l3lab/ntp-mathlib huggingface dataset:
theorem closed_range_toBcf : IsClosed (range (toBcf : C₀(α, β) → α →ᵇ β)) := by
  refine' isClosed_iff_clusterPt.mpr fun f hf => _
  rw [clusterPt_principal_iff] at hf
  have : Tendsto f (cocompact α) (𝓝 0) := by
    refine' Metric.tendsto_nhds.mpr fun ε hε => _
    obtain ⟨_, hg, g, rfl⟩ := hf (ball f (ε / 2)) (ball_mem_nhds f <| half_pos hε)
    refine' (Metric.tendsto_nhds.mp (zero_at_infty g) (ε / 2) (half_pos hε)).mp
      (eventually_of_forall fun x hx => _)
    calc
      dist (f x) 0 ≤ dist (g.toBcf x) (f x) + dist (g x) 0 := dist_triangle_left _ _ _
      _ < dist g.toBcf f + ε / 2 := (add_lt_add_of_le_of_lt (dist_coe_le_dist x) hx)
      _ < ε := by 

Proof state:
F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : MetricSpace β
inst✝¹ : Zero β
inst✝ : ZeroAtInftyContinuousMapClass F α β
C : ℝ
f✝ g✝ : α →C₀ β
f : α →ᵇ β
hf : ∀ U ∈ 𝓝 f, Set.Nonempty (U ∩ range toBcf)
ε : ℝ
hε : ε > 0
g : α →C₀ β
hg : toBcf g ∈ ball f (ε / 2)
x : α
hx : dist (g x) 0 < ε / 2
⊢ dist (toBcf g) f + ε / 2 < ε

Next tactic:
simpa [add_halves ε] using add_lt_add_right (mem_ball.1 hg) (ε / 2)
theorem closed_range_toBcf : IsClosed (range (toBcf : C₀(α, β) → α →ᵇ β)) := by
  refine' isClosed_iff_clusterPt.mpr fun f hf => _
  rw [clusterPt_principal_iff] at hf
  have : Tendsto f (cocompact α) (𝓝 0) := by
    refine' Metric.tendsto_nhds.mpr fun ε hε => _
    obtain ⟨_, hg, g, rfl⟩ := hf (ball f (ε / 2)) (ball_mem_nhds f <| half_pos hε)
    refine' (Metric.tendsto_nhds.mp (zero_at_infty g) (ε / 2) (half_pos hε)).mp
      (eventually_of_forall fun x hx => _)
    calc
      dist (f x) 0 ≤ dist (g.toBcf x) (f x) + dist (g x) 0 := dist_triangle_left _ _ _
      _ < dist g.toBcf f + ε / 2 := (add_lt_add_of_le_of_lt (dist_coe_le_dist x) hx)
      _ < ε := by simpa [add_halves ε] using add_lt_add_right (mem_ball.1 hg) (ε / 2)
  

Proof state:
F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝³ : TopologicalSpace α
inst✝² : MetricSpace β
inst✝¹ : Zero β
inst✝ : ZeroAtInftyContinuousMapClass F α β
C : ℝ
f✝ g : α →C₀ β
f : α →ᵇ β
hf : ∀ U ∈ 𝓝 f, Set.Nonempty (U ∩ range toBcf)
this : Tendsto (⇑f) (cocompact α) (𝓝 0)
⊢ f ∈ range toBcf

Next tactic:
exact ⟨⟨f.toContinuousMap, this⟩, rfl⟩
instance instStar : Star C₀(α, β) where
  star f :=
    { toFun := fun x => star (f x)
      continuous_toFun := (map_continuous f).star
      zero_at_infty' := by
        

Proof state:
F : Type u_1
α : Type u
β : Type v
γ : Type w
inst✝⁴ : TopologicalSpace α
inst✝³ : TopologicalSpace β
inst✝² : AddMonoid β
inst✝¹ : StarAddMonoid β
inst✝ : ContinuousStar β
f : α →C₀ β
⊢ Tendsto (ContinuousMap.mk fun x => star (f x)).toFun (cocompact α) (𝓝 0)

Next tactic:
simpa only [star_zero] using (continuous_star.tendsto (0 : β)).comp (zero_at_infty f)
theorem comm {a b c : ℤ} : Fermat42 a b c ↔ Fermat42 b a c := by
  

