file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Topology/Algebra/Order/IntermediateValue.lean
|
isPreconnected_Iic
|
[] |
[
442,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
441,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean
|
LinearMap.toMatrixAlgEquiv'_mul
|
[] |
[
513,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
510,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/StrictInitial.lean
|
CategoryTheory.Limits.isInitialMul_inv
|
[] |
[
114,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Algebra/Hom/Centroid.lean
|
CentroidHom.cancel_left
|
[
{
"state_after": "no goals",
"state_before": "F : Type ?u.21144\nα : Type u_1\ninst✝ : NonUnitalNonAssocSemiring α\ng f₁ f₂ : CentroidHom α\nhg : Injective ↑g\nh : comp g f₁ = comp g f₂\na : α\n⊢ ↑g (↑f₁ a) = ↑g (↑f₂ a)",
"tactic": "rw [← comp_apply, h, comp_apply]"
}
] |
[
238,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
236,
1
] |
Mathlib/Data/List/BigOperators/Basic.lean
|
List.prod_singleton
|
[] |
[
39,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
38,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
linearIndependent_iff''
|
[
{
"state_after": "case h.e'_2\nι : Type u'\nι' : Type ?u.19688\nR : Type u_1\nK : Type ?u.19694\nM : Type u_2\nM' : Type ?u.19700\nM'' : Type ?u.19703\nV : Type u\nV' : Type ?u.19708\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nH : ∀ (s : Finset ι) (g : ι → R), (∀ (i : ι), ¬i ∈ s → g i = 0) → ∑ i in s, g i • v i = 0 → ∀ (i : ι), g i = 0\ns : Finset ι\ng : ι → R\nhg : ∑ i in s, g i • v i = 0\ni : ι\nhi : i ∈ s\n⊢ g i = if i ∈ s then g i else 0",
"state_before": "ι : Type u'\nι' : Type ?u.19688\nR : Type u_1\nK : Type ?u.19694\nM : Type u_2\nM' : Type ?u.19700\nM'' : Type ?u.19703\nV : Type u\nV' : Type ?u.19708\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nH : ∀ (s : Finset ι) (g : ι → R), (∀ (i : ι), ¬i ∈ s → g i = 0) → ∑ i in s, g i • v i = 0 → ∀ (i : ι), g i = 0\ns : Finset ι\ng : ι → R\nhg : ∑ i in s, g i • v i = 0\ni : ι\nhi : i ∈ s\n⊢ g i = 0",
"tactic": "convert\n H s (fun j => if j ∈ s then g j else 0) (fun j hj => if_neg hj)\n (by simp_rw [ite_smul, zero_smul, Finset.sum_extend_by_zero, hg]) i"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2\nι : Type u'\nι' : Type ?u.19688\nR : Type u_1\nK : Type ?u.19694\nM : Type u_2\nM' : Type ?u.19700\nM'' : Type ?u.19703\nV : Type u\nV' : Type ?u.19708\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nH : ∀ (s : Finset ι) (g : ι → R), (∀ (i : ι), ¬i ∈ s → g i = 0) → ∑ i in s, g i • v i = 0 → ∀ (i : ι), g i = 0\ns : Finset ι\ng : ι → R\nhg : ∑ i in s, g i • v i = 0\ni : ι\nhi : i ∈ s\n⊢ g i = if i ∈ s then g i else 0",
"tactic": "exact (if_pos hi).symm"
},
{
"state_after": "no goals",
"state_before": "ι : Type u'\nι' : Type ?u.19688\nR : Type u_1\nK : Type ?u.19694\nM : Type u_2\nM' : Type ?u.19700\nM'' : Type ?u.19703\nV : Type u\nV' : Type ?u.19708\nv : ι → M\ninst✝⁶ : Semiring R\ninst✝⁵ : AddCommMonoid M\ninst✝⁴ : AddCommMonoid M'\ninst✝³ : AddCommMonoid M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nH : ∀ (s : Finset ι) (g : ι → R), (∀ (i : ι), ¬i ∈ s → g i = 0) → ∑ i in s, g i • v i = 0 → ∀ (i : ι), g i = 0\ns : Finset ι\ng : ι → R\nhg : ∑ i in s, g i • v i = 0\ni : ι\nhi : i ∈ s\n⊢ ∑ i in s, (fun j => if j ∈ s then g j else 0) i • v i = 0",
"tactic": "simp_rw [ite_smul, zero_smul, Finset.sum_extend_by_zero, hg]"
}
] |
[
144,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
Mathlib/NumberTheory/LSeries.lean
|
Nat.ArithmeticFunction.LSeriesSummable_of_bounded_of_one_lt_re
|
[
{
"state_after": "f : ArithmeticFunction ℂ\nm : ℝ\nh : ∀ (n : ℕ), ↑Complex.abs (↑f n) ≤ m\nz : ℂ\nhz : 1 < z.re\n⊢ LSeriesSummable f ↑z.re",
"state_before": "f : ArithmeticFunction ℂ\nm : ℝ\nh : ∀ (n : ℕ), ↑Complex.abs (↑f n) ≤ m\nz : ℂ\nhz : 1 < z.re\n⊢ LSeriesSummable f z",
"tactic": "rw [← LSeriesSummable_iff_of_re_eq_re (Complex.ofReal_re z.re)]"
},
{
"state_after": "f : ArithmeticFunction ℂ\nm : ℝ\nh : ∀ (n : ℕ), ↑Complex.abs (↑f n) ≤ m\nz : ℂ\nhz : 1 < z.re\n⊢ 1 < z.re",
"state_before": "f : ArithmeticFunction ℂ\nm : ℝ\nh : ∀ (n : ℕ), ↑Complex.abs (↑f n) ≤ m\nz : ℂ\nhz : 1 < z.re\n⊢ LSeriesSummable f ↑z.re",
"tactic": "apply LSeriesSummable_of_bounded_of_one_lt_real h"
},
{
"state_after": "no goals",
"state_before": "f : ArithmeticFunction ℂ\nm : ℝ\nh : ∀ (n : ℕ), ↑Complex.abs (↑f n) ≤ m\nz : ℂ\nhz : 1 < z.re\n⊢ 1 < z.re",
"tactic": "exact hz"
}
] |
[
105,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
101,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Type.lean
|
Equiv.Perm.dvd_of_mem_cycleType
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nn : ℕ\nh : n ∈ cycleType σ\n⊢ n ∣ Multiset.lcm (cycleType σ)",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nn : ℕ\nh : n ∈ cycleType σ\n⊢ n ∣ orderOf σ",
"tactic": "rw [← lcm_cycleType]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nn : ℕ\nh : n ∈ cycleType σ\n⊢ n ∣ Multiset.lcm (cycleType σ)",
"tactic": "exact dvd_lcm h"
}
] |
[
183,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Mathlib/Data/Nat/WithBot.lean
|
Nat.WithBot.add_eq_zero_iff
|
[
{
"state_after": "case none.none\n\n⊢ none + none = 0 ↔ none = 0 ∧ none = 0\n\ncase none.some\nval✝ : ℕ\n⊢ none + some val✝ = 0 ↔ none = 0 ∧ some val✝ = 0\n\ncase some.none\nval✝ : ℕ\n⊢ some val✝ + none = 0 ↔ some val✝ = 0 ∧ none = 0\n\ncase some.some\nval✝¹ val✝ : ℕ\n⊢ some val✝¹ + some val✝ = 0 ↔ some val✝¹ = 0 ∧ some val✝ = 0",
"state_before": "n m : WithBot ℕ\n⊢ n + m = 0 ↔ n = 0 ∧ m = 0",
"tactic": "rcases n, m with ⟨_ | _, _ | _⟩"
},
{
"state_after": "case some.none\nval✝ : ℕ\n⊢ some val✝ + none = 0 ↔ some val✝ = 0 ∧ none = 0\n\ncase some.some\nval✝¹ val✝ : ℕ\n⊢ some val✝¹ + some val✝ = 0 ↔ some val✝¹ = 0 ∧ some val✝ = 0",
"state_before": "case none.none\n\n⊢ none + none = 0 ↔ none = 0 ∧ none = 0\n\ncase none.some\nval✝ : ℕ\n⊢ none + some val✝ = 0 ↔ none = 0 ∧ some val✝ = 0\n\ncase some.none\nval✝ : ℕ\n⊢ some val✝ + none = 0 ↔ some val✝ = 0 ∧ none = 0\n\ncase some.some\nval✝¹ val✝ : ℕ\n⊢ some val✝¹ + some val✝ = 0 ↔ some val✝¹ = 0 ∧ some val✝ = 0",
"tactic": "any_goals (exact ⟨fun h => Option.noConfusion h, fun h => Option.noConfusion h.1⟩)"
},
{
"state_after": "case some.some\nval✝¹ val✝ : ℕ\n⊢ some val✝¹ + some val✝ = 0 ↔ some val✝¹ = 0 ∧ some val✝ = 0",
"state_before": "case some.none\nval✝ : ℕ\n⊢ some val✝ + none = 0 ↔ some val✝ = 0 ∧ none = 0\n\ncase some.some\nval✝¹ val✝ : ℕ\n⊢ some val✝¹ + some val✝ = 0 ↔ some val✝¹ = 0 ∧ some val✝ = 0",
"tactic": "exact ⟨fun h => Option.noConfusion h, fun h => Option.noConfusion h.2⟩"
},
{
"state_after": "case some.some\nval✝¹ val✝ : ℕ\n⊢ (fun x x_1 => x + x_1) val✝¹ val✝ = 0 ↔ val✝¹ = 0 ∧ val✝ = 0",
"state_before": "case some.some\nval✝¹ val✝ : ℕ\n⊢ some val✝¹ + some val✝ = 0 ↔ some val✝¹ = 0 ∧ some val✝ = 0",
"tactic": "repeat' erw [WithBot.coe_eq_coe]"
},
{
"state_after": "no goals",
"state_before": "case some.some\nval✝¹ val✝ : ℕ\n⊢ (fun x x_1 => x + x_1) val✝¹ val✝ = 0 ↔ val✝¹ = 0 ∧ val✝ = 0",
"tactic": "exact add_eq_zero_iff' (zero_le _) (zero_le _)"
},
{
"state_after": "case some.some\nval✝¹ val✝ : ℕ\n⊢ some val✝¹ + some val✝ = 0 ↔ some val✝¹ = 0 ∧ some val✝ = 0",
"state_before": "case some.some\nval✝¹ val✝ : ℕ\n⊢ some val✝¹ + some val✝ = 0 ↔ some val✝¹ = 0 ∧ some val✝ = 0",
"tactic": "(exact ⟨fun h => Option.noConfusion h, fun h => Option.noConfusion h.1⟩)"
},
{
"state_after": "case some.some\nval✝¹ val✝ : ℕ\n⊢ some val✝¹ + some val✝ = 0 ↔ some val✝¹ = 0 ∧ some val✝ = 0",
"state_before": "case some.some\nval✝¹ val✝ : ℕ\n⊢ some val✝¹ + some val✝ = 0 ↔ some val✝¹ = 0 ∧ some val✝ = 0",
"tactic": "exact ⟨fun h => Option.noConfusion h, fun h => Option.noConfusion h.1⟩"
},
{
"state_after": "case some.some\nval✝¹ val✝ : ℕ\n⊢ (fun x x_1 => x + x_1) val✝¹ val✝ = 0 ↔ val✝¹ = 0 ∧ val✝ = 0",
"state_before": "case some.some\nval✝¹ val✝ : ℕ\n⊢ (fun x x_1 => x + x_1) val✝¹ val✝ = 0 ↔ val✝¹ = 0 ∧ val✝ = 0",
"tactic": "erw [WithBot.coe_eq_coe]"
}
] |
[
31,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
26,
1
] |
Mathlib/RingTheory/Valuation/Basic.lean
|
Valuation.isEquiv_iff_val_sub_one_lt_one
|
[
{
"state_after": "K : Type u_3\nF : Type ?u.3433435\nR : Type ?u.3433438\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3433450\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\n⊢ (∀ {x : K}, ↑v x < 1 ↔ ↑v' x < 1) ↔ ∀ {x : K}, ↑v (x - 1) < 1 ↔ ↑v' (x - 1) < 1",
"state_before": "K : Type u_3\nF : Type ?u.3433435\nR : Type ?u.3433438\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3433450\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\n⊢ IsEquiv v v' ↔ ∀ {x : K}, ↑v (x - 1) < 1 ↔ ↑v' (x - 1) < 1",
"tactic": "rw [isEquiv_iff_val_lt_one]"
},
{
"state_after": "no goals",
"state_before": "K : Type u_3\nF : Type ?u.3433435\nR : Type ?u.3433438\ninst✝³ : DivisionRing K\nΓ₀ : Type u_1\nΓ'₀ : Type u_2\nΓ''₀ : Type ?u.3433450\ninst✝² : LinearOrderedCommMonoidWithZero Γ''₀\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ'₀\nv : Valuation K Γ₀\nv' : Valuation K Γ'₀\n⊢ (∀ {x : K}, ↑v x < 1 ↔ ↑v' x < 1) ↔ ∀ {x : K}, ↑v (x - 1) < 1 ↔ ↑v' (x - 1) < 1",
"tactic": "exact (Equiv.subRight 1).surjective.forall"
}
] |
[
520,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
516,
1
] |
Mathlib/Algebra/GroupWithZero/Defs.lean
|
pullback_nonzero
|
[
{
"state_after": "G₀ : Type u\nM₀ : Type u_2\nM₀' : Type u_1\nG₀' : Type ?u.9044\ninst✝³ : MulZeroOneClass M₀\ninst✝² : Nontrivial M₀\na b : M₀\ninst✝¹ : Zero M₀'\ninst✝ : One M₀'\nf : M₀' → M₀\nzero : f 0 = 0\none : f 1 = 1\n⊢ ¬0 = 1",
"state_before": "G₀ : Type u\nM₀ : Type u_2\nM₀' : Type u_1\nG₀' : Type ?u.9044\ninst✝³ : MulZeroOneClass M₀\ninst✝² : Nontrivial M₀\na b : M₀\ninst✝¹ : Zero M₀'\ninst✝ : One M₀'\nf : M₀' → M₀\nzero : f 0 = 0\none : f 1 = 1\n⊢ ¬f 0 = f 1",
"tactic": "rw [zero, one]"
},
{
"state_after": "no goals",
"state_before": "G₀ : Type u\nM₀ : Type u_2\nM₀' : Type u_1\nG₀' : Type ?u.9044\ninst✝³ : MulZeroOneClass M₀\ninst✝² : Nontrivial M₀\na b : M₀\ninst✝¹ : Zero M₀'\ninst✝ : One M₀'\nf : M₀' → M₀\nzero : f 0 = 0\none : f 1 = 1\n⊢ ¬0 = 1",
"tactic": "exact zero_ne_one"
}
] |
[
229,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
225,
1
] |
Mathlib/Dynamics/PeriodicPts.lean
|
Function.IsPeriodicPt.mod
|
[] |
[
166,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
11
] |
Mathlib/Data/Vector/Basic.lean
|
Vector.get_replicate
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\nα : Type u_1\na : α\ni : Fin n\n⊢ get (replicate n a) i = a",
"tactic": "apply List.get_replicate"
}
] |
[
130,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Data/Complex/Module.lean
|
Complex.liftAux_apply
|
[] |
[
363,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean
|
hasSum_geometric_of_lt_1
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.68751\nβ : Type ?u.68754\nι : Type ?u.68757\nr : ℝ\nh₁ : 0 ≤ r\nh₂ : r < 1\nthis✝ : r ≠ 1\nthis : Tendsto (fun n => (r ^ n - 1) * (r - 1)⁻¹) atTop (𝓝 ((0 - 1) * (r - 1)⁻¹))\n⊢ Tendsto (fun n => ∑ i in Finset.range n, r ^ i) atTop (𝓝 (1 - r)⁻¹)",
"tactic": "simp_all [neg_inv, geom_sum_eq, div_eq_mul_inv]"
}
] |
[
188,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Finset.prod_dvd_prod_of_dvd
|
[
{
"state_after": "case h₁\nι : Type ?u.807675\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nS : Finset α\ng1 g2 : α → β\nh✝ : ∀ (a : α), a ∈ S → g1 a ∣ g2 a\nh : ∀ (a : α), a ∈ ∅ → g1 a ∣ g2 a\n⊢ Finset.prod ∅ g1 ∣ Finset.prod ∅ g2\n\ncase h₂\nι : Type ?u.807675\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf g : α → β\ninst✝ : CommMonoid β\nS : Finset α\ng1 g2 : α → β\nh✝ : ∀ (a : α), a ∈ S → g1 a ∣ g2 a\na : α\nT : Finset α\n_haS : a ∈ S\n_hTS : T ⊆ S\nhaT : ¬a ∈ T\nIH : (∀ (a : α), a ∈ T → g1 a ∣ g2 a) → Finset.prod T g1 ∣ Finset.prod T g2\nh : ∀ (a_1 : α), a_1 ∈ insert a T → g1 a_1 ∣ g2 a_1\n⊢ Finset.prod (insert a T) g1 ∣ Finset.prod (insert a T) g2",
"state_before": "ι : Type ?u.807675\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nS : Finset α\ng1 g2 : α → β\nh : ∀ (a : α), a ∈ S → g1 a ∣ g2 a\n⊢ Finset.prod S g1 ∣ Finset.prod S g2",
"tactic": "induction' S using Finset.induction_on' with a T _haS _hTS haT IH"
},
{
"state_after": "case h₂\nι : Type ?u.807675\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf g : α → β\ninst✝ : CommMonoid β\nS : Finset α\ng1 g2 : α → β\nh✝ : ∀ (a : α), a ∈ S → g1 a ∣ g2 a\na : α\nT : Finset α\n_haS : a ∈ S\n_hTS : T ⊆ S\nhaT : ¬a ∈ T\nIH : (∀ (a : α), a ∈ T → g1 a ∣ g2 a) → Finset.prod T g1 ∣ Finset.prod T g2\nh : ∀ (a_1 : α), a_1 ∈ insert a T → g1 a_1 ∣ g2 a_1\n⊢ g1 a * ∏ x in T, g1 x ∣ g2 a * ∏ x in T, g2 x",
"state_before": "case h₂\nι : Type ?u.807675\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf g : α → β\ninst✝ : CommMonoid β\nS : Finset α\ng1 g2 : α → β\nh✝ : ∀ (a : α), a ∈ S → g1 a ∣ g2 a\na : α\nT : Finset α\n_haS : a ∈ S\n_hTS : T ⊆ S\nhaT : ¬a ∈ T\nIH : (∀ (a : α), a ∈ T → g1 a ∣ g2 a) → Finset.prod T g1 ∣ Finset.prod T g2\nh : ∀ (a_1 : α), a_1 ∈ insert a T → g1 a_1 ∣ g2 a_1\n⊢ Finset.prod (insert a T) g1 ∣ Finset.prod (insert a T) g2",
"tactic": "rw [Finset.prod_insert haT, Finset.prod_insert haT]"
},
{
"state_after": "no goals",
"state_before": "case h₂\nι : Type ?u.807675\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf g : α → β\ninst✝ : CommMonoid β\nS : Finset α\ng1 g2 : α → β\nh✝ : ∀ (a : α), a ∈ S → g1 a ∣ g2 a\na : α\nT : Finset α\n_haS : a ∈ S\n_hTS : T ⊆ S\nhaT : ¬a ∈ T\nIH : (∀ (a : α), a ∈ T → g1 a ∣ g2 a) → Finset.prod T g1 ∣ Finset.prod T g2\nh : ∀ (a_1 : α), a_1 ∈ insert a T → g1 a_1 ∣ g2 a_1\n⊢ g1 a * ∏ x in T, g1 x ∣ g2 a * ∏ x in T, g2 x",
"tactic": "exact mul_dvd_mul (h a $ T.mem_insert_self a) (IH fun b hb ↦ h b $ Finset.mem_insert_of_mem hb)"
},
{
"state_after": "no goals",
"state_before": "case h₁\nι : Type ?u.807675\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na : α\nf g : α → β\ninst✝ : CommMonoid β\nS : Finset α\ng1 g2 : α → β\nh✝ : ∀ (a : α), a ∈ S → g1 a ∣ g2 a\nh : ∀ (a : α), a ∈ ∅ → g1 a ∣ g2 a\n⊢ Finset.prod ∅ g1 ∣ Finset.prod ∅ g2",
"tactic": "simp"
}
] |
[
1733,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1727,
1
] |
Mathlib/Algebra/Quaternion.lean
|
Quaternion.coe_commute
|
[] |
[
1062,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1061,
1
] |
Mathlib/Algebra/Algebra/Bilinear.lean
|
LinearMap.mulRight_injective
|
[
{
"state_after": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroDivisors A\nx : A\nhx : x ≠ 0\nthis : Nontrivial A := { exists_pair_ne := Exists.intro x (Exists.intro 0 hx) }\n⊢ Function.Injective ↑(mulRight R x)",
"state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroDivisors A\nx : A\nhx : x ≠ 0\n⊢ Function.Injective ↑(mulRight R x)",
"tactic": "letI : Nontrivial A := ⟨⟨x, 0, hx⟩⟩"
},
{
"state_after": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroDivisors A\nx : A\nhx : x ≠ 0\nthis✝ : Nontrivial A := { exists_pair_ne := Exists.intro x (Exists.intro 0 hx) }\nthis : IsDomain A := NoZeroDivisors.to_isDomain A\n⊢ Function.Injective ↑(mulRight R x)",
"state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroDivisors A\nx : A\nhx : x ≠ 0\nthis : Nontrivial A := { exists_pair_ne := Exists.intro x (Exists.intro 0 hx) }\n⊢ Function.Injective ↑(mulRight R x)",
"tactic": "letI := NoZeroDivisors.to_isDomain A"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nA : Type u_1\ninst✝³ : CommSemiring R\ninst✝² : Ring A\ninst✝¹ : Algebra R A\ninst✝ : NoZeroDivisors A\nx : A\nhx : x ≠ 0\nthis✝ : Nontrivial A := { exists_pair_ne := Exists.intro x (Exists.intro 0 hx) }\nthis : IsDomain A := NoZeroDivisors.to_isDomain A\n⊢ Function.Injective ↑(mulRight R x)",
"tactic": "exact mul_left_injective₀ hx"
}
] |
[
249,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
Subsemigroup.prod_mono
|
[] |
[
662,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
660,
1
] |
Mathlib/Analysis/Convex/Slope.lean
|
strictConcaveOn_iff_slope_strict_anti_adjacent
|
[] |
[
237,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
231,
1
] |
Mathlib/Combinatorics/Additive/PluenneckeRuzsa.lean
|
Finset.card_mul_mul_le_card_mul_mul_card_div
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\n⊢ card (A / C⁻¹) * card B ≤ card (A * B) * card (B * C⁻¹)",
"state_before": "α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\n⊢ card (A * C) * card B ≤ card (A * B) * card (B / C)",
"tactic": "rw [← div_inv_eq_mul, div_eq_mul_inv B]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CommGroup α\ninst✝ : DecidableEq α\nA✝ B✝ C✝ A B C : Finset α\n⊢ card (A / C⁻¹) * card B ≤ card (A * B) * card (B * C⁻¹)",
"tactic": "exact card_div_mul_le_card_mul_mul_card_mul _ _ _"
}
] |
[
87,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Topology/UniformSpace/Basic.lean
