awhecmu/canonical-drafter-extract-mathlib huggingface dataset:
-- Lean 4 proof step from Mathlib.Data.Sym.Sym2 (goal)
Goal:
α_1 : Type u_1
β : Type u_2
γ : Type u_3
z : Sym2 α_1
f_1 : α_1 → β
α : Type u_4
R : Type u_5
N : Type u_6
inst : SMul R N
f : { f : α → α → R // ∀ (a₁ a₂ : α), f a₁ a₂ = f a₂ a₁ }
g : { g : α → α → N // ∀ (a₁ a₂ : α), g a₁ a₂ = g a₂ a₁ }
x x_1 : α
⊢ ((↑f : α → α → R) • (↑g : α → α → N)) x x_1 = ((↑f : α → α → R) • (↑g : α → α → N)) x_1 x
have rw : ((↑f : α → α → R) x • (↑g : α → α → N) x) x_1 = ((↑f : α → α → R) • (↑g : α → α → N)) x_1 x
Tactic: rw [Pi.smul_apply', Pi.smul_apply', Pi.smul_apply', Pi.smul_apply', f.prop, g.prop]
-- Lean 4 proof step from Mathlib.Data.Sym.Sym2 (goal)
Goal:
α_1 : Type u_1
β : Type u_2
γ : Type u_3
z : Sym2 α_1
f_1 : α_1 → β
α : Type u_4
R : Type u_5
N : Type u_6
inst : SMul R N
f : { f : α → α → R // ∀ (a₁ a₂ : α), f a₁ a₂ = f a₂ a₁ }
g : { g : α → α → N // ∀ (a₁ a₂ : α), g a₁ a₂ = g a₂ a₁ }
x x_1 : α
rw : ((↑f : α → α → R) x • (↑g : α → α → N) x) x_1 = ((↑f : α → α → R) • (↑g : α → α → N)) x_1 x
⊢ ((↑f : α → α → R) • (↑g : α → α → N)) x x_1 = ((↑f : α → α → R) • (↑g : α → α → N)) x_1 x
have rw₁ : (↑f : α → α → R) x x_1 • (↑g : α → α → N) x x_1 = ((↑f : α → α → R) • (↑g : α → α → N)) x_1 x
Tactic: rw [Pi.smul_apply', Pi.smul_apply', Pi.smul_apply', Pi.smul_apply', f.prop, g.prop]
-- Lean 4 proof step from Mathlib.Data.Sym.Sym2 (goal)
Goal:
α_1 : Type u_1
β : Type u_2
γ : Type u_3
z : Sym2 α_1
f_1 : α_1 → β
α : Type u_4
R : Type u_5
N : Type u_6
inst : SMul R N
f : { f : α → α → R // ∀ (a₁ a₂ : α), f a₁ a₂ = f a₂ a₁ }
g : { g : α → α → N // ∀ (a₁ a₂ : α), g a₁ a₂ = g a₂ a₁ }
x x_1 : α
rw : ((↑f : α → α → R) x • (↑g : α → α → N) x) x_1 = ((↑f : α → α → R) • (↑g : α → α → N)) x_1 x
rw₁ : (↑f : α → α → R) x x_1 • (↑g : α → α → N) x x_1 = ((↑f : α → α → R) • (↑g : α → α → N)) x_1 x
⊢ ((↑f : α → α → R) • (↑g : α → α → N)) x x_1 = ((↑f : α → α → R) • (↑g : α → α → N)) x_1 x
have rw₂ : (↑f : α → α → R) x x_1 • (↑g : α → α → N) x x_1 = ((↑f : α → α → R) x_1 • (↑g : α → α → N) x_1) x
Tactic: rw [Pi.smul_apply', Pi.smul_apply', Pi.smul_apply', Pi.smul_apply', f.prop, g.prop]
-- Lean 4 proof step from Mathlib.Data.Sym.Sym2 (goal)
Goal:
α_1 : Type u_1
β : Type u_2
γ : Type u_3
z : Sym2 α_1
f_1 : α_1 → β
α : Type u_4
R : Type u_5
N : Type u_6
inst : SMul R N
f : { f : α → α → R // ∀ (a₁ a₂ : α), f a₁ a₂ = f a₂ a₁ }
g : { g : α → α → N // ∀ (a₁ a₂ : α), g a₁ a₂ = g a₂ a₁ }
x x_1 : α
rw : ((↑f : α → α → R) x • (↑g : α → α → N) x) x_1 = ((↑f : α → α → R) • (↑g : α → α → N)) x_1 x
rw₁ : (↑f : α → α → R) x x_1 • (↑g : α → α → N) x x_1 = ((↑f : α → α → R) • (↑g : α → α → N)) x_1 x
rw₂ : (↑f : α → α → R) x x_1 • (↑g : α → α → N) x x_1 = ((↑f : α → α → R) x_1 • (↑g : α → α → N) x_1) x
⊢ ((↑f : α → α → R) • (↑g : α → α → N)) x x_1 = ((↑f : α → α → R) • (↑g : α → α → N)) x_1 x
have rw₃ : (↑f : α → α → R) x x_1 • (↑g : α → α → N) x x_1 = (↑f : α → α → R) x_1 x • (↑g : α → α → N) x_1 x
Tactic: rw [Pi.smul_apply', Pi.smul_apply', Pi.smul_apply', Pi.smul_apply', f.prop, g.prop]
-- Lean 4 proof step from Mathlib.Data.Sym.Sym2 (goal)
