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Mathlib/Algebra/Module/Submodule/Lattice.lean | Submodule.mem_sSup_of_mem | [
{
"state_after": "R : Type u_2\nS✝ : Type ?u.168464\nM : Type u_1\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S✝ M\ninst✝¹ : SMul S✝ R\ninst✝ : IsScalarTower S✝ R M\np q : Submodule R M\nS : Set (Submodule R M)\ns : Submodule R M\nhs : s ∈ S\nthis : s ≤ sSup S\n⊢ ∀ {x : M}, x ∈ s → x ∈ sSup S",
"state_before": "R : Type u_2\nS✝ : Type ?u.168464\nM : Type u_1\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S✝ M\ninst✝¹ : SMul S✝ R\ninst✝ : IsScalarTower S✝ R M\np q : Submodule R M\nS : Set (Submodule R M)\ns : Submodule R M\nhs : s ∈ S\n⊢ ∀ {x : M}, x ∈ s → x ∈ sSup S",
"tactic": "have := le_sSup hs"
},
{
"state_after": "R : Type u_2\nS✝ : Type ?u.168464\nM : Type u_1\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S✝ M\ninst✝¹ : SMul S✝ R\ninst✝ : IsScalarTower S✝ R M\np q : Submodule R M\nS : Set (Submodule R M)\ns : Submodule R M\nhs : s ∈ S\nthis : Preorder.toLE.1 s (sSup S)\n⊢ ∀ {x : M}, x ∈ s → x ∈ sSup S",
"state_before": "R : Type u_2\nS✝ : Type ?u.168464\nM : Type u_1\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S✝ M\ninst✝¹ : SMul S✝ R\ninst✝ : IsScalarTower S✝ R M\np q : Submodule R M\nS : Set (Submodule R M)\ns : Submodule R M\nhs : s ∈ S\nthis : s ≤ sSup S\n⊢ ∀ {x : M}, x ∈ s → x ∈ sSup S",
"tactic": "rw [LE.le] at this"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nS✝ : Type ?u.168464\nM : Type u_1\ninst✝⁶ : Semiring R\ninst✝⁵ : Semiring S✝\ninst✝⁴ : AddCommMonoid M\ninst✝³ : Module R M\ninst✝² : Module S✝ M\ninst✝¹ : SMul S✝ R\ninst✝ : IsScalarTower S✝ R M\np q : Submodule R M\nS : Set (Submodule R M)\ns : Submodule R M\nhs : s ∈ S\nthis : Preorder.toLE.1 s (sSup S)\n⊢ ∀ {x : M}, x ∈ s → x ∈ sSup S",
"tactic": "exact this"
}
] | [
324,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
320,
1
] |
Mathlib/Topology/Algebra/UniformGroup.lean | TopologicalGroup.tendstoUniformly_iff | [] | [
618,
54
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
613,
1
] |
Mathlib/Data/Finset/Pointwise.lean | Finset.image_smul_product | [] | [
1275,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1274,
1
] |
Mathlib/Topology/Order.lean | isClosed_induced_iff' | [
{
"state_after": "α : Type u_2\nβ : Type u_1\nt✝ : TopologicalSpace β\nf✝ : α → β\nt : TopologicalSpace β\nf : α → β\ns : Set α\nthis : TopologicalSpace α := induced f t\n⊢ IsClosed s ↔ ∀ (a : α), f a ∈ closure (f '' s) → a ∈ s",
"state_before": "α : Type u_2\nβ : Type u_1\nt✝ : TopologicalSpace β\nf✝ : α → β\nt : TopologicalSpace β\nf : α → β\ns : Set α\n⊢ IsClosed s ↔ ∀ (a : α), f a ∈ closure (f '' s) → a ∈ s",
"tactic": "letI := t.induced f"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nt✝ : TopologicalSpace β\nf✝ : α → β\nt : TopologicalSpace β\nf : α → β\ns : Set α\nthis : TopologicalSpace α := induced f t\n⊢ IsClosed s ↔ ∀ (a : α), f a ∈ closure (f '' s) → a ∈ s",
"tactic": "simp only [← closure_subset_iff_isClosed, subset_def, closure_induced]"
}
] | [
891,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
888,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean | ContDiff.differentiable | [] | [
1477,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1476,
1
] |
Mathlib/Data/Polynomial/HasseDeriv.lean | Polynomial.hasseDeriv_apply | [
{
"state_after": "R : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\n⊢ (sum f fun n r => ↑(monomial (n - k)) (choose n k • r)) = sum f fun i r => ↑(monomial (i - k)) (↑(choose i k) * r)",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\n⊢ ↑(hasseDeriv k) f = sum f fun i r => ↑(monomial (i - k)) (↑(choose i k) * r)",
"tactic": "dsimp [hasseDeriv]"
},
{
"state_after": "case e_f\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\n⊢ (fun n r => ↑(monomial (n - k)) (choose n k • r)) = fun i r => ↑(monomial (i - k)) (↑(choose i k) * r)",
"state_before": "R : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\n⊢ (sum f fun n r => ↑(monomial (n - k)) (choose n k • r)) = sum f fun i r => ↑(monomial (i - k)) (↑(choose i k) * r)",
"tactic": "congr"
},
{
"state_after": "case e_f.h.h.a\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nx✝¹ : ℕ\nx✝ : R\nn✝ : ℕ\n⊢ coeff (↑(monomial (x✝¹ - k)) (choose x✝¹ k • x✝)) n✝ = coeff (↑(monomial (x✝¹ - k)) (↑(choose x✝¹ k) * x✝)) n✝",
"state_before": "case e_f\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\n⊢ (fun n r => ↑(monomial (n - k)) (choose n k • r)) = fun i r => ↑(monomial (i - k)) (↑(choose i k) * r)",
"tactic": "ext"
},
{
"state_after": "case e_f.h.h.a.e_a.h.e_6.h\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nx✝¹ : ℕ\nx✝ : R\nn✝ : ℕ\n⊢ choose x✝¹ k • x✝ = ↑(choose x✝¹ k) * x✝",
"state_before": "case e_f.h.h.a\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nx✝¹ : ℕ\nx✝ : R\nn✝ : ℕ\n⊢ coeff (↑(monomial (x✝¹ - k)) (choose x✝¹ k • x✝)) n✝ = coeff (↑(monomial (x✝¹ - k)) (↑(choose x✝¹ k) * x✝)) n✝",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_f.h.h.a.e_a.h.e_6.h\nR : Type u_1\ninst✝ : Semiring R\nk : ℕ\nf : R[X]\nx✝¹ : ℕ\nx✝ : R\nn✝ : ℕ\n⊢ choose x✝¹ k • x✝ = ↑(choose x✝¹ k) * x✝",
"tactic": "apply nsmul_eq_mul"
}
] | [
69,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
65,
1
] |
Mathlib/LinearAlgebra/Quotient.lean | Submodule.mkQ_apply | [] | [
327,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
326,
1
] |
Mathlib/Dynamics/FixedPoints/Topology.lean | isFixedPt_of_tendsto_iterate | [
{
"state_after": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T2Space α\nf : α → α\nx y : α\nhy : Tendsto (fun n => (f^[n]) x) atTop (𝓝 y)\nhf : ContinuousAt f y\n⊢ Tendsto (fun n => (f^[n + 1]) x) atTop (𝓝 (f y))",
"state_before": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T2Space α\nf : α → α\nx y : α\nhy : Tendsto (fun n => (f^[n]) x) atTop (𝓝 y)\nhf : ContinuousAt f y\n⊢ IsFixedPt f y",
"tactic": "refine' tendsto_nhds_unique ((tendsto_add_atTop_iff_nat 1).1 _) hy"
},
{
"state_after": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T2Space α\nf : α → α\nx y : α\nhy : Tendsto (fun n => (f^[n]) x) atTop (𝓝 y)\nhf : ContinuousAt f y\n⊢ Tendsto (fun n => (f ∘ f^[n]) x) atTop (𝓝 (f y))",
"state_before": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T2Space α\nf : α → α\nx y : α\nhy : Tendsto (fun n => (f^[n]) x) atTop (𝓝 y)\nhf : ContinuousAt f y\n⊢ Tendsto (fun n => (f^[n + 1]) x) atTop (𝓝 (f y))",
"tactic": "simp only [iterate_succ' f]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : TopologicalSpace α\ninst✝ : T2Space α\nf : α → α\nx y : α\nhy : Tendsto (fun n => (f^[n]) x) atTop (𝓝 y)\nhf : ContinuousAt f y\n⊢ Tendsto (fun n => (f ∘ f^[n]) x) atTop (𝓝 (f y))",
"tactic": "exact hf.tendsto.comp hy"
}
] | [
40,
27
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
36,
1
] |
Mathlib/Data/Finsupp/Defs.lean | Finsupp.update_eq_sub_add_single | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.565928\nγ : Type ?u.565931\nι : Type ?u.565934\nM : Type ?u.565937\nM' : Type ?u.565940\nN : Type ?u.565943\nP : Type ?u.565946\nG : Type u_1\nH : Type ?u.565952\nR : Type ?u.565955\nS : Type ?u.565958\ninst✝ : AddGroup G\nf : α →₀ G\na : α\nb : G\n⊢ update f a b = f - single a (↑f a) + single a b",
"tactic": "rw [update_eq_erase_add_single, erase_eq_sub_single]"
}
] | [
1318,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1316,
1
] |
Mathlib/GroupTheory/Perm/List.lean | List.formPerm_pow_length_eq_one_of_nodup | [
{
"state_after": "case H\nα : Type u_1\nβ : Type ?u.842481\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nhl : Nodup l\nx : α\n⊢ ↑(formPerm l ^ length l) x = ↑1 x",
"state_before": "α : Type u_1\nβ : Type ?u.842481\ninst✝ : DecidableEq α\nl : List α\nx : α\nhl : Nodup l\n⊢ formPerm l ^ length l = 1",
"tactic": "ext x"
},
{
"state_after": "case pos\nα : Type u_1\nβ : Type ?u.842481\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nhl : Nodup l\nx : α\nhx : x ∈ l\n⊢ ↑(formPerm l ^ length l) x = ↑1 x\n\ncase neg\nα : Type u_1\nβ : Type ?u.842481\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nhl : Nodup l\nx : α\nhx : ¬x ∈ l\n⊢ ↑(formPerm l ^ length l) x = ↑1 x",
"state_before": "case H\nα : Type u_1\nβ : Type ?u.842481\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nhl : Nodup l\nx : α\n⊢ ↑(formPerm l ^ length l) x = ↑1 x",
"tactic": "by_cases hx : x ∈ l"
},
{
"state_after": "case pos.intro.intro\nα : Type u_1\nβ : Type ?u.842481\ninst✝ : DecidableEq α\nl : List α\nx : α\nhl : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\n⊢ ↑(formPerm l ^ length l) (nthLe l k hk) = ↑1 (nthLe l k hk)",
"state_before": "case pos\nα : Type u_1\nβ : Type ?u.842481\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nhl : Nodup l\nx : α\nhx : x ∈ l\n⊢ ↑(formPerm l ^ length l) x = ↑1 x",
"tactic": "obtain ⟨k, hk, rfl⟩ := nthLe_of_mem hx"
},
{
"state_after": "no goals",
"state_before": "case pos.intro.intro\nα : Type u_1\nβ : Type ?u.842481\ninst✝ : DecidableEq α\nl : List α\nx : α\nhl : Nodup l\nk : ℕ\nhk : k < length l\nhx : nthLe l k hk ∈ l\n⊢ ↑(formPerm l ^ length l) (nthLe l k hk) = ↑1 (nthLe l k hk)",
"tactic": "simp [formPerm_pow_apply_nthLe _ hl, Nat.mod_eq_of_lt hk]"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type ?u.842481\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nhl : Nodup l\nx : α\nhx : ¬x ∈ l\nthis : ¬x ∈ {x | ↑(formPerm l ^ length l) x ≠ x}\n⊢ ↑(formPerm l ^ length l) x = ↑1 x",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.842481\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nhl : Nodup l\nx : α\nhx : ¬x ∈ l\n⊢ ↑(formPerm l ^ length l) x = ↑1 x",
"tactic": "have : x ∉ { x | (l.formPerm ^ l.length) x ≠ x } := by\n intro H\n refine' hx _\n replace H := set_support_zpow_subset l.formPerm l.length H\n simpa using support_formPerm_le' _ H"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nβ : Type ?u.842481\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nhl : Nodup l\nx : α\nhx : ¬x ∈ l\nthis : ¬x ∈ {x | ↑(formPerm l ^ length l) x ≠ x}\n⊢ ↑(formPerm l ^ length l) x = ↑1 x",
"tactic": "simpa using this"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.842481\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nhl : Nodup l\nx : α\nhx : ¬x ∈ l\nH : x ∈ {x | ↑(formPerm l ^ length l) x ≠ x}\n⊢ False",
"state_before": "α : Type u_1\nβ : Type ?u.842481\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nhl : Nodup l\nx : α\nhx : ¬x ∈ l\n⊢ ¬x ∈ {x | ↑(formPerm l ^ length l) x ≠ x}",
"tactic": "intro H"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.842481\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nhl : Nodup l\nx : α\nhx : ¬x ∈ l\nH : x ∈ {x | ↑(formPerm l ^ length l) x ≠ x}\n⊢ x ∈ l",
"state_before": "α : Type u_1\nβ : Type ?u.842481\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nhl : Nodup l\nx : α\nhx : ¬x ∈ l\nH : x ∈ {x | ↑(formPerm l ^ length l) x ≠ x}\n⊢ False",
"tactic": "refine' hx _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.842481\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nhl : Nodup l\nx : α\nhx : ¬x ∈ l\nH : x ∈ {x | ↑(formPerm l) x ≠ x}\n⊢ x ∈ l",
"state_before": "α : Type u_1\nβ : Type ?u.842481\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nhl : Nodup l\nx : α\nhx : ¬x ∈ l\nH : x ∈ {x | ↑(formPerm l ^ length l) x ≠ x}\n⊢ x ∈ l",
"tactic": "replace H := set_support_zpow_subset l.formPerm l.length H"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.842481\ninst✝ : DecidableEq α\nl : List α\nx✝ : α\nhl : Nodup l\nx : α\nhx : ¬x ∈ l\nH : x ∈ {x | ↑(formPerm l) x ≠ x}\n⊢ x ∈ l",
"tactic": "simpa using support_formPerm_le' _ H"
}
] | [
468,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
458,
1
] |
Mathlib/Data/ULift.lean | PLift.up_bijective | [] | [
53,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
52,
1
] |
Mathlib/Algebra/Order/Field/Basic.lean | mul_sub_mul_div_mul_nonpos_iff | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.193207\nα : Type u_1\nβ : Type ?u.193213\ninst✝ : LinearOrderedField α\na b c d : α\nn : ℤ\nhc : c ≠ 0\nhd : d ≠ 0\n⊢ (a * d - b * c) / (c * d) ≤ 0 ↔ a / c ≤ b / d",
"tactic": "rw [mul_comm b c, ← div_sub_div _ _ hc hd, sub_nonpos]"
}
] | [
962,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
960,
1
] |
Mathlib/Algebra/MonoidAlgebra/ToDirectSum.lean | AddMonoidAlgebra.toDirectSum_add | [] | [
131,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
129,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Images.lean | CategoryTheory.Limits.as_factorThruImage | [] | [
335,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
334,
1
] |
Mathlib/Topology/Algebra/ContinuousAffineMap.lean | ContinuousAffineMap.congr_fun | [] | [
