file_path
stringlengths 11
79
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|---|---|---|---|---|---|---|
Mathlib/Order/UpperLower/Basic.lean
|
lowerClosure_iUnion
|
[] |
[
1449,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1448,
1
] |
Mathlib/Data/Fintype/Basic.lean
|
Finset.inter_univ
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.21491\nγ : Type ?u.21494\ninst✝¹ : Fintype α\ns✝ t : Finset α\ninst✝ : DecidableEq α\ns : Finset α\n⊢ s ∩ univ = s",
"tactic": "rw [inter_comm, univ_inter]"
}
] |
[
279,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
279,
1
] |
Mathlib/Topology/Algebra/Order/IntermediateValue.lean
|
setOf_isPreconnected_subset_of_ordered
|
[
{
"state_after": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : s ∈ {s | IsPreconnected s}\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\n⊢ {s | IsPreconnected s} ⊆\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})",
"tactic": "intro s hs"
},
{
"state_after": "case inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : s ∈ {s | IsPreconnected s}\nhs : s = Icc (sInf s) (sSup s)\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})\n\ncase inr.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : s ∈ {s | IsPreconnected s}\nhs : s = Ico (sInf s) (sSup s)\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})\n\ncase inr.inr.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : s ∈ {s | IsPreconnected s}\nhs : s = Ioc (sInf s) (sSup s)\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})\n\ncase inr.inr.inr.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : s ∈ {s | IsPreconnected s}\nhs : s = Ioo (sInf s) (sSup s)\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})\n\ncase inr.inr.inr.inr.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : s ∈ {s | IsPreconnected s}\nhs : s = Ici (sInf s)\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})\n\ncase inr.inr.inr.inr.inr.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : s ∈ {s | IsPreconnected s}\nhs : s = Ioi (sInf s)\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})\n\ncase inr.inr.inr.inr.inr.inr.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : s ∈ {s | IsPreconnected s}\nhs : s = Iic (sSup s)\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})\n\ncase inr.inr.inr.inr.inr.inr.inr.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : s ∈ {s | IsPreconnected s}\nhs : s = Iio (sSup s)\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})\n\ncase inr.inr.inr.inr.inr.inr.inr.inr.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : s ∈ {s | IsPreconnected s}\nhs : s = univ\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})\n\ncase inr.inr.inr.inr.inr.inr.inr.inr.inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : s ∈ {s | IsPreconnected s}\nhs : s ∈ {∅}\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs : s ∈ {s | IsPreconnected s}\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})",
"tactic": "rcases hs.mem_intervals with (hs | hs | hs | hs | hs | hs | hs | hs | hs | hs)"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : s ∈ {s | IsPreconnected s}\nhs : s = Icc (sInf s) (sSup s)\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})",
"tactic": "exact Or.inl <| Or.inl <| Or.inl <| Or.inl ⟨(sInf s, sSup s), hs.symm⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : s ∈ {s | IsPreconnected s}\nhs : s = Ico (sInf s) (sSup s)\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})",
"tactic": "exact Or.inl <| Or.inl <| Or.inl <| Or.inr ⟨(sInf s, sSup s), hs.symm⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : s ∈ {s | IsPreconnected s}\nhs : s = Ioc (sInf s) (sSup s)\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})",
"tactic": "exact Or.inl <| Or.inl <| Or.inr ⟨(sInf s, sSup s), hs.symm⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inr.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : s ∈ {s | IsPreconnected s}\nhs : s = Ioo (sInf s) (sSup s)\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})",
"tactic": "exact Or.inl <| Or.inr ⟨(sInf s, sSup s), hs.symm⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inr.inr.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : s ∈ {s | IsPreconnected s}\nhs : s = Ici (sInf s)\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})",
"tactic": "exact Or.inr <| Or.inl <| Or.inl <| Or.inl <| Or.inl ⟨sInf s, hs.symm⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inr.inr.inr.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : s ∈ {s | IsPreconnected s}\nhs : s = Ioi (sInf s)\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})",
"tactic": "exact Or.inr <| Or.inl <| Or.inl <| Or.inl <| Or.inr ⟨sInf s, hs.symm⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inr.inr.inr.inr.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : s ∈ {s | IsPreconnected s}\nhs : s = Iic (sSup s)\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})",
"tactic": "exact Or.inr <| Or.inl <| Or.inl <| Or.inr ⟨sSup s, hs.symm⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inr.inr.inr.inr.inr.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : s ∈ {s | IsPreconnected s}\nhs : s = Iio (sSup s)\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})",