Goal:
α_1 : Type u_1
β : Type u_2
γ : Type u_3
z : Sym2 α_1
f_1 : α_1 → β
α : Type u_4
R : Type u_5
N : Type u_6
inst : SMul R N
f : { f : α → α → R // ∀ (a₁ a₂ : α), f a₁ a₂ = f a₂ a₁ }
g : { g : α → α → N // ∀ (a₁ a₂ : α), g a₁ a₂ = g a₂ a₁ }
x x_1 : α
rw : ((↑f : α → α → R) x • (↑g : α → α → N) x) x_1 = ((↑f : α → α → R) • (↑g : α → α → N)) x_1 x
rw₁ : (↑f : α → α → R) x x_1 • (↑g : α → α → N) x x_1 = ((↑f : α → α → R) • (↑g : α → α → N)) x_1 x
rw₂ : (↑f : α → α → R) x x_1 • (↑g : α → α → N) x x_1 = ((↑f : α → α → R) x_1 • (↑g : α → α → N) x_1) x
rw₃ : (↑f : α → α → R) x x_1 • (↑g : α → α → N) x x_1 = (↑f : α → α → R) x_1 x • (↑g : α → α → N) x_1 x
⊢ ((↑f : α → α → R) • (↑g : α → α → N)) x x_1 = ((↑f : α → α → R) • (↑g : α → α → N)) x_1 x
have rw₄ : (↑f : α → α → R) x_1 x • (↑g : α → α → N) x x_1 = (↑f : α → α → R) x_1 x • (↑g : α → α → N) x_1 x
Tactic: rw [Pi.smul_apply', Pi.smul_apply', Pi.smul_apply', Pi.smul_apply', f.prop, g.prop]
-- Lean 4 proof step from Mathlib.Data.Sym.Sym2 (goal)
Goal:
α_1 : Type u_1
β : Type u_2
γ : Type u_3
z : Sym2 α_1
f_1 : α_1 → β
α : Type u_4
R : Type u_5
N : Type u_6
inst : SMul R N
f : { f : α → α → R // ∀ (a₁ a₂ : α), f a₁ a₂ = f a₂ a₁ }
g : { g : α → α → N // ∀ (a₁ a₂ : α), g a₁ a₂ = g a₂ a₁ }
x x_1 : α
rw : ((↑f : α → α → R) x • (↑g : α → α → N) x) x_1 = ((↑f : α → α → R) • (↑g : α → α → N)) x_1 x
rw₁ : (↑f : α → α → R) x x_1 • (↑g : α → α → N) x x_1 = ((↑f : α → α → R) • (↑g : α → α → N)) x_1 x
rw₂ : (↑f : α → α → R) x x_1 • (↑g : α → α → N) x x_1 = ((↑f : α → α → R) x_1 • (↑g : α → α → N) x_1) x
rw₃ : (↑f : α → α → R) x x_1 • (↑g : α → α → N) x x_1 = (↑f : α → α → R) x_1 x • (↑g : α → α → N) x_1 x
rw₄ : (↑f : α → α → R) x_1 x • (↑g : α → α → N) x x_1 = (↑f : α → α → R) x_1 x • (↑g : α → α → N) x_1 x
⊢ ((↑f : α → α → R) • (↑g : α → α → N)) x x_1 = ((↑f : α → α → R) • (↑g : α → α → N)) x_1 x
have rw₅ : (↑f : α → α → R) x_1 x • (↑g : α → α → N) x_1 x = (↑f : α → α → R) x_1 x • (↑g : α → α → N) x_1 x
Tactic: rw [Pi.smul_apply', Pi.smul_apply', Pi.smul_apply', Pi.smul_apply', f.prop, g.prop]
-- Lean 4 proof step from Mathlib.Data.Sym.Sym2 (goal)
Goal:
α : Type u_4
R : Type u_5
N : Type u_6
inst : SMul R N
f : { f : α → α → R // ∀ (a₁ a₂ : α), f a₁ a₂ = f a₂ a₁ }
g : { g : α → α → N // ∀ (a₁ a₂ : α), g a₁ a₂ = g a₂ a₁ }
⊢ HSMul.hSMul (α := Sym2 α → R) (β := Sym2 α → N)
      ((lift : (a : { f : α → α → R // ∀ (a₁ a₂ : α), f a₁ a₂ = f a₂ a₁ }) → Sym2 α → R) f : Sym2 α → R)
      ((lift : (a : { f : α → α → N // ∀ (a₁ a₂ : α), f a₁ a₂ = f a₂ a₁ }) → Sym2 α → N) g : Sym2 α → N) =
    (lift : (a : { f : α → α → N // ∀ (a₁ a₂ : α), f a₁ a₂ = f a₂ a₁ }) → Sym2 α → N)
      (Subtype.mk (p := fun (f : α → α → N) ↦ ∀ (a₁ a₂ : α), f a₁ a₂ = f a₂ a₁) ((↑f : α → α → R) • (↑g : α → α → N)) ⋯)
have h : ∀ (x : Sym2 α) (i j : α),
  HSMul.hSMul (α := Sym2 α → R) (β := Sym2 α → N)
      ((lift : (a : { f : α → α → R // ∀ (a₁ a₂ : α), f a₁ a₂ = f a₂ a₁ }) → Sym2 α → R) f : Sym2 α → R)
      ((lift : (a : { f : α → α → N // ∀ (a₁ a₂ : α), f a₁ a₂ = f a₂ a₁ }) → Sym2 α → N) g : Sym2 α → N)