|
tendsto_right_nhds_uniformity
|
[] |
[
860,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
859,
1
] |
Mathlib/GroupTheory/OrderOfElement.lean
|
orderOf_eq_prime_pow
|
[
{
"state_after": "no goals",
"state_before": "G : Type u\nA : Type v\nx y : G\na b : A\nn m : ℕ\ninst✝¹ : Monoid G\ninst✝ : AddMonoid A\np : ℕ\nhp : Fact (Nat.Prime p)\nhnot : ¬x ^ p ^ n = 1\nhfin : x ^ p ^ (n + 1) = 1\n⊢ orderOf x = p ^ (n + 1)",
"tactic": "apply minimalPeriod_eq_prime_pow <;> rwa [isPeriodicPt_mul_iff_pow_eq_one]"
}
] |
[
461,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
459,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iInf_emptyset
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nβ₂ : Type ?u.112728\nγ : Type ?u.112731\nι : Sort ?u.112734\nι' : Sort ?u.112737\nκ : ι → Sort ?u.112742\nκ' : ι' → Sort ?u.112747\ninst✝ : CompleteLattice α\nf✝ g s t : ι → α\na b : α\nf : β → α\n⊢ (⨅ (x : β) (_ : x ∈ ∅), f x) = ⊤",
"tactic": "simp"
}
] |
[
1405,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1405,
1
] |
Mathlib/Probability/ProbabilityMassFunction/Basic.lean
|
Pmf.hasSum_coe_one
|
[] |
[
67,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Algebra/Lie/Matrix.lean
|
lieEquivMatrix'_apply
|
[] |
[
58,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
56,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean
|
one_div_le
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.77173\nα : Type u_1\nβ : Type ?u.77179\ninst✝ : LinearOrderedSemifield α\na b c d e : α\nm n : ℤ\nha : 0 < a\nhb : 0 < b\n⊢ 1 / a ≤ b ↔ 1 / b ≤ a",
"tactic": "simpa using inv_le ha hb"
}
] |
[
436,
100
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
436,
1
] |
Mathlib/CategoryTheory/EqToHom.lean
|
CategoryTheory.eq_conj_eqToHom
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\nX Y : C\nf : X ⟶ Y\n⊢ f = eqToHom (_ : X = X) ≫ f ≫ eqToHom (_ : Y = Y)",
"tactic": "simp only [Category.id_comp, eqToHom_refl, Category.comp_id]"
}
] |
[
295,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
294,
1
] |
Mathlib/Algebra/Regular/Basic.lean
|
IsRegular.ne_zero
|
[] |
[
255,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
|
NonUnitalSubsemiring.map_sup
|
[] |
[
741,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
739,
1
] |
Mathlib/Algebra/Order/Ring/Lemmas.lean
|
posMulReflectLT_iff_contravariant_pos
|
[
{
"state_after": "case inl\nα : Type u_1\na✝ b✝ c✝ d : α\ninst✝¹ : MulZeroClass α\ninst✝ : PartialOrder α\nh✝ : ContravariantClass { x // 0 < x } α (fun x y => ↑x * y) fun x x_1 => x < x_1\na : { x // 0 ≤ x }\nb c : α\nh : ↑a * b < ↑a * c\nha : 0 = ↑a\n⊢ b < c\n\ncase inr\nα : Type u_1\na✝ b✝ c✝ d : α\ninst✝¹ : MulZeroClass α\ninst✝ : PartialOrder α\nh✝ : ContravariantClass { x // 0 < x } α (fun x y => ↑x * y) fun x x_1 => x < x_1\na : { x // 0 ≤ x }\nb c : α\nh : ↑a * b < ↑a * c\nha : 0 < ↑a\n⊢ b < c",
"state_before": "α : Type u_1\na✝ b✝ c✝ d : α\ninst✝¹ : MulZeroClass α\ninst✝ : PartialOrder α\nh✝ : ContravariantClass { x // 0 < x } α (fun x y => ↑x * y) fun x x_1 => x < x_1\na : { x // 0 ≤ x }\nb c : α\nh : ↑a * b < ↑a * c\n⊢ b < c",
"tactic": "obtain ha | ha := a.prop.eq_or_lt"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\na✝ b✝ c✝ d : α\ninst✝¹ : MulZeroClass α\ninst✝ : PartialOrder α\nh✝ : ContravariantClass { x // 0 < x } α (fun x y => ↑x * y) fun x x_1 => x < x_1\na : { x // 0 ≤ x }\nb c : α\nh : ↑a * b < ↑a * c\nha : 0 = ↑a\n⊢ b < c",
"tactic": "simp [←ha] at h"
}
] |
[
473,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
467,
1
] |
Mathlib/LinearAlgebra/FreeModule/Finite/Rank.lean
|
FiniteDimensional.finrank_directSum
|
[
{
"state_after": "R : Type u\nM✝ : Type v\nN : Type w\ninst✝¹⁴ : Ring R\ninst✝¹³ : StrongRankCondition R\ninst✝¹² : AddCommGroup M✝\ninst✝¹¹ : Module R M✝\ninst✝¹⁰ : Module.Free R M✝\ninst✝⁹ : Module.Finite R M✝\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : Module R N\ninst✝⁶ : Module.Free R N\ninst✝⁵ : Module.Finite R N\nι : Type v\ninst✝⁴ : Fintype ι\nM : ι → Type w\ninst✝³ : (i : ι) → AddCommGroup (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : ∀ (i : ι), Module.Free R (M i)\ninst✝ : ∀ (i : ι), Module.Finite R (M i)\nthis : Nontrivial R := nontrivial_of_invariantBasisNumber R\n⊢ finrank R (⨁ (i : ι), M i) = ∑ i : ι, finrank R (M i)",
"state_before": "R : Type u\nM✝ : Type v\nN : Type w\ninst✝¹⁴ : Ring R\ninst✝¹³ : StrongRankCondition R\ninst✝¹² : AddCommGroup M✝\ninst✝¹¹ : Module R M✝\ninst✝¹⁰ : Module.Free R M✝\ninst✝⁹ : Module.Finite R M✝\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : Module R N\ninst✝⁶ : Module.Free R N\ninst✝⁵ : Module.Finite R N\nι : Type v\ninst✝⁴ : Fintype ι\nM : ι → Type w\ninst✝³ : (i : ι) → AddCommGroup (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : ∀ (i : ι), Module.Free R (M i)\ninst✝ : ∀ (i : ι), Module.Finite R (M i)\n⊢ finrank R (⨁ (i : ι), M i) = ∑ i : ι, finrank R (M i)",
"tactic": "letI := nontrivial_of_invariantBasisNumber R"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nM✝ : Type v\nN : Type w\ninst✝¹⁴ : Ring R\ninst✝¹³ : StrongRankCondition R\ninst✝¹² : AddCommGroup M✝\ninst✝¹¹ : Module R M✝\ninst✝¹⁰ : Module.Free R M✝\ninst✝⁹ : Module.Finite R M✝\ninst✝⁸ : AddCommGroup N\ninst✝⁷ : Module R N\ninst✝⁶ : Module.Free R N\ninst✝⁵ : Module.Finite R N\nι : Type v\ninst✝⁴ : Fintype ι\nM : ι → Type w\ninst✝³ : (i : ι) → AddCommGroup (M i)\ninst✝² : (i : ι) → Module R (M i)\ninst✝¹ : ∀ (i : ι), Module.Free R (M i)\ninst✝ : ∀ (i : ι), Module.Finite R (M i)\nthis : Nontrivial R := nontrivial_of_invariantBasisNumber R\n⊢ finrank R (⨁ (i : ι), M i) = ∑ i : ι, finrank R (M i)",
"tactic": "simp only [finrank, fun i => rank_eq_card_chooseBasisIndex R (M i), rank_directSum, ← mk_sigma,\n mk_toNat_eq_card, card_sigma]"
}
] |
[
114,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
109,
1
] |
Mathlib/Topology/FiberBundle/Trivialization.lean
|
Trivialization.source_inter_preimage_target_inter
|
[] |
[
377,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
375,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.ext_iff_degree_le
|
[] |
[
370,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
368,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.bsup_comp
|
[
{
"state_after": "α : Type ?u.362805\nβ : Type ?u.362808\nγ : Type ?u.362811\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no o' : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\nhf : ∀ {i j : Ordinal} (hi : i < o) (hj : j < o), i ≤ j → f i hi ≤ f j hj\ng : (a : Ordinal) → a < o' → Ordinal\nhg : blsub o' g = o\na : Ordinal\nha : a < o'\n⊢ g a ha < blsub o' g",
"state_before": "α : Type ?u.362805\nβ : Type ?u.362808\nγ : Type ?u.362811\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no o' : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\nhf : ∀ {i j : Ordinal} (hi : i < o) (hj : j < o), i ≤ j → f i hi ≤ f j hj\ng : (a : Ordinal) → a < o' → Ordinal\nhg : blsub o' g = o\na : Ordinal\nha : a < o'\n⊢ g a ha < o",
"tactic": "rw [← hg]"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.362805\nβ : Type ?u.362808\nγ : Type ?u.362811\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no o' : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\nhf : ∀ {i j : Ordinal} (hi : i < o) (hj : j < o), i ≤ j → f i hi ≤ f j hj\ng : (a : Ordinal) → a < o' → Ordinal\nhg : blsub o' g = o\na : Ordinal\nha : a < o'\n⊢ g a ha < blsub o' g",
"tactic": "apply lt_blsub"
},
{
"state_after": "case a\nα : Type ?u.362805\nβ : Type ?u.362808\nγ : Type ?u.362811\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no o' : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\nhf : ∀ {i j : Ordinal} (hi : i < o) (hj : j < o), i ≤ j → f i hi ≤ f j hj\ng : (a : Ordinal) → a < o' → Ordinal\nhg : blsub o' g = o\ni : Ordinal\nhi : i < o'\n⊢ f (g i hi) (_ : g i hi < o) ≤ bsup o f\n\ncase a\nα : Type ?u.362805\nβ : Type ?u.362808\nγ : Type ?u.362811\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no o' : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\nhf : ∀ {i j : Ordinal} (hi : i < o) (hj : j < o), i ≤ j → f i hi ≤ f j hj\ng : (a : Ordinal) → a < o' → Ordinal\nhg : blsub o' g = o\ni : Ordinal\nhi : i < o\n⊢ f i hi ≤ bsup o' fun a ha => f (g a ha) (_ : g a ha < o)",
"state_before": "α : Type ?u.362805\nβ : Type ?u.362808\nγ : Type ?u.362811\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no o' : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\nhf : ∀ {i j : Ordinal} (hi : i < o) (hj : j < o), i ≤ j → f i hi ≤ f j hj\ng : (a : Ordinal) → a < o' → Ordinal\nhg : blsub o' g = o\n⊢ (bsup o' fun a ha => f (g a ha) (_ : g a ha < o)) = bsup o f",
"tactic": "apply le_antisymm <;> refine' bsup_le fun i hi => _"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type ?u.362805\nβ : Type ?u.362808\nγ : Type ?u.362811\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no o' : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\nhf : ∀ {i j : Ordinal} (hi : i < o) (hj : j < o), i ≤ j → f i hi ≤ f j hj\ng : (a : Ordinal) → a < o' → Ordinal\nhg : blsub o' g = o\ni : Ordinal\nhi : i < o'\n⊢ f (g i hi) (_ : g i hi < o) ≤ bsup o f",
"tactic": "apply le_bsup"
},
{
"state_after": "case a\nα : Type ?u.362805\nβ : Type ?u.362808\nγ : Type ?u.362811\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no o' : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\nhf : ∀ {i j : Ordinal} (hi : i < o) (hj : j < o), i ≤ j → f i hi ≤ f j hj\ng : (a : Ordinal) → a < o' → Ordinal\nhg : blsub o' g = o\ni : Ordinal\nhi✝ : i < o\nhi : ∃ i_1 hi, i ≤ g i_1 hi\n⊢ f i hi✝ ≤ bsup o' fun a ha => f (g a ha) (_ : g a ha < o)",
"state_before": "case a\nα : Type ?u.362805\nβ : Type ?u.362808\nγ : Type ?u.362811\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no o' : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\nhf : ∀ {i j : Ordinal} (hi : i < o) (hj : j < o), i ≤ j → f i hi ≤ f j hj\ng : (a : Ordinal) → a < o' → Ordinal\nhg : blsub o' g = o\ni : Ordinal\nhi : i < o\n⊢ f i hi ≤ bsup o' fun a ha => f (g a ha) (_ : g a ha < o)",
"tactic": "rw [← hg, lt_blsub_iff] at hi"
},
{
"state_after": "case a.intro.intro\nα : Type ?u.362805\nβ : Type ?u.362808\nγ : Type ?u.362811\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no o' : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\nhf : ∀ {i j : Ordinal} (hi : i < o) (hj : j < o), i ≤ j → f i hi ≤ f j hj\ng : (a : Ordinal) → a < o' → Ordinal\nhg : blsub o' g = o\ni : Ordinal\nhi : i < o\nj : Ordinal\nhj : j < o'\nhj' : i ≤ g j hj\n⊢ f i hi ≤ bsup o' fun a ha => f (g a ha) (_ : g a ha < o)",
"state_before": "case a\nα : Type ?u.362805\nβ : Type ?u.362808\nγ : Type ?u.362811\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no o' : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\nhf : ∀ {i j : Ordinal} (hi : i < o) (hj : j < o), i ≤ j → f i hi ≤ f j hj\ng : (a : Ordinal) → a < o' → Ordinal\nhg : blsub o' g = o\ni : Ordinal\nhi✝ : i < o\nhi : ∃ i_1 hi, i ≤ g i_1 hi\n⊢ f i hi✝ ≤ bsup o' fun a ha => f (g a ha) (_ : g a ha < o)",
"tactic": "rcases hi with ⟨j, hj, hj'⟩"
},
{
"state_after": "no goals",
"state_before": "case a.intro.intro\nα : Type ?u.362805\nβ : Type ?u.362808\nγ : Type ?u.362811\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no o' : Ordinal\nf : (a : Ordinal) → a < o → Ordinal\nhf : ∀ {i j : Ordinal} (hi : i < o) (hj : j < o), i ≤ j → f i hi ≤ f j hj\ng : (a : Ordinal) → a < o' → Ordinal\nhg : blsub o' g = o\ni : Ordinal\nhi : i < o\nj : Ordinal\nhj : j < o'\nhj' : i ≤ g j hj\n⊢ f i hi ≤ bsup o' fun a ha => f (g a ha) (_ : g a ha < o)",
"tactic": "exact (hf _ _ hj').trans (le_bsup _ _ _)"
}
] |
[
1951,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1943,
1
] |
Mathlib/Topology/VectorBundle/Basic.lean
|
Trivialization.linearMapAt_apply
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_4\nB : Type u_1\nF : Type u_2\nE : B → Type u_3\ninst✝⁸ : Semiring R\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace (TotalSpace E)\ne✝ : Trivialization F TotalSpace.proj\nx : TotalSpace E\nb✝ : B\ny✝ : E b✝\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module R F\ninst✝² : (x : B) → AddCommMonoid (E x)\ninst✝¹ : (x : B) → Module R (E x)\ne : Trivialization F TotalSpace.proj\ninst✝ : Trivialization.IsLinear R e\nb : B\ny : E b\n⊢ ↑(Trivialization.linearMapAt R e b) y = if b ∈ e.baseSet then (↑e (totalSpaceMk b y)).snd else 0",
"tactic": "rw [coe_linearMapAt]"
}
] |
[
249,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
247,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.mul_zero
|
[
{
"state_after": "case a.h\nl : Type ?u.264124\nm : Type u_2\nn : Type u_1\no : Type u_3\nm' : o → Type ?u.264138\nn' : o → Type ?u.264143\nR : Type ?u.264146\nS : Type ?u.264149\nα : Type v\nβ : Type w\nγ : Type ?u.264156\ninst✝¹ : NonUnitalNonAssocSemiring α\ninst✝ : Fintype n\nM : Matrix m n α\ni✝ : m\nx✝ : o\n⊢ (M ⬝ 0) i✝ x✝ = OfNat.ofNat 0 i✝ x✝",
"state_before": "l : Type ?u.264124\nm : Type u_2\nn : Type u_1\no : Type u_3\nm' : o → Type ?u.264138\nn' : o → Type ?u.264143\nR : Type ?u.264146\nS : Type ?u.264149\nα : Type v\nβ : Type w\nγ : Type ?u.264156\ninst✝¹ : NonUnitalNonAssocSemiring α\ninst✝ : Fintype n\nM : Matrix m n α\n⊢ M ⬝ 0 = 0",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a.h\nl : Type ?u.264124\nm : Type u_2\nn : Type u_1\no : Type u_3\nm' : o → Type ?u.264138\nn' : o → Type ?u.264143\nR : Type ?u.264146\nS : Type ?u.264149\nα : Type v\nβ : Type w\nγ : Type ?u.264156\ninst✝¹ : NonUnitalNonAssocSemiring α\ninst✝ : Fintype n\nM : Matrix m n α\ni✝ : m\nx✝ : o\n⊢ (M ⬝ 0) i✝ x✝ = OfNat.ofNat 0 i✝ x✝",
"tactic": "apply dotProduct_zero"
}
] |
[
991,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
989,
11
] |
Std/Data/Int/DivMod.lean
|
Int.emod_eq_zero_of_dvd
|
[] |
[
682,
40
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
681,
1
] |
Mathlib/FieldTheory/Subfield.lean
|
Subfield.ext
|
[] |
[
205,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
204,
1
] |
Mathlib/CategoryTheory/Abelian/NonPreadditive.lean
|
CategoryTheory.NonPreadditiveAbelian.neg_neg
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na : X ⟶ Y\n⊢ 0 - (0 - a) = a",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na : X ⟶ Y\n⊢ - -a = a",
"tactic": "rw [neg_def, neg_def]"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na : X ⟶ Y\n⊢ a - a - (0 - a) = a",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na : X ⟶ Y\n⊢ 0 - (0 - a) = a",
"tactic": "conv_lhs =>\n congr; rw [← sub_self a]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : NonPreadditiveAbelian C\nX Y : C\na : X ⟶ Y\n⊢ a - a - (0 - a) = a",
"tactic": "rw [sub_sub_sub, sub_zero, sub_self, sub_zero]"
}
] |
[
386,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
382,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
measurable_nnnorm
|
[] |
[
2049,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2048,
1
] |
Mathlib/SetTheory/Ordinal/Basic.lean
|
Ordinal.lt_one_iff_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.180662\nβ : Type ?u.180665\nγ : Type ?u.180668\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\na : Ordinal\n⊢ a < 1 ↔ a = 0",
"tactic": "simpa using @lt_succ_bot_iff _ _ _ a _ _"
}
] |
[
1093,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1092,
1
] |
Mathlib/Data/Analysis/Topology.lean
|
Ctop.Realizer.isOpen
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.19471\nσ : Type ?u.19474\nτ : Type ?u.19477\ninst✝ : TopologicalSpace α\nF : Realizer α\ns : F.σ\na : α\nm : a ∈ f F.F s\n⊢ 𝓝 a ≤ 𝓟 (f F.F s)",
"tactic": "simpa using F.mem_nhds.2 ⟨s, m, Subset.refl _⟩"
}
] |
[
161,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
11
] |
Std/Logic.lean
|
Decidable.imp_iff_or_not
|
[] |
[
559,
41
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
558,
1
] |
Mathlib/Order/Hom/Basic.lean
|
OrderHom.comp_prod_comp_same
|
[] |
[
418,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
416,
1
] |
Mathlib/Topology/Homotopy/Equiv.lean
|
Homeomorph.coe_toHomotopyEquiv
|
[] |
[
94,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
93,
1
] |
Mathlib/Topology/MetricSpace/Infsep.lean
|
Set.nontrivial_of_einfsep_ne_top
|
[] |
[
97,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/Data/List/Rotate.lean
|
List.rotate_eq_nil_iff
|
[
{
"state_after": "case zero\nα : Type u\nl✝ l : List α\n⊢ rotate l zero = [] ↔ l = []\n\ncase succ\nα : Type u\nl✝ : List α\nn : ℕ\nhn : ∀ {l : List α}, rotate l n = [] ↔ l = []\nl : List α\n⊢ rotate l (succ n) = [] ↔ l = []",
"state_before": "α : Type u\nl : List α\nn : ℕ\n⊢ rotate l n = [] ↔ l = []",
"tactic": "induction' n with n hn generalizing l"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u\nl✝ l : List α\n⊢ rotate l zero = [] ↔ l = []",
"tactic": "simp"
},
{