92,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
91,
1
] |
Mathlib/Order/Lattice.lean | inf_comm | [] | [
496,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
495,
1
] |
Mathlib/Order/CompleteLattice.lean | iSup_const | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.86729\nβ₂ : Type ?u.86732\nγ : Type ?u.86735\nι : Sort u_1\nι' : Sort ?u.86741\nκ : ι → Sort ?u.86746\nκ' : ι' → Sort ?u.86751\ninst✝¹ : CompleteLattice α\nf g s t : ι → α\na b : α\ninst✝ : Nonempty ι\n⊢ (⨆ (x : ι), a) = a",
"tactic": "rw [iSup, range_const, sSup_singleton]"
}
] | [
1044,
96
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1044,
1
] |
Mathlib/Topology/MetricSpace/PiNat.lean | PiNat.exists_disjoint_cylinder | [
{
"state_after": "case inl\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\nx : (n : ℕ) → E n\nhs : IsClosed ∅\nhx : ¬x ∈ ∅\n⊢ ∃ n, Disjoint ∅ (cylinder x n)\n\ncase inr\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx : (n : ℕ) → E n\nhx : ¬x ∈ s\nhne : Set.Nonempty s\n⊢ ∃ n, Disjoint s (cylinder x n)",
"state_before": "E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx : (n : ℕ) → E n\nhx : ¬x ∈ s\n⊢ ∃ n, Disjoint s (cylinder x n)",
"tactic": "rcases eq_empty_or_nonempty s with (rfl | hne)"
},
{
"state_after": "case inr\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx : (n : ℕ) → E n\nhx : ¬x ∈ s\nhne : Set.Nonempty s\nA : 0 < infDist x s\n⊢ ∃ n, Disjoint s (cylinder x n)",
"state_before": "case inr\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx : (n : ℕ) → E n\nhx : ¬x ∈ s\nhne : Set.Nonempty s\n⊢ ∃ n, Disjoint s (cylinder x n)",
"tactic": "have A : 0 < infDist x s := (hs.not_mem_iff_infDist_pos hne).1 hx"
},
{
"state_after": "case inr.intro\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx : (n : ℕ) → E n\nhx : ¬x ∈ s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : ℕ\nhn : (1 / 2) ^ n < infDist x s\n⊢ ∃ n, Disjoint s (cylinder x n)",
"state_before": "case inr\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx : (n : ℕ) → E n\nhx : ¬x ∈ s\nhne : Set.Nonempty s\nA : 0 < infDist x s\n⊢ ∃ n, Disjoint s (cylinder x n)",
"tactic": "obtain ⟨n, hn⟩ : ∃ n, (1 / 2 : ℝ) ^ n < infDist x s := exists_pow_lt_of_lt_one A one_half_lt_one"
},
{
"state_after": "case inr.intro\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx : (n : ℕ) → E n\nhx : ¬x ∈ s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : ℕ\nhn : (1 / 2) ^ n < infDist x s\ny : (n : ℕ) → E n\nys : y ∈ s\nhy : y ∈ cylinder x n\n⊢ False",
"state_before": "case inr.intro\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx : (n : ℕ) → E n\nhx : ¬x ∈ s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : ℕ\nhn : (1 / 2) ^ n < infDist x s\n⊢ ∃ n, Disjoint s (cylinder x n)",
"tactic": "refine' ⟨n, disjoint_left.2 fun y ys hy => ?_⟩"
},
{
"state_after": "case inr.intro\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx : (n : ℕ) → E n\nhx : ¬x ∈ s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : ℕ\nhn : (1 / 2) ^ n < infDist x s\ny : (n : ℕ) → E n\nys : y ∈ s\nhy : y ∈ cylinder x n\n⊢ infDist x s < infDist x s",
"state_before": "case inr.intro\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx : (n : ℕ) → E n\nhx : ¬x ∈ s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : ℕ\nhn : (1 / 2) ^ n < infDist x s\ny : (n : ℕ) → E n\nys : y ∈ s\nhy : y ∈ cylinder x n\n⊢ False",
"tactic": "apply lt_irrefl (infDist x s)"
},
{
"state_after": "no goals",
"state_before": "case inr.intro\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx : (n : ℕ) → E n\nhx : ¬x ∈ s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : ℕ\nhn : (1 / 2) ^ n < infDist x s\ny : (n : ℕ) → E n\nys : y ∈ s\nhy : y ∈ cylinder x n\n⊢ infDist x s < infDist x s",
"tactic": "calc\n infDist x s ≤ dist x y := infDist_le_dist_of_mem ys\n _ ≤ (1 / 2) ^ n := by\n rw [mem_cylinder_comm] at hy\n exact mem_cylinder_iff_dist_le.1 hy\n _ < infDist x s := hn"
},
{
"state_after": "no goals",
"state_before": "case inl\nE : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\nx : (n : ℕ) → E n\nhs : IsClosed ∅\nhx : ¬x ∈ ∅\n⊢ ∃ n, Disjoint ∅ (cylinder x n)",
"tactic": "exact ⟨0, by simp⟩"
},
{
"state_after": "no goals",
"state_before": "E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\nx : (n : ℕ) → E n\nhs : IsClosed ∅\nhx : ¬x ∈ ∅\n⊢ Disjoint ∅ (cylinder x 0)",
"tactic": "simp"
},
{
"state_after": "E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx : (n : ℕ) → E n\nhx : ¬x ∈ s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : ℕ\nhn : (1 / 2) ^ n < infDist x s\ny : (n : ℕ) → E n\nys : y ∈ s\nhy : x ∈ cylinder y n\n⊢ dist x y ≤ (1 / 2) ^ n",
"state_before": "E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx : (n : ℕ) → E n\nhx : ¬x ∈ s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : ℕ\nhn : (1 / 2) ^ n < infDist x s\ny : (n : ℕ) → E n\nys : y ∈ s\nhy : y ∈ cylinder x n\n⊢ dist x y ≤ (1 / 2) ^ n",
"tactic": "rw [mem_cylinder_comm] at hy"
},
{
"state_after": "no goals",
"state_before": "E : ℕ → Type u_1\ninst✝¹ : (n : ℕ) → TopologicalSpace (E n)\ninst✝ : ∀ (n : ℕ), DiscreteTopology (E n)\ns : Set ((n : ℕ) → E n)\nhs : IsClosed s\nx : (n : ℕ) → E n\nhx : ¬x ∈ s\nhne : Set.Nonempty s\nA : 0 < infDist x s\nn : ℕ\nhn : (1 / 2) ^ n < infDist x s\ny : (n : ℕ) → E n\nys : y ∈ s\nhy : x ∈ cylinder y n\n⊢ dist x y ≤ (1 / 2) ^ n",
"tactic": "exact mem_cylinder_iff_dist_le.1 hy"
}
] | [
503,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
490,
1
] |
Mathlib/Topology/UniformSpace/Cauchy.lean | CauchySeq.eventually_eventually | [] | [
250,
53
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
248,
1
] |
Mathlib/RingTheory/AdjoinRoot.lean | AdjoinRoot.isDomain_of_prime | [] | [
375,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
373,
1
] |
Mathlib/Analysis/Calculus/LHopital.lean | deriv.lhopital_zero_atTop_on_Ioi | [
{
"state_after": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf✝ : DifferentiableOn ℝ f (Ioi a)\nhg' : ∀ (x : ℝ), x ∈ Ioi a → deriv g x ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l\nhdf : ∀ (x : ℝ), x ∈ Ioi a → DifferentiableAt ℝ f x\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\nhdf : DifferentiableOn ℝ f (Ioi a)\nhg' : ∀ (x : ℝ), x ∈ Ioi a → deriv g x ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l\n⊢ Tendsto (fun x => f x / g x) atTop l",
"tactic": "have hdf : ∀ x ∈ Ioi a, DifferentiableAt ℝ f x := fun x hx =>\n (hdf x hx).differentiableAt (Ioi_mem_nhds hx)"
},
{
"state_after": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf✝ : DifferentiableOn ℝ f (Ioi a)\nhg' : ∀ (x : ℝ), x ∈ Ioi a → deriv g x ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l\nhdf : ∀ (x : ℝ), x ∈ Ioi a → DifferentiableAt ℝ f x\nhdg : ∀ (x : ℝ), x ∈ Ioi a → DifferentiableAt ℝ g x\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\nhdf✝ : DifferentiableOn ℝ f (Ioi a)\nhg' : ∀ (x : ℝ), x ∈ Ioi a → deriv g x ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l\nhdf : ∀ (x : ℝ), x ∈ Ioi a → DifferentiableAt ℝ f x\n⊢ Tendsto (fun x => f x / g x) atTop l",
"tactic": "have hdg : ∀ x ∈ Ioi a, DifferentiableAt ℝ g x := fun x hx =>\n by_contradiction fun h => hg' x hx (deriv_zero_of_not_differentiableAt h)"
},
{
"state_after": "no goals",
"state_before": "a b : ℝ\nhab : a < b\nl : Filter ℝ\nf f' g g' : ℝ → ℝ\nhdf✝ : DifferentiableOn ℝ f (Ioi a)\nhg' : ∀ (x : ℝ), x ∈ Ioi a → deriv g x ≠ 0\nhftop : Tendsto f atTop (𝓝 0)\nhgtop : Tendsto g atTop (𝓝 0)\nhdiv : Tendsto (fun x => deriv f x / deriv g x) atTop l\nhdf : ∀ (x : ℝ), x ∈ Ioi a → DifferentiableAt ℝ f x\nhdg : ∀ (x : ℝ), x ∈ Ioi a → DifferentiableAt ℝ g x\n⊢ Tendsto (fun x => f x / g x) atTop l",
"tactic": "exact HasDerivAt.lhopital_zero_atTop_on_Ioi (fun x hx => (hdf x hx).hasDerivAt)\n (fun x hx => (hdg x hx).hasDerivAt) hg' hftop hgtop hdiv"
}
] | [
257,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
248,
1
] |
Mathlib/Topology/Instances/TrivSqZeroExt.lean | TrivSqZeroExt.continuous_snd | [] | [
66,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
65,
8
] |
Mathlib/SetTheory/Game/PGame.lean | PGame.lt_neg_iff | [
{
"state_after": "no goals",
"state_before": "x y : PGame\n⊢ y < -x ↔ x < -y",
"tactic": "rw [← neg_neg x, neg_lt_neg_iff, neg_neg]"
}
] | [
1368,
99
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1368,
1
] |
Mathlib/CategoryTheory/Extensive.lean | CategoryTheory.FinitaryExtensive.mono_inl_of_isColimit | [] | [
251,
73
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
249,
1
] |
Mathlib/Algebra/DirectSum/Decomposition.lean | DirectSum.decompose_zero | [] | [
147,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
146,
1
] |
Mathlib/Data/Rat/NNRat.lean | Rat.toNNRat_lt_toNNRat_iff | [] | [
370,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
369,
1
] |
Mathlib/Analysis/Calculus/Deriv/Pow.lean | differentiableAt_pow | [] | [
75,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
74,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Biproducts.lean | CategoryTheory.Limits.biprod.lift_fst | [] | [
1365,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1363,
1
] |
Mathlib/Order/Filter/Pointwise.lean | Filter.comap_mul_comap_le | [] | [
628,
77
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
625,
1
] |
Mathlib/Data/Analysis/Topology.lean | Ctop.Realizer.ext | [] | [
173,
79
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
171,
1
] |
Mathlib/CategoryTheory/Equivalence.lean | CategoryTheory.Equivalence.pow_one | [] | [
461,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
460,
1
] |
Mathlib/Order/RelIso/Basic.lean | RelEmbedding.acc | [
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.41887\nδ : Type ?u.41890\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nu : δ → δ → Prop\nf : r ↪r s\na : α\nb : β\nh : ↑f a = b\n⊢ Acc s b → Acc r a",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.41887\nδ : Type ?u.41890\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nu : δ → δ → Prop\nf : r ↪r s\na : α\n⊢ Acc s (↑f a) → Acc r a",
"tactic": "generalize h : f a = b"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.41887\nδ : Type ?u.41890\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nu : δ → δ → Prop\nf : r ↪r s\na : α\nb : β\nh : ↑f a = b\nac : Acc s b\n⊢ Acc r a",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.41887\nδ : Type ?u.41890\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nu : δ → δ → Prop\nf : r ↪r s\na : α\nb : β\nh : ↑f a = b\n⊢ Acc s b → Acc r a",
"tactic": "intro ac"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.41887\nδ : Type ?u.41890\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nu : δ → δ → Prop\nf : r ↪r s\na✝ : α\nb : β\nh✝ : ↑f a✝ = b\nx✝ : β\nH : ∀ (y : β), s y x✝ → Acc s y\nIH : ∀ (y : β), s y x✝ → ∀ (a : α), ↑f a = y → Acc r a\na : α\nh : ↑f a = x✝\n⊢ Acc r a",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.41887\nδ : Type ?u.41890\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nu : δ → δ → Prop\nf : r ↪r s\na : α\nb : β\nh : ↑f a = b\nac : Acc s b\n⊢ Acc r a",
"tactic": "induction' ac with _ H IH generalizing a"
},
{
"state_after": "case intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.41887\nδ : Type ?u.41890\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nu : δ → δ → Prop\nf : r ↪r s\na✝ : α\nb : β\nh : ↑f a✝ = b\na : α\nH : ∀ (y : β), s y (↑f a) → Acc s y\nIH : ∀ (y : β), s y (↑f a) → ∀ (a : α), ↑f a = y → Acc r a\n⊢ Acc r a",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.41887\nδ : Type ?u.41890\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nu : δ → δ → Prop\nf : r ↪r s\na✝ : α\nb : β\nh✝ : ↑f a✝ = b\nx✝ : β\nH : ∀ (y : β), s y x✝ → Acc s y\nIH : ∀ (y : β), s y x✝ → ∀ (a : α), ↑f a = y → Acc r a\na : α\nh : ↑f a = x✝\n⊢ Acc r a",
"tactic": "subst h"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.41887\nδ : Type ?u.41890\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\nu : δ → δ → Prop\nf : r ↪r s\na✝ : α\nb : β\nh : ↑f a✝ = b\na : α\nH : ∀ (y : β), s y (↑f a) → Acc s y\nIH : ∀ (y : β), s y (↑f a) → ∀ (a : α), ↑f a = y → Acc r a\n⊢ Acc r a",
"tactic": "exact ⟨_, fun a' h => IH (f a') (f.map_rel_iff.2 h) _ rfl⟩"
}
] | [
396,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
391,
11
] |
Mathlib/Computability/NFA.lean | NFA.evalFrom_nil | [] | [
73,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
72,
1
] |
Mathlib/Analysis/Convex/Strict.lean | strictConvex_iff_openSegment_subset | [] | [
57,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
55,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | Real.sinh_neg_iff | [
{
"state_after": "no goals",
"state_before": "x y z : ℝ\n⊢ sinh x < 0 ↔ x < 0",
"tactic": "simpa only [sinh_zero] using @sinh_lt_sinh x 0"