Proof state:
a b c : ℤ
⊢ Fermat42 a b c ↔ Fermat42 b a c

Next tactic:
delta Fermat42
theorem comm {a b c : ℤ} : Fermat42 a b c ↔ Fermat42 b a c := by
  delta Fermat42
  

Proof state:
a b c : ℤ
⊢ a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2 ↔ b ≠ 0 ∧ a ≠ 0 ∧ b ^ 4 + a ^ 4 = c ^ 2

Next tactic:
rw [add_comm]
theorem comm {a b c : ℤ} : Fermat42 a b c ↔ Fermat42 b a c := by
  delta Fermat42
  rw [add_comm]
  

Proof state:
a b c : ℤ
⊢ a ≠ 0 ∧ b ≠ 0 ∧ b ^ 4 + a ^ 4 = c ^ 2 ↔ b ≠ 0 ∧ a ≠ 0 ∧ b ^ 4 + a ^ 4 = c ^ 2

Next tactic:
tauto
theorem mul {a b c k : ℤ} (hk0 : k ≠ 0) :
    Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c) := by
  

Proof state:
a b c k : ℤ
hk0 : k ≠ 0
⊢ Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c)

Next tactic:
delta Fermat42
theorem mul {a b c k : ℤ} (hk0 : k ≠ 0) :
    Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c) := by
  delta Fermat42
  

Proof state:
a b c k : ℤ
hk0 : k ≠ 0
⊢ a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2 ↔ k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2

Next tactic:
constructor
theorem mul {a b c k : ℤ} (hk0 : k ≠ 0) :
    Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c) := by
  delta Fermat42
  constructor
  · 

Proof state:
case mp
a b c k : ℤ
hk0 : k ≠ 0
⊢ a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2 → k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2

Next tactic:
intro f42
theorem mul {a b c k : ℤ} (hk0 : k ≠ 0) :
    Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c) := by
  delta Fermat42
  constructor
  · intro f42
    

Proof state:
case mp
a b c k : ℤ
hk0 : k ≠ 0
f42 : a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2
⊢ k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2

Next tactic:
constructor
theorem mul {a b c k : ℤ} (hk0 : k ≠ 0) :
    Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c) := by
  delta Fermat42
  constructor
  · intro f42
    constructor
    · 

Proof state:
case mp.left
a b c k : ℤ
hk0 : k ≠ 0
f42 : a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2
⊢ k * a ≠ 0

Next tactic:
exact mul_ne_zero hk0 f42.1
theorem mul {a b c k : ℤ} (hk0 : k ≠ 0) :
    Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c) := by
  delta Fermat42
  constructor
  · intro f42
    constructor
    · exact mul_ne_zero hk0 f42.1
    

Proof state:
case mp.right
a b c k : ℤ
hk0 : k ≠ 0
f42 : a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2
⊢ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2

Next tactic:
constructor
theorem mul {a b c k : ℤ} (hk0 : k ≠ 0) :
    Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c) := by
  delta Fermat42
  constructor
  · intro f42
    constructor
    · exact mul_ne_zero hk0 f42.1
    constructor
    · 

Proof state:
case mp.right.left
a b c k : ℤ
hk0 : k ≠ 0
f42 : a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2
⊢ k * b ≠ 0

Next tactic:
exact mul_ne_zero hk0 f42.2.1
theorem mul {a b c k : ℤ} (hk0 : k ≠ 0) :
    Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c) := by
  delta Fermat42
  constructor
  · intro f42
    constructor
    · exact mul_ne_zero hk0 f42.1
    constructor
    · exact mul_ne_zero hk0 f42.2.1
    · 

Proof state:
case mp.right.right
a b c k : ℤ
hk0 : k ≠ 0
f42 : a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2
⊢ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2

Next tactic:
have H : a ^ 4 + b ^ 4 = c ^ 2 := f42.2.2
theorem mul {a b c k : ℤ} (hk0 : k ≠ 0) :
    Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c) := by
  delta Fermat42
  constructor
  · intro f42
    constructor
    · exact mul_ne_zero hk0 f42.1
    constructor
    · exact mul_ne_zero hk0 f42.2.1
    · have H : a ^ 4 + b ^ 4 = c ^ 2 := f42.2.2
      