"state_after": "case succ.nil\nα : Type u\nl : List α\nn : ℕ\nhn : ∀ {l : List α}, rotate l n = [] ↔ l = []\n⊢ rotate [] (succ n) = [] ↔ [] = []\n\ncase succ.cons\nα : Type u\nl : List α\nn : ℕ\nhn : ∀ {l : List α}, rotate l n = [] ↔ l = []\nhd : α\ntl : List α\n⊢ rotate (hd :: tl) (succ n) = [] ↔ hd :: tl = []",
"state_before": "case succ\nα : Type u\nl✝ : List α\nn : ℕ\nhn : ∀ {l : List α}, rotate l n = [] ↔ l = []\nl : List α\n⊢ rotate l (succ n) = [] ↔ l = []",
"tactic": "cases' l with hd tl"
},
{
"state_after": "no goals",
"state_before": "case succ.nil\nα : Type u\nl : List α\nn : ℕ\nhn : ∀ {l : List α}, rotate l n = [] ↔ l = []\n⊢ rotate [] (succ n) = [] ↔ [] = []",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case succ.cons\nα : Type u\nl : List α\nn : ℕ\nhn : ∀ {l : List α}, rotate l n = [] ↔ l = []\nhd : α\ntl : List α\n⊢ rotate (hd :: tl) (succ n) = [] ↔ hd :: tl = []",
"tactic": "simp [rotate_cons_succ, hn]"
}
] |
[
207,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
Mathlib/GroupTheory/Subgroup/Finite.lean
|
Subgroup.noncommProd_mem
|
[] |
[
87,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
1
] |
Mathlib/LinearAlgebra/BilinearForm.lean
|
BilinForm.nondegenerate_iff_ker_eq_bot
|
[
{
"state_after": "R : Type ?u.1436438\nM : Type ?u.1436441\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.1436477\nM₁ : Type ?u.1436480\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type ?u.1437279\nM₃ : Type ?u.1437282\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.1437870\nK : Type ?u.1437873\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1439087\ninst✝¹ : AddCommMonoid M₂'\ninst✝ : Module R₂ M₂'\nB : BilinForm R₂ M₂\n⊢ Nondegenerate B ↔ ∀ (m : M₂), ↑(↑toLin B) m = 0 → m = 0",
"state_before": "R : Type ?u.1436438\nM : Type ?u.1436441\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.1436477\nM₁ : Type ?u.1436480\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type ?u.1437279\nM₃ : Type ?u.1437282\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.1437870\nK : Type ?u.1437873\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1439087\ninst✝¹ : AddCommMonoid M₂'\ninst✝ : Module R₂ M₂'\nB : BilinForm R₂ M₂\n⊢ Nondegenerate B ↔ LinearMap.ker (↑toLin B) = ⊥",
"tactic": "rw [LinearMap.ker_eq_bot']"
},
{
"state_after": "case mp\nR : Type ?u.1436438\nM : Type ?u.1436441\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.1436477\nM₁ : Type ?u.1436480\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type ?u.1437279\nM₃ : Type ?u.1437282\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.1437870\nK : Type ?u.1437873\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1439087\ninst✝¹ : AddCommMonoid M₂'\ninst✝ : Module R₂ M₂'\nB : BilinForm R₂ M₂\nh : Nondegenerate B\n⊢ ∀ (m : M₂), ↑(↑toLin B) m = 0 → m = 0\n\ncase mpr\nR : Type ?u.1436438\nM : Type ?u.1436441\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.1436477\nM₁ : Type ?u.1436480\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type ?u.1437279\nM₃ : Type ?u.1437282\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.1437870\nK : Type ?u.1437873\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1439087\ninst✝¹ : AddCommMonoid M₂'\ninst✝ : Module R₂ M₂'\nB : BilinForm R₂ M₂\nh : ∀ (m : M₂), ↑(↑toLin B) m = 0 → m = 0\n⊢ Nondegenerate B",
"state_before": "R : Type ?u.1436438\nM : Type ?u.1436441\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.1436477\nM₁ : Type ?u.1436480\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type ?u.1437279\nM₃ : Type ?u.1437282\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.1437870\nK : Type ?u.1437873\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1439087\ninst✝¹ : AddCommMonoid M₂'\ninst✝ : Module R₂ M₂'\nB : BilinForm R₂ M₂\n⊢ Nondegenerate B ↔ ∀ (m : M₂), ↑(↑toLin B) m = 0 → m = 0",
"tactic": "constructor <;> intro h"
},
{
"state_after": "case mp\nR : Type ?u.1436438\nM : Type ?u.1436441\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.1436477\nM₁ : Type ?u.1436480\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type ?u.1437279\nM₃ : Type ?u.1437282\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.1437870\nK : Type ?u.1437873\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1439087\ninst✝¹ : AddCommMonoid M₂'\ninst✝ : Module R₂ M₂'\nB : BilinForm R₂ M₂\nh : Nondegenerate B\nm : M₂\nhm : ↑(↑toLin B) m = 0\nx : M₂\n⊢ bilin B m x = 0",
"state_before": "case mp\nR : Type ?u.1436438\nM : Type ?u.1436441\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.1436477\nM₁ : Type ?u.1436480\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type ?u.1437279\nM₃ : Type ?u.1437282\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.1437870\nK : Type ?u.1437873\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1439087\ninst✝¹ : AddCommMonoid M₂'\ninst✝ : Module R₂ M₂'\nB : BilinForm R₂ M₂\nh : Nondegenerate B\n⊢ ∀ (m : M₂), ↑(↑toLin B) m = 0 → m = 0",
"tactic": "refine' fun m hm => h _ fun x => _"
},
{
"state_after": "case mp\nR : Type ?u.1436438\nM : Type ?u.1436441\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.1436477\nM₁ : Type ?u.1436480\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type ?u.1437279\nM₃ : Type ?u.1437282\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.1437870\nK : Type ?u.1437873\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1439087\ninst✝¹ : AddCommMonoid M₂'\ninst✝ : Module R₂ M₂'\nB : BilinForm R₂ M₂\nh : Nondegenerate B\nm : M₂\nhm : ↑(↑toLin B) m = 0\nx : M₂\n⊢ ↑0 x = 0",
"state_before": "case mp\nR : Type ?u.1436438\nM : Type ?u.1436441\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.1436477\nM₁ : Type ?u.1436480\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type ?u.1437279\nM₃ : Type ?u.1437282\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.1437870\nK : Type ?u.1437873\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1439087\ninst✝¹ : AddCommMonoid M₂'\ninst✝ : Module R₂ M₂'\nB : BilinForm R₂ M₂\nh : Nondegenerate B\nm : M₂\nhm : ↑(↑toLin B) m = 0\nx : M₂\n⊢ bilin B m x = 0",
"tactic": "rw [← toLin_apply, hm]"
},
{
"state_after": "no goals",
"state_before": "case mp\nR : Type ?u.1436438\nM : Type ?u.1436441\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.1436477\nM₁ : Type ?u.1436480\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type ?u.1437279\nM₃ : Type ?u.1437282\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.1437870\nK : Type ?u.1437873\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1439087\ninst✝¹ : AddCommMonoid M₂'\ninst✝ : Module R₂ M₂'\nB : BilinForm R₂ M₂\nh : Nondegenerate B\nm : M₂\nhm : ↑(↑toLin B) m = 0\nx : M₂\n⊢ ↑0 x = 0",
"tactic": "rfl"
},
{
"state_after": "case mpr\nR : Type ?u.1436438\nM : Type ?u.1436441\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.1436477\nM₁ : Type ?u.1436480\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type ?u.1437279\nM₃ : Type ?u.1437282\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.1437870\nK : Type ?u.1437873\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1439087\ninst✝¹ : AddCommMonoid M₂'\ninst✝ : Module R₂ M₂'\nB : BilinForm R₂ M₂\nh : ∀ (m : M₂), ↑(↑toLin B) m = 0 → m = 0\nm : M₂\nhm : ∀ (n : M₂), bilin B m n = 0\n⊢ m = 0",
"state_before": "case mpr\nR : Type ?u.1436438\nM : Type ?u.1436441\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.1436477\nM₁ : Type ?u.1436480\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type ?u.1437279\nM₃ : Type ?u.1437282\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.1437870\nK : Type ?u.1437873\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1439087\ninst✝¹ : AddCommMonoid M₂'\ninst✝ : Module R₂ M₂'\nB : BilinForm R₂ M₂\nh : ∀ (m : M₂), ↑(↑toLin B) m = 0 → m = 0\n⊢ Nondegenerate B",
"tactic": "intro m hm"
},
{
"state_after": "case mpr.a\nR : Type ?u.1436438\nM : Type ?u.1436441\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.1436477\nM₁ : Type ?u.1436480\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type ?u.1437279\nM₃ : Type ?u.1437282\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.1437870\nK : Type ?u.1437873\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1439087\ninst✝¹ : AddCommMonoid M₂'\ninst✝ : Module R₂ M₂'\nB : BilinForm R₂ M₂\nh : ∀ (m : M₂), ↑(↑toLin B) m = 0 → m = 0\nm : M₂\nhm : ∀ (n : M₂), bilin B m n = 0\n⊢ ↑(↑toLin B) m = 0",
"state_before": "case mpr\nR : Type ?u.1436438\nM : Type ?u.1436441\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.1436477\nM₁ : Type ?u.1436480\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type ?u.1437279\nM₃ : Type ?u.1437282\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.1437870\nK : Type ?u.1437873\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1439087\ninst✝¹ : AddCommMonoid M₂'\ninst✝ : Module R₂ M₂'\nB : BilinForm R₂ M₂\nh : ∀ (m : M₂), ↑(↑toLin B) m = 0 → m = 0\nm : M₂\nhm : ∀ (n : M₂), bilin B m n = 0\n⊢ m = 0",
"tactic": "apply h"
},
{
"state_after": "case mpr.a.h\nR : Type ?u.1436438\nM : Type ?u.1436441\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.1436477\nM₁ : Type ?u.1436480\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type ?u.1437279\nM₃ : Type ?u.1437282\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.1437870\nK : Type ?u.1437873\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1439087\ninst✝¹ : AddCommMonoid M₂'\ninst✝ : Module R₂ M₂'\nB : BilinForm R₂ M₂\nh : ∀ (m : M₂), ↑(↑toLin B) m = 0 → m = 0\nm : M₂\nhm : ∀ (n : M₂), bilin B m n = 0\nx : M₂\n⊢ ↑(↑(↑toLin B) m) x = ↑0 x",
"state_before": "case mpr.a\nR : Type ?u.1436438\nM : Type ?u.1436441\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.1436477\nM₁ : Type ?u.1436480\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type ?u.1437279\nM₃ : Type ?u.1437282\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.1437870\nK : Type ?u.1437873\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1439087\ninst✝¹ : AddCommMonoid M₂'\ninst✝ : Module R₂ M₂'\nB : BilinForm R₂ M₂\nh : ∀ (m : M₂), ↑(↑toLin B) m = 0 → m = 0\nm : M₂\nhm : ∀ (n : M₂), bilin B m n = 0\n⊢ ↑(↑toLin B) m = 0",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case mpr.a.h\nR : Type ?u.1436438\nM : Type ?u.1436441\ninst✝¹⁶ : Semiring R\ninst✝¹⁵ : AddCommMonoid M\ninst✝¹⁴ : Module R M\nR₁ : Type ?u.1436477\nM₁ : Type ?u.1436480\ninst✝¹³ : Ring R₁\ninst✝¹² : AddCommGroup M₁\ninst✝¹¹ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹⁰ : CommSemiring R₂\ninst✝⁹ : AddCommMonoid M₂\ninst✝⁸ : Module R₂ M₂\nR₃ : Type ?u.1437279\nM₃ : Type ?u.1437282\ninst✝⁷ : CommRing R₃\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R₃ M₃\nV : Type ?u.1437870\nK : Type ?u.1437873\ninst✝⁴ : Field K\ninst✝³ : AddCommGroup V\ninst✝² : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nM₂' : Type ?u.1439087\ninst✝¹ : AddCommMonoid M₂'\ninst✝ : Module R₂ M₂'\nB : BilinForm R₂ M₂\nh : ∀ (m : M₂), ↑(↑toLin B) m = 0 → m = 0\nm : M₂\nhm : ∀ (n : M₂), bilin B m n = 0\nx : M₂\n⊢ ↑(↑(↑toLin B) m) x = ↑0 x",
"tactic": "exact hm x"
}
] |
[
1323,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1313,
1
] |
Mathlib/GroupTheory/Coset.lean
|
Subgroup.card_dvd_of_le
|
[] |
[
820,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
819,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieSubalgebra.topEquiv_apply
|
[] |
[
1272,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1271,
1
] |
Mathlib/Computability/Language.lean
|
Language.iSup_add
|
[] |
[
229,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iInter_plift_up
|
[] |
[
238,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/Order/BooleanAlgebra.lean
|
disjoint_sdiff_self_right
|
[] |
[
217,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/Order/Bounds/Basic.lean
|
IsGreatest.isLeast_image2
|
[] |
[
1496,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1494,
1
] |
Mathlib/FieldTheory/Adjoin.lean
|
IntermediateField.finiteDimensional_iSup_of_finset'
|
[
{
"state_after": "K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nE1 E2 : IntermediateField K L\nι : Type u_1\nf : ι → IntermediateField K L\ns : Finset ι\nh : ∀ (i : ι), i ∈ s → FiniteDimensional K { x // x ∈ f i }\nthis : ∀ (i : { i // i ∈ s }), FiniteDimensional K { x // x ∈ f ↑i }\n⊢ FiniteDimensional K { x // x ∈ ⨆ (i : ι) (_ : i ∈ s), f i }",
"state_before": "K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nE1 E2 : IntermediateField K L\nι : Type u_1\nf : ι → IntermediateField K L\ns : Finset ι\nh : ∀ (i : ι), i ∈ s → FiniteDimensional K { x // x ∈ f i }\n⊢ FiniteDimensional K { x // x ∈ ⨆ (i : ι) (_ : i ∈ s), f i }",
"tactic": "haveI : ∀ i : { i // i ∈ s }, FiniteDimensional K (f i) := fun i => h i i.2"
},
{
"state_after": "K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nE1 E2 : IntermediateField K L\nι : Type u_1\nf : ι → IntermediateField K L\ns : Finset ι\nh : ∀ (i : ι), i ∈ s → FiniteDimensional K { x // x ∈ f i }\nthis✝ : ∀ (i : { i // i ∈ s }), FiniteDimensional K { x // x ∈ f ↑i }\nthis : (⨆ (i : ι) (_ : i ∈ s), f i) = ⨆ (i : { i // i ∈ s }), f ↑i\n⊢ FiniteDimensional K { x // x ∈ ⨆ (i : ι) (_ : i ∈ s), f i }",
"state_before": "K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nE1 E2 : IntermediateField K L\nι : Type u_1\nf : ι → IntermediateField K L\ns : Finset ι\nh : ∀ (i : ι), i ∈ s → FiniteDimensional K { x // x ∈ f i }\nthis : ∀ (i : { i // i ∈ s }), FiniteDimensional K { x // x ∈ f ↑i }\n⊢ FiniteDimensional K { x // x ∈ ⨆ (i : ι) (_ : i ∈ s), f i }",
"tactic": "have : (⨆ i ∈ s, f i) = ⨆ i : { i // i ∈ s }, f i :=\n le_antisymm (iSup_le fun i => iSup_le fun h => le_iSup (fun i : { i // i ∈ s } => f i) ⟨i, h⟩)\n (iSup_le fun i => le_iSup_of_le i (le_iSup_of_le i.2 le_rfl))"
},
{
"state_after": "no goals",
"state_before": "K : Type u_2\nL : Type u_3\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nE1 E2 : IntermediateField K L\nι : Type u_1\nf : ι → IntermediateField K L\ns : Finset ι\nh : ∀ (i : ι), i ∈ s → FiniteDimensional K { x // x ∈ f i }\nthis✝ : ∀ (i : { i // i ∈ s }), FiniteDimensional K { x // x ∈ f ↑i }\nthis : (⨆ (i : ι) (_ : i ∈ s), f i) = ⨆ (i : { i // i ∈ s }), f ↑i\n⊢ FiniteDimensional K { x // x ∈ ⨆ (i : ι) (_ : i ∈ s), f i }",
"tactic": "exact this.symm ▸ IntermediateField.finiteDimensional_iSup_of_finite"
}
] |
[
1250,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1241,
1
] |
Mathlib/Analysis/Normed/Field/InfiniteSum.lean
|
tsum_mul_tsum_eq_tsum_sum_antidiagonal_of_summable_norm
|
[] |
[
99,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.IsSt.unique
|
[
{
"state_after": "case intro\nr s : ℝ\nf : ℕ → ℝ\nhr : IsSt (ofSeq f) r\nhs : IsSt (ofSeq f) s\n⊢ r = s",
"state_before": "x : ℝ*\nr s : ℝ\nhr : IsSt x r\nhs : IsSt x s\n⊢ r = s",
"tactic": "rcases ofSeq_surjective x with ⟨f, rfl⟩"
},
{
"state_after": "case intro\nr s : ℝ\nf : ℕ → ℝ\nhr : Tendsto f (↑(hyperfilter ℕ)) (𝓝 r)\nhs : Tendsto f (↑(hyperfilter ℕ)) (𝓝 s)\n⊢ r = s",
"state_before": "case intro\nr s : ℝ\nf : ℕ → ℝ\nhr : IsSt (ofSeq f) r\nhs : IsSt (ofSeq f) s\n⊢ r = s",
"tactic": "rw [isSt_ofSeq_iff_tendsto] at hr hs"
},
{
"state_after": "no goals",
"state_before": "case intro\nr s : ℝ\nf : ℕ → ℝ\nhr : Tendsto f (↑(hyperfilter ℕ)) (𝓝 r)\nhs : Tendsto f (↑(hyperfilter ℕ)) (𝓝 s)\n⊢ r = s",
"tactic": "exact tendsto_nhds_unique hr hs"
}
] |
[
279,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
276,
1
] |
Mathlib/Topology/OmegaCompletePartialOrder.lean
|
notBelow_isOpen
|
[
{
"state_after": "α : Type u_1\ninst✝ : OmegaCompletePartialOrder α\ny : Scott α\nh : Monotone (notBelow y)\n⊢ IsOpen (notBelow y)",
"state_before": "α : Type u_1\ninst✝ : OmegaCompletePartialOrder α\ny : Scott α\n⊢ IsOpen (notBelow y)",
"tactic": "have h : Monotone (notBelow y) := by\n intro x y' h\n simp only [notBelow, setOf, le_Prop_eq]\n intro h₀ h₁\n apply h₀ (le_trans h h₁)"
},
{
"state_after": "α : Type u_1\ninst✝ : OmegaCompletePartialOrder α\ny : Scott α\nh : Monotone (notBelow y)\n⊢ OmegaCompletePartialOrder.Continuous { toFun := fun x => x ∈ notBelow y, monotone' := h }",
"state_before": "α : Type u_1\ninst✝ : OmegaCompletePartialOrder α\ny : Scott α\nh : Monotone (notBelow y)\n⊢ IsOpen (notBelow y)",
"tactic": "exists h"
},
{
"state_after": "α : Type u_1\ninst✝ : OmegaCompletePartialOrder α\ny : Scott α\nh : Monotone (notBelow y)\nc : Chain α\n⊢ ↑{ toFun := fun x => x ∈ notBelow y, monotone' := h } (ωSup c) =\n ωSup (Chain.map c { toFun := fun x => x ∈ notBelow y, monotone' := h })",
"state_before": "α : Type u_1\ninst✝ : OmegaCompletePartialOrder α\ny : Scott α\nh : Monotone (notBelow y)\n⊢ OmegaCompletePartialOrder.Continuous { toFun := fun x => x ∈ notBelow y, monotone' := h }",
"tactic": "rintro c"
},
{