"tactic": "exact Or.inr <| Or.inl <| Or.inr ⟨sSup s, hs.symm⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inr.inr.inr.inr.inr.inr.inl\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : s ∈ {s | IsPreconnected s}\nhs : s = univ\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})",
"tactic": "exact Or.inr <| Or.inr <| Or.inl hs"
},
{
"state_after": "no goals",
"state_before": "case inr.inr.inr.inr.inr.inr.inr.inr.inr\nα : Type u\nβ : Type v\nγ : Type w\ninst✝⁶ : ConditionallyCompleteLinearOrder α\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : OrderTopology α\ninst✝³ : ConditionallyCompleteLinearOrder β\ninst✝² : TopologicalSpace β\ninst✝¹ : OrderTopology β\ninst✝ : Nonempty γ\ns : Set α\nhs✝ : s ∈ {s | IsPreconnected s}\nhs : s ∈ {∅}\n⊢ s ∈\n range (uncurry Icc) ∪ range (uncurry Ico) ∪ range (uncurry Ioc) ∪ range (uncurry Ioo) ∪\n (range Ici ∪ range Ioi ∪ range Iic ∪ range Iio ∪ {univ, ∅})",
"tactic": "exact Or.inr <| Or.inr <| Or.inr hs"
}
] |
[
331,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
314,
1
] |
Std/Data/Nat/Gcd.lean
|
Nat.gcd_div
|
[
{
"state_after": "no goals",
"state_before": "m n k : Nat\nH1 : k ∣ m\nH2 : k ∣ n\nH0 : k = 0\n⊢ gcd (m / k) (n / k) = gcd m n / k",
"tactic": "simp [H0]"
},
{
"state_after": "m n k : Nat\nH1 : k ∣ m\nH2 : k ∣ n\nH3 : k > 0\n⊢ gcd (m / k) (n / k) * k = gcd m n / k * k",
"state_before": "m n k : Nat\nH1 : k ∣ m\nH2 : k ∣ n\nH3 : k > 0\n⊢ gcd (m / k) (n / k) = gcd m n / k",
"tactic": "apply Nat.eq_of_mul_eq_mul_right H3"
},
{
"state_after": "no goals",
"state_before": "m n k : Nat\nH1 : k ∣ m\nH2 : k ∣ n\nH3 : k > 0\n⊢ gcd (m / k) (n / k) * k = gcd m n / k * k",
"tactic": "rw [Nat.div_mul_cancel (dvd_gcd H1 H2), ← gcd_mul_right,\n Nat.div_mul_cancel H1, Nat.div_mul_cancel H2]"
}
] |
[
123,
54
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
116,
1
] |
Mathlib/Combinatorics/SetFamily/Compression/Down.lean
|
Finset.nonMemberSubfamily_memberSubfamily
|
[
{
"state_after": "case a\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\na✝ : Finset α\n⊢ a✝ ∈ nonMemberSubfamily a (memberSubfamily a 𝒜) ↔ a✝ ∈ memberSubfamily a 𝒜",
"state_before": "α : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\n⊢ nonMemberSubfamily a (memberSubfamily a 𝒜) = memberSubfamily a 𝒜",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\ninst✝ : DecidableEq α\n𝒜 ℬ : Finset (Finset α)\ns : Finset α\na : α\na✝ : Finset α\n⊢ a✝ ∈ nonMemberSubfamily a (memberSubfamily a 𝒜) ↔ a✝ ∈ memberSubfamily a 𝒜",
"tactic": "simp"
}
] |
[
132,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
associated_normalize_iff
|
[] |
[
123,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
122,
1
] |
Mathlib/Algebra/CubicDiscriminant.lean
|
Cubic.degree_of_c_eq_zero'
|
[] |
[
354,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
353,
1
] |
Mathlib/Topology/PathConnected.lean
|
Path.trans_range
|
[
{
"state_after": "X✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx y z : X✝\nι : Type ?u.252948\nγ : Path x y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\n⊢ range\n ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } =\n range ↑γ₁ ∪ range ↑γ₂",
"state_before": "X✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx y z : X✝\nι : Type ?u.252948\nγ : Path x y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\n⊢ range ↑(trans γ₁ γ₂) = range ↑γ₁ ∪ range ↑γ₂",
"tactic": "rw [Path.trans]"
},
{
"state_after": "case a\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx y z : X✝\nι : Type ?u.252948\nγ : Path x y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\n⊢ range\n ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } ⊆\n range ↑γ₁ ∪ range ↑γ₂\n\ncase a\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx y z : X✝\nι : Type ?u.252948\nγ : Path x y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\n⊢ range ↑γ₁ ∪ range ↑γ₂ ⊆\n range\n ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }",
"state_before": "X✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx y z : X✝\nι : Type ?u.252948\nγ : Path x y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\n⊢ range\n ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } =\n range ↑γ₁ ∪ range ↑γ₂",
"tactic": "apply eq_of_subset_of_subset"
},
{
"state_after": "case a.intro.mk.intro\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt :\n ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }\n { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } =\n x\n⊢ x ∈ range ↑γ₁ ∪ range ↑γ₂",
"state_before": "case a\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx y z : X✝\nι : Type ?u.252948\nγ : Path x y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\n⊢ range\n ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) } ⊆\n range ↑γ₁ ∪ range ↑γ₂",
"tactic": "rintro x ⟨⟨t, ht0, ht1⟩, hxt⟩"
},
{
"state_after": "case pos\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt :\n ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }\n { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } =\n x\nh : t ≤ 1 / 2\n⊢ x ∈ range ↑γ₁ ∪ range ↑γ₂\n\ncase neg\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt :\n ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }\n { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } =\n x\nh : ¬t ≤ 1 / 2\n⊢ x ∈ range ↑γ₁ ∪ range ↑γ₂",
