 / (↑i + 2))) n))) (Eq.mpr (id (congrArg (fun _a => _a + 1 / (↑n + 1) * (1 / (↑n + 2)) = 1 - 1 / (↑(n + 1) + 1)) ih)) (Eq.mpr (id (Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.subst_sub (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 1) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n + 1).2 (1, ↑n + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑n) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑n)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n).2 (1, ↑n).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑n))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 1)) (one_mul ((1, ↑n + 1) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑n + 1) ::ᵣ []))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 1 - 1)) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, ↑n + 1 - 1) (mul_one ((-1, ↑n + 1) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))))) (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 1 - 1) (Eq.symm (div_one 1)))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 2) ((1, ↑n + 1 - 1) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = ↑n + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 1 - 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n + 1 - 1).2 (1, ↑n + 1 - 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 1 - 1)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 2)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HMul.hMul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl)) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑n) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑n)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n).2 (1, ↑n).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑n))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 1)) (one_mul ((1, ↑n + 1) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑n + 1) ::ᵣ []))) (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑n) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑n)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n).2 (1, ↑n).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑n))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl 2) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval 2)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 2 (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 2 (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = 2) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, 2) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, 2).2 (1, 2).1)) (Mathlib.Tactic.FieldSimp.zpow'_one 2))))) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 2)) (one_mul ((1, ↑n + 2) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑n + 2) ::ᵣ []))) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (-1, ↑n + 2) (mul_one ((-1, ↑n + 1) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ((↑n + 1 - 1) * (↑n + 2) + 1)) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, (↑n + 1 - 1) * (↑n + 2) + 1) (mul_one ((-1, ↑n + 2) ::ᵣ ((-1, ↑n + 1) ::ᵣ [])).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (add_pos' (Nat.cast_pos'.mpr (Right.add_pos_of_nonneg_of_pos (zero_le n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 ((↑n + 1 - 1) * (↑n + 2) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1) + 1) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 ((↑n + 1 - 1) * (↑n + 2) + 1) (Eq.symm (div_one 1)))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1) + 1) ((1, (↑n + 1 - 1) * (↑n + 2) + 1) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = (↑n + 1 - 1) * (↑n + 2) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, (↑n + 1 - 1) * (↑n + 2) + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, (↑n + 1 - 1) * (↑n + 2) + 1).2 (1, (↑n + 1 - 1) * (↑n + 2) + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ((↑n + 1 - 1) * (↑n + 2) + 1)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑(n + 1) + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.subst_sub (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (add_pos' (Nat.cast_pos'.mpr (Right.add_pos_of_nonneg_of_pos (zero_le n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1) + 1) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑(n + 1) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1) + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑(n + 1) + 1).2 (1, ↑(n + 1) + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑(n + 1) + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑(n + 1)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑(n + 1))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑(n + 1)) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1)) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑(n + 1)) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1)) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑(n + 1)).2 (1, ↑(n + 1)).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑(n + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑(n + 1) + 1)) (one_mul ((1, ↑(n + 1) + 1) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑(n + 1) + 1) ::ᵣ []))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑(n + 1) + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑(n + 1) + 1 - 1)) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, ↑(n + 1) + 1 - 1) (mul_one ((-1, ↑(n + 1) + 1) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑(n + 1) + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑(n + 1) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))))) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1) + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 1) (Eq.symm (div_one 1))))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1) + 1 - 1) ((1, ↑n + 2) ::ᵣ ((1, ↑n + 1) ::ᵣ []))) (congr_arg₂ HMul.hMul (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 2) ((1, ↑n + 1) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = ↑n + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n + 1).2 (1, ↑n + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 1)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 2)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑(n + 1) + 1 - 1)))))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (add_pos' (Nat.cast_pos'.mpr (Right.add_pos_of_nonneg_of_pos (zero_le n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) one_ne_zero))))) (Eq.mpr (id (congr (congrArg (fun x => Eq (((↑n + 1 - 1) * (↑n + 2) + 1) * (x + 1))) (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))) (congrArg (fun x => (↑n + 1) * (↑n + 2) * (x + 1 - 