}
] | [
713,
95
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
713,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | Real.tan_surjective | [] | [
102,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
102,
1
] |
Mathlib/Combinatorics/SimpleGraph/Density.lean | SimpleGraph.mk_mem_interedges_iff | [] | [
335,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
334,
1
] |
Mathlib/Combinatorics/SimpleGraph/Basic.lean | SimpleGraph.mem_incidenceSet | [
{
"state_after": "no goals",
"state_before": "ι : Sort ?u.137668\n𝕜 : Type ?u.137671\nV : Type u\nW : Type v\nX : Type w\nG : SimpleGraph V\nG' : SimpleGraph W\na b c u v✝ w✝ : V\ne : Sym2 V\nv w : V\n⊢ Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ incidenceSet G v ↔ Adj G v w",
"tactic": "simp [incidenceSet]"
}
] | [
959,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
958,
1
] |
Mathlib/LinearAlgebra/FreeModule/Basic.lean | Module.Free.of_basis | [] | [
72,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
71,
1
] |
Mathlib/Order/Filter/AtTopBot.lean | Filter.tendsto_comp_val_Ioi_atTop | [
{
"state_after": "ι : Type ?u.337904\nι' : Type ?u.337907\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.337916\ninst✝¹ : SemilatticeSup α\ninst✝ : NoMaxOrder α\na : α\nf : α → β\nl : Filter β\n⊢ Tendsto (fun x => f ↑x) atTop l ↔ Tendsto (f ∘ Subtype.val) atTop l",
"state_before": "ι : Type ?u.337904\nι' : Type ?u.337907\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.337916\ninst✝¹ : SemilatticeSup α\ninst✝ : NoMaxOrder α\na : α\nf : α → β\nl : Filter β\n⊢ Tendsto (fun x => f ↑x) atTop l ↔ Tendsto f atTop l",
"tactic": "rw [← map_val_Ioi_atTop a, tendsto_map'_iff]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.337904\nι' : Type ?u.337907\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.337916\ninst✝¹ : SemilatticeSup α\ninst✝ : NoMaxOrder α\na : α\nf : α → β\nl : Filter β\n⊢ Tendsto (fun x => f ↑x) atTop l ↔ Tendsto (f ∘ Subtype.val) atTop l",
"tactic": "rfl"
}
] | [
1618,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1616,
1
] |
Mathlib/Analysis/SpecificLimits/Basic.lean | tendsto_factorial_div_pow_self_atTop | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.482700\nβ : Type ?u.482703\nι : Type ?u.482706\nn : ℕ\n⊢ 0 ≤ ↑n !",
"tactic": "exact_mod_cast n.factorial_pos.le"
},
{
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"state_before": "α : Type ?u.482700\nβ : Type ?u.482703\nι : Type ?u.482706\nn : ℕ\n⊢ 0 ≤ ↑n",
"tactic": "exact_mod_cast n.zero_le"
},
{
"state_after": "α : Type ?u.482700\nβ : Type ?u.482703\nι : Type ?u.482706\nn : ℕ\nhn : 0 < n\n⊢ ↑n ! / ↑n ^ n ≤ 1 / ↑n",
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"tactic": "refine' (eventually_gt_atTop 0).mono fun n hn => _"
},
{
"state_after": "case intro\nα : Type ?u.482700\nβ : Type ?u.482703\nι : Type ?u.482706\nk : ℕ\nhn : 0 < succ k\n⊢ ↑(succ k)! / ↑(succ k) ^ succ k ≤ 1 / ↑(succ k)",
"state_before": "α : Type ?u.482700\nβ : Type ?u.482703\nι : Type ?u.482706\nn : ℕ\nhn : 0 < n\n⊢ ↑n ! / ↑n ^ n ≤ 1 / ↑n",
"tactic": "rcases Nat.exists_eq_succ_of_ne_zero hn.ne.symm with ⟨k, rfl⟩"
},
{
"state_after": "case intro\nα : Type ?u.482700\nβ : Type ?u.482703\nι : Type ?u.482706\nk : ℕ\nhn : 0 < succ k\n⊢ (∏ k_1 in Finset.range k, ↑(k_1 + 1 + 1) * (↑k + 1)⁻¹) * (↑(0 + 1) * (↑k + 1)⁻¹) ≤ (↑k + 1)⁻¹",
"state_before": "case intro\nα : Type ?u.482700\nβ : Type ?u.482703\nι : Type ?u.482706\nk : ℕ\nhn : 0 < succ k\n⊢ ↑(succ k)! / ↑(succ k) ^ succ k ≤ 1 / ↑(succ k)",
"tactic": "rw [← prod_range_add_one_eq_factorial, pow_eq_prod_const, div_eq_mul_inv, ← inv_eq_one_div,\n prod_natCast, Nat.cast_succ, ← prod_inv_distrib, ← prod_mul_distrib,\n Finset.prod_range_succ']"
},
{
"state_after": "case intro\nα : Type ?u.482700\nβ : Type ?u.482703\nι : Type ?u.482706\nk : ℕ\nhn : 0 < succ k\n⊢ (∏ x in Finset.range k, (↑x + 1 + 1) * (↑k + 1)⁻¹) * (↑k + 1)⁻¹ ≤ (↑k + 1)⁻¹",
"state_before": "case intro\nα : Type ?u.482700\nβ : Type ?u.482703\nι : Type ?u.482706\nk : ℕ\nhn : 0 < succ k\n⊢ (∏ k_1 in Finset.range k, ↑(k_1 + 1 + 1) * (↑k + 1)⁻¹) * (↑(0 + 1) * (↑k + 1)⁻¹) ≤ (↑k + 1)⁻¹",
"tactic": "simp only [prod_range_succ', one_mul, Nat.cast_add, zero_add, Nat.cast_one]"
},
{
"state_after": "case intro.refine'_1\nα : Type ?u.482700\nβ : Type ?u.482703\nι : Type ?u.482706\nk : ℕ\nhn : 0 < succ k\nx : ℕ\nhx : x < k\n⊢ 0 ≤ (↑x + 1 + 1) * (↑k + 1)⁻¹\n\ncase intro.refine'_2\nα : Type ?u.482700\nβ : Type ?u.482703\nι : Type ?u.482706\nk : ℕ\nhn : 0 < succ k\nx : ℕ\nhx : x < k\n⊢ (↑x + 1 + 1) * (↑k + 1)⁻¹ ≤ 1",
"state_before": "case intro\nα : Type ?u.482700\nβ : Type ?u.482703\nι : Type ?u.482706\nk : ℕ\nhn : 0 < succ k\n⊢ (∏ x in Finset.range k, (↑x + 1 + 1) * (↑k + 1)⁻¹) * (↑k + 1)⁻¹ ≤ (↑k + 1)⁻¹",
"tactic": "refine'\n mul_le_of_le_one_left (inv_nonneg.mpr <| by exact_mod_cast hn.le) (prod_le_one _ _) <;>\n intro x hx <;>\n rw [Finset.mem_range] at hx"
},
{
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"tactic": "exact_mod_cast hn.le"
},
{
"state_after": "no goals",
"state_before": "case intro.refine'_1\nα : Type ?u.482700\nβ : Type ?u.482703\nι : Type ?u.482706\nk : ℕ\nhn : 0 < succ k\nx : ℕ\nhx : x < k\n⊢ 0 ≤ (↑x + 1 + 1) * (↑k + 1)⁻¹",
"tactic": "refine' mul_nonneg _ (inv_nonneg.mpr _) <;> norm_cast <;> linarith"
},
{
"state_after": "case intro.refine'_2\nα : Type ?u.482700\nβ : Type ?u.482703\nι : Type ?u.482706\nk : ℕ\nhn : 0 < succ k\nx : ℕ\nhx : x < k\n⊢ ↑x + 1 + 1 ≤ ↑k + 1",
"state_before": "case intro.refine'_2\nα : Type ?u.482700\nβ : Type ?u.482703\nι : Type ?u.482706\nk : ℕ\nhn : 0 < succ k\nx : ℕ\nhx : x < k\n⊢ (↑x + 1 + 1) * (↑k + 1)⁻¹ ≤ 1",
"tactic": "refine' (div_le_one <| by exact_mod_cast hn).mpr _"
},
{
"state_after": "case intro.refine'_2\nα : Type ?u.482700\nβ : Type ?u.482703\nι : Type ?u.482706\nk : ℕ\nhn : 0 < succ k\nx : ℕ\nhx : x < k\n⊢ x + 1 + 1 ≤ k + 1",
"state_before": "case intro.refine'_2\nα : Type ?u.482700\nβ : Type ?u.482703\nι : Type ?u.482706\nk : ℕ\nhn : 0 < succ k\nx : ℕ\nhx : x < k\n⊢ ↑x + 1 + 1 ≤ ↑k + 1",
"tactic": "norm_cast"
},
{
"state_after": "no goals",
"state_before": "case intro.refine'_2\nα : Type ?u.482700\nβ : Type ?u.482703\nι : Type ?u.482706\nk : ℕ\nhn : 0 < succ k\nx : ℕ\nhx : x < k\n⊢ x + 1 + 1 ≤ k + 1",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.482700\nβ : Type ?u.482703\nι : Type ?u.482706\nk : ℕ\nhn : 0 < succ k\nx : ℕ\nhx : x < k\n⊢ 0 < ↑k + 1",
"tactic": "exact_mod_cast hn"
}
] | [
562,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
541,
1
] |
Mathlib/Order/OmegaCompletePartialOrder.lean | OmegaCompletePartialOrder.ContinuousHom.ωSup_def | [] | [
841,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
840,
1
] |
Mathlib/Data/Real/EReal.lean | EReal.coe_ennreal_nonneg | [] | [
540,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
539,
1
] |
Mathlib/SetTheory/Game/PGame.lean | PGame.ofLists_moveRight | [] | [
200,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
198,
1
] |
Mathlib/LinearAlgebra/Pi.lean | LinearEquiv.piCongrRight_trans | [] | [
375,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
373,
1
] |
Mathlib/Order/Basic.lean | LT.lt.gt | [] | [
304,
4
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
303,
11
] |
Mathlib/Analysis/Complex/Liouville.lean | Differentiable.apply_eq_apply_of_bounded | [
{
"state_after": "E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nhf : Differentiable ℂ f\nhb : Metric.Bounded (range f)\nz w : E\ng : ℂ → F := f ∘ fun t => t • (w - z) + z\n⊢ f z = f w",
"state_before": "E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nhf : Differentiable ℂ f\nhb : Metric.Bounded (range f)\nz w : E\n⊢ f z = f w",
"tactic": "set g : ℂ → F := f ∘ fun t : ℂ => t • (w - z) + z"
},
{
"state_after": "E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nhf : Differentiable ℂ f\nhb : Metric.Bounded (range f)\nz w : E\ng : ℂ → F := f ∘ fun t => t • (w - z) + z\n⊢ g 0 = g 1",
"state_before": "E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nhf : Differentiable ℂ f\nhb : Metric.Bounded (range f)\nz w : E\ng : ℂ → F := f ∘ fun t => t • (w - z) + z\n⊢ f z = f w",
"tactic": "suffices g 0 = g 1 by simpa"
},
{
"state_after": "case hf\nE : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nhf : Differentiable ℂ f\nhb : Metric.Bounded (range f)\nz w : E\ng : ℂ → F := f ∘ fun t => t • (w - z) + z\n⊢ Differentiable ℂ g\n\ncase hb\nE : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nhf : Differentiable ℂ f\nhb : Metric.Bounded (range f)\nz w : E\ng : ℂ → F := f ∘ fun t => t • (w - z) + z\n⊢ Metric.Bounded (range g)",
"state_before": "E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nhf : Differentiable ℂ f\nhb : Metric.Bounded (range f)\nz w : E\ng : ℂ → F := f ∘ fun t => t • (w - z) + z\n⊢ g 0 = g 1",
"tactic": "apply liouville_theorem_aux"
},
{
"state_after": "no goals",
"state_before": "case hf\nE : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nhf : Differentiable ℂ f\nhb : Metric.Bounded (range f)\nz w : E\ng : ℂ → F := f ∘ fun t => t • (w - z) + z\n⊢ Differentiable ℂ g\n\ncase hb\nE : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nhf : Differentiable ℂ f\nhb : Metric.Bounded (range f)\nz w : E\ng : ℂ → F := f ∘ fun t => t • (w - z) + z\n⊢ Metric.Bounded (range g)",
"tactic": "exacts [hf.comp ((differentiable_id.smul_const (w - z)).add_const z),\n hb.mono (range_comp_subset_range _ _)]"
},
{
"state_after": "no goals",
"state_before": "E : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℂ E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℂ F\nf : E → F\nhf : Differentiable ℂ f\nhb : Metric.Bounded (range f)\nz w : E\ng : ℂ → F := f ∘ fun t => t • (w - z) + z\nthis : g 0 = g 1\n⊢ f z = f w",
"tactic": "simpa"
}
] | [
123,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
117,
1
] |
Mathlib/MeasureTheory/Group/Action.lean | MeasureTheory.smul_ae_eq_self_of_mem_zpowers | [
{
"state_after": "case intro\nG : Type u_2\nM : Type ?u.82889\nα : Type u_1\ns : Set α\nm : MeasurableSpace α\ninst✝⁴ : Group G\ninst✝³ : MulAction G α\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\nc : G\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nx : G\nhs : x • s =ᶠ[ae μ] s\nk : ℤ\nhy : x ^ k ∈ Subgroup.zpowers x\n⊢ x ^ k • s =ᶠ[ae μ] s",
"state_before": "G : Type u_2\nM : Type ?u.82889\nα : Type u_1\ns : Set α\nm : MeasurableSpace α\ninst✝⁴ : Group G\ninst✝³ : MulAction G α\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\nc : G\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nx y : G\nhs : x • s =ᶠ[ae μ] s\nhy : y ∈ Subgroup.zpowers x\n⊢ y • s =ᶠ[ae μ] s",
"tactic": "obtain ⟨k, rfl⟩ := Subgroup.mem_zpowers_iff.mp hy"
},
{
"state_after": "case intro\nG : Type u_2\nM : Type ?u.82889\nα : Type u_1\ns : Set α\nm : MeasurableSpace α\ninst✝⁴ : Group G\ninst✝³ : MulAction G α\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\nc : G\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nx : G\nhs : x • s =ᶠ[ae μ] s\nk : ℤ\nhy : x ^ k ∈ Subgroup.zpowers x\ne : α ≃ α := ↑(MulAction.toPermHom G α) x\n⊢ x ^ k • s =ᶠ[ae μ] s",
"state_before": "case intro\nG : Type u_2\nM : Type ?u.82889\nα : Type u_1\ns : Set α\nm : MeasurableSpace α\ninst✝⁴ : Group G\ninst✝³ : MulAction G α\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\nc : G\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nx : G\nhs : x • s =ᶠ[ae μ] s\nk : ℤ\nhy : x ^ k ∈ Subgroup.zpowers x\n⊢ x ^ k • s =ᶠ[ae μ] s",
"tactic": "let e : α ≃ α := MulAction.toPermHom G α x"
},
{
"state_after": "case intro\nG : Type u_2\nM : Type ?u.82889\nα : Type u_1\ns : Set α\nm : MeasurableSpace α\ninst✝⁴ : Group G\ninst✝³ : MulAction G α\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\nc : G\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nx : G\nhs : x • s =ᶠ[ae μ] s\nk : ℤ\nhy : x ^ k ∈ Subgroup.zpowers x\ne : α ≃ α := ↑(MulAction.toPermHom G α) x\nhe : QuasiMeasurePreserving ↑e\n⊢ x ^ k • s =ᶠ[ae μ] s",
"state_before": "case intro\nG : Type u_2\nM : Type ?u.82889\nα : Type u_1\ns : Set α\nm : MeasurableSpace α\ninst✝⁴ : Group G\ninst✝³ : MulAction G α\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\nc : G\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nx : G\nhs : x • s =ᶠ[ae μ] s\nk : ℤ\nhy : x ^ k ∈ Subgroup.zpowers x\ne : α ≃ α := ↑(MulAction.toPermHom G α) x\n⊢ x ^ k • s =ᶠ[ae μ] s",
"tactic": "have he : QuasiMeasurePreserving e μ μ := (measurePreserving_smul x μ).quasiMeasurePreserving"
},
{