"state_before": "case a.intro.mk.intro\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt :\n ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }\n { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } =\n x\n⊢ x ∈ range ↑γ₁ ∪ range ↑γ₂",
"tactic": "by_cases h : t ≤ 1 / 2"
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{
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"tactic": "left"
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{
"state_after": "case pos.h\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt :\n ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }\n { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } =\n x\nh : t ≤ 1 / 2\n⊢ ↑γ₁ { val := 2 * t, property := (_ : 0 ≤ 2 * t ∧ 2 * t ≤ 1) } = x",
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"tactic": "use ⟨2 * t, ⟨by linarith, by linarith⟩⟩"
},
{
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"state_before": "case pos.h\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt :\n ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }\n { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } =\n x\nh : t ≤ 1 / 2\n⊢ ↑γ₁ { val := 2 * t, property := (_ : 0 ≤ 2 * t ∧ 2 * t ≤ 1) } = x",
"tactic": "rw [← γ₁.extend_extends]"
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{
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"tactic": "rwa [coe_mk_mk, Function.comp_apply, if_pos h] at hxt"
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{
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"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "X✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt :\n ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }\n { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } =\n x\nh : t ≤ 1 / 2\n⊢ 2 * t ≤ 1",
"tactic": "linarith"
},
{
"state_after": "case neg.h\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt :\n ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }\n { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } =\n x\nh : ¬t ≤ 1 / 2\n⊢ x ∈ range ↑γ₂",
"state_before": "case neg\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt :\n ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }\n { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } =\n x\nh : ¬t ≤ 1 / 2\n⊢ x ∈ range ↑γ₁ ∪ range ↑γ₂",
"tactic": "right"
},
{
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{
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"tactic": "rw [← γ₂.extend_extends]"
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{
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{
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{
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{
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"tactic": "rintro x (⟨⟨t, ht0, ht1⟩, hxt⟩ | ⟨⟨t, ht0, ht1⟩, hxt⟩)"
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{
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{
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"tactic": "have : t / 2 ≤ 1 / 2 := (div_le_div_right (zero_lt_two : (0 : ℝ) < 2)).mpr ht1"
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{
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{
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},
{
"state_after": "case pos\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t = 0\n⊢ x ∈\n range\n ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }\n\ncase neg\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : ¬t = 0\n⊢ x ∈\n range\n ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }",
"state_before": "case a.inr.intro.mk.intro\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\n⊢ x ∈\n range\n ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }",
"tactic": "by_cases h : t = 0"
},
{
"state_after": "case pos\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t = 0\n⊢ ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }\n { val := 1 / 2, property := (_ : 0 ≤ 1 / 2 ∧ 1 / 2 ≤ 1) } =\n x",
"state_before": "case pos\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t = 0\n⊢ x ∈\n range\n ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }",
"tactic": "use ⟨1 / 2, ⟨by linarith, by linarith⟩⟩"
},
{
"state_after": "case pos\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t = 0\n⊢ extend γ₁ 1 = x",
"state_before": "case pos\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t = 0\n⊢ ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }\n { val := 1 / 2, property := (_ : 0 ≤ 1 / 2 ∧ 1 / 2 ≤ 1) } =\n x",
"tactic": "rw [coe_mk_mk, Function.comp_apply, if_pos le_rfl, Subtype.coe_mk,\n mul_one_div_cancel (two_ne_zero' ℝ)]"
},
{
"state_after": "case pos\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t = 0\n⊢ b = x",
"state_before": "case pos\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t = 0\n⊢ extend γ₁ 1 = x",
"tactic": "rw [γ₁.extend_one]"
},
{
"state_after": "no goals",
"state_before": "case pos\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t = 0\n⊢ b = x",
"tactic": "rwa [← γ₂.extend_extends, h, γ₂.extend_zero] at hxt"
},
{
"state_after": "no goals",
"state_before": "X✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t = 0\n⊢ 0 ≤ 1 / 2",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "X✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t = 0\n⊢ 1 / 2 ≤ 1",