      (Quot.mk (Rel α) (i, j)) =
    (lift : (a : { f : α → α → N // ∀ (a₁ a₂ : α), f a₁ a₂ = f a₂ a₁ }) → Sym2 α → N)
      (Subtype.mk (p := fun (f : α → α → N) ↦ ∀ (a₁ a₂ : α), f a₁ a₂ = f a₂ a₁) ((↑f : α → α → R) • (↑g : α → α → N)) sorry)
      (Quot.mk (Rel α) (i, j))
Tactic: ext ⟨i, j⟩
-- Lean 4 proof step from Mathlib.Data.Sym.Sym2 (goal)
Goal:
α : Type u_1
s : Set α
z : Sym2 α
⊢ z ∈ s.sym2 ↔ (↑z : Set α) ⊆ s
have hf : ∀ (x y : α), s(x, y) ∈ s.sym2 ↔ (↑s(x, y) : Set α) ⊆ s
Tactic: induction z using Sym2.inductionOn
Cleared hypotheses: z
-- Lean 4 proof step from Mathlib.Data.Sym.Sym2 (goal)
Goal:
α : Type u_1
s : Set α
⊢ s.sym2 = uncurry Sym2.mk '' s ×ˢ s
have h : ∀ (x : Sym2 α) (x y : α), Quot.mk (Rel α) (x, y) ∈ s.sym2 ↔ Quot.mk (Rel α) (x, y) ∈ uncurry Sym2.mk '' s ×ˢ s
Tactic: ext ⟨x, y⟩
-- Lean 4 proof step from Mathlib.Data.Sym.Sym2 (goal)
Goal:
α : Type u_1
⊢ Eq.{u_1 + 1} (α := Set (Sym2 α)) (sym2 ∅) ∅
have h : ∀ (x : Sym2 α) (x y : α),
  Membership.mem (γ := Set (Sym2 α)) (sym2 ∅) (Quot.mk (Rel α) (x, y)) ↔
    Membership.mem (γ := Set (Sym2 α)) ∅ (Quot.mk (Rel α) (x, y))
Tactic: ext ⟨x, y⟩
-- Lean 4 proof step from Mathlib.Data.Sym.Sym2 (goal)
Goal:
α : Type u_1
⊢ Eq.{u_1 + 1} (α := Set (Sym2 α)) (sym2 univ) univ
have h : ∀ (x : Sym2 α) (x y : α),
  Membership.mem (γ := Set (Sym2 α)) (sym2 univ) (Quot.mk (Rel α) (x, y)) ↔
    Membership.mem (γ := Set (Sym2 α)) univ (Quot.mk (Rel α) (x, y))
Tactic: ext ⟨x, y⟩
-- Lean 4 proof step from Mathlib.Data.Sym.Sym2 (goal)
Goal:
α : Type u_1
a : α
⊢ Eq (α := Set (Sym2 α)) (sym2 {a}) {s(a, a)}
have h : ∀ (x : Sym2 α) (x y : α),
  Membership.mem (γ := Set (Sym2 α)) (sym2 {a}) (Quot.mk (Rel α) (x, y)) ↔
    Membership.mem (γ := Set (Sym2 α)) {s(a, a)} (Quot.mk (Rel α) (x, y))
Tactic: ext ⟨x, y⟩
-- Lean 4 proof step from Mathlib.Data.Sym.Sym2 (goal)
Goal:
α : Type u_1
a : α
s : Set α
⊢ (insert a s).sym2 = (fun (b : α) ↦ s(a, b)) '' insert a s ∪ s.sym2
have h : ∀ (x : Sym2 α) (x y : α),
  Quot.mk (Rel α) (x, y) ∈ (insert a s).sym2 ↔ Quot.mk (Rel α) (x, y) ∈ (fun (b : α) ↦ s(a, b)) '' insert a s ∪ s.sym2
Tactic: ext ⟨x, y⟩
-- Lean 4 proof step from Mathlib.Data.Sym.Sym2 (goal)
Goal:
α : Type u_1
β : Type u_2
f : α → β
s : Set β
⊢ (f ⁻¹' s).sym2 = Sym2.map f ⁻¹' s.sym2
have h : ∀ (x : Sym2 α) (x y : α), Quot.mk (Rel α) (x, y) ∈ (f ⁻¹' s).sym2 ↔ Quot.mk (Rel α) (x, y) ∈ Sym2.map f ⁻¹' s.sym2
Tactic: ext ⟨x, y⟩
-- Lean 4 proof step from Mathlib.Data.Sym.Sym2 (goal)
Goal:
α : Type u_1
β : Type u_2
f : α → β
s : Set α
⊢ (f '' s).sym2 = Sym2.map f '' s.sym2
have simp_rw : uncurry Sym2.mk '' (f '' s) ×ˢ (f '' s) = Sym2.map f '' uncurry Sym2.mk '' s ×ˢ s
Tactic: simp_rw [sym2_eq_mk_image, prod_image_image_eq, image_image, uncurry, Sym2.map_mk]
-- Lean 4 proof step from Mathlib.Data.Sym.Sym2 (goal)
Goal:
α : Type u_1
β : Type u_2
f : α → β
s : Set α
simp_rw : uncurry Sym2.mk '' (f '' s) ×ˢ (f '' s) = Sym2.map f '' uncurry Sym2.mk '' s ×ˢ s
⊢ (f '' s).sym2 = Sym2.map f '' s.sym2
have simp_rw₁ : uncurry Sym2.mk '' (fun (p : α × α) ↦ (f p.1, f p.2)) '' s ×ˢ s = Sym2.map f '' uncurry Sym2.mk '' s ×ˢ s