"state_after": "case H\nα : Type u_1\ninst✝ : OmegaCompletePartialOrder α\ny : Scott α\nh : Monotone (notBelow y)\nc : Chain α\n⊢ ∀ (c_1 : Prop),\n ↑{ toFun := fun x => x ∈ notBelow y, monotone' := h } (ωSup c) ≤ c_1 ↔\n ωSup (Chain.map c { toFun := fun x => x ∈ notBelow y, monotone' := h }) ≤ c_1",
"state_before": "α : Type u_1\ninst✝ : OmegaCompletePartialOrder α\ny : Scott α\nh : Monotone (notBelow y)\nc : Chain α\n⊢ ↑{ toFun := fun x => x ∈ notBelow y, monotone' := h } (ωSup c) =\n ωSup (Chain.map c { toFun := fun x => x ∈ notBelow y, monotone' := h })",
"tactic": "apply eq_of_forall_ge_iff"
},
{
"state_after": "case H\nα : Type u_1\ninst✝ : OmegaCompletePartialOrder α\ny : Scott α\nh : Monotone (notBelow y)\nc : Chain α\nz : Prop\n⊢ ↑{ toFun := fun x => x ∈ notBelow y, monotone' := h } (ωSup c) ≤ z ↔\n ωSup (Chain.map c { toFun := fun x => x ∈ notBelow y, monotone' := h }) ≤ z",
"state_before": "case H\nα : Type u_1\ninst✝ : OmegaCompletePartialOrder α\ny : Scott α\nh : Monotone (notBelow y)\nc : Chain α\n⊢ ∀ (c_1 : Prop),\n ↑{ toFun := fun x => x ∈ notBelow y, monotone' := h } (ωSup c) ≤ c_1 ↔\n ωSup (Chain.map c { toFun := fun x => x ∈ notBelow y, monotone' := h }) ≤ c_1",
"tactic": "intro z"
},
{
"state_after": "case H\nα : Type u_1\ninst✝ : OmegaCompletePartialOrder α\ny : Scott α\nh : Monotone (notBelow y)\nc : Chain α\nz : Prop\n⊢ ↑{ toFun := fun x => x ∈ notBelow y, monotone' := h } (ωSup c) ≤ z ↔\n ∀ (i : ℕ), ↑(Chain.map c { toFun := fun x => x ∈ notBelow y, monotone' := h }) i ≤ z",
"state_before": "case H\nα : Type u_1\ninst✝ : OmegaCompletePartialOrder α\ny : Scott α\nh : Monotone (notBelow y)\nc : Chain α\nz : Prop\n⊢ ↑{ toFun := fun x => x ∈ notBelow y, monotone' := h } (ωSup c) ≤ z ↔\n ωSup (Chain.map c { toFun := fun x => x ∈ notBelow y, monotone' := h }) ≤ z",
"tactic": "rw [ωSup_le_iff]"
},
{
"state_after": "no goals",
"state_before": "case H\nα : Type u_1\ninst✝ : OmegaCompletePartialOrder α\ny : Scott α\nh : Monotone (notBelow y)\nc : Chain α\nz : Prop\n⊢ ↑{ toFun := fun x => x ∈ notBelow y, monotone' := h } (ωSup c) ≤ z ↔\n ∀ (i : ℕ), ↑(Chain.map c { toFun := fun x => x ∈ notBelow y, monotone' := h }) i ≤ z",
"tactic": "simp only [ωSup_le_iff, notBelow, mem_setOf_eq, le_Prop_eq, OrderHom.coe_fun_mk, Chain.map_coe,\n Function.comp_apply, exists_imp, not_forall]"
},
{
"state_after": "α : Type u_1\ninst✝ : OmegaCompletePartialOrder α\ny x y' : Scott α\nh : x ≤ y'\n⊢ notBelow y x ≤ notBelow y y'",
"state_before": "α : Type u_1\ninst✝ : OmegaCompletePartialOrder α\ny : Scott α\n⊢ Monotone (notBelow y)",
"tactic": "intro x y' h"
},
{
"state_after": "α : Type u_1\ninst✝ : OmegaCompletePartialOrder α\ny x y' : Scott α\nh : x ≤ y'\n⊢ ¬x ≤ y → ¬y' ≤ y",
"state_before": "α : Type u_1\ninst✝ : OmegaCompletePartialOrder α\ny x y' : Scott α\nh : x ≤ y'\n⊢ notBelow y x ≤ notBelow y y'",
"tactic": "simp only [notBelow, setOf, le_Prop_eq]"
},
{
"state_after": "α : Type u_1\ninst✝ : OmegaCompletePartialOrder α\ny x y' : Scott α\nh : x ≤ y'\nh₀ : ¬x ≤ y\nh₁ : y' ≤ y\n⊢ False",
"state_before": "α : Type u_1\ninst✝ : OmegaCompletePartialOrder α\ny x y' : Scott α\nh : x ≤ y'\n⊢ ¬x ≤ y → ¬y' ≤ y",
"tactic": "intro h₀ h₁"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OmegaCompletePartialOrder α\ny x y' : Scott α\nh : x ≤ y'\nh₀ : ¬x ≤ y\nh₁ : y' ≤ y\n⊢ False",
"tactic": "apply h₀ (le_trans h h₁)"
}
] |
[
110,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
98,
1
] |
Mathlib/RingTheory/Ideal/Basic.lean
|
exists_max_ideal_of_mem_nonunits
|
[
{
"state_after": "α : Type u\nβ : Type v\na b : α\ninst✝ : CommSemiring α\nh : a ∈ nonunits α\nthis : Ideal.span {a} ≠ ⊤\n⊢ ∃ I, Ideal.IsMaximal I ∧ a ∈ I",
"state_before": "α : Type u\nβ : Type v\na b : α\ninst✝ : CommSemiring α\nh : a ∈ nonunits α\n⊢ ∃ I, Ideal.IsMaximal I ∧ a ∈ I",
"tactic": "have : Ideal.span ({a} : Set α) ≠ ⊤ := by\n intro H\n rw [Ideal.span_singleton_eq_top] at H\n contradiction"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\na b : α\ninst✝ : CommSemiring α\nh : a ∈ nonunits α\nthis : Ideal.span {a} ≠ ⊤\nI : Ideal α\nImax : Ideal.IsMaximal I\nH : Ideal.span {a} ≤ I\n⊢ ∃ I, Ideal.IsMaximal I ∧ a ∈ I",
"state_before": "α : Type u\nβ : Type v\na b : α\ninst✝ : CommSemiring α\nh : a ∈ nonunits α\nthis : Ideal.span {a} ≠ ⊤\n⊢ ∃ I, Ideal.IsMaximal I ∧ a ∈ I",
"tactic": "rcases Ideal.exists_le_maximal _ this with ⟨I, Imax, H⟩"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\na b : α\ninst✝ : CommSemiring α\nh : a ∈ nonunits α\nthis : Ideal.span {a} ≠ ⊤\nI : Ideal α\nImax : Ideal.IsMaximal I\nH : Ideal.span {a} ≤ I\n⊢ a ∈ I",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\na b : α\ninst✝ : CommSemiring α\nh : a ∈ nonunits α\nthis : Ideal.span {a} ≠ ⊤\nI : Ideal α\nImax : Ideal.IsMaximal I\nH : Ideal.span {a} ≤ I\n⊢ ∃ I, Ideal.IsMaximal I ∧ a ∈ I",
"tactic": "use I, Imax"
},
{
"state_after": "case intro.intro.a\nα : Type u\nβ : Type v\na b : α\ninst✝ : CommSemiring α\nh : a ∈ nonunits α\nthis : Ideal.span {a} ≠ ⊤\nI : Ideal α\nImax : Ideal.IsMaximal I\nH : Ideal.span {a} ≤ I\n⊢ a ∈ Ideal.span {a}",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\na b : α\ninst✝ : CommSemiring α\nh : a ∈ nonunits α\nthis : Ideal.span {a} ≠ ⊤\nI : Ideal α\nImax : Ideal.IsMaximal I\nH : Ideal.span {a} ≤ I\n⊢ a ∈ I",
"tactic": "apply H"
},
{
"state_after": "case intro.intro.a.a\nα : Type u\nβ : Type v\na b : α\ninst✝ : CommSemiring α\nh : a ∈ nonunits α\nthis : Ideal.span {a} ≠ ⊤\nI : Ideal α\nImax : Ideal.IsMaximal I\nH : Ideal.span {a} ≤ I\n⊢ a ∈ {a}",
"state_before": "case intro.intro.a\nα : Type u\nβ : Type v\na b : α\ninst✝ : CommSemiring α\nh : a ∈ nonunits α\nthis : Ideal.span {a} ≠ ⊤\nI : Ideal α\nImax : Ideal.IsMaximal I\nH : Ideal.span {a} ≤ I\n⊢ a ∈ Ideal.span {a}",
"tactic": "apply Ideal.subset_span"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.a.a\nα : Type u\nβ : Type v\na b : α\ninst✝ : CommSemiring α\nh : a ∈ nonunits α\nthis : Ideal.span {a} ≠ ⊤\nI : Ideal α\nImax : Ideal.IsMaximal I\nH : Ideal.span {a} ≤ I\n⊢ a ∈ {a}",
"tactic": "exact Set.mem_singleton a"
},
{
"state_after": "α : Type u\nβ : Type v\na b : α\ninst✝ : CommSemiring α\nh : a ∈ nonunits α\nH : Ideal.span {a} = ⊤\n⊢ False",
"state_before": "α : Type u\nβ : Type v\na b : α\ninst✝ : CommSemiring α\nh : a ∈ nonunits α\n⊢ Ideal.span {a} ≠ ⊤",
"tactic": "intro H"
},
{
"state_after": "α : Type u\nβ : Type v\na b : α\ninst✝ : CommSemiring α\nh : a ∈ nonunits α\nH : IsUnit a\n⊢ False",
"state_before": "α : Type u\nβ : Type v\na b : α\ninst✝ : CommSemiring α\nh : a ∈ nonunits α\nH : Ideal.span {a} = ⊤\n⊢ False",
"tactic": "rw [Ideal.span_singleton_eq_top] at H"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\na b : α\ninst✝ : CommSemiring α\nh : a ∈ nonunits α\nH : IsUnit a\n⊢ False",
"tactic": "contradiction"
}
] |
[
864,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
854,
1
] |
Mathlib/Data/Nat/Parity.lean
|
Nat.bit0_div_two
|
[
{
"state_after": "no goals",
"state_before": "m n : ℕ\n⊢ bit0 n / 2 = n",
"tactic": "rw [← Nat.bit0_eq_bit0, bit0_eq_two_mul, two_mul_div_two_of_even (even_bit0 n)]"
}
] |
[
246,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/Topology/LocalAtTarget.lean
|
Set.restrictPreimage_closedEmbedding
|
[] |
[
64,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
61,
1
] |
Mathlib/Analysis/NormedSpace/Multilinear.lean
|
ContinuousMultilinearMap.norm_constOfIsEmpty
|
[
{
"state_after": "case a\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁸ : Fintype ι\ninst✝¹⁷ : Fintype ι'\ninst✝¹⁶ : NontriviallyNormedField 𝕜\ninst✝¹⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹³ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹² : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝¹¹ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝¹⁰ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁸ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\ninst✝⁵ : NormedAddCommGroup G'\ninst✝⁴ : NormedSpace 𝕜 G'\nc : 𝕜\nf g : ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\n𝕜' : Type ?u.622672\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedSpace 𝕜' G\ninst✝¹ : SMulCommClass 𝕜 𝕜' G\ninst✝ : IsEmpty ι\nx : G\n⊢ ‖constOfIsEmpty 𝕜 E x‖ ≤ ‖x‖\n\ncase a\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁸ : Fintype ι\ninst✝¹⁷ : Fintype ι'\ninst✝¹⁶ : NontriviallyNormedField 𝕜\ninst✝¹⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹³ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹² : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝¹¹ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝¹⁰ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁸ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\ninst✝⁵ : NormedAddCommGroup G'\ninst✝⁴ : NormedSpace 𝕜 G'\nc : 𝕜\nf g : ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\n𝕜' : Type ?u.622672\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedSpace 𝕜' G\ninst✝¹ : SMulCommClass 𝕜 𝕜' G\ninst✝ : IsEmpty ι\nx : G\n⊢ ‖x‖ ≤ ‖constOfIsEmpty 𝕜 E x‖",
"state_before": "𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁸ : Fintype ι\ninst✝¹⁷ : Fintype ι'\ninst✝¹⁶ : NontriviallyNormedField 𝕜\ninst✝¹⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹³ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹² : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝¹¹ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝¹⁰ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁸ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\ninst✝⁵ : NormedAddCommGroup G'\ninst✝⁴ : NormedSpace 𝕜 G'\nc : 𝕜\nf g : ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\n𝕜' : Type ?u.622672\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedSpace 𝕜' G\ninst✝¹ : SMulCommClass 𝕜 𝕜' G\ninst✝ : IsEmpty ι\nx : G\n⊢ ‖constOfIsEmpty 𝕜 E x‖ = ‖x‖",
"tactic": "apply le_antisymm"
},
{
"state_after": "case a\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁸ : Fintype ι\ninst✝¹⁷ : Fintype ι'\ninst✝¹⁶ : NontriviallyNormedField 𝕜\ninst✝¹⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹³ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹² : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝¹¹ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝¹⁰ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁸ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\ninst✝⁵ : NormedAddCommGroup G'\ninst✝⁴ : NormedSpace 𝕜 G'\nc : 𝕜\nf g : ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\n𝕜' : Type ?u.622672\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedSpace 𝕜' G\ninst✝¹ : SMulCommClass 𝕜 𝕜' G\ninst✝ : IsEmpty ι\nx✝ : G\nx : (i : ι) → E i\n⊢ ‖↑(constOfIsEmpty 𝕜 E x✝) x‖ ≤ ‖x✝‖ * ∏ i : ι, ‖x i‖",
"state_before": "case a\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁸ : Fintype ι\ninst✝¹⁷ : Fintype ι'\ninst✝¹⁶ : NontriviallyNormedField 𝕜\ninst✝¹⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹³ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹² : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝¹¹ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝¹⁰ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁸ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\ninst✝⁵ : NormedAddCommGroup G'\ninst✝⁴ : NormedSpace 𝕜 G'\nc : 𝕜\nf g : ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\n𝕜' : Type ?u.622672\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedSpace 𝕜' G\ninst✝¹ : SMulCommClass 𝕜 𝕜' G\ninst✝ : IsEmpty ι\nx : G\n⊢ ‖constOfIsEmpty 𝕜 E x‖ ≤ ‖x‖",
"tactic": "refine' op_norm_le_bound _ (norm_nonneg _) fun x => _"
},
{
"state_after": "no goals",
"state_before": "case a\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁸ : Fintype ι\ninst✝¹⁷ : Fintype ι'\ninst✝¹⁶ : NontriviallyNormedField 𝕜\ninst✝¹⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹³ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹² : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝¹¹ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝¹⁰ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁸ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\ninst✝⁵ : NormedAddCommGroup G'\ninst✝⁴ : NormedSpace 𝕜 G'\nc : 𝕜\nf g : ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\n𝕜' : Type ?u.622672\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedSpace 𝕜' G\ninst✝¹ : SMulCommClass 𝕜 𝕜' G\ninst✝ : IsEmpty ι\nx✝ : G\nx : (i : ι) → E i\n⊢ ‖↑(constOfIsEmpty 𝕜 E x✝) x‖ ≤ ‖x✝‖ * ∏ i : ι, ‖x i‖",
"tactic": "rw [Fintype.prod_empty, mul_one, constOfIsEmpty_apply]"
},
{
"state_after": "no goals",
"state_before": "case a\n𝕜 : Type u\nι : Type v\nι' : Type v'\nn : ℕ\nE : ι → Type wE\nE₁ : ι → Type wE₁\nE' : ι' → Type wE'\nEi : Fin (Nat.succ n) → Type wEi\nG : Type wG\nG' : Type wG'\ninst✝¹⁸ : Fintype ι\ninst✝¹⁷ : Fintype ι'\ninst✝¹⁶ : NontriviallyNormedField 𝕜\ninst✝¹⁵ : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹⁴ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝¹³ : (i : ι) → NormedAddCommGroup (E₁ i)\ninst✝¹² : (i : ι) → NormedSpace 𝕜 (E₁ i)\ninst✝¹¹ : (i : ι') → NormedAddCommGroup (E' i)\ninst✝¹⁰ : (i : ι') → NormedSpace 𝕜 (E' i)\ninst✝⁹ : (i : Fin (Nat.succ n)) → NormedAddCommGroup (Ei i)\ninst✝⁸ : (i : Fin (Nat.succ n)) → NormedSpace 𝕜 (Ei i)\ninst✝⁷ : NormedAddCommGroup G\ninst✝⁶ : NormedSpace 𝕜 G\ninst✝⁵ : NormedAddCommGroup G'\ninst✝⁴ : NormedSpace 𝕜 G'\nc : 𝕜\nf g : ContinuousMultilinearMap 𝕜 E G\nm : (i : ι) → E i\n𝕜' : Type ?u.622672\ninst✝³ : NormedField 𝕜'\ninst✝² : NormedSpace 𝕜' G\ninst✝¹ : SMulCommClass 𝕜 𝕜' G\ninst✝ : IsEmpty ι\nx : G\n⊢ ‖x‖ ≤ ‖constOfIsEmpty 𝕜 E x‖",
"tactic": "simpa using (constOfIsEmpty 𝕜 E x).le_op_norm 0"
}
] |
[
534,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
530,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.coe_pow
|
[] |
[
447,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
446,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
norm_eq_sqrt_inner
|
[] |
[
1000,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
997,
1
] |
Std/Data/RBMap/Alter.lean
|
Std.RBNode.Path.Balanced.insert
|
[] |
[
167,
51
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
163,
11
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean
|
AffineMap.snd_linear
|
[] |
[
372,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
371,
1
] |
Mathlib/Data/Nat/Order/Lemmas.lean
|
Nat.dvd_div_of_mul_dvd
|
[
{
"state_after": "no goals",
"state_before": "a✝ b✝ m n k a b c : ℕ\nh : a * b ∣ c\nha : a = 0\n⊢ b ∣ c / a",
"tactic": "simp [ha]"
},
{
"state_after": "no goals",
"state_before": "a✝ b✝ m n k a b c : ℕ\nh : a * b ∣ c\nha✝ : ¬a = 0\nha : 0 < a\nh1 : ∃ d, c = a * b * d\nd : ℕ\nhd : c = a * b * d\n⊢ c = a * (b * d)",
"tactic": "simpa [mul_assoc] using hd"
}
] |
[
186,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
179,
1
] |
Mathlib/GroupTheory/Coset.lean
|
QuotientGroup.exists_mk
|
[] |
[
507,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
506,
1
] |
Mathlib/Topology/Algebra/Group/Basic.lean
|
isClosedMap_mul_left
|
[] |
[
93,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Topology/Filter.lean
|
Filter.tendsto_nhds
|
[
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.5011\nα : Type u_1\nβ : Type u_2\nX : Type ?u.5020\nY : Type ?u.5023\nla : Filter α\nlb : Filter β\nf : α → Filter β\n⊢ Tendsto f la (𝓝 lb) ↔ ∀ (s : Set β), s ∈ lb → ∀ᶠ (a : α) in la, s ∈ f a",
"tactic": "simp only [nhds_eq', tendsto_lift', mem_setOf_eq]"
}
] |
[
89,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
11
] |
Mathlib/Analysis/NormedSpace/lpSpace.lean
|
lp.hasSum_norm
|
[
{
"state_after": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nf : { x // x ∈ lp E p }\n⊢ HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) (∑' (i : α), ‖↑f i‖ ^ ENNReal.toReal p)",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nf : { x // x ∈ lp E p }\n⊢ HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) (‖f‖ ^ ENNReal.toReal p)",
"tactic": "rw [norm_rpow_eq_tsum hp]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nE : α → Type u_2\np q : ℝ≥0∞\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nhp : 0 < ENNReal.toReal p\nf : { x // x ∈ lp E p }\n⊢ HasSum (fun i => ‖↑f i‖ ^ ENNReal.toReal p) (∑' (i : α), ‖↑f i‖ ^ ENNReal.toReal p)",
"tactic": "exact ((lp.memℓp f).summable hp).hasSum"
}
] |
[
430,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
427,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/RightAngle.lean
|
InnerProductGeometry.cos_angle_sub_of_inner_eq_zero
|
[
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x (-y) = 0\n⊢ Real.cos (angle x (x - y)) = ‖x‖ / ‖x - y‖",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x y = 0\n⊢ Real.cos (angle x (x - y)) = ‖x‖ / ‖x - y‖",
"tactic": "rw [← neg_eq_zero, ← inner_neg_right] at h"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nh : inner x (-y) = 0\n⊢ Real.cos (angle x (x - y)) = ‖x‖ / ‖x - y‖",