"tactic": "linarith"
},
{
"state_after": "case neg\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : ¬t = 0\n⊢ ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }\n { val := (t + 1) / 2, property := (_ : 0 ≤ (t + 1) / 2 ∧ (t + 1) / 2 ≤ 1) } =\n x",
"state_before": "case neg\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : ¬t = 0\n⊢ x ∈\n range\n ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }",
"tactic": "use ⟨(t + 1) / 2, ⟨by linarith, by linarith⟩⟩"
},
{
"state_after": "case neg\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t ≠ 0\n⊢ ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }\n { val := (t + 1) / 2, property := (_ : 0 ≤ (t + 1) / 2 ∧ (t + 1) / 2 ≤ 1) } =\n x",
"state_before": "case neg\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : ¬t = 0\n⊢ ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }\n { val := (t + 1) / 2, property := (_ : 0 ≤ (t + 1) / 2 ∧ (t + 1) / 2 ≤ 1) } =\n x",
"tactic": "replace h : t ≠ 0 := h"
},
{
"state_after": "case neg\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0✝ : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t ≠ 0\nht0 : 0 < t\n⊢ ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }\n { val := (t + 1) / 2, property := (_ : 0 ≤ (t + 1) / 2 ∧ (t + 1) / 2 ≤ 1) } =\n x",
"state_before": "case neg\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t ≠ 0\n⊢ ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }\n { val := (t + 1) / 2, property := (_ : 0 ≤ (t + 1) / 2 ∧ (t + 1) / 2 ≤ 1) } =\n x",
"tactic": "have ht0 := lt_of_le_of_ne ht0 h.symm"
},
{
"state_after": "case neg\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0✝ : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t ≠ 0\nht0 : 0 < t\nthis : ¬(t + 1) / 2 ≤ 1 / 2\n⊢ ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }\n { val := (t + 1) / 2, property := (_ : 0 ≤ (t + 1) / 2 ∧ (t + 1) / 2 ≤ 1) } =\n x",
"state_before": "case neg\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0✝ : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t ≠ 0\nht0 : 0 < t\n⊢ ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }\n { val := (t + 1) / 2, property := (_ : 0 ≤ (t + 1) / 2 ∧ (t + 1) / 2 ≤ 1) } =\n x",
"tactic": "have : ¬(t + 1) / 2 ≤ 1 / 2 := by\n rw [not_le]\n linarith"
},
{
"state_after": "case neg\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0✝ : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t ≠ 0\nht0 : 0 < t\nthis : ¬(t + 1) / 2 ≤ 1 / 2\n⊢ extend γ₂ (2 * ((t + 1) / 2) - 1) = x",
"state_before": "case neg\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0✝ : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t ≠ 0\nht0 : 0 < t\nthis : ¬(t + 1) / 2 ≤ 1 / 2\n⊢ ↑{\n toContinuousMap :=\n ContinuousMap.mk ((fun t => if t ≤ 1 / 2 then extend γ₁ (2 * t) else extend γ₂ (2 * t - 1)) ∘ Subtype.val),\n source' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 0 = a),\n target' :=\n (_ : ((fun x => if x ≤ 1 / 2 then extend γ₁ (2 * x) else extend γ₂ (2 * x - 1)) ∘ Subtype.val) 1 = c) }\n { val := (t + 1) / 2, property := (_ : 0 ≤ (t + 1) / 2 ∧ (t + 1) / 2 ≤ 1) } =\n x",
"tactic": "rw [coe_mk_mk, Function.comp_apply, Subtype.coe_mk, if_neg this]"
},
{
"state_after": "case neg\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0✝ : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t ≠ 0\nht0 : 0 < t\nthis : ¬(t + 1) / 2 ≤ 1 / 2\n⊢ extend γ₂ t = x",
"state_before": "case neg\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0✝ : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t ≠ 0\nht0 : 0 < t\nthis : ¬(t + 1) / 2 ≤ 1 / 2\n⊢ extend γ₂ (2 * ((t + 1) / 2) - 1) = x",
"tactic": "ring_nf"
},
{
"state_after": "no goals",
"state_before": "case neg\nX✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0✝ : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t ≠ 0\nht0 : 0 < t\nthis : ¬(t + 1) / 2 ≤ 1 / 2\n⊢ extend γ₂ t = x",
"tactic": "rwa [γ₂.extend_extends]"
},
{
"state_after": "no goals",
"state_before": "X✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : ¬t = 0\n⊢ 0 ≤ (t + 1) / 2",
"tactic": "linarith"
},
{
"state_after": "no goals",
"state_before": "X✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0 : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : ¬t = 0\n⊢ (t + 1) / 2 ≤ 1",
"tactic": "linarith"
},
{
"state_after": "X✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0✝ : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t ≠ 0\nht0 : 0 < t\n⊢ 1 / 2 < (t + 1) / 2",
"state_before": "X✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0✝ : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t ≠ 0\nht0 : 0 < t\n⊢ ¬(t + 1) / 2 ≤ 1 / 2",
"tactic": "rw [not_le]"
},
{
"state_after": "no goals",
"state_before": "X✝ : Type ?u.252930\nY : Type ?u.252933\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type ?u.252948\nγ : Path x✝ y\nX : Type u_1\ninst✝ : TopologicalSpace X\na b c : X\nγ₁ : Path a b\nγ₂ : Path b c\nx : X\nt : ℝ\nht0✝ : 0 ≤ t\nht1 : t ≤ 1\nhxt : ↑γ₂ { val := t, property := (_ : 0 ≤ t ∧ t ≤ 1) } = x\nh : t ≠ 0\nht0 : 0 < t\n⊢ 1 / 2 < (t + 1) / 2",
"tactic": "linarith"
}
] |
[
413,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
379,
1
] |
Mathlib/Algebra/Order/SMul.lean