Proof state:
case mp.right.right
a b c k : ℤ
hk0 : k ≠ 0
f42 : a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2
H : a ^ 4 + b ^ 4 = c ^ 2
⊢ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2

Next tactic:
linear_combination k ^ 4 * H
theorem mul {a b c k : ℤ} (hk0 : k ≠ 0) :
    Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c) := by
  delta Fermat42
  constructor
  · intro f42
    constructor
    · exact mul_ne_zero hk0 f42.1
    constructor
    · exact mul_ne_zero hk0 f42.2.1
    · have H : a ^ 4 + b ^ 4 = c ^ 2 := f42.2.2
      linear_combination k ^ 4 * H
  · 

Proof state:
case mpr
a b c k : ℤ
hk0 : k ≠ 0
⊢ k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2 → a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2

Next tactic:
intro f42
theorem mul {a b c k : ℤ} (hk0 : k ≠ 0) :
    Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c) := by
  delta Fermat42
  constructor
  · intro f42
    constructor
    · exact mul_ne_zero hk0 f42.1
    constructor
    · exact mul_ne_zero hk0 f42.2.1
    · have H : a ^ 4 + b ^ 4 = c ^ 2 := f42.2.2
      linear_combination k ^ 4 * H
  · intro f42
    

Proof state:
case mpr
a b c k : ℤ
hk0 : k ≠ 0
f42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2
⊢ a ≠ 0 ∧ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2

Next tactic:
constructor
theorem mul {a b c k : ℤ} (hk0 : k ≠ 0) :
    Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c) := by
  delta Fermat42
  constructor
  · intro f42
    constructor
    · exact mul_ne_zero hk0 f42.1
    constructor
    · exact mul_ne_zero hk0 f42.2.1
    · have H : a ^ 4 + b ^ 4 = c ^ 2 := f42.2.2
      linear_combination k ^ 4 * H
  · intro f42
    constructor
    · 

Proof state:
case mpr.left
a b c k : ℤ
hk0 : k ≠ 0
f42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2
⊢ a ≠ 0

Next tactic:
exact right_ne_zero_of_mul f42.1
theorem mul {a b c k : ℤ} (hk0 : k ≠ 0) :
    Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c) := by
  delta Fermat42
  constructor
  · intro f42
    constructor
    · exact mul_ne_zero hk0 f42.1
    constructor
    · exact mul_ne_zero hk0 f42.2.1
    · have H : a ^ 4 + b ^ 4 = c ^ 2 := f42.2.2
      linear_combination k ^ 4 * H
  · intro f42
    constructor
    · exact right_ne_zero_of_mul f42.1
    

Proof state:
case mpr.right
a b c k : ℤ
hk0 : k ≠ 0
f42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2
⊢ b ≠ 0 ∧ a ^ 4 + b ^ 4 = c ^ 2

Next tactic:
constructor
theorem mul {a b c k : ℤ} (hk0 : k ≠ 0) :
    Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c) := by
  delta Fermat42
  constructor
  · intro f42
    constructor
    · exact mul_ne_zero hk0 f42.1
    constructor
    · exact mul_ne_zero hk0 f42.2.1
    · have H : a ^ 4 + b ^ 4 = c ^ 2 := f42.2.2
      linear_combination k ^ 4 * H
  · intro f42
    constructor
    · exact right_ne_zero_of_mul f42.1
    constructor
    · 

Proof state:
case mpr.right.left
a b c k : ℤ
hk0 : k ≠ 0
f42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2
⊢ b ≠ 0

Next tactic:
exact right_ne_zero_of_mul f42.2.1
theorem mul {a b c k : ℤ} (hk0 : k ≠ 0) :
    Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c) := by
  delta Fermat42
  constructor
  · intro f42
    constructor
    · exact mul_ne_zero hk0 f42.1
    constructor
    · exact mul_ne_zero hk0 f42.2.1
    · have H : a ^ 4 + b ^ 4 = c ^ 2 := f42.2.2
      linear_combination k ^ 4 * H
  · intro f42
    constructor
    · exact right_ne_zero_of_mul f42.1
    constructor
    · exact right_ne_zero_of_mul f42.2.1
    

Proof state:
case mpr.right.right
a b c k : ℤ
hk0 : k ≠ 0
f42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2
⊢ a ^ 4 + b ^ 4 = c ^ 2

