l3lab/ntp-mathlib huggingface dataset:
/-- The **Cayley-Hamilton Theorem** for f.g. modules over arbitrary rings states that for each
`R`-endomorphism `φ` of an `R`-module `M` such that `φ(M) ≤ I • M` for some ideal `I`, there
exists some `n` and some `aᵢ ∈ Iⁱ` such that `φⁿ + a₁ φⁿ⁻¹ + ⋯ + aₙ = 0`.

This is the version found in Eisenbud 4.3, which is slightly weaker than Matsumura 2.1
(this lacks the constraint on `n`), and is slightly stronger than Atiyah-Macdonald 2.4.
-/
theorem LinearMap.exists_monic_and_coeff_mem_pow_and_aeval_eq_zero_of_range_le_smul
    [Module.Finite R M] (f : Module.End R M) (I : Ideal R) (hI : LinearMap.range f ≤ I • ⊤) :
    ∃ p : R[X], p.Monic ∧ (∀ k, p.coeff k ∈ I ^ (p.natDegree - k)) ∧ Polynomial.aeval f p = 0 := by
  classical
    cases subsingleton_or_nontrivial R
    · exact ⟨0, Polynomial.monic_of_subsingleton _, by simp⟩
    obtain ⟨s : Finset M, hs : Submodule.span R (s : Set M) = ⊤⟩ :=
      Module.Finite.out (R := R) (M := M)
    -- Porting note: `H` was `rfl`
    obtain ⟨A, H, h⟩ :=
      Matrix.isRepresentation.toEnd_exists_mem_ideal R ((↑) : s → M)
        (by rw [Subtype.range_coe_subtype, Finset.setOf_mem, hs]) f I hI
    rw [← H]
    refine' ⟨A.1.charpoly, A.1.charpoly_monic, _, _⟩
    · 

Proof state:
case inr.intro.intro.intro.refine'_1
ι : Type u_1
inst✝⁴ : Fintype ι
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
I✝ : Ideal R
b : ι → M
hb : Submodule.span R (Set.range b) = ⊤
inst✝ : Module.Finite R M
f : Module.End R M
I : Ideal R
hI : range f ≤ I • ⊤
h✝ : Nontrivial R
s : Finset M
hs : Submodule.span R ↑s = ⊤
A : ↥(Matrix.isRepresentation R Subtype.val)
H : (Matrix.isRepresentation.toEnd R Subtype.val (_ : Submodule.span R (Set.range Subtype.val) = ⊤)) A = f
h : ∀ (i j : { x // x ∈ s }), ↑A i j ∈ I
⊢ ∀ (k : ℕ), coeff (Matrix.charpoly ↑A) k ∈ I ^ (natDegree (Matrix.charpoly ↑A) - k)

Next tactic:
rw [A.1.charpoly_natDegree_eq_dim]
/-- The **Cayley-Hamilton Theorem** for f.g. modules over arbitrary rings states that for each
`R`-endomorphism `φ` of an `R`-module `M` such that `φ(M) ≤ I • M` for some ideal `I`, there
exists some `n` and some `aᵢ ∈ Iⁱ` such that `φⁿ + a₁ φⁿ⁻¹ + ⋯ + aₙ = 0`.

This is the version found in Eisenbud 4.3, which is slightly weaker than Matsumura 2.1
(this lacks the constraint on `n`), and is slightly stronger than Atiyah-Macdonald 2.4.
-/
theorem LinearMap.exists_monic_and_coeff_mem_pow_and_aeval_eq_zero_of_range_le_smul
    [Module.Finite R M] (f : Module.End R M) (I : Ideal R) (hI : LinearMap.range f ≤ I • ⊤) :
    ∃ p : R[X], p.Monic ∧ (∀ k, p.coeff k ∈ I ^ (p.natDegree - k)) ∧ Polynomial.aeval f p = 0 := by
  classical
    cases subsingleton_or_nontrivial R
    · exact ⟨0, Polynomial.monic_of_subsingleton _, by simp⟩
    obtain ⟨s : Finset M, hs : Submodule.span R (s : Set M) = ⊤⟩ :=
      Module.Finite.out (R := R) (M := M)
    -- Porting note: `H` was `rfl`
    obtain ⟨A, H, h⟩ :=
      Matrix.isRepresentation.toEnd_exists_mem_ideal R ((↑) : s → M)
        (by rw [Subtype.range_coe_subtype, Finset.setOf_mem, hs]) f I hI
    rw [← H]
    refine' ⟨A.1.charpoly, A.1.charpoly_monic, _, _⟩
    · rw [A.1.charpoly_natDegree_eq_dim]
      