|
smul_lowerBounds_subset_lowerBounds_smul
|
[] |
[
147,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Combination.lean
|
Finset.weightedVSub_subtype_eq_filter
|
[] |
[
351,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
347,
1
] |
Mathlib/Topology/Algebra/WithZeroTopology.lean
|
WithZeroTopology.hasBasis_nhds_units
|
[] |
[
120,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
118,
1
] |
Mathlib/Data/Int/Cast/Lemmas.lean
|
Int.commute_cast
|
[] |
[
111,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/Data/Nat/ModEq.lean
|
Nat.coprime_of_mul_modEq_one
|
[
{
"state_after": "case intro\nm n✝ a✝ b✝ c d b a n : ℕ\nh : a * b ≡ 1 [MOD n]\ng : ℕ\nhh : n = gcd a n * g\n⊢ coprime a n",
"state_before": "m n✝ a✝ b✝ c d b a n : ℕ\nh : a * b ≡ 1 [MOD n]\n⊢ coprime a n",
"tactic": "obtain ⟨g, hh⟩ := Nat.gcd_dvd_right a n"
},
{
"state_after": "case intro\nm n✝ a✝ b✝ c d b a n : ℕ\nh : a * b ≡ 1 [MOD n]\ng : ℕ\nhh : n = gcd a n * g\n⊢ 1 ≡ 0 [MOD gcd a n]",
"state_before": "case intro\nm n✝ a✝ b✝ c d b a n : ℕ\nh : a * b ≡ 1 [MOD n]\ng : ℕ\nhh : n = gcd a n * g\n⊢ coprime a n",
"tactic": "rw [Nat.coprime_iff_gcd_eq_one, ← Nat.dvd_one, ← Nat.modEq_zero_iff_dvd]"
},
{
"state_after": "no goals",
"state_before": "case intro\nm n✝ a✝ b✝ c d b a n : ℕ\nh : a * b ≡ 1 [MOD n]\ng : ℕ\nhh : n = gcd a n * g\n⊢ 1 ≡ 0 [MOD gcd a n]",
"tactic": "calc\n 1 ≡ a * b [MOD a.gcd n] := (hh ▸ h).symm.of_mul_right g\n _ ≡ 0 * b [MOD a.gcd n] := (Nat.modEq_zero_iff_dvd.mpr (Nat.gcd_dvd_left _ _)).mul_right b\n _ = 0 := by rw [zero_mul]"
},
{
"state_after": "no goals",
"state_before": "m n✝ a✝ b✝ c d b a n : ℕ\nh : a * b ≡ 1 [MOD n]\ng : ℕ\nhh : n = gcd a n * g\n⊢ 0 * b = 0",
"tactic": "rw [zero_mul]"
}
] |
[
397,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
391,
1
] |
Mathlib/Tactic/NormNum/Basic.lean
|
Mathlib.Meta.NormNum.isNat_cast
|
[
{
"state_after": "case mk.refl\nR : Type u_1\ninst✝ : AddMonoidWithOne R\nm : ℕ\n⊢ IsNat (↑↑m) m",
"state_before": "R : Type u_1\ninst✝ : AddMonoidWithOne R\nn m : ℕ\n⊢ IsNat n m → IsNat (↑n) m",
"tactic": "rintro ⟨⟨⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.refl\nR : Type u_1\ninst✝ : AddMonoidWithOne R\nm : ℕ\n⊢ IsNat (↑↑m) m",
"tactic": "exact ⟨rfl⟩"
}
] |
[
72,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Topology/MetricSpace/Antilipschitz.lean
|
antilipschitzWith_iff_le_mul_nndist
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1159\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nK : ℝ≥0\nf : α → β\n⊢ (∀ (x y : α), ↑(nndist x y) ≤ ↑K * ↑(nndist (f x) (f y))) ↔ ∀ (x y : α), nndist x y ≤ K * nndist (f x) (f y)",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1159\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nK : ℝ≥0\nf : α → β\n⊢ AntilipschitzWith K f ↔ ∀ (x y : α), nndist x y ≤ K * nndist (f x) (f y)",
"tactic": "simp only [AntilipschitzWith, edist_nndist]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.1159\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nK : ℝ≥0\nf : α → β\n⊢ (∀ (x y : α), ↑(nndist x y) ≤ ↑K * ↑(nndist (f x) (f y))) ↔ ∀ (x y : α), nndist x y ≤ K * nndist (f x) (f y)",
"tactic": "norm_cast"
}
] |
[
57,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
54,
1
] |
Mathlib/GroupTheory/GroupAction/Basic.lean
|
MulAction.orbitRel.Quotient.mem_orbit
|
[
{
"state_after": "case h\nα : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : Group α\ninst✝ : MulAction α β\nb a✝ : β\n⊢ b ∈ orbit (Quotient.mk'' a✝) ↔ Quotient.mk'' b = Quotient.mk'' a✝",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : Group α\ninst✝ : MulAction α β\nb : β\nx : Quotient α β\n⊢ b ∈ orbit x ↔ Quotient.mk'' b = x",
"tactic": "induction x using Quotient.inductionOn'"
},
{
"state_after": "case h\nα : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : Group α\ninst✝ : MulAction α β\nb a✝ : β\n⊢ b ∈ orbit (Quotient.mk'' a✝) ↔ Setoid.r b a✝",
"state_before": "case h\nα : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : Group α\ninst✝ : MulAction α β\nb a✝ : β\n⊢ b ∈ orbit (Quotient.mk'' a✝) ↔ Quotient.mk'' b = Quotient.mk'' a✝",
"tactic": "rw [Quotient.eq'']"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : Group α\ninst✝ : MulAction α β\nb a✝ : β\n⊢ b ∈ orbit (Quotient.mk'' a✝) ↔ Setoid.r b a✝",
"tactic": "rfl"
}
] |
[
368,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
364,
1
] |
Mathlib/Data/Real/Cardinality.lean
|
Cardinal.mk_Ico_real
|
[] |
[
280,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
279,
1
] |
Mathlib/Order/Heyting/Basic.lean
|
le_himp_himp
|
[] |
[
459,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
458,
1
] |
Mathlib/LinearAlgebra/PiTensorProduct.lean
|
PiTensorProduct.reindex_reindex
|
[] |
[
516,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
514,
1
] |
Mathlib/Topology/VectorBundle/Basic.lean
|
VectorBundleCore.mem_source_at
|
[
{
"state_after": "R : Type u_3\nB : Type u_1\nF : Type u_2\nE : B → Type ?u.424723\ninst✝⁸ : NontriviallyNormedField R\ninst✝⁷ : (x : B) → AddCommMonoid (E x)\ninst✝⁶ : (x : B) → Module R (E x)\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace R F\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace (TotalSpace E)\ninst✝¹ : (x : B) → TopologicalSpace (E x)\ninst✝ : FiberBundle F E\nι : Type u_4\nZ : VectorBundleCore R B F ι\nb : B\na : F\ni j : ι\n⊢ { fst := b, snd := a }.fst ∈ baseSet Z (indexAt Z b)",