1)) (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))))) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 2 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Eq.refl 4)))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 4) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 2) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 3))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 2 * Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 2 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 4) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 2) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 4) (Eq.refl 5)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 4 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))))))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 2 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Eq.refl 3)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 3))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 3))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 3)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 3 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 3) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 3 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 2) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 3))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 2 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 3) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 2) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 3) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 3) (Eq.refl 5)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 4 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))))))))))) ni:ℕ⊢ 0 ≤ 1 / (↑i + 1) * (1 / (↑i + 2)) All goals completed! 🐙 this:∀ (n : ℕ), ∑ i ∈ Finset.range n, 1 / (↑i + 1) * (1 / (↑i + 2)) = 1 - 1 / (↑n + 1) := fun n => Nat.recAux (of_eq_true (Eq.trans (congr (congrArg Eq (Finset.sum_congr (Eq.refl ∅) fun x a => congr (congrArg HMul.hMul (one_div (↑x + 1))) (one_div (↑x + 2)))) (Eq.trans (congrArg (HSub.hSub 1) (Eq.trans (congrArg (HDiv.hDiv 1) (Eq.trans (congrArg (fun x => x + 1) (CharP.cast_eq_zero ℝ 0)) (zero_add 1))) (div_self (of_eq_true (Eq.trans (congrArg Not one_ne_zero._simp_1) not_false_eq_true))))) (_root_.sub_self 1))) (eq_self 0))) (fun n ih => Eq.mpr (id (congrArg (fun _a => _a = 1 - 1 / (↑(n + 1) + 1)) (Finset.sum_range_succ (fun i => 1 / (↑i + 1) * (1 / (↑i + 2))) n))) (Eq.mpr (id (congrArg (fun _a => _a + 1 / (↑n + 1) * (1 / (↑n + 2)) = 1 - 1 / (↑(n + 1) + 1)) ih)) (Eq.mpr (id (Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.subst_sub (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 1) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n + 1).2 (1, ↑n + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑n) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑n)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n).2 (1, ↑n).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑n))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 1)) (one_mul ((1, ↑n + 1) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑n + 1) ::ᵣ []))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 1 - 1)) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, ↑n + 1 - 1) (mul_one ((-1, ↑n + 1) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))))) (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 1 - 1) (Eq.symm (div_one 1)))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 2) ((1, ↑n + 1 - 1) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = ↑n + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 1 - 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n + 1 - 1).2 (1, ↑n + 1 - 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 1 - 1)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 2)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HMul.hMul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl)) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑n) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑n)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n).2 (1, ↑n).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑n))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 1)) (one_mul ((1, ↑n + 1) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑n + 1) ::ᵣ []))) (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑n) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑n)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n).2 (1, ↑n).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑n))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl 2) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval 2)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 2 (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 2 (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = 2) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, 2) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, 2).2 (1, 2).1)) (Mathlib.Tactic.FieldSimp.zpow'_one 2))))) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 2)) (one_mul ((1, ↑n + 2) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑n + 2) ::ᵣ []))) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (-1, ↑n + 2) (mul_one ((-1, ↑n + 1) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ((↑n + 1 - 1) * (↑n + 2) + 1)) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, (↑n + 1 - 1) * (↑n + 2) + 1) (mul_one ((-1, ↑n + 2) ::ᵣ ((-1, ↑n + 1) ::ᵣ [])).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (add_pos' (Nat.cast_pos'.mpr (Right.add_pos_of_nonneg_of_pos (zero_le n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 ((↑n + 1 - 1) * (↑n + 2) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1) + 1) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 ((↑n + 1 - 1) * (↑n + 2) + 1) (Eq.symm (div_one 1)))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1) + 1) ((1, (↑n + 1 - 1) * (↑n + 2) + 1) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = (↑n + 1 - 1) * (↑n + 2) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, (↑n + 1 - 1) * (↑n + 2) + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, (↑n + 1 - 1) * (↑n + 2) + 1).2 (1, (↑n + 1 - 1) * (↑n + 2) + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ((↑n + 1 - 1) * (↑n + 2) + 1)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑(n + 1) + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.subst_sub (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (add_pos' (Nat.cast_pos'.mpr (Right.add_pos_of_nonneg_of_pos (zero_le n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1) + 1) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑(n + 1) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1) + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑(n + 1) + 1).2 (1, ↑(n + 1) + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑(n + 1) + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑(n + 1)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑(n + 1))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑(n + 1)) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1)) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑(n + 1)) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1)) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑(n + 1)).2 (1, ↑(n + 1)).