Next tactic:
apply (mul_right_inj' (pow_ne_zero 4 hk0)).mp
theorem mul {a b c k : ℤ} (hk0 : k ≠ 0) :
    Fermat42 a b c ↔ Fermat42 (k * a) (k * b) (k ^ 2 * c) := by
  delta Fermat42
  constructor
  · intro f42
    constructor
    · exact mul_ne_zero hk0 f42.1
    constructor
    · exact mul_ne_zero hk0 f42.2.1
    · have H : a ^ 4 + b ^ 4 = c ^ 2 := f42.2.2
      linear_combination k ^ 4 * H
  · intro f42
    constructor
    · exact right_ne_zero_of_mul f42.1
    constructor
    · exact right_ne_zero_of_mul f42.2.1
    apply (mul_right_inj' (pow_ne_zero 4 hk0)).mp
    

Proof state:
case mpr.right.right
a b c k : ℤ
hk0 : k ≠ 0
f42 : k * a ≠ 0 ∧ k * b ≠ 0 ∧ (k * a) ^ 4 + (k * b) ^ 4 = (k ^ 2 * c) ^ 2
⊢ k ^ 4 * (a ^ 4 + b ^ 4) = k ^ 4 * c ^ 2

Next tactic:
linear_combination f42.2.2
theorem ne_zero {a b c : ℤ} (h : Fermat42 a b c) : c ≠ 0 := by
  

Proof state:
a b c : ℤ
h : Fermat42 a b c
⊢ c ≠ 0

Next tactic:
apply ne_zero_pow two_ne_zero _
theorem ne_zero {a b c : ℤ} (h : Fermat42 a b c) : c ≠ 0 := by
  apply ne_zero_pow two_ne_zero _; 

Proof state:
a b c : ℤ
h : Fermat42 a b c
⊢ c ^ 2 ≠ 0

Next tactic:
apply ne_of_gt
theorem ne_zero {a b c : ℤ} (h : Fermat42 a b c) : c ≠ 0 := by
  apply ne_zero_pow two_ne_zero _; apply ne_of_gt
  

Proof state:
case h
a b c : ℤ
h : Fermat42 a b c
⊢ 0 < c ^ 2

Next tactic:
rw [← h.2.2, (by ring : a ^ 4 + b ^ 4 = (a ^ 2) ^ 2 + (b ^ 2) ^ 2)]
theorem ne_zero {a b c : ℤ} (h : Fermat42 a b c) : c ≠ 0 := by
  apply ne_zero_pow two_ne_zero _; apply ne_of_gt
  rw [← h.2.2, (by 

Proof state:
a b c : ℤ
h : Fermat42 a b c
⊢ a ^ 4 + b ^ 4 = (a ^ 2) ^ 2 + (b ^ 2) ^ 2

Next tactic:
ring
theorem ne_zero {a b c : ℤ} (h : Fermat42 a b c) : c ≠ 0 := by
  apply ne_zero_pow two_ne_zero _; apply ne_of_gt
  rw [← h.2.2, (by ring : a ^ 4 + b ^ 4 = (a ^ 2) ^ 2 + (b ^ 2) ^ 2)]
  

Proof state:
case h
a b c : ℤ
h : Fermat42 a b c
⊢ 0 < (a ^ 2) ^ 2 + (b ^ 2) ^ 2

Next tactic:
exact
    add_pos (sq_pos_of_ne_zero _ (pow_ne_zero 2 h.1)) (sq_pos_of_ne_zero _ (pow_ne_zero 2 h.2.1))
/-- if we have a solution to `a ^ 4 + b ^ 4 = c ^ 2` then there must be a minimal one. -/
theorem exists_minimal {a b c : ℤ} (h : Fermat42 a b c) : ∃ a0 b0 c0, Minimal a0 b0 c0 := by
  

Proof state:
a b c : ℤ
h : Fermat42 a b c
⊢ ∃ a0 b0 c0, Minimal a0 b0 c0

Next tactic:
let S : Set ℕ := { n | ∃ s : ℤ × ℤ × ℤ, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2 }
/-- if we have a solution to `a ^ 4 + b ^ 4 = c ^ 2` then there must be a minimal one. -/
theorem exists_minimal {a b c : ℤ} (h : Fermat42 a b c) : ∃ a0 b0 c0, Minimal a0 b0 c0 := by
  let S : Set ℕ := { n | ∃ s : ℤ × ℤ × ℤ, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2 }
  