"state_before": "R : Type u_3\nB : Type u_1\nF : Type u_2\nE : B → Type ?u.424723\ninst✝⁸ : NontriviallyNormedField R\ninst✝⁷ : (x : B) → AddCommMonoid (E x)\ninst✝⁶ : (x : B) → Module R (E x)\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace R F\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace (TotalSpace E)\ninst✝¹ : (x : B) → TopologicalSpace (E x)\ninst✝ : FiberBundle F E\nι : Type u_4\nZ : VectorBundleCore R B F ι\nb : B\na : F\ni j : ι\n⊢ { fst := b, snd := a } ∈ (localTrivAt Z b).toLocalHomeomorph.toLocalEquiv.source",
"tactic": "rw [localTrivAt, mem_localTriv_source]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_3\nB : Type u_1\nF : Type u_2\nE : B → Type ?u.424723\ninst✝⁸ : NontriviallyNormedField R\ninst✝⁷ : (x : B) → AddCommMonoid (E x)\ninst✝⁶ : (x : B) → Module R (E x)\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace R F\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace (TotalSpace E)\ninst✝¹ : (x : B) → TopologicalSpace (E x)\ninst✝ : FiberBundle F E\nι : Type u_4\nZ : VectorBundleCore R B F ι\nb : B\na : F\ni j : ι\n⊢ { fst := b, snd := a }.fst ∈ baseSet Z (indexAt Z b)",
"tactic": "exact Z.mem_baseSet_at b"
}
] |
[
749,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
747,
1
] |
Mathlib/Data/PNat/Basic.lean
|
PNat.natPred_add_one
|
[] |
[
44,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
43,
1
] |
Mathlib/Data/Rat/Defs.lean
|
Rat.coe_nat_num
|
[
{
"state_after": "no goals",
"state_before": "a b c : ℚ\nn : ℕ\n⊢ (↑n).num = ↑n",
"tactic": "rw [← Int.cast_ofNat, coe_int_num]"
}
] |
[
534,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
533,
1
] |
Mathlib/Analysis/NormedSpace/Pointwise.lean
|
eventually_singleton_add_smul_subset
|
[
{
"state_after": "case intro.intro\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\n⊢ ∀ᶠ (r : 𝕜) in 𝓝 0, {x} + r • s ⊆ u",
"state_before": "𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\n⊢ ∀ᶠ (r : 𝕜) in 𝓝 0, {x} + r • s ⊆ u",
"tactic": "obtain ⟨ε, εpos, hε⟩ : ∃ ε : ℝ, 0 < ε ∧ closedBall x ε ⊆ u := nhds_basis_closedBall.mem_iff.1 hu"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\n⊢ ∀ᶠ (r : 𝕜) in 𝓝 0, {x} + r • s ⊆ u",
"state_before": "case intro.intro\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\n⊢ ∀ᶠ (r : 𝕜) in 𝓝 0, {x} + r • s ⊆ u",
"tactic": "obtain ⟨R, Rpos, hR⟩ : ∃ R : ℝ, 0 < R ∧ s ⊆ closedBall 0 R := hs.subset_ball_lt 0 0"
},
{
"state_after": "case intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis : closedBall 0 (ε / R) ∈ 𝓝 0\n⊢ ∀ᶠ (r : 𝕜) in 𝓝 0, {x} + r • s ⊆ u",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\n⊢ ∀ᶠ (r : 𝕜) in 𝓝 0, {x} + r • s ⊆ u",
"tactic": "have : Metric.closedBall (0 : 𝕜) (ε / R) ∈ 𝓝 (0 : 𝕜) := closedBall_mem_nhds _ (div_pos εpos Rpos)"
},
{
"state_after": "case h\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis : closedBall 0 (ε / R) ∈ 𝓝 0\nr : 𝕜\nhr : r ∈ closedBall 0 (ε / R)\n⊢ {x} + r • s ⊆ u",
"state_before": "case intro.intro.intro.intro\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis : closedBall 0 (ε / R) ∈ 𝓝 0\n⊢ ∀ᶠ (r : 𝕜) in 𝓝 0, {x} + r • s ⊆ u",
"tactic": "filter_upwards [this]with r hr"
},
{
"state_after": "case h\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis : closedBall 0 (ε / R) ∈ 𝓝 0\nr : 𝕜\nhr : r ∈ closedBall 0 (ε / R)\n⊢ (fun x_1 => -x + x_1) ⁻¹' (r • s) ⊆ u",
"state_before": "case h\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis : closedBall 0 (ε / R) ∈ 𝓝 0\nr : 𝕜\nhr : r ∈ closedBall 0 (ε / R)\n⊢ {x} + r • s ⊆ u",
"tactic": "simp only [image_add_left, singleton_add]"
},
{
"state_after": "case h\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis : closedBall 0 (ε / R) ∈ 𝓝 0\nr : 𝕜\nhr : r ∈ closedBall 0 (ε / R)\ny : E\nhy : y ∈ (fun x_1 => -x + x_1) ⁻¹' (r • s)\n⊢ y ∈ u",
"state_before": "case h\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis : closedBall 0 (ε / R) ∈ 𝓝 0\nr : 𝕜\nhr : r ∈ closedBall 0 (ε / R)\n⊢ (fun x_1 => -x + x_1) ⁻¹' (r • s) ⊆ u",
"tactic": "intro y hy"
},
{
"state_after": "case h.intro.intro\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis : closedBall 0 (ε / R) ∈ 𝓝 0\nr : 𝕜\nhr : r ∈ closedBall 0 (ε / R)\ny : E\nhy : y ∈ (fun x_1 => -x + x_1) ⁻¹' (r • s)\nz : E\nzs : z ∈ s\nhz : r • z = -x + y\n⊢ y ∈ u",
"state_before": "case h\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis : closedBall 0 (ε / R) ∈ 𝓝 0\nr : 𝕜\nhr : r ∈ closedBall 0 (ε / R)\ny : E\nhy : y ∈ (fun x_1 => -x + x_1) ⁻¹' (r • s)\n⊢ y ∈ u",
"tactic": "obtain ⟨z, zs, hz⟩ : ∃ z : E, z ∈ s ∧ r • z = -x + y := by simpa [mem_smul_set] using hy"
},
{
"state_after": "case h.intro.intro\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis : closedBall 0 (ε / R) ∈ 𝓝 0\nr : 𝕜\nhr : r ∈ closedBall 0 (ε / R)\ny : E\nhy : y ∈ (fun x_1 => -x + x_1) ⁻¹' (r • s)\nz : E\nzs : z ∈ s\nhz : r • z = -x + y\nI : ‖r • z‖ ≤ ε\n⊢ y ∈ u",
"state_before": "case h.intro.intro\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis : closedBall 0 (ε / R) ∈ 𝓝 0\nr : 𝕜\nhr : r ∈ closedBall 0 (ε / R)\ny : E\nhy : y ∈ (fun x_1 => -x + x_1) ⁻¹' (r • s)\nz : E\nzs : z ∈ s\nhz : r • z = -x + y\n⊢ y ∈ u",