Proof state:
case inr.intro.intro.intro.refine'_1
ι : Type u_1
inst✝⁴ : Fintype ι
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
I✝ : Ideal R
b : ι → M
hb : Submodule.span R (Set.range b) = ⊤
inst✝ : Module.Finite R M
f : Module.End R M
I : Ideal R
hI : range f ≤ I • ⊤
h✝ : Nontrivial R
s : Finset M
hs : Submodule.span R ↑s = ⊤
A : ↥(Matrix.isRepresentation R Subtype.val)
H : (Matrix.isRepresentation.toEnd R Subtype.val (_ : Submodule.span R (Set.range Subtype.val) = ⊤)) A = f
h : ∀ (i j : { x // x ∈ s }), ↑A i j ∈ I
⊢ ∀ (k : ℕ), coeff (Matrix.charpoly ↑A) k ∈ I ^ (Fintype.card { x // x ∈ s } - k)

Next tactic:
exact coeff_charpoly_mem_ideal_pow h
/-- The **Cayley-Hamilton Theorem** for f.g. modules over arbitrary rings states that for each
`R`-endomorphism `φ` of an `R`-module `M` such that `φ(M) ≤ I • M` for some ideal `I`, there
exists some `n` and some `aᵢ ∈ Iⁱ` such that `φⁿ + a₁ φⁿ⁻¹ + ⋯ + aₙ = 0`.

This is the version found in Eisenbud 4.3, which is slightly weaker than Matsumura 2.1
(this lacks the constraint on `n`), and is slightly stronger than Atiyah-Macdonald 2.4.
-/
theorem LinearMap.exists_monic_and_coeff_mem_pow_and_aeval_eq_zero_of_range_le_smul
    [Module.Finite R M] (f : Module.End R M) (I : Ideal R) (hI : LinearMap.range f ≤ I • ⊤) :
    ∃ p : R[X], p.Monic ∧ (∀ k, p.coeff k ∈ I ^ (p.natDegree - k)) ∧ Polynomial.aeval f p = 0 := by
  classical
    cases subsingleton_or_nontrivial R
    · exact ⟨0, Polynomial.monic_of_subsingleton _, by simp⟩
    obtain ⟨s : Finset M, hs : Submodule.span R (s : Set M) = ⊤⟩ :=
      Module.Finite.out (R := R) (M := M)
    -- Porting note: `H` was `rfl`
    obtain ⟨A, H, h⟩ :=
      Matrix.isRepresentation.toEnd_exists_mem_ideal R ((↑) : s → M)
        (by rw [Subtype.range_coe_subtype, Finset.setOf_mem, hs]) f I hI
    rw [← H]
    refine' ⟨A.1.charpoly, A.1.charpoly_monic, _, _⟩
    · rw [A.1.charpoly_natDegree_eq_dim]
      exact coeff_charpoly_mem_ideal_pow h
    · 

Proof state:
case inr.intro.intro.intro.refine'_2
ι : Type u_1
inst✝⁴ : Fintype ι
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
I✝ : Ideal R
b : ι → M
hb : Submodule.span R (Set.range b) = ⊤
inst✝ : Module.Finite R M
f : Module.End R M
I : Ideal R
hI : range f ≤ I • ⊤
h✝ : Nontrivial R
s : Finset M
hs : Submodule.span R ↑s = ⊤
A : ↥(Matrix.isRepresentation R Subtype.val)
H : (Matrix.isRepresentation.toEnd R Subtype.val (_ : Submodule.span R (Set.range Subtype.val) = ⊤)) A = f
h : ∀ (i j : { x // x ∈ s }), ↑A i j ∈ I
⊢ (aeval ((Matrix.isRepresentation.toEnd R Subtype.val (_ : Submodule.span R (Set.range Subtype.val) = ⊤)) A))
      (Matrix.charpoly ↑A) =
    0

Next tactic:
rw [Polynomial.aeval_algHom_apply,
        ← map_zero (Matrix.isRepresentation.toEnd R ((↑) : s → M) _)]
/-- The **Cayley-Hamilton Theorem** for f.g. modules over arbitrary rings states that for each
`R`-endomorphism `φ` of an `R`-module `M` such that `φ(M) ≤ I • M` for some ideal `I`, there
exists some `n` and some `aᵢ ∈ Iⁱ` such that `φⁿ + a₁ φⁿ⁻¹ + ⋯ + aₙ = 0`.