Proof state:
a b c : ℤ
h : Fermat42 a b c
S : Set ℕ := {n | ∃ s, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2}
⊢ ∃ a0 b0 c0, Minimal a0 b0 c0

Next tactic:
have S_nonempty : S.Nonempty := by
    use Int.natAbs c
    rw [Set.mem_setOf_eq]
    use ⟨a, ⟨b, c⟩⟩
/-- if we have a solution to `a ^ 4 + b ^ 4 = c ^ 2` then there must be a minimal one. -/
theorem exists_minimal {a b c : ℤ} (h : Fermat42 a b c) : ∃ a0 b0 c0, Minimal a0 b0 c0 := by
  let S : Set ℕ := { n | ∃ s : ℤ × ℤ × ℤ, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2 }
  have S_nonempty : S.Nonempty := by
    

Proof state:
a b c : ℤ
h : Fermat42 a b c
S : Set ℕ := {n | ∃ s, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2}
⊢ Set.Nonempty S

Next tactic:
use Int.natAbs c
/-- if we have a solution to `a ^ 4 + b ^ 4 = c ^ 2` then there must be a minimal one. -/
theorem exists_minimal {a b c : ℤ} (h : Fermat42 a b c) : ∃ a0 b0 c0, Minimal a0 b0 c0 := by
  let S : Set ℕ := { n | ∃ s : ℤ × ℤ × ℤ, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2 }
  have S_nonempty : S.Nonempty := by
    use Int.natAbs c
    

Proof state:
case h
a b c : ℤ
h : Fermat42 a b c
S : Set ℕ := {n | ∃ s, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2}
⊢ Int.natAbs c ∈ S

Next tactic:
rw [Set.mem_setOf_eq]
/-- if we have a solution to `a ^ 4 + b ^ 4 = c ^ 2` then there must be a minimal one. -/
theorem exists_minimal {a b c : ℤ} (h : Fermat42 a b c) : ∃ a0 b0 c0, Minimal a0 b0 c0 := by
  let S : Set ℕ := { n | ∃ s : ℤ × ℤ × ℤ, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2 }
  have S_nonempty : S.Nonempty := by
    use Int.natAbs c
    rw [Set.mem_setOf_eq]
    

Proof state:
case h
a b c : ℤ
h : Fermat42 a b c
S : Set ℕ := {n | ∃ s, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2}
⊢ ∃ s, Fermat42 s.1 s.2.1 s.2.2 ∧ Int.natAbs c = Int.natAbs s.2.2

Next tactic:
use ⟨a, ⟨b, c⟩⟩
/-- if we have a solution to `a ^ 4 + b ^ 4 = c ^ 2` then there must be a minimal one. -/
theorem exists_minimal {a b c : ℤ} (h : Fermat42 a b c) : ∃ a0 b0 c0, Minimal a0 b0 c0 := by
  let S : Set ℕ := { n | ∃ s : ℤ × ℤ × ℤ, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2 }
  have S_nonempty : S.Nonempty := by
    use Int.natAbs c
    rw [Set.mem_setOf_eq]
    use ⟨a, ⟨b, c⟩⟩
  

Proof state:
a b c : ℤ
h : Fermat42 a b c
S : Set ℕ := {n | ∃ s, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2}
S_nonempty : Set.Nonempty S
⊢ ∃ a0 b0 c0, Minimal a0 b0 c0

Next tactic:
let m : ℕ := Nat.find S_nonempty
/-- if we have a solution to `a ^ 4 + b ^ 4 = c ^ 2` then there must be a minimal one. -/
theorem exists_minimal {a b c : ℤ} (h : Fermat42 a b c) : ∃ a0 b0 c0, Minimal a0 b0 c0 := by
  let S : Set ℕ := { n | ∃ s : ℤ × ℤ × ℤ, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2 }
  have S_nonempty : S.Nonempty := by
    use Int.natAbs c
    rw [Set.mem_setOf_eq]
    use ⟨a, ⟨b, c⟩⟩
  let m : ℕ := Nat.find S_nonempty
  