"state_after": "case intro\nG : Type u_2\nM : Type ?u.82889\nα : Type u_1\ns : Set α\nm : MeasurableSpace α\ninst✝⁴ : Group G\ninst✝³ : MulAction G α\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\nc : G\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nx : G\nhs : x • s =ᶠ[ae μ] s\nk : ℤ\nhy : x ^ k ∈ Subgroup.zpowers x\ne : α ≃ α := ↑(MulAction.toPermHom G α) x\nhe : QuasiMeasurePreserving ↑e\nhe' : QuasiMeasurePreserving ↑e.symm\n⊢ x ^ k • s =ᶠ[ae μ] s",
"state_before": "case intro\nG : Type u_2\nM : Type ?u.82889\nα : Type u_1\ns : Set α\nm : MeasurableSpace α\ninst✝⁴ : Group G\ninst✝³ : MulAction G α\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\nc : G\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nx : G\nhs : x • s =ᶠ[ae μ] s\nk : ℤ\nhy : x ^ k ∈ Subgroup.zpowers x\ne : α ≃ α := ↑(MulAction.toPermHom G α) x\nhe : QuasiMeasurePreserving ↑e\n⊢ x ^ k • s =ᶠ[ae μ] s",
"tactic": "have he' : QuasiMeasurePreserving e.symm μ μ :=\n (measurePreserving_smul x⁻¹ μ).quasiMeasurePreserving"
},
{
"state_after": "case intro\nG : Type u_2\nM : Type ?u.82889\nα : Type u_1\ns : Set α\nm : MeasurableSpace α\ninst✝⁴ : Group G\ninst✝³ : MulAction G α\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\nc : G\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nx : G\nhs : x • s =ᶠ[ae μ] s\nk : ℤ\nhy : x ^ k ∈ Subgroup.zpowers x\ne : α ≃ α := ↑(MulAction.toPermHom G α) x\nhe : QuasiMeasurePreserving ↑e\nhe' : QuasiMeasurePreserving ↑e.symm\nh : ↑(e ^ k) '' s =ᶠ[ae μ] s\n⊢ x ^ k • s =ᶠ[ae μ] s",
"state_before": "case intro\nG : Type u_2\nM : Type ?u.82889\nα : Type u_1\ns : Set α\nm : MeasurableSpace α\ninst✝⁴ : Group G\ninst✝³ : MulAction G α\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\nc : G\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nx : G\nhs : x • s =ᶠ[ae μ] s\nk : ℤ\nhy : x ^ k ∈ Subgroup.zpowers x\ne : α ≃ α := ↑(MulAction.toPermHom G α) x\nhe : QuasiMeasurePreserving ↑e\nhe' : QuasiMeasurePreserving ↑e.symm\n⊢ x ^ k • s =ᶠ[ae μ] s",
"tactic": "have h := he.image_zpow_ae_eq he' k hs"
},
{
"state_after": "case intro\nG : Type u_2\nM : Type ?u.82889\nα : Type u_1\ns : Set α\nm : MeasurableSpace α\ninst✝⁴ : Group G\ninst✝³ : MulAction G α\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\nc : G\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nx : G\nhs : x • s =ᶠ[ae μ] s\nk : ℤ\nhy : x ^ k ∈ Subgroup.zpowers x\ne : α ≃ α := ↑(MulAction.toPermHom G α) x\nhe : QuasiMeasurePreserving ↑e\nhe' : QuasiMeasurePreserving ↑e.symm\nh : (fun a => ↑(↑(MulAction.toPermHom G α) (x ^ k)) a) '' s =ᶠ[ae μ] s\n⊢ x ^ k • s =ᶠ[ae μ] s",
"state_before": "case intro\nG : Type u_2\nM : Type ?u.82889\nα : Type u_1\ns : Set α\nm : MeasurableSpace α\ninst✝⁴ : Group G\ninst✝³ : MulAction G α\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\nc : G\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nx : G\nhs : x • s =ᶠ[ae μ] s\nk : ℤ\nhy : x ^ k ∈ Subgroup.zpowers x\ne : α ≃ α := ↑(MulAction.toPermHom G α) x\nhe : QuasiMeasurePreserving ↑e\nhe' : QuasiMeasurePreserving ↑e.symm\nh : ↑(e ^ k) '' s =ᶠ[ae μ] s\n⊢ x ^ k • s =ᶠ[ae μ] s",
"tactic": "simp only [← MonoidHom.map_zpow] at h"
},
{
"state_after": "no goals",
"state_before": "case intro\nG : Type u_2\nM : Type ?u.82889\nα : Type u_1\ns : Set α\nm : MeasurableSpace α\ninst✝⁴ : Group G\ninst✝³ : MulAction G α\ninst✝² : MeasurableSpace G\ninst✝¹ : MeasurableSMul G α\nc : G\nμ : Measure α\ninst✝ : SMulInvariantMeasure G α μ\nx : G\nhs : x • s =ᶠ[ae μ] s\nk : ℤ\nhy : x ^ k ∈ Subgroup.zpowers x\ne : α ≃ α := ↑(MulAction.toPermHom G α) x\nhe : QuasiMeasurePreserving ↑e\nhe' : QuasiMeasurePreserving ↑e.symm\nh : (fun a => ↑(↑(MulAction.toPermHom G α) (x ^ k)) a) '' s =ᶠ[ae μ] s\n⊢ x ^ k • s =ᶠ[ae μ] s",
"tactic": "simpa only [MulAction.toPermHom_apply, MulAction.toPerm_apply, image_smul] using h"
}
] | [
274,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
265,
1
] |
Mathlib/Logic/Basic.lean | Decidable.not_ball | [] | [
1090,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1087,
11
] |
Mathlib/Topology/ContinuousFunction/Basic.lean | ContinuousMap.restrict_apply_mk | [] | [
378,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
376,
1
] |
Mathlib/Algebra/Algebra/Hom.lean | AlgHom.map_zero | [] | [
250,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
249,
11
] |
Mathlib/Data/Real/Sqrt.lean | Real.continuous_sqrt | [] | [
189,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
188,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Midpoint.lean | midpoint_self | [] | [
200,
25
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
199,
1
] |
Mathlib/Algebra/Order/Monoid/WithTop.lean | WithBot.one_le_coe | [] | [
509,
13
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
508,
1
] |
Mathlib/Data/Polynomial/Div.lean | Polynomial.zero_divByMonic | [
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Ring R\np✝ q p : R[X]\n⊢ (if h : Monic p then\n (if h_1 : degree p ≤ degree 0 ∧ 0 ≠ 0 then\n let z := ↑C (leadingCoeff 0) * X ^ (natDegree 0 - natDegree p);\n let_fun _wf := (_ : degree (0 - ↑C (leadingCoeff 0) * X ^ (natDegree 0 - natDegree p) * p) < degree 0);\n let dm := divModByMonicAux (0 - z * p) (_ : Monic p);\n (z + dm.fst, dm.snd)\n else (0, 0)).fst\n else 0) =\n 0",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Ring R\np✝ q p : R[X]\n⊢ 0 /ₘ p = 0",
"tactic": "unfold divByMonic divModByMonicAux"
},
{
"state_after": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Ring R\np✝ q p : R[X]\n⊢ (if h : Monic p then\n (if degree p ≤ ⊥ ∧ ¬0 = 0 then\n (↑C 0 * X ^ (0 - natDegree p) + (divModByMonicAux (0 - ↑C 0 * X ^ (0 - natDegree p) * p) (_ : Monic p)).fst,\n (divModByMonicAux (0 - ↑C 0 * X ^ (0 - natDegree p) * p) (_ : Monic p)).snd)\n else (0, 0)).fst\n else 0) =\n 0",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Ring R\np✝ q p : R[X]\n⊢ (if h : Monic p then\n (if h_1 : degree p ≤ degree 0 ∧ 0 ≠ 0 then\n let z := ↑C (leadingCoeff 0) * X ^ (natDegree 0 - natDegree p);\n let_fun _wf := (_ : degree (0 - ↑C (leadingCoeff 0) * X ^ (natDegree 0 - natDegree p) * p) < degree 0);\n let dm := divModByMonicAux (0 - z * p) (_ : Monic p);\n (z + dm.fst, dm.snd)\n else (0, 0)).fst\n else 0) =\n 0",
"tactic": "dsimp"
},
{
"state_after": "case pos\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Ring R\np✝ q p : R[X]\nhp : Monic p\n⊢ (if h : Monic p then\n (if degree p ≤ ⊥ ∧ ¬0 = 0 then\n (↑C 0 * X ^ (0 - natDegree p) + (divModByMonicAux (0 - ↑C 0 * X ^ (0 - natDegree p) * p) (_ : Monic p)).fst,\n (divModByMonicAux (0 - ↑C 0 * X ^ (0 - natDegree p) * p) (_ : Monic p)).snd)\n else (0, 0)).fst\n else 0) =\n 0\n\ncase neg\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Ring R\np✝ q p : R[X]\nhp : ¬Monic p\n⊢ (if h : Monic p then\n (if degree p ≤ ⊥ ∧ ¬0 = 0 then\n (↑C 0 * X ^ (0 - natDegree p) + (divModByMonicAux (0 - ↑C 0 * X ^ (0 - natDegree p) * p) (_ : Monic p)).fst,\n (divModByMonicAux (0 - ↑C 0 * X ^ (0 - natDegree p) * p) (_ : Monic p)).snd)\n else (0, 0)).fst\n else 0) =\n 0",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Ring R\np✝ q p : R[X]\n⊢ (if h : Monic p then\n (if degree p ≤ ⊥ ∧ ¬0 = 0 then\n (↑C 0 * X ^ (0 - natDegree p) + (divModByMonicAux (0 - ↑C 0 * X ^ (0 - natDegree p) * p) (_ : Monic p)).fst,\n (divModByMonicAux (0 - ↑C 0 * X ^ (0 - natDegree p) * p) (_ : Monic p)).snd)\n else (0, 0)).fst\n else 0) =\n 0",
"tactic": "by_cases hp : Monic p"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Ring R\np✝ q p : R[X]\nhp : Monic p\n⊢ (if h : Monic p then\n (if degree p ≤ ⊥ ∧ ¬0 = 0 then\n (↑C 0 * X ^ (0 - natDegree p) + (divModByMonicAux (0 - ↑C 0 * X ^ (0 - natDegree p) * p) (_ : Monic p)).fst,\n (divModByMonicAux (0 - ↑C 0 * X ^ (0 - natDegree p) * p) (_ : Monic p)).snd)\n else (0, 0)).fst\n else 0) =\n 0",
"tactic": "rw [dif_pos hp, if_neg (mt And.right (not_not_intro rfl))]"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\nS : Type v\nT : Type w\nA : Type z\na b : R\nn : ℕ\ninst✝ : Ring R\np✝ q p : R[X]\nhp : ¬Monic p\n⊢ (if h : Monic p then\n (if degree p ≤ ⊥ ∧ ¬0 = 0 then\n (↑C 0 * X ^ (0 - natDegree p) + (divModByMonicAux (0 - ↑C 0 * X ^ (0 - natDegree p) * p) (_ : Monic p)).fst,\n (divModByMonicAux (0 - ↑C 0 * X ^ (0 - natDegree p) * p) (_ : Monic p)).snd)\n else (0, 0)).fst\n else 0) =\n 0",
"tactic": "rw [dif_neg hp]"
}
] | [
187,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
182,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean | PowerSeries.rescale_neg_one_X | [
{
"state_after": "no goals",
"state_before": "R : Type ?u.3551663\nA : Type u_1\ninst✝ : CommRing A\n⊢ ↑(rescale (-1)) X = -X",
"tactic": "rw [rescale_X, map_neg, map_one, neg_one_mul]"
}
] | [
1964,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1963,
1
] |
Mathlib/Topology/UniformSpace/Equicontinuity.lean | Equicontinuous.closure' | [] | [
410,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
408,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean | MeasureTheory.Measure.exists_measure_inter_spanningSets_pos | [
{
"state_after": "α : Type u_1\nβ : Type ?u.733391\nγ : Type ?u.733394\nδ : Type ?u.733397\nι : Type ?u.733400\nR : Type ?u.733403\nR' : Type ?u.733406\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\ns : Set α\n⊢ (¬∃ n, 0 < ↑↑μ (s ∩ spanningSets μ n)) ↔ ¬0 < ↑↑μ s",
"state_before": "α : Type u_1\nβ : Type ?u.733391\nγ : Type ?u.733394\nδ : Type ?u.733397\nι : Type ?u.733400\nR : Type ?u.733403\nR' : Type ?u.733406\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\ns : Set α\n⊢ (∃ n, 0 < ↑↑μ (s ∩ spanningSets μ n)) ↔ 0 < ↑↑μ s",
"tactic": "rw [← not_iff_not]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.733391\nγ : Type ?u.733394\nδ : Type ?u.733397\nι : Type ?u.733400\nR : Type ?u.733403\nR' : Type ?u.733406\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\ns : Set α\n⊢ (∀ (x : ℕ), ↑↑μ (s ∩ spanningSets μ x) = 0) ↔ ↑↑μ s = 0",
"state_before": "α : Type u_1\nβ : Type ?u.733391\nγ : Type ?u.733394\nδ : Type ?u.733397\nι : Type ?u.733400\nR : Type ?u.733403\nR' : Type ?u.733406\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\ns : Set α\n⊢ (¬∃ n, 0 < ↑↑μ (s ∩ spanningSets μ n)) ↔ ¬0 < ↑↑μ s",
"tactic": "simp only [not_exists, not_lt, nonpos_iff_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.733391\nγ : Type ?u.733394\nδ : Type ?u.733397\nι : Type ?u.733400\nR : Type ?u.733403\nR' : Type ?u.733406\nm0 : MeasurableSpace α\ninst✝³ : MeasurableSpace β\ninst✝² : MeasurableSpace γ\nμ✝ μ₁ μ₂ μ₃ ν ν' ν₁ ν₂ : Measure α\ns✝ s' t : Set α\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : SigmaFinite μ\ns : Set α\n⊢ (∀ (x : ℕ), ↑↑μ (s ∩ spanningSets μ x) = 0) ↔ ↑↑μ s = 0",
"tactic": "exact forall_measure_inter_spanningSets_eq_zero s"
}
] | [
3569,
52
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
3565,
1
] |
Mathlib/Data/Stream/Init.lean | Stream'.odd_eq | [] | [
461,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
460,
1
] |
Mathlib/Data/Polynomial/Degree/TrailingDegree.lean | Polynomial.le_natTrailingDegree | [
{
"state_after": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nhp : p ≠ 0\nhn : ∀ (m : ℕ), m < n → coeff p m = 0\n⊢ n ≤ min' (support p) (_ : Finset.Nonempty (support p))",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nhp : p ≠ 0\nhn : ∀ (m : ℕ), m < n → coeff p m = 0\n⊢ n ≤ natTrailingDegree p",
"tactic": "rw [natTrailingDegree_eq_support_min' hp]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b : R\nn m : ℕ\ninst✝ : Semiring R\np q r : R[X]\nhp : p ≠ 0\nhn : ∀ (m : ℕ), m < n → coeff p m = 0\n⊢ n ≤ min' (support p) (_ : Finset.Nonempty (support p))",
"tactic": "exact Finset.le_min' _ _ _ fun m hm => not_lt.1 fun hmn => mem_support_iff.1 hm <| hn _ hmn"
}
] | [
335,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
332,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean | ZFSet.mem_wf | [] | [
1189,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1187,
1
] |
Mathlib/Order/RelClasses.lean | antisymm_of' | [] | [
61,
12
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
60,
1
] |
Mathlib/Data/Holor.lean | Holor.cprankMax_nil | [
{
"state_after": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝¹ : Monoid α\ninst✝ : AddMonoid α\nx : Holor α []\nh : CPRankMax (0 + 1) (x + 0)\n⊢ CPRankMax 1 x",
"state_before": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝¹ : Monoid α\ninst✝ : AddMonoid α\nx : Holor α []\n⊢ CPRankMax 1 x",
"tactic": "have h := CPRankMax.succ 0 x 0 (CPRankMax1.nil x) CPRankMax.zero"
},
{
"state_after": "no goals",
"state_before": "α : Type\nd : ℕ\nds ds₁ ds₂ ds₃ : List ℕ\ninst✝¹ : Monoid α\ninst✝ : AddMonoid α\nx : Holor α []\nh : CPRankMax (0 + 1) (x + 0)\n⊢ CPRankMax 1 x",
"tactic": "rwa [add_zero x, zero_add] at h"
}
] | [
329,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
327,
1
] |
Mathlib/Data/List/Intervals.lean | List.Ico.succ_top | [
{