This is the version found in Eisenbud 4.3, which is slightly weaker than Matsumura 2.1
(this lacks the constraint on `n`), and is slightly stronger than Atiyah-Macdonald 2.4.
-/
theorem LinearMap.exists_monic_and_coeff_mem_pow_and_aeval_eq_zero_of_range_le_smul
    [Module.Finite R M] (f : Module.End R M) (I : Ideal R) (hI : LinearMap.range f ≤ I • ⊤) :
    ∃ p : R[X], p.Monic ∧ (∀ k, p.coeff k ∈ I ^ (p.natDegree - k)) ∧ Polynomial.aeval f p = 0 := by
  classical
    cases subsingleton_or_nontrivial R
    · exact ⟨0, Polynomial.monic_of_subsingleton _, by simp⟩
    obtain ⟨s : Finset M, hs : Submodule.span R (s : Set M) = ⊤⟩ :=
      Module.Finite.out (R := R) (M := M)
    -- Porting note: `H` was `rfl`
    obtain ⟨A, H, h⟩ :=
      Matrix.isRepresentation.toEnd_exists_mem_ideal R ((↑) : s → M)
        (by rw [Subtype.range_coe_subtype, Finset.setOf_mem, hs]) f I hI
    rw [← H]
    refine' ⟨A.1.charpoly, A.1.charpoly_monic, _, _⟩
    · rw [A.1.charpoly_natDegree_eq_dim]
      exact coeff_charpoly_mem_ideal_pow h
    · rw [Polynomial.aeval_algHom_apply,
        ← map_zero (Matrix.isRepresentation.toEnd R ((↑) : s → M) _)]
      

Proof state:
case inr.intro.intro.intro.refine'_2
ι : Type u_1
inst✝⁴ : Fintype ι
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
I✝ : Ideal R
b : ι → M
hb : Submodule.span R (Set.range b) = ⊤
inst✝ : Module.Finite R M
f : Module.End R M
I : Ideal R
hI : range f ≤ I • ⊤
h✝ : Nontrivial R
s : Finset M
hs : Submodule.span R ↑s = ⊤
A : ↥(Matrix.isRepresentation R Subtype.val)
H : (Matrix.isRepresentation.toEnd R Subtype.val (_ : Submodule.span R (Set.range Subtype.val) = ⊤)) A = f
h : ∀ (i j : { x // x ∈ s }), ↑A i j ∈ I
⊢ (Matrix.isRepresentation.toEnd R Subtype.val (_ : Submodule.span R (Set.range Subtype.val) = ⊤))
      ((aeval A) (Matrix.charpoly ↑A)) =
    (Matrix.isRepresentation.toEnd R Subtype.val ?m.303073) 0
ι : Type u_1
inst✝⁴ : Fintype ι
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
I✝ : Ideal R
b : ι → M
hb : Submodule.span R (Set.range b) = ⊤
inst✝ : Module.Finite R M
f : Module.End R M
I : Ideal R
hI : range f ≤ I • ⊤
h✝ : Nontrivial R
s : Finset M
hs : Submodule.span R ↑s = ⊤
A : ↥(Matrix.isRepresentation R Subtype.val)
H : (Matrix.isRepresentation.toEnd R Subtype.val (_ : Submodule.span R (Set.range Subtype.val) = ⊤)) A = f
h : ∀ (i j : { x // x ∈ s }), ↑A i j ∈ I
⊢ Submodule.span R (Set.range Subtype.val) = ⊤

Next tactic:
congr 1
/-- The **Cayley-Hamilton Theorem** for f.g. modules over arbitrary rings states that for each
`R`-endomorphism `φ` of an `R`-module `M` such that `φ(M) ≤ I • M` for some ideal `I`, there
exists some `n` and some `aᵢ ∈ Iⁱ` such that `φⁿ + a₁ φⁿ⁻¹ + ⋯ + aₙ = 0`.