Proof state:
a b c : ℤ
h : Fermat42 a b c
S : Set ℕ := {n | ∃ s, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2}
S_nonempty : Set.Nonempty S
m : ℕ := Nat.find S_nonempty
⊢ ∃ a0 b0 c0, Minimal a0 b0 c0

Next tactic:
have m_mem : m ∈ S := Nat.find_spec S_nonempty
/-- if we have a solution to `a ^ 4 + b ^ 4 = c ^ 2` then there must be a minimal one. -/
theorem exists_minimal {a b c : ℤ} (h : Fermat42 a b c) : ∃ a0 b0 c0, Minimal a0 b0 c0 := by
  let S : Set ℕ := { n | ∃ s : ℤ × ℤ × ℤ, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2 }
  have S_nonempty : S.Nonempty := by
    use Int.natAbs c
    rw [Set.mem_setOf_eq]
    use ⟨a, ⟨b, c⟩⟩
  let m : ℕ := Nat.find S_nonempty
  have m_mem : m ∈ S := Nat.find_spec S_nonempty
  

Proof state:
a b c : ℤ
h : Fermat42 a b c
S : Set ℕ := {n | ∃ s, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2}
S_nonempty : Set.Nonempty S
m : ℕ := Nat.find S_nonempty
m_mem : m ∈ S
⊢ ∃ a0 b0 c0, Minimal a0 b0 c0

Next tactic:
rcases m_mem with ⟨s0, hs0, hs1⟩
/-- if we have a solution to `a ^ 4 + b ^ 4 = c ^ 2` then there must be a minimal one. -/
theorem exists_minimal {a b c : ℤ} (h : Fermat42 a b c) : ∃ a0 b0 c0, Minimal a0 b0 c0 := by
  let S : Set ℕ := { n | ∃ s : ℤ × ℤ × ℤ, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2 }
  have S_nonempty : S.Nonempty := by
    use Int.natAbs c
    rw [Set.mem_setOf_eq]
    use ⟨a, ⟨b, c⟩⟩
  let m : ℕ := Nat.find S_nonempty
  have m_mem : m ∈ S := Nat.find_spec S_nonempty
  rcases m_mem with ⟨s0, hs0, hs1⟩
  

Proof state:
case intro.intro
a b c : ℤ
h : Fermat42 a b c
S : Set ℕ := {n | ∃ s, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2}
S_nonempty : Set.Nonempty S
m : ℕ := Nat.find S_nonempty
s0 : ℤ × ℤ × ℤ
hs0 : Fermat42 s0.1 s0.2.1 s0.2.2
hs1 : m = Int.natAbs s0.2.2
⊢ ∃ a0 b0 c0, Minimal a0 b0 c0

Next tactic:
use s0.1, s0.2.1, s0.2.2, hs0
/-- if we have a solution to `a ^ 4 + b ^ 4 = c ^ 2` then there must be a minimal one. -/
theorem exists_minimal {a b c : ℤ} (h : Fermat42 a b c) : ∃ a0 b0 c0, Minimal a0 b0 c0 := by
  let S : Set ℕ := { n | ∃ s : ℤ × ℤ × ℤ, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2 }
  have S_nonempty : S.Nonempty := by
    use Int.natAbs c
    rw [Set.mem_setOf_eq]
    use ⟨a, ⟨b, c⟩⟩
  let m : ℕ := Nat.find S_nonempty
  have m_mem : m ∈ S := Nat.find_spec S_nonempty
  rcases m_mem with ⟨s0, hs0, hs1⟩
  use s0.1, s0.2.1, s0.2.2, hs0
  

Proof state:
case right
a b c : ℤ
h : Fermat42 a b c
S : Set ℕ := {n | ∃ s, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2}
S_nonempty : Set.Nonempty S
m : ℕ := Nat.find S_nonempty
s0 : ℤ × ℤ × ℤ
hs0 : Fermat42 s0.1 s0.2.1 s0.2.2
hs1 : m = Int.natAbs s0.2.2
⊢ ∀ (a1 b1 c1 : ℤ), Fermat42 a1 b1 c1 → Int.natAbs s0.2.2 ≤ Int.natAbs c1