"tactic": "have I : ‖r • z‖ ≤ ε :=\n calc\n ‖r • z‖ = ‖r‖ * ‖z‖ := norm_smul _ _\n _ ≤ ε / R * R :=\n (mul_le_mul (mem_closedBall_zero_iff.1 hr) (mem_closedBall_zero_iff.1 (hR zs))\n (norm_nonneg _) (div_pos εpos Rpos).le)\n _ = ε := by field_simp [Rpos.ne']"
},
{
"state_after": "case h.intro.intro\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis✝ : closedBall 0 (ε / R) ∈ 𝓝 0\nr : 𝕜\nhr : r ∈ closedBall 0 (ε / R)\ny : E\nhy : y ∈ (fun x_1 => -x + x_1) ⁻¹' (r • s)\nz : E\nzs : z ∈ s\nhz : r • z = -x + y\nI : ‖r • z‖ ≤ ε\nthis : y = x + r • z\n⊢ y ∈ u",
"state_before": "case h.intro.intro\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis : closedBall 0 (ε / R) ∈ 𝓝 0\nr : 𝕜\nhr : r ∈ closedBall 0 (ε / R)\ny : E\nhy : y ∈ (fun x_1 => -x + x_1) ⁻¹' (r • s)\nz : E\nzs : z ∈ s\nhz : r • z = -x + y\nI : ‖r • z‖ ≤ ε\n⊢ y ∈ u",
"tactic": "have : y = x + r • z := by simp only [hz, add_neg_cancel_left]"
},
{
"state_after": "case h.intro.intro.a\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis✝ : closedBall 0 (ε / R) ∈ 𝓝 0\nr : 𝕜\nhr : r ∈ closedBall 0 (ε / R)\ny : E\nhy : y ∈ (fun x_1 => -x + x_1) ⁻¹' (r • s)\nz : E\nzs : z ∈ s\nhz : r • z = -x + y\nI : ‖r • z‖ ≤ ε\nthis : y = x + r • z\n⊢ y ∈ closedBall x ε",
"state_before": "case h.intro.intro\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis✝ : closedBall 0 (ε / R) ∈ 𝓝 0\nr : 𝕜\nhr : r ∈ closedBall 0 (ε / R)\ny : E\nhy : y ∈ (fun x_1 => -x + x_1) ⁻¹' (r • s)\nz : E\nzs : z ∈ s\nhz : r • z = -x + y\nI : ‖r • z‖ ≤ ε\nthis : y = x + r • z\n⊢ y ∈ u",
"tactic": "apply hε"
},
{
"state_after": "no goals",
"state_before": "case h.intro.intro.a\n𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis✝ : closedBall 0 (ε / R) ∈ 𝓝 0\nr : 𝕜\nhr : r ∈ closedBall 0 (ε / R)\ny : E\nhy : y ∈ (fun x_1 => -x + x_1) ⁻¹' (r • s)\nz : E\nzs : z ∈ s\nhz : r • z = -x + y\nI : ‖r • z‖ ≤ ε\nthis : y = x + r • z\n⊢ y ∈ closedBall x ε",
"tactic": "simpa only [this, dist_eq_norm, add_sub_cancel', mem_closedBall] using I"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis : closedBall 0 (ε / R) ∈ 𝓝 0\nr : 𝕜\nhr : r ∈ closedBall 0 (ε / R)\ny : E\nhy : y ∈ (fun x_1 => -x + x_1) ⁻¹' (r • s)\n⊢ ∃ z, z ∈ s ∧ r • z = -x + y",
"tactic": "simpa [mem_smul_set] using hy"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis : closedBall 0 (ε / R) ∈ 𝓝 0\nr : 𝕜\nhr : r ∈ closedBall 0 (ε / R)\ny : E\nhy : y ∈ (fun x_1 => -x + x_1) ⁻¹' (r • s)\nz : E\nzs : z ∈ s\nhz : r • z = -x + y\n⊢ ε / R * R = ε",
"tactic": "field_simp [Rpos.ne']"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : E\ns : Set E\nhs : Metric.Bounded s\nu : Set E\nhu : u ∈ 𝓝 x\nε : ℝ\nεpos : 0 < ε\nhε : closedBall x ε ⊆ u\nR : ℝ\nRpos : 0 < R\nhR : s ⊆ closedBall 0 R\nthis : closedBall 0 (ε / R) ∈ 𝓝 0\nr : 𝕜\nhr : r ∈ closedBall 0 (ε / R)\ny : E\nhy : y ∈ (fun x_1 => -x + x_1) ⁻¹' (r • s)\nz : E\nzs : z ∈ s\nhz : r • z = -x + y\nI : ‖r • z‖ ≤ ε\n⊢ y = x + r • z",
"tactic": "simp only [hz, add_neg_cancel_left]"
}
] |
[
144,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
126,
1
] |
Mathlib/Algebra/Homology/LocalCohomology.lean
|
Ideal.exists_pow_le_of_le_radical_of_fG
|
[
{
"state_after": "case intro\nR : Type u\ninst✝ : CommRing R\nI J : Ideal R\nhIJ : I ≤ radical J\nhJ : FG (radical J)\nk : ℕ\nhk : radical J ^ k ≤ J\n⊢ ∃ k, I ^ k ≤ J",
"state_before": "R : Type u\ninst✝ : CommRing R\nI J : Ideal R\nhIJ : I ≤ radical J\nhJ : FG (radical J)\n⊢ ∃ k, I ^ k ≤ J",
"tactic": "obtain ⟨k, hk⟩ := J.exists_radical_pow_le_of_fg hJ"
},
{
"state_after": "case intro\nR : Type u\ninst✝ : CommRing R\nI J : Ideal R\nhIJ : I ≤ radical J\nhJ : FG (radical J)\nk : ℕ\nhk : radical J ^ k ≤ J\n⊢ I ^ k ≤ J",
"state_before": "case intro\nR : Type u\ninst✝ : CommRing R\nI J : Ideal R\nhIJ : I ≤ radical J\nhJ : FG (radical J)\nk : ℕ\nhk : radical J ^ k ≤ J\n⊢ ∃ k, I ^ k ≤ J",
"tactic": "use k"
},
{
"state_after": "no goals",
"state_before": "case intro\nR : Type u\ninst✝ : CommRing R\nI J : Ideal R\nhIJ : I ≤ radical J\nhJ : FG (radical J)\nk : ℕ\nhk : radical J ^ k ≤ J\n⊢ I ^ k ≤ J",
"tactic": "calc\n I ^ k ≤ J.radical ^ k := Ideal.pow_mono hIJ _\n _ ≤ J := hk"
}
] |
[
212,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
206,
1
] |
src/lean/Init/Data/Nat/SOM.lean
|
Nat.SOM.Poly.mul_denote
|
[
{
"state_after": "ctx : Context\np₁ p₂ : Poly\n⊢ denote ctx [] + denote ctx p₁ * denote ctx p₂ = denote ctx p₁ * denote ctx p₂",
"state_before": "ctx : Context\np₁ p₂ : Poly\n⊢ denote ctx (mul p₁ p₂) = denote ctx p₁ * denote ctx p₂",
"tactic": "simp [mul, go]"
},
{
"state_after": "no goals",
"state_before": "ctx : Context\np₁ p₂ : Poly\n⊢ denote ctx [] + denote ctx p₁ * denote ctx p₂ = denote ctx p₁ * denote ctx p₂",
"tactic": "simp!"
},
{
"state_after": "no goals",
"state_before": "ctx : Context\np₁✝ p₂ p₁ acc : Poly\n⊢ denote ctx (mul.go p₂ p₁ acc) = denote ctx acc + denote ctx p₁ * denote ctx p₂",
"tactic": "match p₁ with\n| [] => simp!\n| (k, m) :: p₁ =>\n simp! [go p₁, Nat.left_distrib, add_denote, mulMon_denote,\n Nat.add_assoc, Nat.add_comm, Nat.add_left_comm,\n Nat.mul_assoc, Nat.mul_comm, Nat.mul_left_comm]"
},
{
"state_after": "no goals",
"state_before": "ctx : Context\np₁✝ p₂ p₁ acc : Poly\n⊢ denote ctx (mul.go p₂ [] acc) = denote ctx acc + denote ctx [] * denote ctx p₂",
"tactic": "simp!"