This is the version found in Eisenbud 4.3, which is slightly weaker than Matsumura 2.1
(this lacks the constraint on `n`), and is slightly stronger than Atiyah-Macdonald 2.4.
-/
theorem LinearMap.exists_monic_and_coeff_mem_pow_and_aeval_eq_zero_of_range_le_smul
    [Module.Finite R M] (f : Module.End R M) (I : Ideal R) (hI : LinearMap.range f ≤ I • ⊤) :
    ∃ p : R[X], p.Monic ∧ (∀ k, p.coeff k ∈ I ^ (p.natDegree - k)) ∧ Polynomial.aeval f p = 0 := by
  classical
    cases subsingleton_or_nontrivial R
    · exact ⟨0, Polynomial.monic_of_subsingleton _, by simp⟩
    obtain ⟨s : Finset M, hs : Submodule.span R (s : Set M) = ⊤⟩ :=
      Module.Finite.out (R := R) (M := M)
    -- Porting note: `H` was `rfl`
    obtain ⟨A, H, h⟩ :=
      Matrix.isRepresentation.toEnd_exists_mem_ideal R ((↑) : s → M)
        (by rw [Subtype.range_coe_subtype, Finset.setOf_mem, hs]) f I hI
    rw [← H]
    refine' ⟨A.1.charpoly, A.1.charpoly_monic, _, _⟩
    · rw [A.1.charpoly_natDegree_eq_dim]
      exact coeff_charpoly_mem_ideal_pow h
    · rw [Polynomial.aeval_algHom_apply,
        ← map_zero (Matrix.isRepresentation.toEnd R ((↑) : s → M) _)]
      congr 1
      

Proof state:
case inr.intro.intro.intro.refine'_2.h.e_6.h
ι : Type u_1
inst✝⁴ : Fintype ι
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
I✝ : Ideal R
b : ι → M
hb : Submodule.span R (Set.range b) = ⊤
inst✝ : Module.Finite R M
f : Module.End R M
I : Ideal R
hI : range f ≤ I • ⊤
h✝ : Nontrivial R
s : Finset M
hs : Submodule.span R ↑s = ⊤
A : ↥(Matrix.isRepresentation R Subtype.val)
H : (Matrix.isRepresentation.toEnd R Subtype.val (_ : Submodule.span R (Set.range Subtype.val) = ⊤)) A = f
h : ∀ (i j : { x // x ∈ s }), ↑A i j ∈ I
⊢ (aeval A) (Matrix.charpoly ↑A) = 0

Next tactic:
ext1
/-- The **Cayley-Hamilton Theorem** for f.g. modules over arbitrary rings states that for each
`R`-endomorphism `φ` of an `R`-module `M` such that `φ(M) ≤ I • M` for some ideal `I`, there
exists some `n` and some `aᵢ ∈ Iⁱ` such that `φⁿ + a₁ φⁿ⁻¹ + ⋯ + aₙ = 0`.

This is the version found in Eisenbud 4.3, which is slightly weaker than Matsumura 2.1
(this lacks the constraint on `n`), and is slightly stronger than Atiyah-Macdonald 2.4.
-/
theorem LinearMap.exists_monic_and_coeff_mem_pow_and_aeval_eq_zero_of_range_le_smul
    [Module.Finite R M] (f : Module.End R M) (I : Ideal R) (hI : LinearMap.range f ≤ I • ⊤) :
    ∃ p : R[X], p.Monic ∧ (∀ k, p.coeff k ∈ I ^ (p.natDegree - k)) ∧ Polynomial.aeval f p = 0 := by
  classical
    cases subsingleton_or_nontrivial R
    · exact ⟨0, Polynomial.monic_of_subsingleton _, by simp⟩
    obtain ⟨s : Finset M, hs : Submodule.span R (s : Set M) = ⊤⟩ :=
      Module.Finite.out (R := R) (M := M)
    -- Porting note: `H` was `rfl`
    obtain ⟨A, H, h⟩ :=
      Matrix.isRepresentation.toEnd_exists_mem_ideal R ((↑) : s → M)
        (by rw [Subtype.range_coe_subtype, Finset.setOf_mem, hs]) f I hI
    rw [← H]
    refine' ⟨A.1.charpoly, A.1.charpoly_monic, _, _⟩
    · rw [A.1.charpoly_natDegree_eq_dim]
      exact coeff_charpoly_mem_ideal_pow h
    · rw [Polynomial.aeval_algHom_apply,
        ← map_zero (Matrix.isRepresentation.toEnd R ((↑) : s → M) _)]
      congr 1
      ext1
      

Proof state:
case inr.intro.intro.intro.refine'_2.h.e_6.h.a
ι : Type u_1
inst✝⁴ : Fintype ι
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
I✝ : Ideal R
b : ι → M
hb : Submodule.span R (Set.range b) = ⊤
inst✝ : Module.Finite R M
f : Module.End R M
I : Ideal R
hI : range f ≤ I • ⊤
h✝ : Nontrivial R
s : Finset M
hs : Submodule.span R ↑s = ⊤
A : ↥(Matrix.isRepresentation R Subtype.val)
H : (Matrix.isRepresentation.toEnd R Subtype.val (_ : Submodule.span R (Set.range Subtype.val) = ⊤)) A = f
h : ∀ (i j : { x // x ∈ s }), ↑A i j ∈ I
⊢ ↑((aeval A) (Matrix.charpoly ↑A)) = ↑0