"tactic": "rw [sub_eq_add_neg, cos_angle_add_of_inner_eq_zero h]"
}
] |
[
285,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
282,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.disjUnion_empty
|
[] |
[
1018,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1016,
1
] |
Mathlib/Probability/Integration.lean
|
ProbabilityTheory.IndepFun.integrable_right_of_integrable_mul
|
[
{
"state_after": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\n⊢ HasFiniteIntegral Y",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\n⊢ Integrable Y",
"tactic": "refine' ⟨hY, _⟩"
},
{
"state_after": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) ≠ 0\n⊢ HasFiniteIntegral Y",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\n⊢ HasFiniteIntegral Y",
"tactic": "have I : (∫⁻ ω, ‖X ω‖₊ ∂μ) ≠ 0 := fun H ↦ by\n have I : (fun ω => ‖X ω‖₊ : Ω → ℝ≥0∞) =ᵐ[μ] 0 := (lintegral_eq_zero_iff' hX.ennnorm).1 H\n apply h'X\n filter_upwards [I] with ω hω\n simpa using hω"
},
{
"state_after": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖Y a‖₊ ∂μ) = ⊤\n⊢ False",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) ≠ 0\n⊢ HasFiniteIntegral Y",
"tactic": "refine lt_top_iff_ne_top.2 fun H => ?_"
},
{
"state_after": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖Y a‖₊ ∂μ) = ⊤\nJ : IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊\n⊢ False",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖Y a‖₊ ∂μ) = ⊤\n⊢ False",
"tactic": "have J : IndepFun (fun ω => ‖X ω‖₊ : Ω → ℝ≥0∞) (fun ω => ‖Y ω‖₊ : Ω → ℝ≥0∞) μ := by\n have M : Measurable fun x : β => (‖x‖₊ : ℝ≥0∞) := measurable_nnnorm.coe_nnreal_ennreal\n apply IndepFun.comp hXY M M"
},
{
"state_after": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖Y a‖₊ ∂μ) = ⊤\nJ : IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊\nA : (∫⁻ (ω : Ω), ↑‖X ω * Y ω‖₊ ∂μ) < ⊤\n⊢ False",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖Y a‖₊ ∂μ) = ⊤\nJ : IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊\n⊢ False",
"tactic": "have A : (∫⁻ ω, ‖X ω * Y ω‖₊ ∂μ) < ∞ := h'XY.2"
},
{
"state_after": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖Y a‖₊ ∂μ) = ⊤\nJ : IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊\nA : (∫⁻ (ω : Ω), ↑‖X ω‖₊ * ↑‖Y ω‖₊ ∂μ) < ⊤\n⊢ False",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖Y a‖₊ ∂μ) = ⊤\nJ : IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊\nA : (∫⁻ (ω : Ω), ↑‖X ω * Y ω‖₊ ∂μ) < ⊤\n⊢ False",
"tactic": "simp only [nnnorm_mul, ENNReal.coe_mul] at A"
},
{
"state_after": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖Y a‖₊ ∂μ) = ⊤\nJ : IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊\nA : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) * ⊤ < ⊤\n⊢ False",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖Y a‖₊ ∂μ) = ⊤\nJ : IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊\nA : (∫⁻ (ω : Ω), ↑‖X ω‖₊ * ↑‖Y ω‖₊ ∂μ) < ⊤\n⊢ False",
"tactic": "rw [lintegral_mul_eq_lintegral_mul_lintegral_of_indepFun'' hX.ennnorm hY.ennnorm J, H] at A"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖Y a‖₊ ∂μ) = ⊤\nJ : IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊\nA : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) * ⊤ < ⊤\n⊢ False",
"tactic": "simp only [ENNReal.mul_top I] at A"
},
{
"state_after": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nH : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) = 0\nI : (fun ω => ↑‖X ω‖₊) =ᵐ[μ] 0\n⊢ False",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nH : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) = 0\n⊢ False",
"tactic": "have I : (fun ω => ‖X ω‖₊ : Ω → ℝ≥0∞) =ᵐ[μ] 0 := (lintegral_eq_zero_iff' hX.ennnorm).1 H"
},
{
"state_after": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nH : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) = 0\nI : (fun ω => ↑‖X ω‖₊) =ᵐ[μ] 0\n⊢ X =ᵐ[μ] 0",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nH : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) = 0\nI : (fun ω => ↑‖X ω‖₊) =ᵐ[μ] 0\n⊢ False",
"tactic": "apply h'X"
},
{
"state_after": "case h\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nH : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) = 0\nI : (fun ω => ↑‖X ω‖₊) =ᵐ[μ] 0\nω : Ω\nhω : ↑‖X ω‖₊ = OfNat.ofNat 0 ω\n⊢ X ω = OfNat.ofNat 0 ω",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nH : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) = 0\nI : (fun ω => ↑‖X ω‖₊) =ᵐ[μ] 0\n⊢ X =ᵐ[μ] 0",
"tactic": "filter_upwards [I] with ω hω"
},
{
"state_after": "no goals",
"state_before": "case h\nΩ : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nH : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) = 0\nI : (fun ω => ↑‖X ω‖₊) =ᵐ[μ] 0\nω : Ω\nhω : ↑‖X ω‖₊ = OfNat.ofNat 0 ω\n⊢ X ω = OfNat.ofNat 0 ω",
"tactic": "simpa using hω"
},
{
"state_after": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖Y a‖₊ ∂μ) = ⊤\nM : Measurable fun x => ↑‖x‖₊\n⊢ IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖Y a‖₊ ∂μ) = ⊤\n⊢ IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊",
"tactic": "have M : Measurable fun x : β => (‖x‖₊ : ℝ≥0∞) := measurable_nnnorm.coe_nnreal_ennreal"
},
{
"state_after": "no goals",
"state_before": "Ω : Type u_2\nmΩ : MeasurableSpace Ω\nμ : MeasureTheory.Measure Ω\nf g : Ω → ℝ≥0∞\nX✝ Y✝ : Ω → ℝ\nβ : Type u_1\ninst✝² : MeasurableSpace β\nX Y : Ω → β\ninst✝¹ : NormedDivisionRing β\ninst✝ : BorelSpace β\nhXY : IndepFun X Y\nh'XY : Integrable (X * Y)\nhX : AEStronglyMeasurable X μ\nhY : AEStronglyMeasurable Y μ\nh'X : ¬X =ᵐ[μ] 0\nI : (∫⁻ (ω : Ω), ↑‖X ω‖₊ ∂μ) ≠ 0\nH : (∫⁻ (a : Ω), ↑‖Y a‖₊ ∂μ) = ⊤\nM : Measurable fun x => ↑‖x‖₊\n⊢ IndepFun (fun ω => ↑‖X ω‖₊) fun ω => ↑‖Y ω‖₊",
"tactic": "apply IndepFun.comp hXY M M"
}
] |
[
199,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
182,
1
] |
Std/Data/Int/DivMod.lean
|
Int.sub_emod
|
[
{
"state_after": "a b n : Int\n⊢ (a - b + b) % n = (a % n - b % n + b) % n",
"state_before": "a b n : Int\n⊢ (a - b) % n = (a % n - b % n) % n",
"tactic": "apply (emod_add_cancel_right b).mp"
},
{
"state_after": "no goals",
"state_before": "a b n : Int\n⊢ (a - b + b) % n = (a % n - b % n + b) % n",
"tactic": "rw [Int.sub_add_cancel, ← Int.add_emod_emod, Int.sub_add_cancel, emod_emod]"
}
] |
[
499,
78
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
497,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
Real.cosh_lt_cosh
|
[] |
[
736,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
735,
1
] |
Mathlib/Data/Real/CauSeq.lean
|
CauSeq.mul_apply
|
[] |
[
304,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
303,
1
] |
Mathlib/Data/Multiset/LocallyFinite.lean
|
Multiset.map_add_left_Icc
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ map ((fun x x_1 => x + x_1) c) (Icc a b) = Icc (c + a) (c + b)",
"tactic": "classical rw [Icc, Icc, ← Finset.image_add_left_Icc, Finset.image_val,\n ((Finset.nodup _).map <| add_right_injective c).dedup]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝² : OrderedCancelAddCommMonoid α\ninst✝¹ : ExistsAddOfLE α\ninst✝ : LocallyFiniteOrder α\na b c : α\n⊢ map ((fun x x_1 => x + x_1) c) (Icc a b) = Icc (c + a) (c + b)",
"tactic": "rw [Icc, Icc, ← Finset.image_add_left_Icc, Finset.image_val,\n((Finset.nodup _).map <| add_right_injective c).dedup]"
}
] |
[
291,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
289,
1
] |
Mathlib/Data/Rat/Defs.lean
|
Rat.divInt_neg_den
|
[] |
[
169,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
169,
1
] |
Mathlib/Analysis/NormedSpace/Units.lean
|
Ideal.IsMaximal.closure_eq
|
[] |
[
265,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
264,
1
] |
Mathlib/MeasureTheory/Function/EssSup.lean
|
essInf_const'
|
[] |
[
72,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Algebra/Group/Units.lean
|
IsUnit.mul_right_injective
|
[] |
[
794,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
793,
11
] |
Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean
|
GromovHausdorff.HD_lipschitz_aux1
|
[
{
"state_after": "case intro\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\n⊢ (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ (⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y)) + dist f g",
"state_before": "X : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\n⊢ (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ (⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y)) + dist f g",
"tactic": "rcases (Real.bounded_iff_bddBelow_bddAbove.1 g.bounded_range).1 with ⟨cg, hcg⟩"
},
{
"state_after": "case intro\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\n⊢ (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ (⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y)) + dist f g",
"state_before": "case intro\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\n⊢ (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ (⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y)) + dist f g",
"tactic": "have Hcg : ∀ x, cg ≤ g x := fun x => hcg (mem_range_self x)"
},
{
"state_after": "case intro.intro\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\n⊢ (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ (⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y)) + dist f g",
"state_before": "case intro\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\n⊢ (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ (⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y)) + dist f g",
"tactic": "rcases (Real.bounded_iff_bddBelow_bddAbove.1 f.bounded_range).1 with ⟨cf, hcf⟩"
},
{
"state_after": "case intro.intro\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\n⊢ (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ (⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y)) + dist f g",
"state_before": "case intro.intro\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\n⊢ (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ (⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y)) + dist f g",
"tactic": "have Hcf : ∀ x, cf ≤ f x := fun x => hcf (mem_range_self x)"
},
{
"state_after": "case intro.intro\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\nZ : (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ ⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\n⊢ (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ (⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y)) + dist f g",
"state_before": "case intro.intro\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\n⊢ (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ (⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y)) + dist f g",
"tactic": "have Z : (⨆ x, ⨅ y, f (inl x, inr y)) ≤ ⨆ x, ⨅ y, g (inl x, inr y) + dist f g :=\n ciSup_mono (HD_bound_aux1 _ (dist f g)) fun x =>\n ciInf_mono ⟨cf, forall_range_iff.2 fun i => Hcf _⟩ fun y => coe_le_coe_add_dist"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\nZ : (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ ⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nE1 : ∀ (x : X), (⨅ (y : Y), ↑g (inl x, inr y)) + dist f g = ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nE2 : (⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y)) + dist f g = ⨆ (x : X), (⨅ (y : Y), ↑g (inl x, inr y)) + dist f g\n⊢ (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ (⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y)) + dist f g",
"tactic": "simpa [E2, E1, Function.comp]"
},
{
"state_after": "X : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx✝ y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\nZ : (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ ⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nx : X\n⊢ (⨅ (y : Y), ↑g (inl x, inr y)) + dist f g = ⨅ (y : Y), ↑g (inl x, inr y) + dist f g",
"state_before": "X : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\nZ : (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ ⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\n⊢ ∀ (x : X), (⨅ (y : Y), ↑g (inl x, inr y)) + dist f g = ⨅ (y : Y), ↑g (inl x, inr y) + dist f g",
"tactic": "intro x"
},
{
"state_after": "case refine'_1\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx✝ y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\nZ : (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ ⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nx : X\n⊢ Monotone fun x => id x + dist f g\n\ncase refine'_2\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx✝ y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\nZ : (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ ⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nx : X\n⊢ BddBelow (range fun y => ↑g (inl x, inr y))",
"state_before": "X : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx✝ y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\nZ : (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ ⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nx : X\n⊢ (⨅ (y : Y), ↑g (inl x, inr y)) + dist f g = ⨅ (y : Y), ↑g (inl x, inr y) + dist f g",
"tactic": "refine' Monotone.map_ciInf_of_continuousAt (continuousAt_id.add continuousAt_const) _ _"
},
{
"state_after": "case refine'_1\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx✝¹ y✝ z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\nZ : (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ ⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nx✝ : X\nx y : ℝ\nhx : x ≤ y\n⊢ (fun x => id x + dist f g) x ≤ (fun x => id x + dist f g) y",
"state_before": "case refine'_1\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx✝ y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\nZ : (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ ⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nx : X\n⊢ Monotone fun x => id x + dist f g",
"tactic": "intro x y hx"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx✝¹ y✝ z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\nZ : (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ ⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nx✝ : X\nx y : ℝ\nhx : x ≤ y\n⊢ (fun x => id x + dist f g) x ≤ (fun x => id x + dist f g) y",
"tactic": "simpa"
},
{
"state_after": "case refine'_2\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx✝ y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\nZ : (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ ⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nx : X\n⊢ BddBelow (range fun y => ↑g (inl x, inr y))",
"state_before": "case refine'_2\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx✝ y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\nZ : (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ ⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nx : X\n⊢ BddBelow (range fun y => ↑g (inl x, inr y))",
"tactic": "show BddBelow (range fun y : Y => g (inl x, inr y))"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx✝ y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\nZ : (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ ⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nx : X\n⊢ BddBelow (range fun y => ↑g (inl x, inr y))",
"tactic": "exact ⟨cg, forall_range_iff.2 fun i => Hcg _⟩"
},
{
"state_after": "case refine'_1\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\nZ : (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ ⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nE1 : ∀ (x : X), (⨅ (y : Y), ↑g (inl x, inr y)) + dist f g = ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\n⊢ Monotone fun x => id x + dist f g\n\ncase refine'_2\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\nZ : (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ ⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nE1 : ∀ (x : X), (⨅ (y : Y), ↑g (inl x, inr y)) + dist f g = ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\n⊢ BddAbove (range fun x => ⨅ (y : Y), ↑g (inl x, inr y))",
"state_before": "X : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\nZ : (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ ⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nE1 : ∀ (x : X), (⨅ (y : Y), ↑g (inl x, inr y)) + dist f g = ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\n⊢ (⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y)) + dist f g = ⨆ (x : X), (⨅ (y : Y), ↑g (inl x, inr y)) + dist f g",
"tactic": "refine' Monotone.map_ciSup_of_continuousAt (continuousAt_id.add continuousAt_const) _ _"
},
{
"state_after": "case refine'_1\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx✝ y✝ z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\nZ : (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ ⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nE1 : ∀ (x : X), (⨅ (y : Y), ↑g (inl x, inr y)) + dist f g = ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nx y : ℝ\nhx : x ≤ y\n⊢ (fun x => id x + dist f g) x ≤ (fun x => id x + dist f g) y",