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑(n + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑(n + 1) + 1)) (one_mul ((1, ↑(n + 1) + 1) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑(n + 1) + 1) ::ᵣ []))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑(n + 1) + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑(n + 1) + 1 - 1)) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, ↑(n + 1) + 1 - 1) (mul_one ((-1, ↑(n + 1) + 1) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑(n + 1) + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑(n + 1) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))))) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1) + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 1) (Eq.symm (div_one 1))))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1) + 1 - 1) ((1, ↑n + 2) ::ᵣ ((1, ↑n + 1) ::ᵣ []))) (congr_arg₂ HMul.hMul (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 2) ((1, ↑n + 1) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = ↑n + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n + 1).2 (1, ↑n + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 1)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 2)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑(n + 1) + 1 - 1)))))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (add_pos' (Nat.cast_pos'.mpr (Right.add_pos_of_nonneg_of_pos (zero_le n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) one_ne_zero))))) (Eq.mpr (id (congr (congrArg (fun x => Eq (((↑n + 1 - 1) * (↑n + 2) + 1) * (x + 1))) (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))) (congrArg (fun x => (↑n + 1) * (↑n + 2) * (x + 1 - 1)) (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))))) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 2 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Eq.refl 4)))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 4) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 2) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 3))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 2 * Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 2 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 4) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 2) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 4) (Eq.refl 5)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 4 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))))))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 2 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Eq.refl 3)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 3))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 3))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 3)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 3 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 3) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 3 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 2) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 3))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 2 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 3) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 2) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 3) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 3) (Eq.refl 5)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 4 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))))))))))) n⊢ Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, 1 / (↑i + 1) * (1 / (↑i + 2))) Filter.atTop (nhds 1) simp_rw this:∀ (n : ℕ), ∑ i ∈ Finset.range n, 1 / (↑i + 1) * (1 / (↑i + 2)) = 1 - 1 / (↑n + 1) := fun n => Nat.recAux (of_eq_true (Eq.trans (congr (congrArg Eq (Finset.sum_congr (Eq.refl ∅) fun x a => congr (congrArg HMul.hMul (one_div (↑x + 1))) (one_div (↑x + 2)))) (Eq.trans (congrArg (HSub.hSub 1) (Eq.trans (congrArg (HDiv.hDiv 1) (Eq.trans (congrArg (fun x => x + 1) (CharP.cast_eq_zero ℝ 0)) (zero_add 1))) (div_self (of_eq_true (Eq.trans (congrArg Not one_ne_zero._simp_1) not_false_eq_true))))) (_root_.sub_self 1))) (eq_self 0))) (fun n ih => Eq.mpr (id (congrArg (fun _a => _a = 1 - 1 / (↑(n + 1) + 1)) (Finset.sum_range_succ (fun i => 1 / (↑i + 1) * (1 / (↑i + 2))) n))) (Eq.mpr (id (congrArg (fun _a => _a + 1 / (↑n + 1) * (1 / (↑n + 2)) = 1 - 1 / (↑(n + 1) + 1)) ih)) (Eq.mpr (id (Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.subst_sub (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 1) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n + 1).2 (1, ↑n + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑n) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑n)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n).2 (1, ↑n).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑n))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 1)) (one_mul ((1, ↑n + 1) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑n + 1) ::ᵣ []))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 1 - 1)) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, ↑n + 1 - 1) (mul_one ((-1, ↑n + 1) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))))) (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 1 - 1) (Eq.symm (div_one 1)))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 2) ((1, ↑n + 1 - 1) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = ↑n + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 1 - 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n + 1 - 1).2 (1, ↑n + 1 - 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 1 - 1)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 2)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HMul.hMul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl)) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑n) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑n)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n).2 (1, ↑n).