Next tactic:
rw [Polynomial.aeval_subalgebra_coe, Matrix.aeval_self_charpoly, Subalgebra.coe_zero]
theorem LinearMap.exists_monic_and_aeval_eq_zero [Module.Finite R M] (f : Module.End R M) :
    ∃ p : R[X], p.Monic ∧ Polynomial.aeval f p = 0 :=
  (LinearMap.exists_monic_and_coeff_mem_pow_and_aeval_eq_zero_of_range_le_smul R f ⊤ (by 

Proof state:
ι : Type u_1
inst✝⁴ : Fintype ι
M : Type u_2
inst✝³ : AddCommGroup M
R : Type u_3
inst✝² : CommRing R
inst✝¹ : Module R M
I : Ideal R
b : ι → M
hb : Submodule.span R (Set.range b) = ⊤
inst✝ : Module.Finite R M
f : Module.End R M
⊢ range f ≤ ⊤ • ⊤

Next tactic:
simp
theorem isNat_eq_false [AddMonoidWithOne α] [CharZero α] : {a b : α} → {a' b' : ℕ} →
    IsNat a a' → IsNat b b' → Nat.beq a' b' = false → ¬a = b
  | _, _, _, _, ⟨rfl⟩, ⟨rfl⟩, h => by 

Proof state:
α : Type u_1
inst✝¹ : AddMonoidWithOne α
inst✝ : CharZero α
n✝¹ n✝ : ℕ
h : Nat.beq n✝¹ n✝ = false
⊢ ¬↑n✝¹ = ↑n✝

Next tactic:
simp
theorem isNat_eq_false [AddMonoidWithOne α] [CharZero α] : {a b : α} → {a' b' : ℕ} →
    IsNat a a' → IsNat b b' → Nat.beq a' b' = false → ¬a = b
  | _, _, _, _, ⟨rfl⟩, ⟨rfl⟩, h => by simp; 

Proof state:
α : Type u_1
inst✝¹ : AddMonoidWithOne α
inst✝ : CharZero α
n✝¹ n✝ : ℕ
h : Nat.beq n✝¹ n✝ = false
⊢ ¬n✝¹ = n✝

Next tactic:
exact Nat.ne_of_beq_eq_false h
theorem isInt_eq_false [Ring α] [CharZero α] : {a b : α} → {a' b' : ℤ} →
    IsInt a a' → IsInt b b' → decide (a' = b') = false → ¬a = b
  | _, _, _, _, ⟨rfl⟩, ⟨rfl⟩, h => by 

Proof state:
α : Type u_1
inst✝¹ : Ring α
inst✝ : CharZero α
n✝¹ n✝ : ℤ
h : decide (n✝¹ = n✝) = false
⊢ ¬↑n✝¹ = ↑n✝

Next tactic:
simp
theorem isInt_eq_false [Ring α] [CharZero α] : {a b : α} → {a' b' : ℤ} →
    IsInt a a' → IsInt b b' → decide (a' = b') = false → ¬a = b
  | _, _, _, _, ⟨rfl⟩, ⟨rfl⟩, h => by simp; 

Proof state:
α : Type u_1
inst✝¹ : Ring α
inst✝ : CharZero α
n✝¹ n✝ : ℤ
h : decide (n✝¹ = n✝) = false
⊢ ¬n✝¹ = n✝

Next tactic:
exact of_decide_eq_false h
theorem Rat.invOf_denom_swap [Ring α] (n₁ n₂ : ℤ) (a₁ a₂ : α)
    [Invertible a₁] [Invertible a₂] : n₁ * ⅟a₁ = n₂ * ⅟a₂ ↔ n₁ * a₂ = n₂ * a₁ := by
  

Proof state:
α : Type u_1
inst✝² : Ring α
n₁ n₂ : ℤ
a₁ a₂ : α
inst✝¹ : Invertible a₁
inst✝ : Invertible a₂
⊢ ↑n₁ * ⅟a₁ = ↑n₂ * ⅟a₂ ↔ ↑n₁ * a₂ = ↑n₂ * a₁

Next tactic:
rw [mul_invOf_eq_iff_eq_mul_right, ← Int.commute_cast, mul_assoc,
    ← mul_left_eq_iff_eq_invOf_mul, Int.commute_cast]
theorem isRat_eq_false [Ring α] [CharZero α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
    IsRat a na da → IsRat b nb db →
    decide (Int.mul na (.ofNat db) = Int.mul nb (.ofNat da)) = false → ¬a = b
  | _, _, _, _, _, _, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
    