Tactic: simp_rw [sym2_eq_mk_image, prod_image_image_eq, image_image, uncurry, Sym2.map_mk]
-- Lean 4 proof step from Mathlib.Data.Sym.Sym2 (goal)
Goal:
α : Type u_1
β : Type u_2
f : α → β
s : Set α
simp_rw : uncurry Sym2.mk '' (f '' s) ×ˢ (f '' s) = Sym2.map f '' uncurry Sym2.mk '' s ×ˢ s
simp_rw₁ : uncurry Sym2.mk '' (fun (p : α × α) ↦ (f p.1, f p.2)) '' s ×ˢ s = Sym2.map f '' uncurry Sym2.mk '' s ×ˢ s
⊢ (f '' s).sym2 = Sym2.map f '' s.sym2
have simp_rw₂ : Eq.{u_2 + 1} (α := Set (Sym2 β)) ((fun (x : α × α) ↦ uncurry Sym2.mk (f x.1, f x.2)) '' s ×ˢ s)
  ((fun (x : α × α) ↦ Sym2.map f (uncurry Sym2.mk x)) '' s ×ˢ s)
Tactic: simp_rw [sym2_eq_mk_image, prod_image_image_eq, image_image, uncurry, Sym2.map_mk]
-- Lean 4 proof step from Mathlib.Data.Sym.Sym2 (goal)
Goal:
α : Type u_1
β : Type u_2
f : α → β
s : Set α
simp_rw : uncurry Sym2.mk '' (f '' s) ×ˢ (f '' s) = Sym2.map f '' uncurry Sym2.mk '' s ×ˢ s
simp_rw₁ : uncurry Sym2.mk '' (fun (p : α × α) ↦ (f p.1, f p.2)) '' s ×ˢ s = Sym2.map f '' uncurry Sym2.mk '' s ×ˢ s
simp_rw₂ :
  Eq.{u_2 + 1} (α := Set (Sym2 β)) ((fun (x : α × α) ↦ uncurry Sym2.mk (f x.1, f x.2)) '' s ×ˢ s)
    ((fun (x : α × α) ↦ Sym2.map f (uncurry Sym2.mk x)) '' s ×ˢ s)
⊢ (f '' s).sym2 = Sym2.map f '' s.sym2
have simp_rw₃ : Eq.{u_2 + 1} (α := Set (Sym2 β)) ((fun (a : α × α) ↦ s(f a.1, f a.2)) '' s ×ˢ s)
  ((fun (a : α × α) ↦ Sym2.map f s(a.1, a.2)) '' s ×ˢ s)
Tactic: simp_rw [sym2_eq_mk_image, prod_image_image_eq, image_image, uncurry, Sym2.map_mk]
-- Lean 4 proof step from Mathlib.Data.Sym.Sym2 (goal)
Goal:
α : Type u_1
ι : Type u_4
f : ι → Set α
⊢ Eq.{u_1 + 1} (α := Set (Sym2 α)) (sym2 (⋂ (i : ι), f i)) (⋂ (i : ι), (f i).sym2)
have h : ∀ (x : Sym2 α) (x y : α),
  Membership.mem (γ := Set (Sym2 α)) (sym2 (⋂ (i : ι), f i)) (Quot.mk (Rel α) (x, y)) ↔
    Membership.mem (γ := Set (Sym2 α)) (⋂ (i : ι), (f i).sym2) (Quot.mk (Rel α) (x, y))
Tactic: ext ⟨x, y⟩
-- Lean 4 proof step from Mathlib.Dynamics.TopologicalEntropy.CoverEntropy (goal)
Goal:
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
n : ℕ
inst : U.IsSymm
s : Finset X
h : IsDynCoverOf T F U n (↑s : Set X)
⊢ ∃ (t : Finset X), IsDynCoverOf T F U n (↑t : Set X) ∧ #t ≤ #s ∧ ∀ x ∈ t, (ball x (dynEntourage T U n) ∩ F).Nonempty
have h₁ : IsDynCoverOf T F U n (↑({x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}) : Set X) ∧
  #({x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}) ≤ #s ∧
    ∀ x ∈ {x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}, (ball x (dynEntourage T U n) ∩ F).Nonempty
Tactic: use {x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}
-- Lean 4 proof step from Mathlib.Dynamics.TopologicalEntropy.CoverEntropy (goal)
Goal:
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
n : ℕ
inst : U.IsSymm
s : Finset X
h : IsDynCoverOf T F U n (↑s : Set X)
⊢ IsDynCoverOf T F U n (↑({x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}) : Set X) ∧
    #({x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}) ≤ #s ∧
      ∀ x ∈ {x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}, (ball x (dynEntourage T U n) ∩ F).Nonempty