Next tactic:
intro a1 b1 c1 h1
/-- if we have a solution to `a ^ 4 + b ^ 4 = c ^ 2` then there must be a minimal one. -/
theorem exists_minimal {a b c : ℤ} (h : Fermat42 a b c) : ∃ a0 b0 c0, Minimal a0 b0 c0 := by
  let S : Set ℕ := { n | ∃ s : ℤ × ℤ × ℤ, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2 }
  have S_nonempty : S.Nonempty := by
    use Int.natAbs c
    rw [Set.mem_setOf_eq]
    use ⟨a, ⟨b, c⟩⟩
  let m : ℕ := Nat.find S_nonempty
  have m_mem : m ∈ S := Nat.find_spec S_nonempty
  rcases m_mem with ⟨s0, hs0, hs1⟩
  use s0.1, s0.2.1, s0.2.2, hs0
  intro a1 b1 c1 h1
  

Proof state:
case right
a b c : ℤ
h : Fermat42 a b c
S : Set ℕ := {n | ∃ s, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2}
S_nonempty : Set.Nonempty S
m : ℕ := Nat.find S_nonempty
s0 : ℤ × ℤ × ℤ
hs0 : Fermat42 s0.1 s0.2.1 s0.2.2
hs1 : m = Int.natAbs s0.2.2
a1 b1 c1 : ℤ
h1 : Fermat42 a1 b1 c1
⊢ Int.natAbs s0.2.2 ≤ Int.natAbs c1

Next tactic:
rw [← hs1]
/-- if we have a solution to `a ^ 4 + b ^ 4 = c ^ 2` then there must be a minimal one. -/
theorem exists_minimal {a b c : ℤ} (h : Fermat42 a b c) : ∃ a0 b0 c0, Minimal a0 b0 c0 := by
  let S : Set ℕ := { n | ∃ s : ℤ × ℤ × ℤ, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2 }
  have S_nonempty : S.Nonempty := by
    use Int.natAbs c
    rw [Set.mem_setOf_eq]
    use ⟨a, ⟨b, c⟩⟩
  let m : ℕ := Nat.find S_nonempty
  have m_mem : m ∈ S := Nat.find_spec S_nonempty
  rcases m_mem with ⟨s0, hs0, hs1⟩
  use s0.1, s0.2.1, s0.2.2, hs0
  intro a1 b1 c1 h1
  rw [← hs1]
  

Proof state:
case right
a b c : ℤ
h : Fermat42 a b c
S : Set ℕ := {n | ∃ s, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2}
S_nonempty : Set.Nonempty S
m : ℕ := Nat.find S_nonempty
s0 : ℤ × ℤ × ℤ
hs0 : Fermat42 s0.1 s0.2.1 s0.2.2
hs1 : m = Int.natAbs s0.2.2
a1 b1 c1 : ℤ
h1 : Fermat42 a1 b1 c1
⊢ m ≤ Int.natAbs c1

Next tactic:
apply Nat.find_min'
/-- if we have a solution to `a ^ 4 + b ^ 4 = c ^ 2` then there must be a minimal one. -/
theorem exists_minimal {a b c : ℤ} (h : Fermat42 a b c) : ∃ a0 b0 c0, Minimal a0 b0 c0 := by
  let S : Set ℕ := { n | ∃ s : ℤ × ℤ × ℤ, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2 }
  have S_nonempty : S.Nonempty := by
    use Int.natAbs c
    rw [Set.mem_setOf_eq]
    use ⟨a, ⟨b, c⟩⟩
  let m : ℕ := Nat.find S_nonempty
  have m_mem : m ∈ S := Nat.find_spec S_nonempty
  rcases m_mem with ⟨s0, hs0, hs1⟩
  use s0.1, s0.2.1, s0.2.2, hs0
  intro a1 b1 c1 h1
  rw [← hs1]
  apply Nat.find_min'
  

Proof state:
case right.h
a b c : ℤ
h : Fermat42 a b c
S : Set ℕ := {n | ∃ s, Fermat42 s.1 s.2.1 s.2.2 ∧ n = Int.natAbs s.2.2}
S_nonempty : Set.Nonempty S
m : ℕ := Nat.find S_nonempty
s0 : ℤ × ℤ × ℤ
hs0 : Fermat42 s0.1 s0.2.1 s0.2.2
hs1 : m = Int.natAbs s0.2.2
a1 b1 c1 : ℤ
h1 : Fermat42 a1 b1 c1
⊢ Int.natAbs c1 ∈ S