Proof state:
α : Type u_1
inst✝¹ : Ring α
inst✝ : CharZero α
num✝¹ num✝ : ℤ
denom✝¹ denom✝ : ℕ
inv✝¹ : Invertible ↑denom✝¹
inv✝ : Invertible ↑denom✝
h : decide (Int.mul num✝¹ (Int.ofNat denom✝) = Int.mul num✝ (Int.ofNat denom✝¹)) = false
⊢ ¬↑num✝¹ * ⅟↑denom✝¹ = ↑num✝ * ⅟↑denom✝

Next tactic:
rw [Rat.invOf_denom_swap]
theorem isRat_eq_false [Ring α] [CharZero α] : {a b : α} → {na nb : ℤ} → {da db : ℕ} →
    IsRat a na da → IsRat b nb db →
    decide (Int.mul na (.ofNat db) = Int.mul nb (.ofNat da)) = false → ¬a = b
  | _, _, _, _, _, _, ⟨_, rfl⟩, ⟨_, rfl⟩, h => by
    rw [Rat.invOf_denom_swap]; 

Proof state:
α : Type u_1
inst✝¹ : Ring α
inst✝ : CharZero α
num✝¹ num✝ : ℤ
denom✝¹ denom✝ : ℕ
inv✝¹ : Invertible ↑denom✝¹
inv✝ : Invertible ↑denom✝
h : decide (Int.mul num✝¹ (Int.ofNat denom✝) = Int.mul num✝ (Int.ofNat denom✝¹)) = false
⊢ ¬↑num✝¹ * ↑denom✝ = ↑num✝ * ↑denom✝¹

Next tactic:
exact mod_cast of_decide_eq_false h
theorem pi_gt_sqrtTwoAddSeries (n : ℕ) :
    (2 : ℝ) ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) < π := by
  

Proof state:
n : ℕ
⊢ 2 ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) < π

Next tactic:
have : sqrt (2 - sqrtTwoAddSeries 0 n) / (2 : ℝ) * (2 : ℝ) ^ (n + 2) < π := by
    rw [← lt_div_iff, ← sin_pi_over_two_pow_succ]; apply sin_lt; apply div_pos pi_pos
    all_goals apply pow_pos; norm_num
theorem pi_gt_sqrtTwoAddSeries (n : ℕ) :
    (2 : ℝ) ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) < π := by
  have : sqrt (2 - sqrtTwoAddSeries 0 n) / (2 : ℝ) * (2 : ℝ) ^ (n + 2) < π := by
    

Proof state:
n : ℕ
⊢ sqrt (2 - sqrtTwoAddSeries 0 n) / 2 * 2 ^ (n + 2) < π

Next tactic:
rw [← lt_div_iff, ← sin_pi_over_two_pow_succ]
theorem pi_gt_sqrtTwoAddSeries (n : ℕ) :
    (2 : ℝ) ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) < π := by
  have : sqrt (2 - sqrtTwoAddSeries 0 n) / (2 : ℝ) * (2 : ℝ) ^ (n + 2) < π := by
    rw [← lt_div_iff, ← sin_pi_over_two_pow_succ]; 

Proof state:
n : ℕ
⊢ sin (π / 2 ^ (n + 2)) < π / 2 ^ (n + 2)
n : ℕ ⊢ 0 < 2 ^ (n + 2)

Next tactic:
apply sin_lt
theorem pi_gt_sqrtTwoAddSeries (n : ℕ) :
    (2 : ℝ) ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) < π := by
  have : sqrt (2 - sqrtTwoAddSeries 0 n) / (2 : ℝ) * (2 : ℝ) ^ (n + 2) < π := by
    rw [← lt_div_iff, ← sin_pi_over_two_pow_succ]; apply sin_lt; 

Proof state:
case h
n : ℕ
⊢ 0 < π / 2 ^ (n + 2)
n : ℕ ⊢ 0 < 2 ^ (n + 2)

Next tactic:
apply div_pos pi_pos
theorem pi_gt_sqrtTwoAddSeries (n : ℕ) :
    (2 : ℝ) ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) < π := by
  have : sqrt (2 - sqrtTwoAddSeries 0 n) / (2 : ℝ) * (2 : ℝ) ^ (n + 2) < π := by
    rw [← lt_div_iff, ← sin_pi_over_two_pow_succ]; apply sin_lt; apply div_pos pi_pos
    

Proof state:
case h
n : ℕ
⊢ 0 < 2 ^ (n + 2)
n : ℕ ⊢ 0 < 2 ^ (n + 2)