have h₁ : IsDynCoverOf T F U n {x : X | x ∈ s ∧ (ball x (dynEntourage T U n) ∩ F).Nonempty} ∧
  #({x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}) ≤ #s
Tactic: simp only [Finset.coe_filter, Finset.mem_filter, and_imp, imp_self, implies_true, and_true]
-- Lean 4 proof step from Mathlib.Dynamics.TopologicalEntropy.CoverEntropy (goal)
Goal:
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
n : ℕ
inst : U.IsSymm
s : Finset X
h : IsDynCoverOf T F U n (↑s : Set X)
⊢ IsDynCoverOf T F U n {x : X | x ∈ s ∧ (ball x (dynEntourage T U n) ∩ F).Nonempty} ∧
    #({x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}) ≤ #s
have h₁ : ∀ (y : X) (y_F : y ∈ F),
  ∃ (y_1 : X),
    Membership.mem (γ := Set X) {x : X | x ∈ s ∧ (ball x (dynEntourage T U n) ∩ F).Nonempty} y_1 ∧
      (y, y_1) ∈ dynEntourage T U n
Tactic: refine ⟨fun y y_F ↦ ?_, Finset.card_mono (Finset.filter_subset _ s)⟩
-- Lean 4 proof step from Mathlib.Dynamics.TopologicalEntropy.CoverEntropy (goal)
Goal:
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
n : ℕ
inst : U.IsSymm
s : Finset X
h : IsDynCoverOf T F U n (↑s : Set X)
y : X
y_F : y ∈ F
⊢ ∃ (y_1 : X),
    Membership.mem (γ := Set X) {x : X | x ∈ s ∧ (ball x (dynEntourage T U n) ∩ F).Nonempty} y_1 ∧
      (y, y_1) ∈ dynEntourage T U n
have h₁ : ∀ (z : X) (z_s : z ∈ (↑s : Set X)) (y_Bz : (y, z) ∈ dynEntourage T U n),
  ∃ (y_1 : X),
    Membership.mem (γ := Set X) {x : X | x ∈ s ∧ (ball x (dynEntourage T U n) ∩ F).Nonempty} y_1 ∧
      (y, y_1) ∈ dynEntourage T U n
Tactic: obtain ⟨z, z_s, y_Bz⟩ := h y_F
-- Lean 4 proof step from Mathlib.Dynamics.TopologicalEntropy.CoverEntropy (goal)
Goal:
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
n : ℕ
inst : U.IsSymm
s : Finset X
h : IsDynCoverOf T F U n (↑s : Set X)
w : Finset X
⊢ ∃ (t : Finset X), IsDynCoverOf T F U n (↑t : Set X) ∧ #t ≤ #s ∧ ∀ x ∈ t, (ball x (dynEntourage T U n) ∩ F).Nonempty
have h₁ : IsDynCoverOf T F U n (↑({x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}) : Set X) ∧
  #({x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}) ≤ #s ∧
    ∀ x ∈ {x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}, (ball x (dynEntourage T U n) ∩ F).Nonempty
Tactic: classical
  use {x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}
  simp only [Finset.coe_filter, Finset.mem_filter, and_imp, imp_self, implies_true, and_true]
  refine ⟨fun y y_F ↦ ?_, Finset.card_mono (Finset.filter_subset _ s)⟩
  obtain ⟨z, z_s, y_Bz⟩ := h y_F
  exact ⟨z, ⟨z_s, _, (dynEntourage T U n).symm y_Bz, y_F⟩, y_Bz⟩
-- Lean 4 proof step from Mathlib.Dynamics.TopologicalEntropy.CoverEntropy (goal)
Goal:
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
n : ℕ
inst : U.IsSymm
s : Finset X
h : IsDynCoverOf T F U n (↑s : Set X)
w : Finset X
h₁ h₂ :
  IsDynCoverOf T F U n (↑({x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}) : Set X) ∧
    #({x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}) ≤ #s ∧
      ∀ x ∈ {x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}, (ball x (dynEntourage T U n) ∩ F).Nonempty