"state_after": "case hml\nn m : ℕ\nh : n ≤ m\n⊢ m ≤ m + 1",
"state_before": "n m : ℕ\nh : n ≤ m\n⊢ Ico n (m + 1) = Ico n m ++ [m]",
"tactic": "rwa [← succ_singleton, append_consecutive]"
},
{
"state_after": "no goals",
"state_before": "case hml\nn m : ℕ\nh : n ≤ m\n⊢ m ≤ m + 1",
"tactic": "exact Nat.le_succ _"
}
] | [
129,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
127,
1
] |
Mathlib/Data/Nat/Pow.lean | Nat.pow_le_iff_le_right | [] | [
109,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
108,
1
] |
Mathlib/Algebra/Lie/Subalgebra.lean | LieSubalgebra.smul_mem | [] | [
145,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
144,
1
] |
Mathlib/Data/Real/NNReal.lean | NNReal.inv_lt_inv | [] | [
911,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
910,
1
] |
Mathlib/RingTheory/PowerSeries/Basic.lean | MvPowerSeries.trunc_c | [
{
"state_after": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\na : R\nm : σ →₀ ℕ\n⊢ (if m < n then if m = 0 then a else 0 else 0) = if 0 = m then a else 0",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\na : R\nm : σ →₀ ℕ\n⊢ MvPolynomial.coeff m (↑(trunc R n) (↑(C σ R) a)) = MvPolynomial.coeff m (↑MvPolynomial.C a)",
"tactic": "rw [coeff_trunc, coeff_C, MvPolynomial.coeff_C]"
},
{
"state_after": "case inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\na : R\nm : σ →₀ ℕ\nhnn : ¬n = m\nH : ¬m < n\nh✝ : 0 = m\n⊢ 0 = a",
"state_before": "σ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\na : R\nm : σ →₀ ℕ\n⊢ (if m < n then if m = 0 then a else 0 else 0) = if 0 = m then a else 0",
"tactic": "split_ifs with H <;> first |rfl|try simp_all"
},
{
"state_after": "case inr.inl.h\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\na : R\nm : σ →₀ ℕ\nhnn : ¬n = m\nH : ¬m < n\nh✝ : 0 = m\n⊢ False",
"state_before": "case inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\na : R\nm : σ →₀ ℕ\nhnn : ¬n = m\nH : ¬m < n\nh✝ : 0 = m\n⊢ 0 = a",
"tactic": "exfalso"
},
{
"state_after": "case inr.inl.h\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\na : R\nm : σ →₀ ℕ\nhnn : ¬n = m\nH : ¬m < n\nh✝ : 0 = m\n⊢ m < n",
"state_before": "case inr.inl.h\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\na : R\nm : σ →₀ ℕ\nhnn : ¬n = m\nH : ¬m < n\nh✝ : 0 = m\n⊢ False",
"tactic": "apply H"
},
{
"state_after": "case inr.inl.h\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\na : R\nhnn : ¬n = 0\nH : ¬0 < n\n⊢ 0 < n",
"state_before": "case inr.inl.h\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\na : R\nm : σ →₀ ℕ\nhnn : ¬n = m\nH : ¬m < n\nh✝ : 0 = m\n⊢ m < n",
"tactic": "subst m"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.h\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\na : R\nhnn : ¬n = 0\nH : ¬0 < n\n⊢ 0 < n",
"tactic": "exact Ne.bot_lt hnn"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\na : R\nm : σ →₀ ℕ\nH : ¬m < n\nh✝ : ¬0 = m\n⊢ 0 = 0",
"tactic": "rfl"
},
{
"state_after": "case inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\na : R\nm : σ →₀ ℕ\nhnn : ¬n = m\nH : ¬m < n\nh✝ : 0 = m\n⊢ 0 = a",
"state_before": "case inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\nhnn : n ≠ 0\na : R\nm : σ →₀ ℕ\nH : ¬m < n\nh✝ : 0 = m\n⊢ 0 = a",
"tactic": "try simp_all"
},
{
"state_after": "case inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝ : CommSemiring R\nn✝ n : σ →₀ ℕ\na : R\nm : σ →₀ ℕ\nhnn : ¬n = m\nH : ¬m < n\nh✝ : 0 = m\n⊢ 0 = a",
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] | [
730,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
726,
1
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Mathlib/LinearAlgebra/Finsupp.lean | Finsupp.mapRange.linearMap_toAddMonoidHom | [] | [
844,
32
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
841,
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src/lean/Init/Control/StateCps.lean | StateCpsT.runK_bind_pure | [] | [
64,
166
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
64,
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Mathlib/Analysis/BoxIntegral/Partition/Tagged.lean | BoxIntegral.TaggedPrepartition.isPartition_iff_iUnion_eq | [] | [
101,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
100,
1
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Mathlib/Topology/Algebra/UniformGroup.lean | uniformity_translate_mul | [] | [
178,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
169,
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Mathlib/Combinatorics/SimpleGraph/Subgraph.lean | SimpleGraph.neighborSet_subgraphOfAdj_of_ne_of_ne | [
{
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] | [
960,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
957,
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Mathlib/Data/List/AList.lean | AList.lookup_to_alist | [
{
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"tactic": "rw [List.toAList, lookup, dlookup_dedupKeys]"
}
] | [
316,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
315,
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Mathlib/Algebra/Star/StarAlgHom.lean | NonUnitalStarAlgHom.coe_comp | [] | [
212,
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] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
211,
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Mathlib/Data/Int/GCD.lean | Int.gcd_dvd_iff | [
{
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},
{
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},
{
"state_after": "case mp\na b : ℤ\nn : ℕ\nh : gcd a b ∣ n\n⊢ ∃ x y, a * (gcdA a b * ↑(n / gcd a b)) + b * (gcdB a b * ↑(n / gcd a b)) = a * x + b * y",
"state_before": "case mp\na b : ℤ\nn : ℕ\nh : gcd a b ∣ n\n⊢ ∃ x y, ↑n = a * x + b * y",
"tactic": "rw [← Nat.mul_div_cancel' h, Int.ofNat_mul, gcd_eq_gcd_ab, add_mul, mul_assoc, mul_assoc]"
},
{
"state_after": "no goals",
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"tactic": "exact ⟨_, _, rfl⟩"
},
{
"state_after": "case mpr.intro.intro\na b : ℤ\nn : ℕ\nx y : ℤ\nh : ↑n = a * x + b * y\n⊢ gcd a b ∣ n",
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},
{
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"tactic": "rw [← Int.coe_nat_dvd, h]"
},
{
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"tactic": "exact\n dvd_add (dvd_mul_of_dvd_left (gcd_dvd_left a b) _) (dvd_mul_of_dvd_left (gcd_dvd_right a b) y)"
}
] | [
399,
101
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
391,
1
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Mathlib/Algebra/Ring/BooleanRing.lean | ofBoolRing_toBoolRing | [] | [
419,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
418,
1
] |
Mathlib/MeasureTheory/Group/Measure.lean | MeasureTheory.eventually_mul_right_iff | [
{
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"tactic": "conv_rhs => rw [Filter.Eventually, ← map_mul_right_ae μ t]; rfl"
}
] | [
329,
66
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
327,
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Mathlib/Data/Stream/Init.lean | Stream'.nth_interleave_right | [
{
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},
{
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"tactic": "rw [nth_succ, nth_succ, interleave_eq, tail_cons, tail_cons,\n nth_interleave_right n (tail s₁) (tail s₂)]"
},
{
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"tactic": "rfl"
}
] | [
449,
8
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
442,
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Mathlib/Topology/Instances/Matrix.lean | continuous_matrix_diag | [] | [
187,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
186,
1
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Mathlib/Order/Filter/AtTopBot.lean | Filter.tendsto_atBot | [] | [
370,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
368,
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Mathlib/Data/Vector.lean | Vector.head_cons | [] | [
66,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
65,
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Mathlib/LinearAlgebra/Matrix/Nondegenerate.lean | Matrix.nondegenerate_of_det_ne_zero | [
{
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"tactic": "intro v hv"
},
{
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"tactic": "ext i"
},
{
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"tactic": "specialize hv (M.cramer (Pi.single i 1))"
},
{
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"tactic": "refine' (mul_eq_zero.mp _).resolve_right hM"
},
{
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"tactic": "convert hv"
},
{
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"tactic": "simp only [mulVec_cramer M (Pi.single i 1), dotProduct, Pi.smul_apply, smul_eq_mul]"
},
{
"state_after": "case h.e'_2.h₀\nm : Type u_1\nR : Type ?u.4064\nA : Type u_2\ninst✝⁴ : Fintype m\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : DecidableEq m\nM : Matrix m m A\nhM : det M ≠ 0\nv : m → A\ni : m\nhv : v ⬝ᵥ mulVec M (↑(cramer M) (Pi.single i 1)) = 0\n⊢ ∀ (b : m), b ∈ Finset.univ → b ≠ i → v b * (det M * Pi.single i 1 b) = 0\n\ncase h.e'_2.h₁\nm : Type u_1\nR : Type ?u.4064\nA : Type u_2\ninst✝⁴ : Fintype m\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : DecidableEq m\nM : Matrix m m A\nhM : det M ≠ 0\nv : m → A\ni : m\nhv : v ⬝ᵥ mulVec M (↑(cramer M) (Pi.single i 1)) = 0\n⊢ ¬i ∈ Finset.univ → v i * (det M * Pi.single i 1 i) = 0",
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"tactic": "rw [Finset.sum_eq_single i, Pi.single_eq_same, mul_one]"
},
{
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"tactic": "intro j _ hj"
},
{
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"tactic": "simp [hj]"
},
{
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"tactic": "intros"
},
{
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"tactic": "have := Finset.mem_univ i"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.h₁\nm : Type u_1\nR : Type ?u.4064\nA : Type u_2\ninst✝⁴ : Fintype m\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : DecidableEq m\nM : Matrix m m A\nhM : det M ≠ 0\nv : m → A\ni : m\nhv : v ⬝ᵥ mulVec M (↑(cramer M) (Pi.single i 1)) = 0\na✝ : ¬i ∈ Finset.univ\nthis : i ∈ Finset.univ\n⊢ v i * (det M * Pi.single i 1 i) = 0",
"tactic": "contradiction"
}
] | [
66,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
53,
1
] |
Mathlib/Analysis/SpecialFunctions/Log/Base.lean | Real.b_pos | [
{
"state_after": "no goals",
"state_before": "b x y : ℝ\nhb : 1 < b\n⊢ 0 < b",
"tactic": "linarith"
}
] | [
142,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
142,
9
] |
Mathlib/MeasureTheory/Constructions/Polish.lean | MeasureTheory.measurablySeparable_range_of_disjoint | [
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\n⊢ False",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\n⊢ MeasurablySeparable (range f) (range g)",
"tactic": "by_contra hfg"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n⊢ False",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\n⊢ False",
"tactic": "have I : ∀ n x y, ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y', x' ∈ cylinder x n ∧ y' ∈ cylinder y n ∧\n ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1)) := by\n intro n x y\n contrapose!\n intro H\n rw [← iUnion_cylinder_update x n, ← iUnion_cylinder_update y n, image_iUnion, image_iUnion]\n refine' MeasurablySeparable.iUnion fun i j => _\n exact H _ _ (update_mem_cylinder _ _ _) (update_mem_cylinder _ _ _)"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\n⊢ False",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n⊢ False",
"tactic": "let A :=\n { p : ℕ × (ℕ → ℕ) × (ℕ → ℕ) //\n ¬MeasurablySeparable (f '' cylinder p.2.1 p.1) (g '' cylinder p.2.2 p.1) }"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nthis :\n ∀ (p : A),\n ∃ q,\n (↑q).fst = (↑p).fst + 1 ∧\n (↑q).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst ∧ (↑q).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\n⊢ False",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\n⊢ False",
"tactic": "have : ∀ p : A, ∃ q : A,\n q.1.1 = p.1.1 + 1 ∧ q.1.2.1 ∈ cylinder p.1.2.1 p.1.1 ∧ q.1.2.2 ∈ cylinder p.1.2.2 p.1.1 := by\n rintro ⟨⟨n, x, y⟩, hp⟩\n rcases I n x y hp with ⟨x', y', hx', hy', h'⟩\n exact ⟨⟨⟨n + 1, x', y'⟩, h'⟩, rfl, hx', hy'⟩"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\n⊢ False",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nthis :\n ∀ (p : A),\n ∃ q,\n (↑q).fst = (↑p).fst + 1 ∧\n (↑q).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst ∧ (↑q).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\n⊢ False",
"tactic": "choose F hFn hFx hFy using this"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\n⊢ False",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\n⊢ False",