Next tactic:
use ⟨a1, ⟨b1, c1⟩⟩
/-- a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` must have `a` and `b` coprime. -/
theorem coprime_of_minimal {a b c : ℤ} (h : Minimal a b c) : IsCoprime a b := by
  

Proof state:
a b c : ℤ
h : Minimal a b c
⊢ IsCoprime a b

Next tactic:
apply Int.gcd_eq_one_iff_coprime.mp
/-- a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` must have `a` and `b` coprime. -/
theorem coprime_of_minimal {a b c : ℤ} (h : Minimal a b c) : IsCoprime a b := by
  apply Int.gcd_eq_one_iff_coprime.mp
  

Proof state:
a b c : ℤ
h : Minimal a b c
⊢ Int.gcd a b = 1

Next tactic:
by_contra hab
/-- a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` must have `a` and `b` coprime. -/
theorem coprime_of_minimal {a b c : ℤ} (h : Minimal a b c) : IsCoprime a b := by
  apply Int.gcd_eq_one_iff_coprime.mp
  by_contra hab
  

Proof state:
a b c : ℤ
h : Minimal a b c
hab : ¬Int.gcd a b = 1
⊢ False

Next tactic:
obtain ⟨p, hp, hpa, hpb⟩ := Nat.Prime.not_coprime_iff_dvd.mp hab
/-- a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` must have `a` and `b` coprime. -/
theorem coprime_of_minimal {a b c : ℤ} (h : Minimal a b c) : IsCoprime a b := by
  apply Int.gcd_eq_one_iff_coprime.mp
  by_contra hab
  obtain ⟨p, hp, hpa, hpb⟩ := Nat.Prime.not_coprime_iff_dvd.mp hab
  

Proof state:
case intro.intro.intro
a b c : ℤ
h : Minimal a b c
hab : ¬Int.gcd a b = 1
p : ℕ
hp : Nat.Prime p
hpa : p ∣ Int.natAbs a
hpb : p ∣ Int.natAbs b
⊢ False

Next tactic:
obtain ⟨a1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpa
/-- a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` must have `a` and `b` coprime. -/
theorem coprime_of_minimal {a b c : ℤ} (h : Minimal a b c) : IsCoprime a b := by
  apply Int.gcd_eq_one_iff_coprime.mp
  by_contra hab
  obtain ⟨p, hp, hpa, hpb⟩ := Nat.Prime.not_coprime_iff_dvd.mp hab
  obtain ⟨a1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpa
  

Proof state:
case intro.intro.intro.intro
b c : ℤ
p : ℕ
hp : Nat.Prime p
hpb : p ∣ Int.natAbs b
a1 : ℤ
h : Minimal (↑p * a1) b c
hab : ¬Int.gcd (↑p * a1) b = 1
hpa : p ∣ Int.natAbs (↑p * a1)
⊢ False

Next tactic:
obtain ⟨b1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpb
/-- a minimal solution to `a ^ 4 + b ^ 4 = c ^ 2` must have `a` and `b` coprime. -/
theorem coprime_of_minimal {a b c : ℤ} (h : Minimal a b c) : IsCoprime a b := by
  apply Int.gcd_eq_one_iff_coprime.mp
  by_contra hab
  obtain ⟨p, hp, hpa, hpb⟩ := Nat.Prime.not_coprime_iff_dvd.mp hab
  obtain ⟨a1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpa
  obtain ⟨b1, rfl⟩ := Int.coe_nat_dvd_left.mpr hpb
  

Proof state:
case intro.intro.intro.intro.intro
c : ℤ
p : ℕ
hp : Nat.Prime p
a1 : ℤ
hpa : p ∣ Int.natAbs (↑p * a1)
b1 : ℤ
hpb : p ∣ Int.natAbs (↑p * b1)
h : Minimal (↑p * a1) (↑p * b1) c
hab : ¬Int.gcd (↑p * a1) (↑p * b1) = 1
⊢ False

Next tactic:
have hpc : (p : ℤ) ^ 2 ∣ c := by
    rw [← Int.pow_dvd_pow_iff zero_lt_two, ← h.1.2.2]
    apply Dvd.intro (a1 ^ 4 + b1 ^ 4)
    ring