"state_before": "case refine'_1\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\nZ : (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ ⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nE1 : ∀ (x : X), (⨅ (y : Y), ↑g (inl x, inr y)) + dist f g = ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\n⊢ Monotone fun x => id x + dist f g",
"tactic": "intro x y hx"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx✝ y✝ z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\nZ : (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ ⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nE1 : ∀ (x : X), (⨅ (y : Y), ↑g (inl x, inr y)) + dist f g = ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nx y : ℝ\nhx : x ≤ y\n⊢ (fun x => id x + dist f g) x ≤ (fun x => id x + dist f g) y",
"tactic": "simpa"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf✝ : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\nf g : GromovHausdorff.Cb X Y\ncg : ℝ\nhcg : cg ∈ lowerBounds (range ↑g)\nHcg : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cg ≤ ↑g x\ncf : ℝ\nhcf : cf ∈ lowerBounds (range ↑f)\nHcf : ∀ (x : (X ⊕ Y) × (X ⊕ Y)), cf ≤ ↑f x\nZ : (⨆ (x : X), ⨅ (y : Y), ↑f (inl x, inr y)) ≤ ⨆ (x : X), ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\nE1 : ∀ (x : X), (⨅ (y : Y), ↑g (inl x, inr y)) + dist f g = ⨅ (y : Y), ↑g (inl x, inr y) + dist f g\n⊢ BddAbove (range fun x => ⨅ (y : Y), ↑g (inl x, inr y))",
"tactic": "simpa using HD_bound_aux1 _ 0"
}
] |
[
377,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
351,
9
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisInsertion.l_sup_u
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nι : Sort x\nκ : ι → Sort ?u.37330\na✝ a₁ a₂ : α\nb✝ b₁ b₂ : β\nl : α → β\nu : β → α\ninst✝¹ : SemilatticeSup α\ninst✝ : SemilatticeSup β\ngi : GaloisInsertion l u\na b : β\n⊢ l (u a) ⊔ l (u b) = a ⊔ b",
"tactic": "simp only [gi.l_u_eq]"
}
] |
[
537,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
533,
1
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iInter_of_empty
|
[] |
[
1336,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1335,
1
] |
Mathlib/Order/Cover.lean
|
toDual_wcovby_toDual_iff
|
[] |
[
139,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
138,
1
] |
Mathlib/Topology/Perfect.lean
|
Perfect.exists_nat_bool_injection
|
[
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\n⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Injective f",
"state_before": "α : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\n⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Injective f",
"tactic": "obtain ⟨u, -, upos', hu⟩ := exists_seq_strictAnti_tendsto' (zero_lt_one' ℝ≥0∞)"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\n⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Injective f",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\n⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Injective f",
"tactic": "have upos := fun n => (upos' n).1"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\n⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Injective f",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\n⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Injective f",
"tactic": "let P := Subtype fun E : Set α => Perfect E ∧ E.Nonempty"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\n⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Injective f",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\n⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Injective f",
"tactic": "choose C0 C1 h0 h1 hdisj using\n fun {C : Set α} (hC : Perfect C) (hnonempty : C.Nonempty) {ε : ℝ≥0∞} (hε : 0 < ε) =>\n hC.small_diam_splitting hnonempty hε"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\n⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Injective f",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\n⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Injective f",
"tactic": "let D : List Bool → Set α := fun l => (DP l).val"
},
{
"state_after": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : ∀ {x : ℕ → Bool}, x ∈ (inducedMap D).fst\n⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Injective f",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\n⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Injective f",
"tactic": "have hdom : ∀ {x : ℕ → Bool}, x ∈ (inducedMap D).1 := fun {x} => by\n rw [hanti.map_of_vanishingDiam hdiam fun l => (DP l).property.2]\n apply mem_univ"
},
{
"state_after": "case intro.intro.intro.refine'_1\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : ∀ {x : ℕ → Bool}, x ∈ (inducedMap D).fst\n⊢ (range fun x => Sigma.snd (inducedMap D) { val := x, property := (_ : x ∈ (inducedMap D).fst) }) ⊆ C\n\ncase intro.intro.intro.refine'_2\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : ∀ {x : ℕ → Bool}, x ∈ (inducedMap D).fst\n⊢ Continuous fun x => Sigma.snd (inducedMap D) { val := x, property := (_ : x ∈ (inducedMap D).fst) }\n\ncase intro.intro.intro.refine'_3\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : ∀ {x : ℕ → Bool}, x ∈ (inducedMap D).fst\n⊢ Injective fun x => Sigma.snd (inducedMap D) { val := x, property := (_ : x ∈ (inducedMap D).fst) }",
"state_before": "case intro.intro.intro\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : ∀ {x : ℕ → Bool}, x ∈ (inducedMap D).fst\n⊢ ∃ f, range f ⊆ C ∧ Continuous f ∧ Injective f",
"tactic": "refine' ⟨fun x => (inducedMap D).2 ⟨x, hdom⟩, _, _, _⟩"
},
{
"state_after": "case intro.intro.intro.refine'_3\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : ∀ {x : ℕ → Bool}, x ∈ (inducedMap D).fst\nx y : ℕ → Bool\nhxy :\n (fun x => Sigma.snd (inducedMap D) { val := x, property := (_ : x ∈ (inducedMap D).fst) }) x =\n (fun x => Sigma.snd (inducedMap D) { val := x, property := (_ : x ∈ (inducedMap D).fst) }) y\n⊢ x = y",
"state_before": "case intro.intro.intro.refine'_3\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : ∀ {x : ℕ → Bool}, x ∈ (inducedMap D).fst\n⊢ Injective fun x => Sigma.snd (inducedMap D) { val := x, property := (_ : x ∈ (inducedMap D).fst) }",
"tactic": "intro x y hxy"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.refine'_3\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : ∀ {x : ℕ → Bool}, x ∈ (inducedMap D).fst\nx y : ℕ → Bool\nhxy :\n (fun x => Sigma.snd (inducedMap D) { val := x, property := (_ : x ∈ (inducedMap D).fst) }) x =\n (fun x => Sigma.snd (inducedMap D) { val := x, property := (_ : x ∈ (inducedMap D).fst) }) y\n⊢ x = y",
"tactic": "simpa only [← Subtype.val_inj] using hdisj'.map_injective hxy"
},
{
"state_after": "case nil\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\n⊢ P\n\ncase cons\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\na : Bool\nl : List Bool\nih : P\n⊢ P",
"state_before": "α : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nl : List Bool\n⊢ P",
"tactic": "induction' l with a l ih"
},
{
"state_after": "case cons.false\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nl : List Bool\nih : P\n⊢ P\n\ncase cons.true\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nl : List Bool\nih : P\n⊢ P",
"state_before": "case cons\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\na : Bool\nl : List Bool\nih : P\n⊢ P",
"tactic": "cases a"
},
{
"state_after": "case cons.true\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nl : List Bool\nih : P\n⊢ Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))",
"state_before": "case cons.true\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nl : List Bool\nih : P\n⊢ P",
"tactic": "use C1 ih.property.1 ih.property.2 (upos l.length.succ)"
},
{
"state_after": "no goals",
"state_before": "case cons.true\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nl : List Bool\nih : P\n⊢ Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))",
"tactic": "exact ⟨(h1 _ _ _).1, (h1 _ _ _).2.1⟩"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\n⊢ P",
"tactic": "exact ⟨C, ⟨hC, hnonempty⟩⟩"
},
{
"state_after": "case cons.false\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nl : List Bool\nih : P\n⊢ Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))",
"state_before": "case cons.false\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nl : List Bool\nih : P\n⊢ P",
"tactic": "use C0 ih.property.1 ih.property.2 (upos l.length.succ)"
},
{
"state_after": "no goals",
"state_before": "case cons.false\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nl : List Bool\nih : P\n⊢ Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))",
"tactic": "exact ⟨(h0 _ _ _).1, (h0 _ _ _).2.1⟩"
},
{
"state_after": "α : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\n⊢ CantorScheme.Antitone D",
"state_before": "α : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\n⊢ ClosureAntitone D",
"tactic": "refine' Antitone.closureAntitone _ fun l => (DP l).property.1.closed"
},
{
"state_after": "α : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nl : List Bool\na : Bool\n⊢ D (a :: l) ⊆ D l",
"state_before": "α : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\n⊢ CantorScheme.Antitone D",
"tactic": "intro l a"
},
{
"state_after": "case false\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nl : List Bool\n⊢ D (false :: l) ⊆ D l\n\ncase true\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nl : List Bool\n⊢ D (true :: l) ⊆ D l",
"state_before": "α : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nl : List Bool\na : Bool\n⊢ D (a :: l) ⊆ D l",
"tactic": "cases a"
},
{
"state_after": "no goals",
"state_before": "case true\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nl : List Bool\n⊢ D (true :: l) ⊆ D l",
"tactic": "exact (h1 _ _ _).2.2.1"
},
{
"state_after": "no goals",
"state_before": "case false\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nl : List Bool\n⊢ D (false :: l) ⊆ D l",
"tactic": "exact (h0 _ _ _).2.2.1"
},
{
"state_after": "α : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\n⊢ Tendsto (fun n => EMetric.diam (D (PiNat.res x n))) atTop (𝓝 0)",
"state_before": "α : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\n⊢ VanishingDiam D",
"tactic": "intro x"
},
{
"state_after": "case hgf\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\n⊢ ∀ᶠ (b : ℕ) in atTop, 0 ≤ EMetric.diam (D (PiNat.res x b))\n\ncase hfh\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\n⊢ ∀ᶠ (b : ℕ) in atTop, EMetric.diam (D (PiNat.res x b)) ≤ u b",
"state_before": "α : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\n⊢ Tendsto (fun n => EMetric.diam (D (PiNat.res x n))) atTop (𝓝 0)",
"tactic": "apply tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds hu"
},
{
"state_after": "case hfh\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\n⊢ ∃ a, ∀ (b : ℕ), b ≥ a → EMetric.diam (D (PiNat.res x b)) ≤ u b",
"state_before": "case hfh\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\n⊢ ∀ᶠ (b : ℕ) in atTop, EMetric.diam (D (PiNat.res x b)) ≤ u b",
"tactic": "rw [eventually_atTop]"
},
{
"state_after": "case hfh\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\nm : ℕ\nhm : 1 ≤ m\n⊢ EMetric.diam (D (PiNat.res x m)) ≤ u m",
"state_before": "case hfh\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\n⊢ ∃ a, ∀ (b : ℕ), b ≥ a → EMetric.diam (D (PiNat.res x b)) ≤ u b",
"tactic": "refine' ⟨1, fun m (hm : 1 ≤ m) => _⟩"
},
{
"state_after": "case hfh\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\nm : ℕ\nhm : m ≠ 0\n⊢ EMetric.diam (D (PiNat.res x m)) ≤ u m",
"state_before": "case hfh\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\nm : ℕ\nhm : 1 ≤ m\n⊢ EMetric.diam (D (PiNat.res x m)) ≤ u m",
"tactic": "rw [Nat.one_le_iff_ne_zero] at hm"
},
{
"state_after": "case hfh.intro\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\nn : ℕ\nhm : Nat.succ n ≠ 0\n⊢ EMetric.diam (D (PiNat.res x (Nat.succ n))) ≤ u (Nat.succ n)",
"state_before": "case hfh\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\nm : ℕ\nhm : m ≠ 0\n⊢ EMetric.diam (D (PiNat.res x m)) ≤ u m",
"tactic": "rcases Nat.exists_eq_succ_of_ne_zero hm with ⟨n, rfl⟩"
},
{
"state_after": "case hfh.intro\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\nn : ℕ\nhm : Nat.succ n ≠ 0\n⊢ EMetric.diam\n ↑(Bool.rec (motive := fun t => x n = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = x n) ▸\n {\n val :=\n C0\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))),\n property :=\n (_ :\n Perfect\n (C0\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n))))) ∧\n Set.Nonempty\n (C0\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))))) })\n (fun h =>\n (_ : true = x n) ▸\n {\n val :=\n C1\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))),\n property :=\n (_ :\n Perfect\n (C1\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n))))) ∧\n Set.Nonempty\n (C1\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))))) })\n (x n) (_ : x n = x n)) ≤\n u (Nat.succ n)",
"state_before": "case hfh.intro\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\nn : ℕ\nhm : Nat.succ n ≠ 0\n⊢ EMetric.diam (D (PiNat.res x (Nat.succ n))) ≤ u (Nat.succ n)",
"tactic": "dsimp"
},
{
"state_after": "case hfh.intro.false\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\nn : ℕ\nhm : Nat.succ n ≠ 0\n⊢ EMetric.diam\n ↑(Bool.rec (motive := fun t => false = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = false) ▸\n {\n val :=\n C0\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))),\n property :=\n (_ :\n Perfect\n (C0\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n))))) ∧\n Set.Nonempty\n (C0\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))))) })\n (fun h =>\n (_ : true = false) ▸\n {\n val :=\n C1\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))),\n property :=\n (_ :\n Perfect\n (C1\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n))))) ∧\n Set.Nonempty\n (C1\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))))) })\n false (_ : false = false)) ≤\n u (Nat.succ n)\n\ncase hfh.intro.true\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\nn : ℕ\nhm : Nat.succ n ≠ 0\n⊢ EMetric.diam\n ↑(Bool.rec (motive := fun t => true = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = true) ▸\n {\n val :=\n C0\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))),\n property :=\n (_ :\n Perfect\n (C0\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n))))) ∧\n Set.Nonempty\n (C0\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))))) })\n (fun h =>\n (_ : true = true) ▸\n {\n val :=\n C1\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))),\n property :=\n (_ :\n Perfect\n (C1\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n))))) ∧\n Set.Nonempty\n (C1\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))))) })\n true (_ : true = true)) ≤\n u (Nat.succ n)",
"state_before": "case hfh.intro\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\nn : ℕ\nhm : Nat.succ n ≠ 0\n⊢ EMetric.diam\n ↑(Bool.rec (motive := fun t => x n = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = x n) ▸\n {\n val :=\n C0\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))),\n property :=\n (_ :\n Perfect\n (C0\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n))))) ∧\n Set.Nonempty\n (C0\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))))) })\n (fun h =>\n (_ : true = x n) ▸\n {\n val :=\n C1\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))),\n property :=\n (_ :\n Perfect\n (C1\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n))))) ∧\n Set.Nonempty\n (C1\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))))) })\n (x n) (_ : x n = x n)) ≤\n u (Nat.succ n)",
"tactic": "cases x n"
},
{
"state_after": "case h.e'_4.h.e'_1.h.e'_1\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\nn : ℕ\nhm : Nat.succ n ≠ 0\n⊢ n = List.length (PiNat.res x n)",
"state_before": "case hfh.intro.true\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\nn : ℕ\nhm : Nat.succ n ≠ 0\n⊢ EMetric.diam\n ↑(Bool.rec (motive := fun t => true = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = true) ▸\n {\n val :=\n C0\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))),\n property :=\n (_ :\n Perfect\n (C0\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n))))) ∧\n Set.Nonempty\n (C0\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))))) })\n (fun h =>\n (_ : true = true) ▸\n {\n val :=\n C1\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))),\n property :=\n (_ :\n Perfect\n (C1\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n))))) ∧\n Set.Nonempty\n (C1\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))))) })\n true (_ : true = true)) ≤\n u (Nat.succ n)",