"tactic": "let p0 : A := ⟨⟨0, fun _ => 0, fun _ => 0⟩, by simp [hfg]⟩"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\n⊢ False",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\n⊢ False",
"tactic": "let p : ℕ → A := fun n => (F^[n]) p0"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\n⊢ False",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\n⊢ False",
"tactic": "have prec : ∀ n, p (n + 1) = F (p n) := fun n => by simp only [iterate_succ', Function.comp]"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\n⊢ False",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\n⊢ False",
"tactic": "set x : ℕ → ℕ := fun n => (p (n + 1)).1.2.1 n with hx"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\n⊢ False",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\n⊢ False",
"tactic": "set y : ℕ → ℕ := fun n => (p (n + 1)).1.2.2 n with hy"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\n⊢ False",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\n⊢ False",
"tactic": "obtain ⟨u, v, u_open, v_open, xu, yv, huv⟩ :\n ∃ u v : Set α, IsOpen u ∧ IsOpen v ∧ f x ∈ u ∧ g y ∈ v ∧ Disjoint u v := by\n apply t2_separation\n exact disjoint_iff_forall_ne.1 h (mem_range_self _) (mem_range_self _)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\n⊢ False",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\n⊢ False",
"tactic": "letI : MetricSpace (ℕ → ℕ) := metricSpaceNatNat"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\n⊢ False",
"state_before": "case intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\n⊢ False",
"tactic": "obtain ⟨εx, εxpos, hεx⟩ : ∃ (εx : ℝ), εx > 0 ∧ Metric.ball x εx ⊆ f ⁻¹' u := by\n apply Metric.mem_nhds_iff.1\n exact hf.continuousAt.preimage_mem_nhds (u_open.mem_nhds xu)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\n⊢ False",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\n⊢ False",
"tactic": "obtain ⟨εy, εypos, hεy⟩ : ∃ (εy : ℝ), εy > 0 ∧ Metric.ball y εy ⊆ g ⁻¹' v := by\n apply Metric.mem_nhds_iff.1\n exact hg.continuousAt.preimage_mem_nhds (v_open.mem_nhds yv)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\n⊢ False",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\n⊢ False",
"tactic": "obtain ⟨n, hn⟩ : ∃ n : ℕ, (1 / 2 : ℝ) ^ n < min εx εy :=\n exists_pow_lt_of_lt_one (lt_min εxpos εypos) (by norm_num)"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\nB : MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\n⊢ False",
"tactic": "exact M n B"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nn : ℕ\nx y : ℕ → ℕ\n⊢ ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧ y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\n⊢ ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))",
"tactic": "intro n x y"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nn : ℕ\nx y : ℕ → ℕ\n⊢ (∀ (x' y' : ℕ → ℕ),\n x' ∈ cylinder x n →\n y' ∈ cylinder y n → MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))) →\n MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nn : ℕ\nx y : ℕ → ℕ\n⊢ ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧ y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))",
"tactic": "contrapose!"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nn : ℕ\nx y : ℕ → ℕ\nH :\n ∀ (x' y' : ℕ → ℕ),\n x' ∈ cylinder x n → y' ∈ cylinder y n → MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n⊢ MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nn : ℕ\nx y : ℕ → ℕ\n⊢ (∀ (x' y' : ℕ → ℕ),\n x' ∈ cylinder x n →\n y' ∈ cylinder y n → MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))) →\n MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)",
"tactic": "intro H"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nn : ℕ\nx y : ℕ → ℕ\nH :\n ∀ (x' y' : ℕ → ℕ),\n x' ∈ cylinder x n → y' ∈ cylinder y n → MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n⊢ MeasurablySeparable (⋃ (i : ℕ), f '' cylinder (update x n i) (n + 1))\n (⋃ (i : ℕ), g '' cylinder (update y n i) (n + 1))",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nn : ℕ\nx y : ℕ → ℕ\nH :\n ∀ (x' y' : ℕ → ℕ),\n x' ∈ cylinder x n → y' ∈ cylinder y n → MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n⊢ MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)",
"tactic": "rw [← iUnion_cylinder_update x n, ← iUnion_cylinder_update y n, image_iUnion, image_iUnion]"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nn : ℕ\nx y : ℕ → ℕ\nH :\n ∀ (x' y' : ℕ → ℕ),\n x' ∈ cylinder x n → y' ∈ cylinder y n → MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\ni j : ℕ\n⊢ MeasurablySeparable (f '' cylinder (update x n i) (n + 1)) (g '' cylinder (update y n j) (n + 1))",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nn : ℕ\nx y : ℕ → ℕ\nH :\n ∀ (x' y' : ℕ → ℕ),\n x' ∈ cylinder x n → y' ∈ cylinder y n → MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n⊢ MeasurablySeparable (⋃ (i : ℕ), f '' cylinder (update x n i) (n + 1))\n (⋃ (i : ℕ), g '' cylinder (update y n i) (n + 1))",
"tactic": "refine' MeasurablySeparable.iUnion fun i j => _"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nn : ℕ\nx y : ℕ → ℕ\nH :\n ∀ (x' y' : ℕ → ℕ),\n x' ∈ cylinder x n → y' ∈ cylinder y n → MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\ni j : ℕ\n⊢ MeasurablySeparable (f '' cylinder (update x n i) (n + 1)) (g '' cylinder (update y n j) (n + 1))",
"tactic": "exact H _ _ (update_mem_cylinder _ _ _) (update_mem_cylinder _ _ _)"
},
{
"state_after": "case mk.mk.mk\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nn : ℕ\nx y : ℕ → ℕ\nhp :\n ¬MeasurablySeparable (f '' cylinder (n, x, y).snd.fst (n, x, y).fst) (g '' cylinder (n, x, y).snd.snd (n, x, y).fst)\n⊢ ∃ q,\n (↑q).fst = (↑{ val := (n, x, y), property := hp }).fst + 1 ∧\n (↑q).snd.fst ∈\n cylinder (↑{ val := (n, x, y), property := hp }).snd.fst (↑{ val := (n, x, y), property := hp }).fst ∧\n (↑q).snd.snd ∈\n cylinder (↑{ val := (n, x, y), property := hp }).snd.snd (↑{ val := (n, x, y), property := hp }).fst",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\n⊢ ∀ (p : A),\n ∃ q,\n (↑q).fst = (↑p).fst + 1 ∧\n (↑q).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst ∧ (↑q).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst",
"tactic": "rintro ⟨⟨n, x, y⟩, hp⟩"
},
{
"state_after": "case mk.mk.mk.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nn : ℕ\nx y : ℕ → ℕ\nhp :\n ¬MeasurablySeparable (f '' cylinder (n, x, y).snd.fst (n, x, y).fst) (g '' cylinder (n, x, y).snd.snd (n, x, y).fst)\nx' y' : ℕ → ℕ\nhx' : x' ∈ cylinder x n\nhy' : y' ∈ cylinder y n\nh' : ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n⊢ ∃ q,\n (↑q).fst = (↑{ val := (n, x, y), property := hp }).fst + 1 ∧\n (↑q).snd.fst ∈\n cylinder (↑{ val := (n, x, y), property := hp }).snd.fst (↑{ val := (n, x, y), property := hp }).fst ∧\n (↑q).snd.snd ∈\n cylinder (↑{ val := (n, x, y), property := hp }).snd.snd (↑{ val := (n, x, y), property := hp }).fst",
"state_before": "case mk.mk.mk\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nn : ℕ\nx y : ℕ → ℕ\nhp :\n ¬MeasurablySeparable (f '' cylinder (n, x, y).snd.fst (n, x, y).fst) (g '' cylinder (n, x, y).snd.snd (n, x, y).fst)\n⊢ ∃ q,\n (↑q).fst = (↑{ val := (n, x, y), property := hp }).fst + 1 ∧\n (↑q).snd.fst ∈\n cylinder (↑{ val := (n, x, y), property := hp }).snd.fst (↑{ val := (n, x, y), property := hp }).fst ∧\n (↑q).snd.snd ∈\n cylinder (↑{ val := (n, x, y), property := hp }).snd.snd (↑{ val := (n, x, y), property := hp }).fst",
"tactic": "rcases I n x y hp with ⟨x', y', hx', hy', h'⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.mk.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nn : ℕ\nx y : ℕ → ℕ\nhp :\n ¬MeasurablySeparable (f '' cylinder (n, x, y).snd.fst (n, x, y).fst) (g '' cylinder (n, x, y).snd.snd (n, x, y).fst)\nx' y' : ℕ → ℕ\nhx' : x' ∈ cylinder x n\nhy' : y' ∈ cylinder y n\nh' : ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\n⊢ ∃ q,\n (↑q).fst = (↑{ val := (n, x, y), property := hp }).fst + 1 ∧\n (↑q).snd.fst ∈\n cylinder (↑{ val := (n, x, y), property := hp }).snd.fst (↑{ val := (n, x, y), property := hp }).fst ∧\n (↑q).snd.snd ∈\n cylinder (↑{ val := (n, x, y), property := hp }).snd.snd (↑{ val := (n, x, y), property := hp }).fst",
"tactic": "exact ⟨⟨⟨n + 1, x', y'⟩, h'⟩, rfl, hx', hy'⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\n⊢ ¬MeasurablySeparable (f '' cylinder (0, fun x => 0, fun x => 0).snd.fst (0, fun x => 0, fun x => 0).fst)\n (g '' cylinder (0, fun x => 0, fun x => 0).snd.snd (0, fun x => 0, fun x => 0).fst)",
"tactic": "simp [hfg]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nn : ℕ\n⊢ p (n + 1) = F (p n)",
"tactic": "simp only [iterate_succ', Function.comp]"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\nn : ℕ\n⊢ (↑(p n)).fst = n",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\n⊢ ∀ (n : ℕ), (↑(p n)).fst = n",
"tactic": "intro n"
},
{
"state_after": "case zero\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\n⊢ (↑(p Nat.zero)).fst = Nat.zero\n\ncase succ\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\nn : ℕ\nIH : (↑(p n)).fst = n\n⊢ (↑(p (Nat.succ n))).fst = Nat.succ n",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\nn : ℕ\n⊢ (↑(p n)).fst = n",
"tactic": "induction' n with n IH"
},
{
"state_after": "no goals",
"state_before": "case zero\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\n⊢ (↑(p Nat.zero)).fst = Nat.zero",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\nn : ℕ\nIH : (↑(p n)).fst = n\n⊢ (↑(p (Nat.succ n))).fst = Nat.succ n",
"tactic": "simp only [prec, hFn, IH]"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nm : ℕ\n⊢ ∀ (n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\n⊢ ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m",
"tactic": "intro m"
},
{
"state_after": "case base\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nm : ℕ\n⊢ Prod.fst (↑(p (m + 1))).snd m = Prod.fst (↑(p (m + 1))).snd m\n\ncase succ\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nm : ℕ\n⊢ ∀ (n : ℕ),\n m + 1 ≤ n →\n Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m →\n Prod.fst (↑(p (n + 1))).snd m = Prod.fst (↑(p (m + 1))).snd m",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nm : ℕ\n⊢ ∀ (n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m",
"tactic": "apply Nat.le_induction"
},
{
"state_after": "case succ\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nm n : ℕ\nhmn : m + 1 ≤ n\nIH : Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\n⊢ Prod.fst (↑(p (n + 1))).snd m = Prod.fst (↑(p (m + 1))).snd m",
"state_before": "case succ\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nm : ℕ\n⊢ ∀ (n : ℕ),\n m + 1 ≤ n →\n Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m →\n Prod.fst (↑(p (n + 1))).snd m = Prod.fst (↑(p (m + 1))).snd m",
"tactic": "intro n hmn IH"
},
{
"state_after": "case succ\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI✝ :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nm n : ℕ\nhmn : m + 1 ≤ n\nIH : Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nI : Prod.fst (↑(F (p n))).snd m = Prod.fst (↑(p n)).snd m\n⊢ Prod.fst (↑(p (n + 1))).snd m = Prod.fst (↑(p (m + 1))).snd m",
"state_before": "case succ\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nm n : ℕ\nhmn : m + 1 ≤ n\nIH : Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\n⊢ Prod.fst (↑(p (n + 1))).snd m = Prod.fst (↑(p (m + 1))).snd m",
"tactic": "have I : (F (p n)).val.snd.fst m = (p n).val.snd.fst m := by\n apply hFx (p n) m\n rw [pn_fst]\n exact hmn"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI✝ :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nm n : ℕ\nhmn : m + 1 ≤ n\nIH : Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nI : Prod.fst (↑(F (p n))).snd m = Prod.fst (↑(p n)).snd m\n⊢ Prod.fst (↑(p (n + 1))).snd m = Prod.fst (↑(p (m + 1))).snd m",
"tactic": "rw [prec, I, IH]"
},
{
"state_after": "no goals",
"state_before": "case base\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nm : ℕ\n⊢ Prod.fst (↑(p (m + 1))).snd m = Prod.fst (↑(p (m + 1))).snd m",
"tactic": "rfl"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nm n : ℕ\nhmn : m + 1 ≤ n\nIH : Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\n⊢ m < (↑(p n)).fst",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nm n : ℕ\nhmn : m + 1 ≤ n\nIH : Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\n⊢ Prod.fst (↑(F (p n))).snd m = Prod.fst (↑(p n)).snd m",
"tactic": "apply hFx (p n) m"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nm n : ℕ\nhmn : m + 1 ≤ n\nIH : Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\n⊢ m < n",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nm n : ℕ\nhmn : m + 1 ≤ n\nIH : Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\n⊢ m < (↑(p n)).fst",