Next tactic:
all_goals apply pow_pos; norm_num
theorem pi_gt_sqrtTwoAddSeries (n : ℕ) :
    (2 : ℝ) ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) < π := by
  have : sqrt (2 - sqrtTwoAddSeries 0 n) / (2 : ℝ) * (2 : ℝ) ^ (n + 2) < π := by
    rw [← lt_div_iff, ← sin_pi_over_two_pow_succ]; apply sin_lt; apply div_pos pi_pos
    all_goals 

Proof state:
case h
n : ℕ
⊢ 0 < 2 ^ (n + 2)

Next tactic:
apply pow_pos
theorem pi_gt_sqrtTwoAddSeries (n : ℕ) :
    (2 : ℝ) ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) < π := by
  have : sqrt (2 - sqrtTwoAddSeries 0 n) / (2 : ℝ) * (2 : ℝ) ^ (n + 2) < π := by
    rw [← lt_div_iff, ← sin_pi_over_two_pow_succ]; apply sin_lt; apply div_pos pi_pos
    all_goals apply pow_pos; 

Proof state:
case h.H
n : ℕ
⊢ 0 < 2

Next tactic:
norm_num
theorem pi_gt_sqrtTwoAddSeries (n : ℕ) :
    (2 : ℝ) ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) < π := by
  have : sqrt (2 - sqrtTwoAddSeries 0 n) / (2 : ℝ) * (2 : ℝ) ^ (n + 2) < π := by
    rw [← lt_div_iff, ← sin_pi_over_two_pow_succ]; apply sin_lt; apply div_pos pi_pos
    all_goals 

Proof state:
n : ℕ
⊢ 0 < 2 ^ (n + 2)

Next tactic:
apply pow_pos
theorem pi_gt_sqrtTwoAddSeries (n : ℕ) :
    (2 : ℝ) ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) < π := by
  have : sqrt (2 - sqrtTwoAddSeries 0 n) / (2 : ℝ) * (2 : ℝ) ^ (n + 2) < π := by
    rw [← lt_div_iff, ← sin_pi_over_two_pow_succ]; apply sin_lt; apply div_pos pi_pos
    all_goals apply pow_pos; 

Proof state:
case H
n : ℕ
⊢ 0 < 2

Next tactic:
norm_num
theorem pi_gt_sqrtTwoAddSeries (n : ℕ) :
    (2 : ℝ) ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) < π := by
  have : sqrt (2 - sqrtTwoAddSeries 0 n) / (2 : ℝ) * (2 : ℝ) ^ (n + 2) < π := by
    rw [← lt_div_iff, ← sin_pi_over_two_pow_succ]; apply sin_lt; apply div_pos pi_pos
    all_goals apply pow_pos; norm_num
  

Proof state:
n : ℕ
this : sqrt (2 - sqrtTwoAddSeries 0 n) / 2 * 2 ^ (n + 2) < π
⊢ 2 ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) < π

Next tactic:
apply lt_of_le_of_lt (le_of_eq _) this
theorem pi_gt_sqrtTwoAddSeries (n : ℕ) :
    (2 : ℝ) ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) < π := by
  have : sqrt (2 - sqrtTwoAddSeries 0 n) / (2 : ℝ) * (2 : ℝ) ^ (n + 2) < π := by
    rw [← lt_div_iff, ← sin_pi_over_two_pow_succ]; apply sin_lt; apply div_pos pi_pos
    all_goals apply pow_pos; norm_num
  apply lt_of_le_of_lt (le_of_eq _) this
  

Proof state:
n : ℕ
this : sqrt (2 - sqrtTwoAddSeries 0 n) / 2 * 2 ^ (n + 2) < π
⊢ 2 ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) = sqrt (2 - sqrtTwoAddSeries 0 n) / 2 * 2 ^ (n + 2)

Next tactic:
rw [pow_succ _ (n + 1), ← mul_assoc, div_mul_cancel, mul_comm]
theorem pi_gt_sqrtTwoAddSeries (n : ℕ) :
    (2 : ℝ) ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) < π := by
  have : sqrt (2 - sqrtTwoAddSeries 0 n) / (2 : ℝ) * (2 : ℝ) ^ (n + 2) < π := by
    rw [← lt_div_iff, ← sin_pi_over_two_pow_succ]; apply sin_lt; apply div_pos pi_pos
    all_goals apply pow_pos; norm_num
  apply lt_of_le_of_lt (le_of_eq _) this
  rw [pow_succ _ (n + 1), ← mul_assoc, div_mul_cancel, mul_comm]; 