"tactic": "convert(h1 _ _ _).2.2.2"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4.h.e'_1.h.e'_1\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\nn : ℕ\nhm : Nat.succ n ≠ 0\n⊢ n = List.length (PiNat.res x n)",
"tactic": "rw [PiNat.res_length]"
},
{
"state_after": "no goals",
"state_before": "case hgf\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\n⊢ ∀ᶠ (b : ℕ) in atTop, 0 ≤ EMetric.diam (D (PiNat.res x b))",
"tactic": "simp"
},
{
"state_after": "case h.e'_4.h.e'_1.h.e'_1\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\nn : ℕ\nhm : Nat.succ n ≠ 0\n⊢ n = List.length (PiNat.res x n)",
"state_before": "case hfh.intro.false\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\nn : ℕ\nhm : Nat.succ n ≠ 0\n⊢ EMetric.diam\n ↑(Bool.rec (motive := fun t => false = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = false) ▸\n {\n val :=\n C0\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))),\n property :=\n (_ :\n Perfect\n (C0\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n))))) ∧\n Set.Nonempty\n (C0\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))))) })\n (fun h =>\n (_ : true = false) ▸\n {\n val :=\n C1\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))),\n property :=\n (_ :\n Perfect\n (C1\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n))))) ∧\n Set.Nonempty\n (C1\n (_ :\n Perfect\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ :\n Set.Nonempty\n ↑(List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.rec (motive := fun t => a = t → { E // Perfect E ∧ Set.Nonempty E })\n (fun h =>\n (_ : false = a) ▸\n {\n val :=\n C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n {\n val :=\n C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih)\n (_ : 0 < u (Nat.succ (List.length l))))) })\n a (_ : a = a))\n (PiNat.res x n)))\n (_ : 0 < u (Nat.succ (List.length (PiNat.res x n)))))) })\n false (_ : false = false)) ≤\n u (Nat.succ n)",
"tactic": "convert(h0 _ _ _).2.2.2"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4.h.e'_1.h.e'_1\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nx : ℕ → Bool\nn : ℕ\nhm : Nat.succ n ≠ 0\n⊢ n = List.length (PiNat.res x n)",
"tactic": "rw [PiNat.res_length]"
},
{
"state_after": "case false.true\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nl : List Bool\nhab : false ≠ true\n⊢ Disjoint (D (false :: l)) (D (true :: l))\n\ncase true.false\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nl : List Bool\nhab : true ≠ false\n⊢ Disjoint (D (true :: l)) (D (false :: l))",
"state_before": "α : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\n⊢ CantorScheme.Disjoint D",
"tactic": "rintro l (a | a) (b | b) hab <;> try contradiction"
},
{
"state_after": "no goals",
"state_before": "case true.false\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nl : List Bool\nhab : true ≠ false\n⊢ Disjoint (D (true :: l)) (D (false :: l))",
"tactic": "exact (hdisj _ _ _).symm"
},
{
"state_after": "no goals",
"state_before": "case true.true\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nl : List Bool\nhab : true ≠ true\n⊢ Disjoint (D (true :: l)) (D (true :: l))",
"tactic": "contradiction"
},
{
"state_after": "no goals",
"state_before": "case false.true\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nl : List Bool\nhab : false ≠ true\n⊢ Disjoint (D (false :: l)) (D (true :: l))",
"tactic": "exact hdisj _ _ _"
},
{
"state_after": "α : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nx : ℕ → Bool\n⊢ x ∈ univ",
"state_before": "α : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nx : ℕ → Bool\n⊢ x ∈ (inducedMap D).fst",
"tactic": "rw [hanti.map_of_vanishingDiam hdiam fun l => (DP l).property.2]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nx : ℕ → Bool\n⊢ x ∈ univ",
"tactic": "apply mem_univ"
},
{
"state_after": "case intro.intro.intro.refine'_1.intro\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : ∀ {x : ℕ → Bool}, x ∈ (inducedMap D).fst\nx : ℕ → Bool\n⊢ (fun x => Sigma.snd (inducedMap D) { val := x, property := (_ : x ∈ (inducedMap D).fst) }) x ∈ C",
"state_before": "case intro.intro.intro.refine'_1\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : ∀ {x : ℕ → Bool}, x ∈ (inducedMap D).fst\n⊢ (range fun x => Sigma.snd (inducedMap D) { val := x, property := (_ : x ∈ (inducedMap D).fst) }) ⊆ C",
"tactic": "rintro y ⟨x, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.refine'_1.intro\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : ∀ {x : ℕ → Bool}, x ∈ (inducedMap D).fst\nx : ℕ → Bool\n⊢ (fun x => Sigma.snd (inducedMap D) { val := x, property := (_ : x ∈ (inducedMap D).fst) }) x ∈ C",
"tactic": "exact map_mem ⟨_, hdom⟩ 0"
},
{
"state_after": "case intro.intro.intro.refine'_2\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : ∀ {x : ℕ → Bool}, x ∈ (inducedMap D).fst\n⊢ Continuous fun x => { val := x, property := (_ : x ∈ (inducedMap D).fst) }",
"state_before": "case intro.intro.intro.refine'_2\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : ∀ {x : ℕ → Bool}, x ∈ (inducedMap D).fst\n⊢ Continuous fun x => Sigma.snd (inducedMap D) { val := x, property := (_ : x ∈ (inducedMap D).fst) }",
"tactic": "apply hdiam.map_continuous.comp"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.refine'_2\nα : Type u_1\ninst✝¹ : MetricSpace α\nC : Set α\nhC : Perfect C\nε : ℝ≥0∞\nhnonempty : Set.Nonempty C\ninst✝ : CompleteSpace α\nu : ℕ → ℝ≥0∞\nupos' : ∀ (n : ℕ), u n ∈ Ioo 0 1\nhu : Tendsto u atTop (𝓝 0)\nupos : ∀ (n : ℕ), 0 < u n\nP : Type u_1 := { E // Perfect E ∧ Set.Nonempty E }\nC0 C1 : {C : Set α} → Perfect C → Set.Nonempty C → {ε : ℝ≥0∞} → 0 < ε → Set α\nh0 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C0 hC hnonempty hε) ∧\n Set.Nonempty (C0 hC hnonempty hε) ∧ C0 hC hnonempty hε ⊆ C ∧ EMetric.diam (C0 hC hnonempty hε) ≤ ε\nh1 :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Perfect (C1 hC hnonempty hε) ∧\n Set.Nonempty (C1 hC hnonempty hε) ∧ C1 hC hnonempty hε ⊆ C ∧ EMetric.diam (C1 hC hnonempty hε) ≤ ε\nhdisj :\n ∀ {C : Set α} (hC : Perfect C) (hnonempty : Set.Nonempty C) {ε : ℝ≥0∞} (hε : 0 < ε),\n Disjoint (C0 hC hnonempty hε) (C1 hC hnonempty hε)\nDP : List Bool → P :=\n fun l =>\n List.rec { val := C, property := (_ : Perfect C ∧ Set.Nonempty C) }\n (fun a l ih =>\n Bool.casesOn (motive := fun t => a = t → P) a\n (fun h =>\n (_ : false = a) ▸\n { val := C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C0 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (fun h =>\n (_ : true = a) ▸\n { val := C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))),\n property :=\n (_ :\n Perfect (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l)))) ∧\n Set.Nonempty\n (C1 (_ : Perfect ↑ih) (_ : Set.Nonempty ↑ih) (_ : 0 < u (Nat.succ (List.length l))))) })\n (_ : a = a))\n l\nD : List Bool → Set α := fun l => ↑(DP l)\nhanti : ClosureAntitone D\nhdiam : VanishingDiam D\nhdisj' : CantorScheme.Disjoint D\nhdom : ∀ {x : ℕ → Bool}, x ∈ (inducedMap D).fst\n⊢ Continuous fun x => { val := x, property := (_ : x ∈ (inducedMap D).fst) }",
"tactic": "continuity"
}
] |
[
317,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
268,
1
] |
Mathlib/Topology/MetricSpace/Infsep.lean
|
Set.Nontrivial.infsep_lt_iff
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.59186\ninst✝ : PseudoMetricSpace α\nx y z : α\ns t : Set α\nd : ℝ\nhs : Set.Nontrivial s\n⊢ ¬infsep s < d ↔ ¬∃ x x_1 y x_2 _hxy, dist x y < d",
"state_before": "α : Type u_1\nβ : Type ?u.59186\ninst✝ : PseudoMetricSpace α\nx y z : α\ns t : Set α\nd : ℝ\nhs : Set.Nontrivial s\n⊢ infsep s < d ↔ ∃ x x_1 y x_2 _hxy, dist x y < d",
"tactic": "rw [← not_iff_not]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.59186\ninst✝ : PseudoMetricSpace α\nx y z : α\ns t : Set α\nd : ℝ\nhs : Set.Nontrivial s\n⊢ d ≤ infsep s ↔ ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → x ≠ y → d ≤ dist x y",
"state_before": "α : Type u_1\nβ : Type ?u.59186\ninst✝ : PseudoMetricSpace α\nx y z : α\ns t : Set α\nd : ℝ\nhs : Set.Nontrivial s\n⊢ ¬infsep s < d ↔ ¬∃ x x_1 y x_2 _hxy, dist x y < d",
"tactic": "push_neg"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.59186\ninst✝ : PseudoMetricSpace α\nx y z : α\ns t : Set α\nd : ℝ\nhs : Set.Nontrivial s\n⊢ d ≤ infsep s ↔ ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → x ≠ y → d ≤ dist x y",
"tactic": "exact hs.le_infsep_iff"
}
] |
[
408,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
404,
1
] |
Mathlib/Algebra/Order/Ring/Canonical.lean
|
mul_add_mul_le_mul_add_mul'
|
[
{
"state_after": "α : Type u\nβ : Type ?u.5132\ninst✝¹ : StrictOrderedSemiring α\na b c d : α\ninst✝ : ExistsAddOfLE α\nhba : b ≤ a\nhdc : d ≤ c\n⊢ b • c + a • d ≤ b • d + a • c",
"state_before": "α : Type u\nβ : Type ?u.5132\ninst✝¹ : StrictOrderedSemiring α\na b c d : α\ninst✝ : ExistsAddOfLE α\nhba : b ≤ a\nhdc : d ≤ c\n⊢ a • d + b • c ≤ a • c + b • d",
"tactic": "rw [add_comm (a • d), add_comm (a • c)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type ?u.5132\ninst✝¹ : StrictOrderedSemiring α\na b c d : α\ninst✝ : ExistsAddOfLE α\nhba : b ≤ a\nhdc : d ≤ c\n⊢ b • c + a • d ≤ b • d + a • c",
"tactic": "exact mul_add_mul_le_mul_add_mul hba hdc"
}
] |
[
66,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Infinite.exists_ne_map_eq_of_mapsTo
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns : Set α\nt : Set β\nf : α → β\nhs : Set.Infinite s\nhf : MapsTo f s t\nht : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → x ≠ y → f x ≠ f y\n⊢ ¬Set.Finite t",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns : Set α\nt : Set β\nf : α → β\nhs : Set.Infinite s\nhf : MapsTo f s t\nht : Set.Finite t\n⊢ ∃ x, x ∈ s ∧ ∃ y, y ∈ s ∧ x ≠ y ∧ f x = f y",
"tactic": "contrapose! ht"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\nγ : Type x\ns : Set α\nt : Set β\nf : α → β\nhs : Set.Infinite s\nhf : MapsTo f s t\nht : ∀ (x : α), x ∈ s → ∀ (y : α), y ∈ s → x ≠ y → f x ≠ f y\n⊢ ¬Set.Finite t",
"tactic": "exact infinite_of_injOn_mapsTo (fun x hx y hy => not_imp_not.1 (ht x hx y hy)) hf hs"
}
] |
[
1369,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1366,
1
] |
Mathlib/Topology/FiberBundle/Trivialization.lean
|
Trivialization.proj_symm_apply
|
[] |
[
395,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
394,
1
] |
Mathlib/NumberTheory/Liouville/LiouvilleWith.lean
|
LiouvilleWith.add_rat_iff
|
[
{
"state_after": "no goals",
"state_before": "p q x y : ℝ\nr : ℚ\nm : ℤ\nn : ℕ\nh : LiouvilleWith p (x + ↑r)\n⊢ LiouvilleWith p x",
"tactic": "simpa using h.add_rat (-r)"
}
] |
[
203,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
202,
1
] |
src/lean/Init/Data/Array/BasicAux.lean
|
List.of_toArrayAux_eq_toArrayAux
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nas bs : List α\ncs ds : Array α\nh : toArrayAux as cs = toArrayAux bs ds\nhlen : Array.size cs = Array.size ds\n⊢ as = bs ∧ cs = ds",
"tactic": "match as, bs with\n| [], [] => simp [toArrayAux] at h; simp [h]\n| a::as, [] => simp [toArrayAux] at h; rw [← h] at hlen; simp_arith [size_toArrayAux] at hlen\n| [], b::bs => simp [toArrayAux] at h; rw [h] at hlen; simp_arith [size_toArrayAux] at hlen\n| a::as, b::bs =>\n simp [toArrayAux] at h\n have : (cs.push a).size = (ds.push b).size := by simp [*]\n have ⟨ih₁, ih₂⟩ := of_toArrayAux_eq_toArrayAux h this\n simp [ih₁]\n have := Array.of_push_eq_push ih₂\n simp [this]"
},
{
"state_after": "α : Type u_1\nas bs : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\nh : cs = ds\n⊢ [] = [] ∧ cs = ds",
"state_before": "α : Type u_1\nas bs : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\nh : toArrayAux [] cs = toArrayAux [] ds\n⊢ [] = [] ∧ cs = ds",
"tactic": "simp [toArrayAux] at h"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nas bs : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\nh : cs = ds\n⊢ [] = [] ∧ cs = ds",
"tactic": "simp [h]"
},
{
"state_after": "α : Type u_1\nas✝ bs : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\na : α\nas : List α\nh : toArrayAux as (Array.push cs a) = ds\n⊢ a :: as = [] ∧ cs = ds",
"state_before": "α : Type u_1\nas✝ bs : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\na : α\nas : List α\nh : toArrayAux (a :: as) cs = toArrayAux [] ds\n⊢ a :: as = [] ∧ cs = ds",
"tactic": "simp [toArrayAux] at h"
},
{
"state_after": "α : Type u_1\nas✝ bs : List α\ncs ds : Array α\na : α\nas : List α\nhlen : Array.size cs = Array.size (toArrayAux as (Array.push cs a))\nh : toArrayAux as (Array.push cs a) = ds\n⊢ a :: as = [] ∧ cs = ds",
"state_before": "α : Type u_1\nas✝ bs : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\na : α\nas : List α\nh : toArrayAux as (Array.push cs a) = ds\n⊢ a :: as = [] ∧ cs = ds",
"tactic": "rw [← h] at hlen"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nas✝ bs : List α\ncs ds : Array α\na : α\nas : List α\nhlen : Array.size cs = Array.size (toArrayAux as (Array.push cs a))\nh : toArrayAux as (Array.push cs a) = ds\n⊢ a :: as = [] ∧ cs = ds",
"tactic": "simp_arith [size_toArrayAux] at hlen"
},
{
"state_after": "α : Type u_1\nas bs✝ : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\nb : α\nbs : List α\nh : cs = toArrayAux bs (Array.push ds b)\n⊢ [] = b :: bs ∧ cs = ds",
"state_before": "α : Type u_1\nas bs✝ : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\nb : α\nbs : List α\nh : toArrayAux [] cs = toArrayAux (b :: bs) ds\n⊢ [] = b :: bs ∧ cs = ds",
"tactic": "simp [toArrayAux] at h"
},
{
"state_after": "α : Type u_1\nas bs✝ : List α\ncs ds : Array α\nb : α\nbs : List α\nhlen : Array.size (toArrayAux bs (Array.push ds b)) = Array.size ds\nh : cs = toArrayAux bs (Array.push ds b)\n⊢ [] = b :: bs ∧ cs = ds",
"state_before": "α : Type u_1\nas bs✝ : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\nb : α\nbs : List α\nh : cs = toArrayAux bs (Array.push ds b)\n⊢ [] = b :: bs ∧ cs = ds",
"tactic": "rw [h] at hlen"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nas bs✝ : List α\ncs ds : Array α\nb : α\nbs : List α\nhlen : Array.size (toArrayAux bs (Array.push ds b)) = Array.size ds\nh : cs = toArrayAux bs (Array.push ds b)\n⊢ [] = b :: bs ∧ cs = ds",
"tactic": "simp_arith [size_toArrayAux] at hlen"
},
{
"state_after": "α : Type u_1\nas✝ bs✝ : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\na : α\nas : List α\nb : α\nbs : List α\nh : toArrayAux as (Array.push cs a) = toArrayAux bs (Array.push ds b)\n⊢ a :: as = b :: bs ∧ cs = ds",
"state_before": "α : Type u_1\nas✝ bs✝ : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\na : α\nas : List α\nb : α\nbs : List α\nh : toArrayAux (a :: as) cs = toArrayAux (b :: bs) ds\n⊢ a :: as = b :: bs ∧ cs = ds",
"tactic": "simp [toArrayAux] at h"
},
{
"state_after": "α : Type u_1\nas✝ bs✝ : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\na : α\nas : List α\nb : α\nbs : List α\nh : toArrayAux as (Array.push cs a) = toArrayAux bs (Array.push ds b)\nthis : Array.size (Array.push cs a) = Array.size (Array.push ds b)\n⊢ a :: as = b :: bs ∧ cs = ds",
"state_before": "α : Type u_1\nas✝ bs✝ : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\na : α\nas : List α\nb : α\nbs : List α\nh : toArrayAux as (Array.push cs a) = toArrayAux bs (Array.push ds b)\n⊢ a :: as = b :: bs ∧ cs = ds",
"tactic": "have : (cs.push a).size = (ds.push b).size := by simp [*]"
},
{
"state_after": "α : Type u_1\nas✝ bs✝ : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\na : α\nas : List α\nb : α\nbs : List α\nh : toArrayAux as (Array.push cs a) = toArrayAux bs (Array.push ds b)\nthis : Array.size (Array.push cs a) = Array.size (Array.push ds b)\nih₁ : as = bs\nih₂ : Array.push cs a = Array.push ds b\n⊢ a :: as = b :: bs ∧ cs = ds",
"state_before": "α : Type u_1\nas✝ bs✝ : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\na : α\nas : List α\nb : α\nbs : List α\nh : toArrayAux as (Array.push cs a) = toArrayAux bs (Array.push ds b)\nthis : Array.size (Array.push cs a) = Array.size (Array.push ds b)\n⊢ a :: as = b :: bs ∧ cs = ds",
"tactic": "have ⟨ih₁, ih₂⟩ := of_toArrayAux_eq_toArrayAux h this"
},
{
"state_after": "α : Type u_1\nas✝ bs✝ : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\na : α\nas : List α\nb : α\nbs : List α\nh : toArrayAux as (Array.push cs a) = toArrayAux bs (Array.push ds b)\nthis : Array.size (Array.push cs a) = Array.size (Array.push ds b)\nih₁ : as = bs\nih₂ : Array.push cs a = Array.push ds b\n⊢ a = b ∧ cs = ds",
"state_before": "α : Type u_1\nas✝ bs✝ : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\na : α\nas : List α\nb : α\nbs : List α\nh : toArrayAux as (Array.push cs a) = toArrayAux bs (Array.push ds b)\nthis : Array.size (Array.push cs a) = Array.size (Array.push ds b)\nih₁ : as = bs\nih₂ : Array.push cs a = Array.push ds b\n⊢ a :: as = b :: bs ∧ cs = ds",