"tactic": "rw [pn_fst]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nm n : ℕ\nhmn : m + 1 ≤ n\nIH : Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\n⊢ m < n",
"tactic": "exact hmn"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nm : ℕ\n⊢ ∀ (n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\n⊢ ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m",
"tactic": "intro m"
},
{
"state_after": "case base\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nm : ℕ\n⊢ Prod.snd (↑(p (m + 1))).snd m = Prod.snd (↑(p (m + 1))).snd m\n\ncase succ\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nm : ℕ\n⊢ ∀ (n : ℕ),\n m + 1 ≤ n →\n Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m →\n Prod.snd (↑(p (n + 1))).snd m = Prod.snd (↑(p (m + 1))).snd m",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nm : ℕ\n⊢ ∀ (n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m",
"tactic": "apply Nat.le_induction"
},
{
"state_after": "case succ\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nm n : ℕ\nhmn : m + 1 ≤ n\nIH : Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\n⊢ Prod.snd (↑(p (n + 1))).snd m = Prod.snd (↑(p (m + 1))).snd m",
"state_before": "case succ\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nm : ℕ\n⊢ ∀ (n : ℕ),\n m + 1 ≤ n →\n Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m →\n Prod.snd (↑(p (n + 1))).snd m = Prod.snd (↑(p (m + 1))).snd m",
"tactic": "intro n hmn IH"
},
{
"state_after": "case succ\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI✝ :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nm n : ℕ\nhmn : m + 1 ≤ n\nIH : Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nI : Prod.snd (↑(F (p n))).snd m = Prod.snd (↑(p n)).snd m\n⊢ Prod.snd (↑(p (n + 1))).snd m = Prod.snd (↑(p (m + 1))).snd m",
"state_before": "case succ\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nm n : ℕ\nhmn : m + 1 ≤ n\nIH : Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\n⊢ Prod.snd (↑(p (n + 1))).snd m = Prod.snd (↑(p (m + 1))).snd m",
"tactic": "have I : (F (p n)).val.snd.snd m = (p n).val.snd.snd m := by\n apply hFy (p n) m\n rw [pn_fst]\n exact hmn"
},
{
"state_after": "no goals",
"state_before": "case succ\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI✝ :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nm n : ℕ\nhmn : m + 1 ≤ n\nIH : Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nI : Prod.snd (↑(F (p n))).snd m = Prod.snd (↑(p n)).snd m\n⊢ Prod.snd (↑(p (n + 1))).snd m = Prod.snd (↑(p (m + 1))).snd m",
"tactic": "rw [prec, I, IH]"
},
{
"state_after": "no goals",
"state_before": "case base\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nm : ℕ\n⊢ Prod.snd (↑(p (m + 1))).snd m = Prod.snd (↑(p (m + 1))).snd m",
"tactic": "rfl"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nm n : ℕ\nhmn : m + 1 ≤ n\nIH : Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\n⊢ m < (↑(p n)).fst",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nm n : ℕ\nhmn : m + 1 ≤ n\nIH : Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\n⊢ Prod.snd (↑(F (p n))).snd m = Prod.snd (↑(p n)).snd m",
"tactic": "apply hFy (p n) m"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nm n : ℕ\nhmn : m + 1 ≤ n\nIH : Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\n⊢ m < n",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nm n : ℕ\nhmn : m + 1 ≤ n\nIH : Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\n⊢ m < (↑(p n)).fst",
"tactic": "rw [pn_fst]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nm n : ℕ\nhmn : m + 1 ≤ n\nIH : Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\n⊢ m < n",
"tactic": "exact hmn"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nn : ℕ\n⊢ ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\n⊢ ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)",
"tactic": "intro n"
},
{
"state_after": "case h.e'_1.h.e'_3.h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nn : ℕ\n⊢ cylinder x n = cylinder (↑(p n)).snd.fst (↑(p n)).fst\n\ncase h.e'_1.h.e'_4.h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nn : ℕ\n⊢ cylinder y n = cylinder (↑(p n)).snd.snd (↑(p n)).fst",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nn : ℕ\n⊢ ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)",
"tactic": "convert(p n).2 using 3"
},
{
"state_after": "case h.e'_1.h.e'_3.h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nn : ℕ\n⊢ ∀ (i : ℕ), i < n → x i = Prod.fst (↑(p n)).snd i",
"state_before": "case h.e'_1.h.e'_3.h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nn : ℕ\n⊢ cylinder x n = cylinder (↑(p n)).snd.fst (↑(p n)).fst",
"tactic": "rw [pn_fst, ← mem_cylinder_iff_eq, mem_cylinder_iff]"
},
{
"state_after": "case h.e'_1.h.e'_3.h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nn i : ℕ\nhi : i < n\n⊢ x i = Prod.fst (↑(p n)).snd i",
"state_before": "case h.e'_1.h.e'_3.h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nn : ℕ\n⊢ ∀ (i : ℕ), i < n → x i = Prod.fst (↑(p n)).snd i",
"tactic": "intro i hi"
},
{
"state_after": "case h.e'_1.h.e'_3.h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nn i : ℕ\nhi : i < n\n⊢ (fun n => Prod.fst (↑(p (n + 1))).snd n) i = Prod.fst (↑(p n)).snd i",
"state_before": "case h.e'_1.h.e'_3.h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nn i : ℕ\nhi : i < n\n⊢ x i = Prod.fst (↑(p n)).snd i",
"tactic": "rw [hx]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_1.h.e'_3.h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nn i : ℕ\nhi : i < n\n⊢ (fun n => Prod.fst (↑(p (n + 1))).snd n) i = Prod.fst (↑(p n)).snd i",
"tactic": "exact (Ix i n hi).symm"
},
{
"state_after": "case h.e'_1.h.e'_4.h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nn : ℕ\n⊢ ∀ (i : ℕ), i < n → y i = Prod.snd (↑(p n)).snd i",
"state_before": "case h.e'_1.h.e'_4.h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nn : ℕ\n⊢ cylinder y n = cylinder (↑(p n)).snd.snd (↑(p n)).fst",
"tactic": "rw [pn_fst, ← mem_cylinder_iff_eq, mem_cylinder_iff]"
},
{
"state_after": "case h.e'_1.h.e'_4.h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nn i : ℕ\nhi : i < n\n⊢ y i = Prod.snd (↑(p n)).snd i",
"state_before": "case h.e'_1.h.e'_4.h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nn : ℕ\n⊢ ∀ (i : ℕ), i < n → y i = Prod.snd (↑(p n)).snd i",
"tactic": "intro i hi"
},
{
"state_after": "case h.e'_1.h.e'_4.h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nn i : ℕ\nhi : i < n\n⊢ (fun n => Prod.snd (↑(p (n + 1))).snd n) i = Prod.snd (↑(p n)).snd i",
"state_before": "case h.e'_1.h.e'_4.h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nn i : ℕ\nhi : i < n\n⊢ y i = Prod.snd (↑(p n)).snd i",
"tactic": "rw [hy]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_1.h.e'_4.h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nn i : ℕ\nhi : i < n\n⊢ (fun n => Prod.snd (↑(p (n + 1))).snd n) i = Prod.snd (↑(p n)).snd i",
"tactic": "exact (Iy i n hi).symm"
},
{
"state_after": "case h\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\n⊢ f x ≠ g y",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\n⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ f x ∈ u ∧ g y ∈ v ∧ Disjoint u v",
"tactic": "apply t2_separation"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\n⊢ f x ≠ g y",
"tactic": "exact disjoint_iff_forall_ne.1 h (mem_range_self _) (mem_range_self _)"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\n⊢ f ⁻¹' u ∈ 𝓝 x",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\n⊢ ∃ εx, εx > 0 ∧ ball x εx ⊆ f ⁻¹' u",
"tactic": "apply Metric.mem_nhds_iff.1"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\n⊢ f ⁻¹' u ∈ 𝓝 x",
"tactic": "exact hf.continuousAt.preimage_mem_nhds (u_open.mem_nhds xu)"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\n⊢ g ⁻¹' v ∈ 𝓝 y",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\n⊢ ∃ εy, εy > 0 ∧ ball y εy ⊆ g ⁻¹' v",
"tactic": "apply Metric.mem_nhds_iff.1"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\n⊢ g ⁻¹' v ∈ 𝓝 y",
"tactic": "exact hg.continuousAt.preimage_mem_nhds (v_open.mem_nhds yv)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\n⊢ 1 / 2 < 1",
"tactic": "norm_num"
},
{
"state_after": "case refine'_1\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\n⊢ f '' cylinder x n ⊆ u\n\ncase refine'_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\n⊢ Disjoint (g '' cylinder y n) u",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\n⊢ MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)",
"tactic": "refine' ⟨u, _, _, u_open.measurableSet⟩"
},
{
"state_after": "case refine'_1\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\n⊢ cylinder x n ⊆ f ⁻¹' u",
"state_before": "case refine'_1\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\n⊢ f '' cylinder x n ⊆ u",
"tactic": "rw [image_subset_iff]"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\n⊢ cylinder x n ⊆ ball x εx",
"state_before": "case refine'_1\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\n⊢ cylinder x n ⊆ f ⁻¹' u",
"tactic": "apply Subset.trans _ hεx"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\nz : ℕ → ℕ\nhz : z ∈ cylinder x n\n⊢ z ∈ ball x εx",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\n⊢ cylinder x n ⊆ ball x εx",
"tactic": "intro z hz"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\nz : ℕ → ℕ\nhz : dist z x ≤ (1 / 2) ^ n\n⊢ z ∈ ball x εx",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\nz : ℕ → ℕ\nhz : z ∈ cylinder x n\n⊢ z ∈ ball x εx",
"tactic": "rw [mem_cylinder_iff_dist_le] at hz"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\nz : ℕ → ℕ\nhz : dist z x ≤ (1 / 2) ^ n\n⊢ z ∈ ball x εx",
"tactic": "exact hz.trans_lt (hn.trans_le (min_le_left _ _))"
},
{
"state_after": "case refine'_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\n⊢ g '' cylinder y n ≤ v",
"state_before": "case refine'_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\n⊢ Disjoint (g '' cylinder y n) u",
"tactic": "refine' Disjoint.mono_left _ huv.symm"
},
{
"state_after": "case refine'_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\n⊢ g '' cylinder y n ⊆ v",
"state_before": "case refine'_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\n⊢ g '' cylinder y n ≤ v",
"tactic": "change g '' cylinder y n ⊆ v"
},
{
"state_after": "case refine'_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\n⊢ cylinder y n ⊆ g ⁻¹' v",
"state_before": "case refine'_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\n⊢ g '' cylinder y n ⊆ v",
"tactic": "rw [image_subset_iff]"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\n⊢ cylinder y n ⊆ ball y εy",
"state_before": "case refine'_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\n⊢ cylinder y n ⊆ g ⁻¹' v",
"tactic": "apply Subset.trans _ hεy"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\nz : ℕ → ℕ\nhz : z ∈ cylinder y n\n⊢ z ∈ ball y εy",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\n⊢ cylinder y n ⊆ ball y εy",
"tactic": "intro z hz"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\nz : ℕ → ℕ\nhz : dist z y ≤ (1 / 2) ^ n\n⊢ z ∈ ball y εy",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\nz : ℕ → ℕ\nhz : z ∈ cylinder y n\n⊢ z ∈ ball y εy",
"tactic": "rw [mem_cylinder_iff_dist_le] at hz"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\nι : Type ?u.68523\ninst✝² : T2Space α\ninst✝¹ : MeasurableSpace α\ninst✝ : BorelSpace α\nf g : (ℕ → ℕ) → α\nhf : Continuous f\nhg : Continuous g\nh : Disjoint (range f) (range g)\nhfg : ¬MeasurablySeparable (range f) (range g)\nI :\n ∀ (n : ℕ) (x y : ℕ → ℕ),\n ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n) →\n ∃ x' y',\n x' ∈ cylinder x n ∧\n y' ∈ cylinder y n ∧ ¬MeasurablySeparable (f '' cylinder x' (n + 1)) (g '' cylinder y' (n + 1))\nA : Type := { p // ¬MeasurablySeparable (f '' cylinder p.snd.fst p.fst) (g '' cylinder p.snd.snd p.fst) }\nF : A → A\nhFn : ∀ (p : A), (↑(F p)).fst = (↑p).fst + 1\nhFx : ∀ (p : A), (↑(F p)).snd.fst ∈ cylinder (↑p).snd.fst (↑p).fst\nhFy : ∀ (p : A), (↑(F p)).snd.snd ∈ cylinder (↑p).snd.snd (↑p).fst\np0 : A :=\n { val := (0, fun x => 0, fun x => 0),\n property := (_ : ¬MeasurablySeparable (f '' cylinder (fun x => 0) 0) (g '' cylinder (fun x => 0) 0)) }\np : ℕ → A := fun n => (F^[n]) p0\nprec : ∀ (n : ℕ), p (n + 1) = F (p n)\npn_fst : ∀ (n : ℕ), (↑(p n)).fst = n\nIx : ∀ (m n : ℕ), m + 1 ≤ n → Prod.fst (↑(p n)).snd m = Prod.fst (↑(p (m + 1))).snd m\nIy : ∀ (m n : ℕ), m + 1 ≤ n → Prod.snd (↑(p n)).snd m = Prod.snd (↑(p (m + 1))).snd m\nx : ℕ → ℕ := fun n => Prod.fst (↑(p (n + 1))).snd n\nhx : x = fun n => Prod.fst (↑(p (n + 1))).snd n\ny : ℕ → ℕ := fun n => Prod.snd (↑(p (n + 1))).snd n\nhy : y = fun n => Prod.snd (↑(p (n + 1))).snd n\nM : ∀ (n : ℕ), ¬MeasurablySeparable (f '' cylinder x n) (g '' cylinder y n)\nu v : Set α\nu_open : IsOpen u\nv_open : IsOpen v\nxu : f x ∈ u\nyv : g y ∈ v\nhuv : Disjoint u v\nthis : MetricSpace (ℕ → ℕ) := metricSpaceNatNat\nεx : ℝ\nεxpos : εx > 0\nhεx : ball x εx ⊆ f ⁻¹' u\nεy : ℝ\nεypos : εy > 0\nhεy : ball y εy ⊆ g ⁻¹' v\nn : ℕ\nhn : (1 / 2) ^ n < min εx εy\nz : ℕ → ℕ\nhz : dist z y ≤ (1 / 2) ^ n\n⊢ z ∈ ball y εy",