Proof state:
case h
n : ℕ
this : sqrt (2 - sqrtTwoAddSeries 0 n) / 2 * 2 ^ (n + 2) < π
⊢ 2 ≠ 0

Next tactic:
norm_num
theorem pi_lt_sqrtTwoAddSeries (n : ℕ) :
    π < (2 : ℝ) ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) + 1 / (4 : ℝ) ^ n := by
  

Proof state:
n : ℕ
⊢ π < 2 ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) + 1 / 4 ^ n

Next tactic:
have : π <
      (sqrt (2 - sqrtTwoAddSeries 0 n) / (2 : ℝ) + (1 : ℝ) / ((2 : ℝ) ^ n) ^ 3 / 4) *
      (2 : ℝ) ^ (n + 2) := by
    rw [← div_lt_iff, ← sin_pi_over_two_pow_succ]
    refine' lt_of_lt_of_le (lt_add_of_sub_right_lt (sin_gt_sub_cube _ _)) _
    · apply div_pos pi_pos; apply pow_pos; norm_num
    · rw [div_le_iff']
      · refine' le_trans pi_le_four _
        simp only [show (4 : ℝ) = (2 : ℝ) ^ 2 by norm_num, mul_one]
        apply pow_le_pow_right; norm_num; apply le_add_of_nonneg_left; apply Nat.zero_le
      · apply pow_pos; norm_num
    apply add_le_add_left; rw [div_le_div_right]
    rw [le_div_iff, ← mul_pow]
    refine' le_trans _ (le_of_eq (one_pow 3)); apply pow_le_pow_left
    · apply le_of_lt; apply mul_pos; apply div_pos pi_pos; apply pow_pos; norm_num; apply pow_pos
      norm_num
    rw [← le_div_iff]
    refine' le_trans ((div_le_div_right _).mpr pi_le_four) _; apply pow_pos; norm_num
    rw [pow_succ, pow_succ, ← mul_assoc, ← div_div]
    -- Porting note: removed `convert le_rfl`
    all_goals (repeat' apply pow_pos); norm_num
theorem pi_lt_sqrtTwoAddSeries (n : ℕ) :
    π < (2 : ℝ) ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) + 1 / (4 : ℝ) ^ n := by
  have : π <
      (sqrt (2 - sqrtTwoAddSeries 0 n) / (2 : ℝ) + (1 : ℝ) / ((2 : ℝ) ^ n) ^ 3 / 4) *
      (2 : ℝ) ^ (n + 2) := by
    

Proof state:
n : ℕ
⊢ π < (sqrt (2 - sqrtTwoAddSeries 0 n) / 2 + 1 / (2 ^ n) ^ 3 / 4) * 2 ^ (n + 2)

Next tactic:
rw [← div_lt_iff, ← sin_pi_over_two_pow_succ]
theorem pi_lt_sqrtTwoAddSeries (n : ℕ) :
    π < (2 : ℝ) ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) + 1 / (4 : ℝ) ^ n := by
  have : π <
      (sqrt (2 - sqrtTwoAddSeries 0 n) / (2 : ℝ) + (1 : ℝ) / ((2 : ℝ) ^ n) ^ 3 / 4) *
      (2 : ℝ) ^ (n + 2) := by
    rw [← div_lt_iff, ← sin_pi_over_two_pow_succ]
    

Proof state:
n : ℕ
⊢ π / 2 ^ (n + 2) < sin (π / 2 ^ (n + 2)) + 1 / (2 ^ n) ^ 3 / 4
n : ℕ ⊢ 0 < 2 ^ (n + 2)

Next tactic:
refine' lt_of_lt_of_le (lt_add_of_sub_right_lt (sin_gt_sub_cube _ _)) _
theorem pi_lt_sqrtTwoAddSeries (n : ℕ) :
    π < (2 : ℝ) ^ (n + 1) * sqrt (2 - sqrtTwoAddSeries 0 n) + 1 / (4 : ℝ) ^ n := by
  have : π <
      (sqrt (2 - sqrtTwoAddSeries 0 n) / (2 : ℝ) + (1 : ℝ) / ((2 : ℝ) ^ n) ^ 3 / 4) *
      (2 : ℝ) ^ (n + 2) := by
    rw [← div_lt_iff, ← sin_pi_over_two_pow_succ]
    refine' lt_of_lt_of_le (lt_add_of_sub_right_lt (sin_gt_sub_cube _ _)) _
    · 

Proof state:
case refine'_1
n : ℕ
⊢ 0 < π / 2 ^ (n + 2)

Next tactic:
apply div_pos pi_pos