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑n))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 1)) (one_mul ((1, ↑n + 1) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑n + 1) ::ᵣ []))) (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑n) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑n)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n).2 (1, ↑n).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑n))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl 2) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval 2)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 2 (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 2 (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = 2) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, 2) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, 2).2 (1, 2).1)) (Mathlib.Tactic.FieldSimp.zpow'_one 2))))) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 2)) (one_mul ((1, ↑n + 2) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑n + 2) ::ᵣ []))) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (-1, ↑n + 2) (mul_one ((-1, ↑n + 1) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ((↑n + 1 - 1) * (↑n + 2) + 1)) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, (↑n + 1 - 1) * (↑n + 2) + 1) (mul_one ((-1, ↑n + 2) ::ᵣ ((-1, ↑n + 1) ::ᵣ [])).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (add_pos' (Nat.cast_pos'.mpr (Right.add_pos_of_nonneg_of_pos (zero_le n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 ((↑n + 1 - 1) * (↑n + 2) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1) + 1) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 ((↑n + 1 - 1) * (↑n + 2) + 1) (Eq.symm (div_one 1)))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1) + 1) ((1, (↑n + 1 - 1) * (↑n + 2) + 1) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = (↑n + 1 - 1) * (↑n + 2) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, (↑n + 1 - 1) * (↑n + 2) + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, (↑n + 1 - 1) * (↑n + 2) + 1).2 (1, (↑n + 1 - 1) * (↑n + 2) + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ((↑n + 1 - 1) * (↑n + 2) + 1)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑(n + 1) + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.subst_sub (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (add_pos' (Nat.cast_pos'.mpr (Right.add_pos_of_nonneg_of_pos (zero_le n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1) + 1) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑(n + 1) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1) + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑(n + 1) + 1).2 (1, ↑(n + 1) + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑(n + 1) + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑(n + 1)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑(n + 1))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑(n + 1)) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1)) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑(n + 1)) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1)) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑(n + 1)).2 (1, ↑(n + 1)).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑(n + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑(n + 1) + 1)) (one_mul ((1, ↑(n + 1) + 1) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑(n + 1) + 1) ::ᵣ []))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑(n + 1) + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑(n + 1) + 1 - 1)) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, ↑(n + 1) + 1 - 1) (mul_one ((-1, ↑(n + 1) + 1) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑(n + 1) + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑(n + 1) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))))) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1) + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 1) (Eq.symm (div_one 1))))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1) + 1 - 1) ((1, ↑n + 2) ::ᵣ ((1, ↑n + 1) ::ᵣ []))) (congr_arg₂ HMul.hMul (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 2) ((1, ↑n + 1) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = ↑n + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n + 1).2 (1, ↑n + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 1)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 2)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑(n + 1) + 1 - 1)))))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (add_pos' (Nat.cast_pos'.mpr (Right.add_pos_of_nonneg_of_pos (zero_le n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) one_ne_zero))))) (Eq.mpr (id (congr (congrArg (fun x => Eq (((↑n + 1 - 1) * (↑n + 2) + 1) * (x + 1))) (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))) (congrArg (fun x => (↑n + 1) * (↑n + 2) * (x + 1 - 1)) (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))))) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 2 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Eq.refl 4)))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 4) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 2) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 3))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 2 * Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 2 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 4) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 2) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 4) (Eq.refl 5)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 4 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))))))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 2 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Eq.refl 3)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 3))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 3))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 3)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 3 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 3) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 3 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 2) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 3))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 2 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 3) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 2) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 3) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 3) (Eq.refl 5)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 4 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))))))))))) n⊢ Filter.Tendsto (fun n => ∑ i ∈ Finset.range n, 1 / (↑i + 1) * (1 / (↑i + 2))) Filter.atTop (nhds 1)this] this:∀ (n : ℕ), ∑ i ∈ Finset.range n, 1 / (↑i + 1) * (1 / (↑i + 2)) = 1 - 1 / (↑n + 1) := fun n => Nat.recAux (of_eq_true (Eq.trans (congr (congrArg Eq (Finset.sum_congr (Eq.refl ∅) fun x a => congr (congrArg HMul.hMul (one_div (↑x + 1))) (one_div (↑x + 2)))) (Eq.trans (congrArg (HSub.hSub 1) (Eq.trans (congrArg (HDiv.hDiv 1) (Eq.trans (congrArg (fun x => x + 1) (CharP.cast_eq_zero ℝ 0)) (zero_add 1))) (div_self (of_eq_true (Eq.trans (congrArg Not one_ne_zero._simp_1) not_false_eq_true))))) (_root_.sub_self 1))) (eq_self 0))) (fun n ih => Eq.mpr (id (congrArg (fun _a => _a = 1 - 1 / (↑(n + 1) + 1)) (Finset.sum_range_succ (fun i => 1 / (↑i + 1) * (1 / (↑i + 2))) n))) (Eq.mpr (id (congrArg (fun _a => _a + 1 / (↑n + 1) * (1 / (↑n + 2)) = 1 - 1 / (↑(n + 1) + 1)) ih)) (Eq.mpr (id (Mathlib.Tactic.FieldSimp.eq_eq_cancel_eq (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.subst_sub (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 1) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n + 1).2 (1, ↑n + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑n) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑n)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n).2 (1, ↑n).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑n))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 1)) (one_mul ((1, ↑n + 1) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑n + 1) ::ᵣ []))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 1 - 1)) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, ↑n + 1 - 1) (mul_one ((-1, ↑n + 1) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))))) (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 1 - 1) (Eq.symm (div_one 1)))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 2) ((1, ↑n + 1 - 1) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = ↑n + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 1 - 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n + 1 - 1).2 (1, ↑n + 1 - 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 1 - 1)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 2)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HMul.hMul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl)) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑n) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑n)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n).2 (1, ↑n).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑n))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 1)) (one_mul ((1, ↑n + 1) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑n + 1) ::ᵣ []))) (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑n) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑n)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑n) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑n) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n).2 (1, ↑n).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑n))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl 2) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval 2)) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 2 (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 2 (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = 2) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, 2) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, 2).2 (1, 2).1)) (Mathlib.Tactic.FieldSimp.zpow'_one 2))))) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑n + 2)) (one_mul ((1, ↑n + 2) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑n + 2) ::ᵣ []))) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (-1, ↑n + 2) (mul_one ((-1, ↑n + 1) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ((↑n + 1 - 1) * (↑n + 2) + 1)) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, (↑n + 1 - 1) * (↑n + 2) + 1) (mul_one ((-1, ↑n + 2) ::ᵣ ((-1, ↑n + 1) ::ᵣ [])).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (add_pos' (Nat.cast_pos'.mpr (Right.add_pos_of_nonneg_of_pos (zero_le n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 ((↑n + 1 - 1) * (↑n + 2) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑n + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1) + 1) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 ((↑n + 1 - 1) * (↑n + 2) + 1) (Eq.symm (div_one 1)))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1) + 1) ((1, (↑n + 1 - 1) * (↑n + 2) + 1) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = (↑n + 1 - 1) * (↑n + 2) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, (↑n + 1 - 1) * (↑n + 2) + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, (↑n + 1 - 1) * (↑n + 2) + 1).2 (1, (↑n + 1 - 1) * (↑n + 2) + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ((↑n + 1 - 1) * (↑n + 2) + 1)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑(n + 1) + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.subst_sub (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (add_pos' (Nat.cast_pos'.mpr (Right.add_pos_of_nonneg_of_pos (zero_le n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1) + 1) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑(n + 1) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1) + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑(n + 1) + 1).2 (1, ↑(n + 1) + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑(n + 1) + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul (Eq.trans (congr_arg₂ HDiv.hDiv rfl rfl) rfl) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.div_eq_eval (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Mathlib.Tactic.FieldSimp.subst_add (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Eq.trans (Eq.refl ↑(n + 1)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval ↑(n + 1))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑(n + 1)) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1)) (Eq.symm (div_one 1))) (Eq.mpr (id (congrArg (fun _a => _a = ↑(n + 1)) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1)) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑(n + 1)).2 (1, ↑(n + 1)).1)) (Mathlib.Tactic.FieldSimp.zpow'_one ↑(n + 1)))))) (Mathlib.Tactic.FieldSimp.eq_mul_of_eq_eq_eq_mul rfl (Eq.trans (Mathlib.Tactic.FieldSimp.NF.one_eq_eval ℝ) (Eq.symm (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑(n + 1) + 1)) (one_mul ((1, ↑(n + 1) + 1) ::ᵣ []).eval)) (Mathlib.Tactic.FieldSimp.NF.one_div_eq_eval ((1, ↑(n + 1) + 1) ::ᵣ []))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑(n + 1) + 1) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval []))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Eq.symm (div_one 1)) rfl)) (Mathlib.Tactic.FieldSimp.NF.atom_eq_eval (↑(n + 1) + 1 - 1)) (Mathlib.Tactic.FieldSimp.NF.mul_eq_eval₃ (1, ↑(n + 1) + 1 - 1) (mul_one ((-1, ↑(n + 1) + 1) ::ᵣ []).eval))) (Eq.symm (Mathlib.Tactic.FieldSimp.NF.eval_mul_eval_cons 1 (↑(n + 1) + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval (-1) (↑(n + 1) + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))))) (Mathlib.Tactic.FieldSimp.NF.eval_cons_mul_eval_cons_neg (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (one_mul (Mathlib.Tactic.FieldSimp.NF.eval [])))))))) rfl (Mathlib.Tactic.FieldSimp.eq_div_of_eq_one_of_subst (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑(n + 1) + 1 - 1) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 2) (Mathlib.Tactic.FieldSimp.NF.cons_eq_div_of_eq_div 1 (↑n + 1) (Eq.symm (div_one 1))))) (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑(n + 1) + 1 - 1) ((1, ↑n + 2) ::ᵣ ((1, ↑n + 1) ::ᵣ []))) (congr_arg₂ HMul.hMul (Eq.trans (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 2) ((1, ↑n + 1) ::ᵣ [])) (congr_arg₂ HMul.hMul (Eq.mpr (id (congrArg (fun _a => _a = ↑n + 1) (Mathlib.Tactic.FieldSimp.NF.eval_cons (1, ↑n + 1) []))) (Eq.trans (one_mul (Mathlib.Tactic.FieldSimp.zpow' (1, ↑n + 1).2 (1, ↑n + 1).1)) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 1)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑n + 2)))) (Mathlib.Tactic.FieldSimp.zpow'_one (↑(n + 1) + 1 - 1)))))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (add_pos' (Nat.cast_pos'.mpr (Right.add_pos_of_nonneg_of_pos (zero_le n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℕ (Eq.refl 1)) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2)) (Eq.refl (Nat.ble 1 2))))) (Mathlib.Tactic.FieldSimp.NF.cons_ne_zero (-1) (ne_of_gt (Right.add_pos_of_nonneg_of_pos (Nat.cast_nonneg' n) (Mathlib.Meta.Positivity.pos_of_isNat (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one) (Eq.refl (Nat.ble 1 1))))) one_ne_zero))))) (Eq.mpr (id (congr (congrArg (fun x => Eq (((↑n + 1 - 1) * (↑n + 2) + 1) * (x + 1))) (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))) (congrArg (fun x => (↑n + 1) * (↑n + 2) * (x + 1 - 1)) (Eq.trans (Nat.cast_add n 1) (congrArg (HAdd.hAdd ↑n) Nat.cast_one))))) (Mathlib.Tactic.Ring.of_eq (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap_zero (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 0)))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 2 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Eq.refl 4)))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 4) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 2) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 3))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 2 * Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 2 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 4) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 2) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 4) (Eq.refl 5)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 4 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))))))) (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.mul_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 2 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Eq.refl 3)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + 0)))))) (Mathlib.Tactic.Ring.sub_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.add_congr (Mathlib.Tactic.Ring.atom_pf ↑n) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_gt (Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.cast_pos (Mathlib.Meta.NormNum.isNat_ofNat ℝ Nat.cast_one)) (Mathlib.Tactic.Ring.sub_pf (Mathlib.Tactic.Ring.neg_add (Mathlib.Tactic.Ring.neg_one_mul (Mathlib.Meta.NormNum.IsInt.to_raw_eq (Mathlib.Meta.NormNum.isInt_mul (Eq.refl HMul.hMul) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1)) (Eq.refl (Int.negOfNat 1))))) Mathlib.Tactic.Ring.neg_zero) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.IsInt.to_isNat (Mathlib.Meta.NormNum.isInt_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.to_isInt (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2)) (Mathlib.Meta.NormNum.IsInt.of_raw ℝ (Int.negOfNat 1)) (Eq.refl (Int.ofNat 1))))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2)) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_right (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 2))) (Mathlib.Tactic.Ring.mul_zero (Nat.rawCast 2)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 1 * Nat.rawCast 2 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 1) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 3))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 2))) (Mathlib.Tactic.Ring.mul_one (Nat.rawCast 3))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 1 * Nat.rawCast 3)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 3 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 3) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 3 + 0)))) (Mathlib.Tactic.Ring.add_mul (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pf_left (↑n) (Nat.rawCast 2) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_add (Mathlib.Tactic.Ring.mul_pp_pf_overlap (↑n) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℕ 1) (Eq.refl 3))) (Mathlib.Tactic.Ring.one_mul (Nat.rawCast 1))) (Mathlib.Tactic.Ring.mul_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1)) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 2 * Nat.rawCast 1) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.zero_mul (Nat.rawCast 1 + (↑n ^ Nat.rawCast 1 * Nat.rawCast 1 + 0))) (Mathlib.Tactic.Ring.add_pf_add_zero (↑n ^ Nat.rawCast 2 * Nat.rawCast 1 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))) (Mathlib.Tactic.Ring.add_pf_add_lt (↑n ^ Nat.rawCast 1 * Nat.rawCast 3) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 2) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 3) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 1) (Eq.refl 4)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0))))) (Mathlib.Tactic.Ring.add_pf_add_lt (Nat.rawCast 2) (Mathlib.Tactic.Ring.add_pf_add_overlap (Mathlib.Tactic.Ring.add_overlap_pf (↑n) (Nat.rawCast 1) (Mathlib.Meta.NormNum.IsNat.to_raw_eq (Mathlib.Meta.NormNum.isNat_add (Eq.refl HAdd.hAdd) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 2) (Mathlib.Meta.NormNum.IsNat.of_raw ℝ 3) (Eq.refl 5)))) (Mathlib.Tactic.Ring.add_pf_zero_add (↑n ^ Nat.rawCast 2 * Nat.rawCast 4 + (↑n ^ Nat.rawCast 3 * Nat.rawCast 1 + 0)))))))))))) n⊢ nhds 1 = nhds (1 - 0) All goals completed! 🐙