"tactic": "exact hz.trans_lt (hn.trans_le (min_le_right _ _))"
}
] | [
402,
14
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
293,
1
] |
Mathlib/Topology/Basic.lean | frontier_closure_subset | [] | [
723,
74
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
722,
1
] |
Mathlib/Topology/Perfect.lean | Preperfect.open_inter | [
{
"state_after": "case intro\nα : Type u_1\ninst✝ : TopologicalSpace α\nC U : Set α\nhC : Preperfect C\nhU : IsOpen U\nx : α\nxU : x ∈ U\nxC : x ∈ C\n⊢ AccPt x (𝓟 (U ∩ C))",
"state_before": "α : Type u_1\ninst✝ : TopologicalSpace α\nC U : Set α\nhC : Preperfect C\nhU : IsOpen U\n⊢ Preperfect (U ∩ C)",
"tactic": "rintro x ⟨xU, xC⟩"
},
{
"state_after": "case intro\nα : Type u_1\ninst✝ : TopologicalSpace α\nC U : Set α\nhC : Preperfect C\nhU : IsOpen U\nx : α\nxU : x ∈ U\nxC : x ∈ C\n⊢ U ∈ 𝓝 x",
"state_before": "case intro\nα : Type u_1\ninst✝ : TopologicalSpace α\nC U : Set α\nhC : Preperfect C\nhU : IsOpen U\nx : α\nxU : x ∈ U\nxC : x ∈ C\n⊢ AccPt x (𝓟 (U ∩ C))",
"tactic": "apply (hC _ xC).nhds_inter"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_1\ninst✝ : TopologicalSpace α\nC U : Set α\nhC : Preperfect C\nhU : IsOpen U\nx : α\nxU : x ∈ U\nxC : x ∈ C\n⊢ U ∈ 𝓝 x",
"tactic": "exact hU.mem_nhds xU"
}
] | [
98,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
94,
1
] |
Mathlib/Order/WellFoundedSet.lean | IsAntichain.finite_of_partiallyWellOrderedOn | [
{
"state_after": "ι : Type ?u.49249\nα : Type u_1\nβ : Type ?u.49255\nr : α → α → Prop\nr' : β → β → Prop\nf : α → β\ns t : Set α\na : α\nha : IsAntichain r s\nhp : PartiallyWellOrderedOn s r\nhi : Set.Infinite s\n⊢ False",
"state_before": "ι : Type ?u.49249\nα : Type u_1\nβ : Type ?u.49255\nr : α → α → Prop\nr' : β → β → Prop\nf : α → β\ns t : Set α\na : α\nha : IsAntichain r s\nhp : PartiallyWellOrderedOn s r\n⊢ Set.Finite s",
"tactic": "refine' not_infinite.1 fun hi => _"
},
{
"state_after": "case intro.intro.intro\nι : Type ?u.49249\nα : Type u_1\nβ : Type ?u.49255\nr : α → α → Prop\nr' : β → β → Prop\nf : α → β\ns t : Set α\na : α\nha : IsAntichain r s\nhp : PartiallyWellOrderedOn s r\nhi : Set.Infinite s\nm n : ℕ\nhmn : m < n\nh : r ↑(↑(Infinite.natEmbedding s hi) m) ↑(↑(Infinite.natEmbedding s hi) n)\n⊢ False",
"state_before": "ι : Type ?u.49249\nα : Type u_1\nβ : Type ?u.49255\nr : α → α → Prop\nr' : β → β → Prop\nf : α → β\ns t : Set α\na : α\nha : IsAntichain r s\nhp : PartiallyWellOrderedOn s r\nhi : Set.Infinite s\n⊢ False",
"tactic": "obtain ⟨m, n, hmn, h⟩ := hp (fun n => hi.natEmbedding _ n) fun n => (hi.natEmbedding _ n).2"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nι : Type ?u.49249\nα : Type u_1\nβ : Type ?u.49255\nr : α → α → Prop\nr' : β → β → Prop\nf : α → β\ns t : Set α\na : α\nha : IsAntichain r s\nhp : PartiallyWellOrderedOn s r\nhi : Set.Infinite s\nm n : ℕ\nhmn : m < n\nh : r ↑(↑(Infinite.natEmbedding s hi) m) ↑(↑(Infinite.natEmbedding s hi) n)\n⊢ False",
"tactic": "exact hmn.ne ((hi.natEmbedding _).injective <| Subtype.val_injective <|\n ha.eq (hi.natEmbedding _ m).2 (hi.natEmbedding _ n).2 h)"
}
] | [
287,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
282,
1
] |
Mathlib/Analysis/NormedSpace/Pointwise.lean | smul_unitBall_of_pos | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.198360\nE : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace ℝ E\nx y z : E\nδ ε r : ℝ\nhr : 0 < r\n⊢ r • ball 0 1 = ball 0 r",
"tactic": "rw [smul_unitBall hr.ne', Real.norm_of_nonneg hr.le]"
}
] | [
152,
55
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
151,
1
] |
Mathlib/Analysis/Quaternion.lean | Quaternion.coeComplex_re | [] | [
117,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
116,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.isBigOWith_norm_norm | [] | [
809,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
807,
1
] |
Mathlib/Analysis/Asymptotics/Asymptotics.lean | Asymptotics.isLittleO_const_left | [
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.609377\nE : Type ?u.609380\nF : Type ?u.609383\nG : Type ?u.609386\nE' : Type ?u.609389\nF' : Type ?u.609392\nG' : Type ?u.609395\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.609404\nR : Type ?u.609407\nR' : Type ?u.609410\n𝕜 : Type ?u.609413\n𝕜' : Type ?u.609416\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\n⊢ (fun _x => 0) =o[l] g'' ↔ 0 = 0 ∨ Tendsto (norm ∘ g'') l atTop\n\ncase inr\nα : Type u_1\nβ : Type ?u.609377\nE : Type ?u.609380\nF : Type ?u.609383\nG : Type ?u.609386\nE' : Type ?u.609389\nF' : Type ?u.609392\nG' : Type ?u.609395\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.609404\nR : Type ?u.609407\nR' : Type ?u.609410\n𝕜 : Type ?u.609413\n𝕜' : Type ?u.609416\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc✝ c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nc : E''\nhc : c ≠ 0\n⊢ (fun _x => c) =o[l] g'' ↔ c = 0 ∨ Tendsto (norm ∘ g'') l atTop",
"state_before": "α : Type u_1\nβ : Type ?u.609377\nE : Type ?u.609380\nF : Type ?u.609383\nG : Type ?u.609386\nE' : Type ?u.609389\nF' : Type ?u.609392\nG' : Type ?u.609395\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.609404\nR : Type ?u.609407\nR' : Type ?u.609410\n𝕜 : Type ?u.609413\n𝕜' : Type ?u.609416\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc✝ c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nc : E''\n⊢ (fun _x => c) =o[l] g'' ↔ c = 0 ∨ Tendsto (norm ∘ g'') l atTop",
"tactic": "rcases eq_or_ne c 0 with (rfl | hc)"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.609377\nE : Type ?u.609380\nF : Type ?u.609383\nG : Type ?u.609386\nE' : Type ?u.609389\nF' : Type ?u.609392\nG' : Type ?u.609395\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.609404\nR : Type ?u.609407\nR' : Type ?u.609410\n𝕜 : Type ?u.609413\n𝕜' : Type ?u.609416\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\n⊢ (fun _x => 0) =o[l] g'' ↔ 0 = 0 ∨ Tendsto (norm ∘ g'') l atTop",
"tactic": "simp only [isLittleO_zero, eq_self_iff_true, true_or_iff]"
},
{
"state_after": "case inr\nα : Type u_1\nβ : Type ?u.609377\nE : Type ?u.609380\nF : Type ?u.609383\nG : Type ?u.609386\nE' : Type ?u.609389\nF' : Type ?u.609392\nG' : Type ?u.609395\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.609404\nR : Type ?u.609407\nR' : Type ?u.609410\n𝕜 : Type ?u.609413\n𝕜' : Type ?u.609416\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc✝ c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nc : E''\nhc : c ≠ 0\n⊢ Tendsto (fun x => ‖g'' x‖) l atTop ↔ Tendsto (norm ∘ g'') l atTop",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.609377\nE : Type ?u.609380\nF : Type ?u.609383\nG : Type ?u.609386\nE' : Type ?u.609389\nF' : Type ?u.609392\nG' : Type ?u.609395\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.609404\nR : Type ?u.609407\nR' : Type ?u.609410\n𝕜 : Type ?u.609413\n𝕜' : Type ?u.609416\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc✝ c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nc : E''\nhc : c ≠ 0\n⊢ (fun _x => c) =o[l] g'' ↔ c = 0 ∨ Tendsto (norm ∘ g'') l atTop",
"tactic": "simp only [hc, false_or_iff, isLittleO_const_left_of_ne hc]"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.609377\nE : Type ?u.609380\nF : Type ?u.609383\nG : Type ?u.609386\nE' : Type ?u.609389\nF' : Type ?u.609392\nG' : Type ?u.609395\nE'' : Type u_2\nF'' : Type u_3\nG'' : Type ?u.609404\nR : Type ?u.609407\nR' : Type ?u.609410\n𝕜 : Type ?u.609413\n𝕜' : Type ?u.609416\ninst✝¹² : Norm E\ninst✝¹¹ : Norm F\ninst✝¹⁰ : Norm G\ninst✝⁹ : SeminormedAddCommGroup E'\ninst✝⁸ : SeminormedAddCommGroup F'\ninst✝⁷ : SeminormedAddCommGroup G'\ninst✝⁶ : NormedAddCommGroup E''\ninst✝⁵ : NormedAddCommGroup F''\ninst✝⁴ : NormedAddCommGroup G''\ninst✝³ : SeminormedRing R\ninst✝² : SeminormedRing R'\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedField 𝕜'\nc✝ c' c₁ c₂ : ℝ\nf : α → E\ng : α → F\nk : α → G\nf' : α → E'\ng' : α → F'\nk' : α → G'\nf'' : α → E''\ng'' : α → F''\nk'' : α → G''\nl l' : Filter α\nc : E''\nhc : c ≠ 0\n⊢ Tendsto (fun x => ‖g'' x‖) l atTop ↔ Tendsto (norm ∘ g'') l atTop",
"tactic": "rfl"
}
] | [
1860,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1856,
1
] |
Mathlib/Data/Rat/NNRat.lean | NNRat.num_div_den | [
{
"state_after": "case a\np q✝ q : ℚ≥0\n⊢ ↑(↑(num q) / ↑(den q)) = ↑q",
"state_before": "p q✝ q : ℚ≥0\n⊢ ↑(num q) / ↑(den q) = q",
"tactic": "ext1"
},
{
"state_after": "case a\np q✝ q : ℚ≥0\n⊢ ↑(↑q).num / ↑(den q) = ↑q",
"state_before": "case a\np q✝ q : ℚ≥0\n⊢ ↑(↑(num q) / ↑(den q)) = ↑q",
"tactic": "rw [coe_div, coe_natCast, coe_natCast, num, ← Int.cast_ofNat,\n Int.natAbs_of_nonneg (Rat.num_nonneg_iff_zero_le.2 q.prop)]"
},
{
"state_after": "no goals",
"state_before": "case a\np q✝ q : ℚ≥0\n⊢ ↑(↑q).num / ↑(den q) = ↑q",
"tactic": "exact Rat.num_div_den q"
}
] | [
489,
26
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
485,
1
] |
Mathlib/Order/Basic.lean | le_of_eq_of_le' | [] | [
195,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
194,
1
] |
Std/Data/List/Lemmas.lean | List.head!_of_head? | [] | [
439,
23
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
438,
1
] |
Mathlib/Order/Filter/Pi.lean | Filter.coprodᵢ_neBot_iff' | [
{
"state_after": "no goals",
"state_before": "ι : Type u_2\nα : ι → Type u_1\nf f₁ f₂ : (i : ι) → Filter (α i)\ns : (i : ι) → Set (α i)\n⊢ NeBot (Filter.coprodᵢ f) ↔ (∀ (i : ι), Nonempty (α i)) ∧ ∃ d, NeBot (f d)",
"tactic": "simp only [Filter.coprodᵢ, iSup_neBot, ← exists_and_left, ← comap_eval_neBot_iff']"
}
] | [
215,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
213,
1
] |
Mathlib/Data/Finset/Image.lean | Finset.map_map | [
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf✝ : α ↪ β\ns✝ : Finset α\nf : α ↪ β\ng : β ↪ γ\ns : Finset α\n⊢ Multiset.map ((fun x => ↑g x) ∘ fun x => ↑f x) s.val = Multiset.map (fun x => ↑(Embedding.trans f g) x) s.val",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf✝ : α ↪ β\ns✝ : Finset α\nf : α ↪ β\ng : β ↪ γ\ns : Finset α\n⊢ (map g (map f s)).val = (map (Embedding.trans f g) s).val",
"tactic": "simp only [map_val, Multiset.map_map]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf✝ : α ↪ β\ns✝ : Finset α\nf : α ↪ β\ng : β ↪ γ\ns : Finset α\n⊢ Multiset.map ((fun x => ↑g x) ∘ fun x => ↑f x) s.val = Multiset.map (fun x => ↑(Embedding.trans f g) x) s.val",
"tactic": "rfl"
}
] | [
138,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
137,
1
] |
Mathlib/MeasureTheory/MeasurableSpaceDef.lean | MeasurableSpace.measurableSpace_iSup_eq | [
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.36119\nγ : Type ?u.36122\nδ : Type ?u.36125\nδ' : Type ?u.36128\nι : Sort u_2\ns✝ t u : Set α\nm : ι → MeasurableSpace α\ns : Set α\n⊢ MeasurableSet s ↔ MeasurableSet s",
"state_before": "α : Type u_1\nβ : Type ?u.36119\nγ : Type ?u.36122\nδ : Type ?u.36125\nδ' : Type ?u.36128\nι : Sort u_2\ns t u : Set α\nm : ι → MeasurableSpace α\n⊢ (⨆ (n : ι), m n) = generateFrom {s | ∃ n, MeasurableSet s}",
"tactic": "ext s"
},
{
"state_after": "case h\nα : Type u_1\nβ : Type ?u.36119\nγ : Type ?u.36122\nδ : Type ?u.36125\nδ' : Type ?u.36128\nι : Sort u_2\ns✝ t u : Set α\nm : ι → MeasurableSpace α\ns : Set α\n⊢ GenerateMeasurable {s | ∃ i, MeasurableSet s} s ↔ MeasurableSet s",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.36119\nγ : Type ?u.36122\nδ : Type ?u.36125\nδ' : Type ?u.36128\nι : Sort u_2\ns✝ t u : Set α\nm : ι → MeasurableSpace α\ns : Set α\n⊢ MeasurableSet s ↔ MeasurableSet s",
"tactic": "rw [measurableSet_iSup]"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\nβ : Type ?u.36119\nγ : Type ?u.36122\nδ : Type ?u.36125\nδ' : Type ?u.36128\nι : Sort u_2\ns✝ t u : Set α\nm : ι → MeasurableSpace α\ns : Set α\n⊢ GenerateMeasurable {s | ∃ i, MeasurableSet s} s ↔ MeasurableSet s",
"tactic": "rfl"
}
] | [
517,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
513,
1
] |
Mathlib/ModelTheory/Satisfiability.lean | FirstOrder.Language.Theory.IsSatisfiable.isFinitelySatisfiable | [] | [
105,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
104,
1
] |
Mathlib/Order/Directed.lean | directedOn_singleton | [] | [
243,
57
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
242,
1
] |
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