⊢ ∃ (t : Finset X), IsDynCoverOf T F U n (↑t : Set X) ∧ #t ≤ #s ∧ ∀ x ∈ t, (ball x (dynEntourage T U n) ∩ F).Nonempty
have h₃ : IsDynCoverOf T F U n {x : X | x ∈ s ∧ (ball x (dynEntourage T U n) ∩ F).Nonempty} ∧
  #({x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}) ≤ #s
Tactic: classical
  use {x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}
  simp only [Finset.coe_filter, Finset.mem_filter, and_imp, imp_self, implies_true, and_true]
  refine ⟨fun y y_F ↦ ?_, Finset.card_mono (Finset.filter_subset _ s)⟩
  obtain ⟨z, z_s, y_Bz⟩ := h y_F
  exact ⟨z, ⟨z_s, _, (dynEntourage T U n).symm y_Bz, y_F⟩, y_Bz⟩
-- Lean 4 proof step from Mathlib.Dynamics.TopologicalEntropy.CoverEntropy (goal)
Goal:
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
n : ℕ
inst : U.IsSymm
s : Finset X
h : IsDynCoverOf T F U n (↑s : Set X)
w : Finset X
h₁ h₂ :
  IsDynCoverOf T F U n (↑({x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}) : Set X) ∧
    #({x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}) ≤ #s ∧
      ∀ x ∈ {x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}, (ball x (dynEntourage T U n) ∩ F).Nonempty
h₃ :
  IsDynCoverOf T F U n {x : X | x ∈ s ∧ (ball x (dynEntourage T U n) ∩ F).Nonempty} ∧
    #({x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}) ≤ #s
⊢ ∃ (t : Finset X), IsDynCoverOf T F U n (↑t : Set X) ∧ #t ≤ #s ∧ ∀ x ∈ t, (ball x (dynEntourage T U n) ∩ F).Nonempty
have h₄ : ∀ (y : X) (y_F : y ∈ F),
  ∃ (y_1 : X),
    Membership.mem (γ := Set X) {x : X | x ∈ s ∧ (ball x (dynEntourage T U n) ∩ F).Nonempty} y_1 ∧
      (y, y_1) ∈ dynEntourage T U n
Tactic: classical
  use {x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}
  simp only [Finset.coe_filter, Finset.mem_filter, and_imp, imp_self, implies_true, and_true]
  refine ⟨fun y y_F ↦ ?_, Finset.card_mono (Finset.filter_subset _ s)⟩
  obtain ⟨z, z_s, y_Bz⟩ := h y_F
  exact ⟨z, ⟨z_s, _, (dynEntourage T U n).symm y_Bz, y_F⟩, y_Bz⟩
-- Lean 4 proof step from Mathlib.Dynamics.TopologicalEntropy.CoverEntropy (goal)
Goal:
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
n : ℕ
inst : U.IsSymm
s : Finset X
h : IsDynCoverOf T F U n (↑s : Set X)
w : Finset X
h₁ h₂ :
  IsDynCoverOf T F U n (↑({x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}) : Set X) ∧
    #({x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}) ≤ #s ∧
      ∀ x ∈ {x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}, (ball x (dynEntourage T U n) ∩ F).Nonempty
h₃ :
  IsDynCoverOf T F U n {x : X | x ∈ s ∧ (ball x (dynEntourage T U n) ∩ F).Nonempty} ∧
    #({x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}) ≤ #s
h₄ :
  ∀ (y : X) (y_F : y ∈ F),
    ∃ (y_1 : X),
      Membership.mem (γ := Set X) {x : X | x ∈ s ∧ (ball x (dynEntourage T U n) ∩ F).Nonempty} y_1 ∧
        (y, y_1) ∈ dynEntourage T U n
⊢ ∃ (t : Finset X), IsDynCoverOf T F U n (↑t : Set X) ∧ #t ≤ #s ∧ ∀ x ∈ t, (ball x (dynEntourage T U n) ∩ F).Nonempty
have h₅ : ∀ (y : X) (y_F : y ∈ F) (z : X) (z_s : z ∈ (↑s : Set X)) (y_Bz : (y, z) ∈ dynEntourage T U n),
  ∃ (y_1 : X),
    Membership.mem (γ := Set X) {x : X | x ∈ s ∧ (ball x (dynEntourage T U n) ∩ F).Nonempty} y_1 ∧