A configuration of rectangles of sides 1 / (n + 1) and 1 / (n + 2).

structure Configuration : Type where rect (n : ℕ) : Rectangle rect_width (n : ℕ) : (rect n).width = 1 / (n + 1) rect_height (n : ℕ) : (rect n).height = 1 / (n + 2)

A "packing" means that the interiors of any two rectangles are disjoint.

def Configuration.IsPacking (c : Configuration) : Prop := Pairwise fun m n ↦ interior (c.rect m).toSet ∩ interior (c.rect n).toSet = ∅

Can a unit square be covered by rectangles of width 1 / (n + 1) and height 1 / (n + 2)?

@[category research open, AMS 51] theorem declaration uses 'sorry'rectangles_cover_unit_square : answer(sorry) ↔ ∃ c : Configuration, ∀ p ∈ unitSquare, ∃ n, p ∈ (c.rect n).toSet := ⊢ True ↔ ∃ c, ∀ p ∈ unitSquare, ∃ n, p ∈ (c.rect n).toSet All goals completed! 🐙

Equivalently, can a unit square be packed with rectangles of width 1 / (n + 1) and height 1 / (n + 2)?

@[category research open, AMS 51] theorem declaration uses 'sorry'rectangles_pack_unit_square : answer(sorry) ↔ ∃ c : Configuration, (∀ n, (c.rect n).toSet ⊆ unitSquare) ∧ c.IsPacking := ⊢ True ↔ ∃ c, (∀ (n : ℕ), (c.rect n).toSet ⊆ unitSquare) ∧ c.IsPacking All goals completed! 🐙

It is known that packing the rectangles into a square of side length 133/132 is possible.

Reference: https://www.sciencedirect.com/science/article/pii/0097316594901163

@[category research solved, AMS 51] theorem declaration uses 'sorry'rectangles_pack_square_133_div_132 : (∃ c : Configuration, (∀ n, (c.rect n).toSet ⊆ Rectangle.toSet ⟨133/132, 133/132, (0, 0), 0⟩) ∧ c.IsPacking) := ⊢ ∃ c, (∀ (n : ℕ), (c.rect n).toSet ⊆ { width := 133 / 132, height := 133 / 132, start := (0, 0), rotation := 0 }.toSet) ∧ c.IsPacking All goals completed! 🐙

It is known that packing the rectangles into a square of side length 501/500 is possible.

Reference: https://www.sciencedirect.com/science/article/pii/S0167506008706009

@[category research solved, AMS 51] theorem declaration uses 'sorry'rectangles_pack_square_501_div_500 : (∃ c : Configuration, (∀ n, (c.rect n).toSet ⊆ Rectangle.toSet ⟨501/500, 501/500, (0, 0), 0⟩) ∧ c.IsPacking) := ⊢ ∃ c, (∀ (n : ℕ), (c.rect n).toSet ⊆ { width := 501 / 500, height := 501 / 500, start := (0, 0), rotation := 0 }.toSet) ∧ c.IsPacking All goals completed! 🐙 end Mathoverflow34145