"tactic": "simp [ih₁]"
},
{
"state_after": "α : Type u_1\nas✝ bs✝ : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\na : α\nas : List α\nb : α\nbs : List α\nh : toArrayAux as (Array.push cs a) = toArrayAux bs (Array.push ds b)\nthis✝ : Array.size (Array.push cs a) = Array.size (Array.push ds b)\nih₁ : as = bs\nih₂ : Array.push cs a = Array.push ds b\nthis : cs = ds ∧ a = b\n⊢ a = b ∧ cs = ds",
"state_before": "α : Type u_1\nas✝ bs✝ : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\na : α\nas : List α\nb : α\nbs : List α\nh : toArrayAux as (Array.push cs a) = toArrayAux bs (Array.push ds b)\nthis : Array.size (Array.push cs a) = Array.size (Array.push ds b)\nih₁ : as = bs\nih₂ : Array.push cs a = Array.push ds b\n⊢ a = b ∧ cs = ds",
"tactic": "have := Array.of_push_eq_push ih₂"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nas✝ bs✝ : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\na : α\nas : List α\nb : α\nbs : List α\nh : toArrayAux as (Array.push cs a) = toArrayAux bs (Array.push ds b)\nthis✝ : Array.size (Array.push cs a) = Array.size (Array.push ds b)\nih₁ : as = bs\nih₂ : Array.push cs a = Array.push ds b\nthis : cs = ds ∧ a = b\n⊢ a = b ∧ cs = ds",
"tactic": "simp [this]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nas✝ bs✝ : List α\ncs ds : Array α\nhlen : Array.size cs = Array.size ds\na : α\nas : List α\nb : α\nbs : List α\nh : toArrayAux as (Array.push cs a) = toArrayAux bs (Array.push ds b)\n⊢ Array.size (Array.push cs a) = Array.size (Array.push ds b)",
"tactic": "simp [*]"
}
] |
[
33,
16
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
22,
9
] |
Mathlib/SetTheory/Lists.lean
|
Lists'.cons_subset
|
[
{
"state_after": "α : Type u_1\na : Lists α\nl₁ l₂ : Lists' α true\nh : cons a l₁ ⊆ l₂\n⊢ a ∈ l₂ ∧ l₁ ⊆ l₂",
"state_before": "α : Type u_1\na : Lists α\nl₁ l₂ : Lists' α true\n⊢ cons a l₁ ⊆ l₂ ↔ a ∈ l₂ ∧ l₁ ⊆ l₂",
"tactic": "refine' ⟨fun h => _, fun ⟨⟨a', m, e⟩, s⟩ => Subset.cons e m s⟩"
},
{
"state_after": "α : Type u_1\na : Lists α\nl₁ l₂ l₁' : Lists' α true\nh' : cons a l₁ = l₁'\nh : l₁' ⊆ l₂\n⊢ a ∈ l₂ ∧ l₁ ⊆ l₂",
"state_before": "α : Type u_1\na : Lists α\nl₁ l₂ : Lists' α true\nh : cons a l₁ ⊆ l₂\n⊢ a ∈ l₂ ∧ l₁ ⊆ l₂",
"tactic": "generalize h' : Lists'.cons a l₁ = l₁' at h"
},
{
"state_after": "case nil\nα : Type u_1\na : Lists α\nl₁ l₂ : Lists' α true\nh' : cons a l₁ = nil\n⊢ a ∈ l₂ ∧ l₁ ⊆ l₂\n\ncase cons\nα : Type u_1\na : Lists α\nl₁ l₂ : Lists' α true\na' a'' : Lists α\nl : Lists' α true\ne : a' ~ a''\nh' : cons a l₁ = cons a' l\nm : a'' ∈ toList l₂\ns : Subset l l₂\n⊢ a ∈ l₂ ∧ l₁ ⊆ l₂",
"state_before": "α : Type u_1\na : Lists α\nl₁ l₂ l₁' : Lists' α true\nh' : cons a l₁ = l₁'\nh : l₁' ⊆ l₂\n⊢ a ∈ l₂ ∧ l₁ ⊆ l₂",
"tactic": "cases' h with l a' a'' l l' e m s"
},
{
"state_after": "case cons.mk\nα : Type u_1\nl₁ l₂ : Lists' α true\na' a'' : Lists α\nl : Lists' α true\ne : a' ~ a''\nm : a'' ∈ toList l₂\ns : Subset l l₂\nfst✝ : Bool\nsnd✝ : Lists' α fst✝\nh' : cons { fst := fst✝, snd := snd✝ } l₁ = cons a' l\n⊢ { fst := fst✝, snd := snd✝ } ∈ l₂ ∧ l₁ ⊆ l₂",
"state_before": "case cons\nα : Type u_1\na : Lists α\nl₁ l₂ : Lists' α true\na' a'' : Lists α\nl : Lists' α true\ne : a' ~ a''\nh' : cons a l₁ = cons a' l\nm : a'' ∈ toList l₂\ns : Subset l l₂\n⊢ a ∈ l₂ ∧ l₁ ⊆ l₂",
"tactic": "cases a"
},
{
"state_after": "case cons.mk.mk\nα : Type u_1\nl₁ l₂ : Lists' α true\na'' : Lists α\nl : Lists' α true\nm : a'' ∈ toList l₂\ns : Subset l l₂\nfst✝¹ : Bool\nsnd✝¹ : Lists' α fst✝¹\nfst✝ : Bool\nsnd✝ : Lists' α fst✝\ne : { fst := fst✝, snd := snd✝ } ~ a''\nh' : cons { fst := fst✝¹, snd := snd✝¹ } l₁ = cons { fst := fst✝, snd := snd✝ } l\n⊢ { fst := fst✝¹, snd := snd✝¹ } ∈ l₂ ∧ l₁ ⊆ l₂",
"state_before": "case cons.mk\nα : Type u_1\nl₁ l₂ : Lists' α true\na' a'' : Lists α\nl : Lists' α true\ne : a' ~ a''\nm : a'' ∈ toList l₂\ns : Subset l l₂\nfst✝ : Bool\nsnd✝ : Lists' α fst✝\nh' : cons { fst := fst✝, snd := snd✝ } l₁ = cons a' l\n⊢ { fst := fst✝, snd := snd✝ } ∈ l₂ ∧ l₁ ⊆ l₂",
"tactic": "cases a'"
},
{
"state_after": "case cons.mk.mk.refl\nα : Type u_1\nl₁ l₂ : Lists' α true\na'' : Lists α\nm : a'' ∈ toList l₂\nfst✝ : Bool\nsnd✝ : Lists' α fst✝\ne : { fst := fst✝, snd := snd✝ } ~ a''\ns : Subset l₁ l₂\n⊢ { fst := fst✝, snd := snd✝ } ∈ l₂ ∧ l₁ ⊆ l₂",
"state_before": "case cons.mk.mk\nα : Type u_1\nl₁ l₂ : Lists' α true\na'' : Lists α\nl : Lists' α true\nm : a'' ∈ toList l₂\ns : Subset l l₂\nfst✝¹ : Bool\nsnd✝¹ : Lists' α fst✝¹\nfst✝ : Bool\nsnd✝ : Lists' α fst✝\ne : { fst := fst✝, snd := snd✝ } ~ a''\nh' : cons { fst := fst✝¹, snd := snd✝¹ } l₁ = cons { fst := fst✝, snd := snd✝ } l\n⊢ { fst := fst✝¹, snd := snd✝¹ } ∈ l₂ ∧ l₁ ⊆ l₂",
"tactic": "cases h'"
},
{
"state_after": "no goals",
"state_before": "case cons.mk.mk.refl\nα : Type u_1\nl₁ l₂ : Lists' α true\na'' : Lists α\nm : a'' ∈ toList l₂\nfst✝ : Bool\nsnd✝ : Lists' α fst✝\ne : { fst := fst✝, snd := snd✝ } ~ a''\ns : Subset l₁ l₂\n⊢ { fst := fst✝, snd := snd✝ } ∈ l₂ ∧ l₁ ⊆ l₂",
"tactic": "exact ⟨⟨_, m, e⟩, s⟩"
},
{
"state_after": "case nil.mk\nα : Type u_1\nl₁ l₂ : Lists' α true\nfst✝ : Bool\nsnd✝ : Lists' α fst✝\nh' : cons { fst := fst✝, snd := snd✝ } l₁ = nil\n⊢ { fst := fst✝, snd := snd✝ } ∈ l₂ ∧ l₁ ⊆ l₂",
"state_before": "case nil\nα : Type u_1\na : Lists α\nl₁ l₂ : Lists' α true\nh' : cons a l₁ = nil\n⊢ a ∈ l₂ ∧ l₁ ⊆ l₂",
"tactic": "cases a"
},
{
"state_after": "no goals",
"state_before": "case nil.mk\nα : Type u_1\nl₁ l₂ : Lists' α true\nfst✝ : Bool\nsnd✝ : Lists' α fst✝\nh' : cons { fst := fst✝, snd := snd✝ } l₁ = nil\n⊢ { fst := fst✝, snd := snd✝ } ∈ l₂ ∧ l₁ ⊆ l₂",
"tactic": "cases h'"
}
] |
[
174,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/Analysis/Calculus/LHopital.lean
|
HasDerivAt.lhopital_zero_atTop
|
[
{
"state_after": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∃ v, v ∈ atTop ∧ ∀ (y : ℝ), y ∈ v → HasDerivAt f (f' y) y\nhgg' : ∃ v, v ∈ atTop ∧ ∀ (y : ℝ), y ∈ v → HasDerivAt g (g' y) y\nhg' : ∃ v, v ∈ atTop ∧ ∀ (y : ℝ), y ∈ v → g' y ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\n⊢ Tendsto (fun x => f x / g x) atTop l",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∀ᶠ (x : ℝ) in atTop, HasDerivAt f (f' x) x\nhgg' : ∀ᶠ (x : ℝ) in atTop, HasDerivAt g (g' x) x\nhg' : ∀ᶠ (x : ℝ) in atTop, g' x ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\n⊢ Tendsto (fun x => f x / g x) atTop l",
"tactic": "rw [eventually_iff_exists_mem] at *"
},
{
"state_after": "case intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhgg' : ∃ v, v ∈ atTop ∧ ∀ (y : ℝ), y ∈ v → HasDerivAt g (g' y) y\nhg' : ∃ v, v ∈ atTop ∧ ∀ (y : ℝ), y ∈ v → g' y ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atTop\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\n⊢ Tendsto (fun x => f x / g x) atTop l",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhff' : ∃ v, v ∈ atTop ∧ ∀ (y : ℝ), y ∈ v → HasDerivAt f (f' y) y\nhgg' : ∃ v, v ∈ atTop ∧ ∀ (y : ℝ), y ∈ v → HasDerivAt g (g' y) y\nhg' : ∃ v, v ∈ atTop ∧ ∀ (y : ℝ), y ∈ v → g' y ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\n⊢ Tendsto (fun x => f x / g x) atTop l",
"tactic": "rcases hff' with ⟨s₁, hs₁, hff'⟩"
},
{
"state_after": "case intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhg' : ∃ v, v ∈ atTop ∧ ∀ (y : ℝ), y ∈ v → g' y ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atTop\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atTop\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\n⊢ Tendsto (fun x => f x / g x) atTop l",
"state_before": "case intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhgg' : ∃ v, v ∈ atTop ∧ ∀ (y : ℝ), y ∈ v → HasDerivAt g (g' y) y\nhg' : ∃ v, v ∈ atTop ∧ ∀ (y : ℝ), y ∈ v → g' y ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atTop\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\n⊢ Tendsto (fun x => f x / g x) atTop l",
"tactic": "rcases hgg' with ⟨s₂, hs₂, hgg'⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atTop\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atTop\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atTop\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\n⊢ Tendsto (fun x => f x / g x) atTop l",
"state_before": "case intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhg' : ∃ v, v ∈ atTop ∧ ∀ (y : ℝ), y ∈ v → g' y ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atTop\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atTop\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\n⊢ Tendsto (fun x => f x / g x) atTop l",
"tactic": "rcases hg' with ⟨s₃, hs₃, hg'⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atTop\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atTop\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atTop\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\n⊢ Tendsto (fun x => f x / g x) atTop l",
"state_before": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atTop\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atTop\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atTop\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\n⊢ Tendsto (fun x => f x / g x) atTop l",
"tactic": "let s := s₁ ∩ s₂ ∩ s₃"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atTop\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atTop\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atTop\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nhs : s ∈ atTop\n⊢ Tendsto (fun x => f x / g x) atTop l",
"state_before": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atTop\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atTop\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atTop\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\n⊢ Tendsto (fun x => f x / g x) atTop l",
"tactic": "have hs : s ∈ atTop := inter_mem (inter_mem hs₁ hs₂) hs₃"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atTop\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atTop\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atTop\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nhs : ∃ a, ∀ (b : ℝ), b ≥ a → b ∈ s\n⊢ Tendsto (fun x => f x / g x) atTop l",
"state_before": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atTop\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atTop\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atTop\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nhs : s ∈ atTop\n⊢ Tendsto (fun x => f x / g x) atTop l",
"tactic": "rw [mem_atTop_sets] at hs"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl✝ : Filter ℝ\nf f' g g' : ℝ → ℝ\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l✝\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atTop\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atTop\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atTop\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nl : ℝ\nhl : ∀ (b : ℝ), b ≥ l → b ∈ s\n⊢ Tendsto (fun x => f x / g x) atTop l✝",
"state_before": "case intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atTop\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atTop\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atTop\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nhs : ∃ a, ∀ (b : ℝ), b ≥ a → b ∈ s\n⊢ Tendsto (fun x => f x / g x) atTop l",
"tactic": "rcases hs with ⟨l, hl⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl✝ : Filter ℝ\nf f' g g' : ℝ → ℝ\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l✝\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atTop\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atTop\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atTop\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nl : ℝ\nhl : ∀ (b : ℝ), b ≥ l → b ∈ s\nhl' : Ioi l ⊆ s\n⊢ Tendsto (fun x => f x / g x) atTop l✝",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl✝ : Filter ℝ\nf f' g g' : ℝ → ℝ\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l✝\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atTop\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atTop\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atTop\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nl : ℝ\nhl : ∀ (b : ℝ), b ≥ l → b ∈ s\n⊢ Tendsto (fun x => f x / g x) atTop l✝",
"tactic": "have hl' : Ioi l ⊆ s := fun x hx => hl x (le_of_lt hx)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\na b : ℝ\nhab : a < b\nl✝ : Filter ℝ\nf f' g g' : ℝ → ℝ\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l✝\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atTop\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atTop\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atTop\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nl : ℝ\nhl : ∀ (b : ℝ), b ≥ l → b ∈ s\nhl' : Ioi l ⊆ s\n⊢ Tendsto (fun x => f x / g x) atTop l✝",
"tactic": "refine' lhopital_zero_atTop_on_Ioi _ _ (fun x hx => hg' x <| (hl' hx).2) hftop hgtop hdiv <;>\n intro x hx <;> apply_assumption <;> first | exact (hl' hx).1.1| exact (hl' hx).1.2"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.refine'_2.a\na b : ℝ\nhab : a < b\nl✝ : Filter ℝ\nf f' g g' : ℝ → ℝ\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l✝\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atTop\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atTop\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atTop\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nl : ℝ\nhl : ∀ (b : ℝ), b ≥ l → b ∈ s\nhl' : Ioi l ⊆ s\nx : ℝ\nhx : x ∈ Ioi l\n⊢ x ∈ s₂",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.refine'_2.a\na b : ℝ\nhab : a < b\nl✝ : Filter ℝ\nf f' g g' : ℝ → ℝ\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l✝\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atTop\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atTop\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atTop\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nl : ℝ\nhl : ∀ (b : ℝ), b ≥ l → b ∈ s\nhl' : Ioi l ⊆ s\nx : ℝ\nhx : x ∈ Ioi l\n⊢ x ∈ s₂",
"tactic": "exact (hl' hx).1.1"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.refine'_2.a\na b : ℝ\nhab : a < b\nl✝ : Filter ℝ\nf f' g g' : ℝ → ℝ\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => f' x / g' x) atTop l✝\ns₁ : Set ℝ\nhs₁ : s₁ ∈ atTop\nhff' : ∀ (y : ℝ), y ∈ s₁ → HasDerivAt f (f' y) y\ns₂ : Set ℝ\nhs₂ : s₂ ∈ atTop\nhgg' : ∀ (y : ℝ), y ∈ s₂ → HasDerivAt g (g' y) y\ns₃ : Set ℝ\nhs₃ : s₃ ∈ atTop\nhg' : ∀ (y : ℝ), y ∈ s₃ → g' y ≠ 0\ns : Set ℝ := s₁ ∩ s₂ ∩ s₃\nl : ℝ\nhl : ∀ (b : ℝ), b ≥ l → b ∈ s\nhl' : Ioi l ⊆ s\nx : ℝ\nhx : x ∈ Ioi l\n⊢ x ∈ s₂",
"tactic": "exact (hl' hx).1.2"
}
] |
[
357,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
1
] |
Mathlib/Topology/Algebra/Order/LiminfLimsup.lean
|
iUnion_Iic_eq_Iio_of_lt_of_tendsto
|
[] |
[
450,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
447,
1
] |
Mathlib/Topology/MetricSpace/PiNat.lean
|
PiNat.apply_eq_of_lt_firstDiff
|
[
{
"state_after": "E : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhn : n < if h : x ≠ y then Nat.find (_ : ∃ a, x a ≠ y a) else 0\n⊢ x n = y n",
"state_before": "E : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhn : n < firstDiff x y\n⊢ x n = y n",
"tactic": "rw [firstDiff_def] at hn"
},
{
"state_after": "case inl\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nh : x ≠ y\nhn : n < Nat.find (_ : ∃ a, x a ≠ y a)\n⊢ x n = y n\n\ncase inr\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nh : ¬x ≠ y\nhn : n < 0\n⊢ x n = y n",
"state_before": "E : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nhn : n < if h : x ≠ y then Nat.find (_ : ∃ a, x a ≠ y a) else 0\n⊢ x n = y n",
"tactic": "split_ifs at hn with h"
},
{
"state_after": "case a\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nh : x ≠ y\nhn : n < Nat.find (_ : ∃ a, x a ≠ y a)\n⊢ x n = y n ↔ ¬x n ≠ y n",
"state_before": "case inl\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nh : x ≠ y\nhn : n < Nat.find (_ : ∃ a, x a ≠ y a)\n⊢ x n = y n",
"tactic": "convert Nat.find_min (ne_iff.1 h) hn"
},
{
"state_after": "no goals",
"state_before": "case a\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nh : x ≠ y\nhn : n < Nat.find (_ : ∃ a, x a ≠ y a)\n⊢ x n = y n ↔ ¬x n ≠ y n",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case inr\nE : ℕ → Type u_1\nx y : (n : ℕ) → E n\nn : ℕ\nh : ¬x ≠ y\nhn : n < 0\n⊢ x n = y n",
"tactic": "exact (not_lt_zero' hn).elim"
}
] |
[
87,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.preimage_sub_const_Ioi
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedAddCommGroup α\na b c : α\n⊢ (fun x => x - a) ⁻¹' Ioi b = Ioi (b + a)",
"tactic": "simp [sub_eq_add_neg]"
}
] |
[
182,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
181,
1
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.