      (y, y_1) ∈ dynEntourage T U n
Tactic: classical
  use {x ∈ s | (ball x (dynEntourage T U n) ∩ F).Nonempty}
  simp only [Finset.coe_filter, Finset.mem_filter, and_imp, imp_self, implies_true, and_true]
  refine ⟨fun y y_F ↦ ?_, Finset.card_mono (Finset.filter_subset _ s)⟩
  obtain ⟨z, z_s, y_Bz⟩ := h y_F
  exact ⟨z, ⟨z_s, _, (dynEntourage T U n).symm y_Bz, y_F⟩, y_Bz⟩
-- Lean 4 proof step from Mathlib.Dynamics.TopologicalEntropy.CoverEntropy (goal)
Goal:
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
m : ℕ
F_inv : MapsTo T F F
inst : U.IsSymm
n : ℕ
s : Finset X
h : IsDynCoverOf T F U m (↑s : Set X)
⊢ ∃ (t : Finset X), IsDynCoverOf T F (U ○ U) (m * n) (↑t : Set X) ∧ #t ≤ #s ^ n
have inl : ∀ (F_inv : MapsTo T ∅ ∅) (h : IsDynCoverOf T ∅ U m (↑s : Set X)),
  ∃ (t : Finset X), IsDynCoverOf T ∅ (U ○ U) (m * n) (↑t : Set X) ∧ #t ≤ #s ^ n
Tactic: rcases F.eq_empty_or_nonempty with rfl | F_nemp
Cleared hypotheses: F, F_inv, h
-- Lean 4 proof step from Mathlib.Dynamics.TopologicalEntropy.CoverEntropy (goal)
Goal:
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
m : ℕ
F_inv : MapsTo T F F
inst : U.IsSymm
n : ℕ
s : Finset X
h : IsDynCoverOf T F U m (↑s : Set X)
inl :
  ∀ (F_inv : MapsTo T ∅ ∅) (h : IsDynCoverOf T ∅ U m (↑s : Set X)),
    ∃ (t : Finset X), IsDynCoverOf T ∅ (U ○ U) (m * n) (↑t : Set X) ∧ #t ≤ #s ^ n
⊢ ∃ (t : Finset X), IsDynCoverOf T F (U ○ U) (m * n) (↑t : Set X) ∧ #t ≤ #s ^ n
have inr : ∀ (F_nemp : F.Nonempty), ∃ (t : Finset X), IsDynCoverOf T F (U ○ U) (m * n) (↑t : Set X) ∧ #t ≤ #s ^ n
Tactic: rcases F.eq_empty_or_nonempty with rfl | F_nemp
-- Lean 4 proof step from Mathlib.Dynamics.TopologicalEntropy.CoverEntropy (hypothesis)
Goal:
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
m : ℕ
F_inv : MapsTo T F F
inst : U.IsSymm
n : ℕ
s : Finset X
h : IsDynCoverOf T F U m (↑s : Set X)
F_nemp : F.Nonempty
⊢ ∃ (t : Finset X), IsDynCoverOf T F (U ○ U) (m * n) (↑t : Set X) ∧ #t ≤ #s ^ n
have x : Nonempty X
Tactic: have _ : Nonempty X := F_nemp.nonempty
-- Lean 4 proof step from Mathlib.Dynamics.TopologicalEntropy.CoverEntropy (hypothesis)
Goal:
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
m : ℕ
F_inv : MapsTo T F F
inst : U.IsSymm
n : ℕ
s : Finset X
h : IsDynCoverOf T F U m (↑s : Set X)
F_nemp : F.Nonempty
h_1 : Nonempty X
⊢ ∃ (t : Finset X), IsDynCoverOf T F (U ○ U) (m * n) (↑t : Set X) ∧ #t ≤ #s ^ n
have s_nemp : (↑s : Set X).Nonempty
Tactic: have s_nemp := h.nonempty F_nemp
-- Lean 4 proof step from Mathlib.Dynamics.TopologicalEntropy.CoverEntropy (goal)
Goal:
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
m : ℕ
F_inv : MapsTo T F F
inst : U.IsSymm
n : ℕ
s : Finset X
h : IsDynCoverOf T F U m (↑s : Set X)
F_nemp : F.Nonempty
h_1 : Nonempty X
s_nemp : (↑s : Set X).Nonempty
⊢ ∃ (t : Finset X), IsDynCoverOf T F (U ○ U) (m * n) (↑t : Set X) ∧ #t ≤ #s ^ n
have inr : ∀ (x : X) (x_F : x ∈ F), ∃ (t : Finset X), IsDynCoverOf T F (U ○ U) (m * n) (↑t : Set X) ∧ #t ≤ #s ^ n
Tactic: obtain ⟨x, x_F⟩ := F_nemp
Cleared hypotheses: F_nemp
-- Lean 4 proof step from Mathlib.Dynamics.TopologicalEntropy.CoverEntropy (goal)
Goal:
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
m : ℕ
F_inv : MapsTo T F F
inst : U.IsSymm
n : ℕ
s : Finset X
h : IsDynCoverOf T F U m (↑s : Set X)
h_1 : Nonempty X
s_nemp : (↑s : Set X).Nonempty
x : X
x_F : x ∈ F
⊢ ∃ (t : Finset X), IsDynCoverOf T F (U ○ U) (m * n) (↑t : Set X) ∧ #t ≤ #s ^ n
have inr : ∀ (h : IsDynCoverOf T F U 0 (↑s : Set X)), ∃ (t : Finset X), IsDynCoverOf T F (U ○ U) (0 * n) (↑t : Set X) ∧ #t ≤ #s ^ n
Tactic: rcases m.eq_zero_or_pos with rfl | m_pos
Cleared hypotheses: m, h
-- Lean 4 proof step from Mathlib.Dynamics.TopologicalEntropy.CoverEntropy (goal)
Goal:
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
m : ℕ
F_inv : MapsTo T F F
inst : U.IsSymm
n : ℕ
s : Finset X
h : IsDynCoverOf T F U m (↑s : Set X)
h_1 : Nonempty X
s_nemp : (↑s : Set X).Nonempty
x : X
x_F : x ∈ F
inr :
  ∀ (h : IsDynCoverOf T F U 0 (↑s : Set X)),
    ∃ (t : Finset X), IsDynCoverOf T F (U ○ U) (0 * n) (↑t : Set X) ∧ #t ≤ #s ^ n
⊢ ∃ (t : Finset X), IsDynCoverOf T F (U ○ U) (m * n) (↑t : Set X) ∧ #t ≤ #s ^ n
have inr₁ : ∀ (m_pos : m > 0), ∃ (t : Finset X), IsDynCoverOf T F (U ○ U) (m * n) (↑t : Set X) ∧ #t ≤ #s ^ n
Tactic: rcases m.eq_zero_or_pos with rfl | m_pos
-- Lean 4 proof step from Mathlib.Dynamics.TopologicalEntropy.CoverEntropy (goal)
Goal:
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
F_inv : MapsTo T F F
inst : U.IsSymm
n : ℕ
s : Finset X
h_1 : Nonempty X
s_nemp : (↑s : Set X).Nonempty
x : X
x_F : x ∈ F
h : IsDynCoverOf T F U 0 (↑s : Set X)
⊢ ∃ (t : Finset X), IsDynCoverOf T F (U ○ U) (0 * n) (↑t : Set X) ∧ #t ≤ #s ^ n
have h₁ : IsDynCoverOf T F (U ○ U) (0 * n) (↑{x} : Set X) ∧ Finset.card (α := X) {x} ≤ #s ^ n
Tactic: use {x}
-- Lean 4 proof step from Mathlib.Dynamics.TopologicalEntropy.CoverEntropy (goal)
Goal:
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
F_inv : MapsTo T F F
inst : U.IsSymm
n : ℕ
s : Finset X
h_1 : Nonempty X
s_nemp : (↑s : Set X).Nonempty
x : X
x_F : x ∈ F
h : IsDynCoverOf T F U 0 (↑s : Set X)
⊢ IsDynCoverOf T F (U ○ U) (0 * n) (↑{x} : Set X) ∧ Finset.card (α := X) {x} ≤ #s ^ n
have h₁ : IsDynCoverOf T F (U ○ U) 0 {x} ∧ 1 ≤ #s ^ n
Tactic: simp only [zero_mul, Finset.coe_singleton, Finset.card_singleton]
-- Lean 4 proof step from Mathlib.Dynamics.TopologicalEntropy.CoverEntropy (goal)
Goal:
X : Type u_1
T : X → X
U : SetRel X X
F : Set X
m : ℕ
F_inv : MapsTo T F F
inst : U.IsSymm
n : ℕ
s : Finset X
h : IsDynCoverOf T F U m (↑s : Set X)
h_1 : Nonempty X
s_nemp : (↑s : Set X).Nonempty
x : X
x_F : x ∈ F
m_pos : m > 0
t : Fin n → ↥s
⊢ ∃ (y : X),
    ⋂ (k : Fin n), T^[m * (↑k : ℕ)] ⁻¹' ball (↑(t k) : X) (dynEntourage T U m) ⊆ ball y (dynEntourage T (U ○ U) (m * n))
have inl : ∀
  (inter_empt :
    Eq.{u_1 + 1} (α := Set X) (⋂ (k : Fin n), T^[m * (↑k : ℕ)] ⁻¹' ball (↑(t k) : X) (dynEntourage T U m)) ∅),
  ∃ (y : X),
    ⋂ (k : Fin n), T^[m * (↑k : ℕ)] ⁻¹' ball (↑(t k) : X) (dynEntourage T U m) ⊆ ball y (dynEntourage T (U ○ U) (m * n))
Tactic: rcases (⋂ k : Fin n, T^[m * k] ⁻¹' ball (t k) (dynEntourage T U m)).eq_empty_or_nonempty
      with inter_empt | inter_nemp