},
{
"state_after": "no goals",
"state_before": "ctx : Context\np₁✝¹ p₂ p₁✝ acc : Poly\nk : Nat\nm : Mon\np₁ : List (Nat × Mon)\n⊢ denote ctx (mul.go p₂ ((k, m) :: p₁) acc) = denote ctx acc + denote ctx ((k, m) :: p₁) * denote ctx p₂",
"tactic": "simp! [go p₁, Nat.left_distrib, add_denote, mulMon_denote,\n Nat.add_assoc, Nat.add_comm, Nat.add_left_comm,\n Nat.mul_assoc, Nat.mul_comm, Nat.mul_left_comm]"
}
] |
[
169,
61
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
160,
1
] |
Mathlib/GroupTheory/Index.lean
|
Subgroup.card_mul_index
|
[
{
"state_after": "G : Type u_1\ninst✝ : Group G\nH K L : Subgroup G\n⊢ relindex ⊥ H * index H = index ⊥",
"state_before": "G : Type u_1\ninst✝ : Group G\nH K L : Subgroup G\n⊢ Nat.card { x // x ∈ H } * index H = Nat.card G",
"tactic": "rw [← relindex_bot_left, ← index_bot]"
},
{
"state_after": "no goals",
"state_before": "G : Type u_1\ninst✝ : Group G\nH K L : Subgroup G\n⊢ relindex ⊥ H * index H = index ⊥",
"tactic": "exact relindex_mul_index bot_le"
}
] |
[
294,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
292,
1
] |
Std/Data/Nat/Lemmas.lean
|
Nat.sub_lt_left_of_lt_add
|
[
{
"state_after": "n k m : Nat\nH : n ≤ k\nh : k < n + m\nthis : succ k - n ≤ n + m - n\n⊢ k - n < m",
"state_before": "n k m : Nat\nH : n ≤ k\nh : k < n + m\n⊢ k - n < m",
"tactic": "have := Nat.sub_le_sub_right (succ_le_of_lt h) n"
},
{
"state_after": "no goals",
"state_before": "n k m : Nat\nH : n ≤ k\nh : k < n + m\nthis : succ k - n ≤ n + m - n\n⊢ k - n < m",
"tactic": "rwa [Nat.add_sub_cancel_left, Nat.succ_sub H] at this"
}
] |
[
459,
56
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
457,
11
] |
Mathlib/Topology/Constructions.lean
|
isOpen_prod_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type ?u.47317\nδ : Type ?u.47320\nε : Type ?u.47323\nζ : Type ?u.47326\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : TopologicalSpace β\ninst✝³ : TopologicalSpace γ\ninst✝² : TopologicalSpace δ\ninst✝¹ : TopologicalSpace ε\ninst✝ : TopologicalSpace ζ\ns : Set (α × β)\n⊢ (∀ (a : α × β), a ∈ s → s ∈ 𝓝 a) ↔\n ∀ (a : α) (b : β), (a, b) ∈ s → ∃ u v, IsOpen u ∧ IsOpen v ∧ a ∈ u ∧ b ∈ v ∧ u ×ˢ v ⊆ s",
"tactic": "simp_rw [Prod.forall, mem_nhds_prod_iff', and_left_comm]"
}
] |
[
665,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
662,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Continuity.lean
|
Real.continuousAt_rpow_of_ne
|
[
{
"state_after": "p : ℝ × ℝ\nhp✝ : p.fst ≠ 0\nhp : p.fst < 0 ∨ p.fst > 0\n⊢ ContinuousAt (fun p => p.fst ^ p.snd) p",
"state_before": "p : ℝ × ℝ\nhp : p.fst ≠ 0\n⊢ ContinuousAt (fun p => p.fst ^ p.snd) p",
"tactic": "rw [ne_iff_lt_or_gt] at hp"
},
{
"state_after": "case inl\np : ℝ × ℝ\nhp✝ : p.fst ≠ 0\nhp : p.fst < 0\n⊢ ContinuousAt (fun x => exp (log x.fst * x.snd) * cos (x.snd * π)) p",
"state_before": "case inl\np : ℝ × ℝ\nhp✝ : p.fst ≠ 0\nhp : p.fst < 0\n⊢ ContinuousAt (fun p => p.fst ^ p.snd) p",
"tactic": "rw [continuousAt_congr (rpow_eq_nhds_of_neg hp)]"
},
{
"state_after": "case inl.refine'_1\np : ℝ × ℝ\nhp✝ : p.fst ≠ 0\nhp : p.fst < 0\n⊢ ContinuousAt (fun x => exp (log x.fst * x.snd)) p\n\ncase inl.refine'_2\np : ℝ × ℝ\nhp✝ : p.fst ≠ 0\nhp : p.fst < 0\n⊢ ContinuousAt (fun x => x.snd * π) p",
"state_before": "case inl\np : ℝ × ℝ\nhp✝ : p.fst ≠ 0\nhp : p.fst < 0\n⊢ ContinuousAt (fun x => exp (log x.fst * x.snd) * cos (x.snd * π)) p",
"tactic": "refine' ContinuousAt.mul _ (continuous_cos.continuousAt.comp _)"
},
{
"state_after": "case inl.refine'_1\np : ℝ × ℝ\nhp✝ : p.fst ≠ 0\nhp : p.fst < 0\n⊢ ContinuousAt (fun x => log x.fst) p",
"state_before": "case inl.refine'_1\np : ℝ × ℝ\nhp✝ : p.fst ≠ 0\nhp : p.fst < 0\n⊢ ContinuousAt (fun x => exp (log x.fst * x.snd)) p",
"tactic": "refine' continuous_exp.continuousAt.comp (ContinuousAt.mul _ continuous_snd.continuousAt)"
},
{
"state_after": "case inl.refine'_1\np : ℝ × ℝ\nhp✝ : p.fst ≠ 0\nhp : p.fst < 0\n⊢ p.fst ≠ 0",
"state_before": "case inl.refine'_1\np : ℝ × ℝ\nhp✝ : p.fst ≠ 0\nhp : p.fst < 0\n⊢ ContinuousAt (fun x => log x.fst) p",
"tactic": "refine' (continuousAt_log _).comp continuous_fst.continuousAt"
},
{
"state_after": "no goals",
"state_before": "case inl.refine'_1\np : ℝ × ℝ\nhp✝ : p.fst ≠ 0\nhp : p.fst < 0\n⊢ p.fst ≠ 0",
"tactic": "exact hp.ne"
},
{
"state_after": "no goals",
"state_before": "case inl.refine'_2\np : ℝ × ℝ\nhp✝ : p.fst ≠ 0\nhp : p.fst < 0\n⊢ ContinuousAt (fun x => x.snd * π) p",
"tactic": "exact continuous_snd.continuousAt.mul continuousAt_const"
},
{
"state_after": "case inr\np : ℝ × ℝ\nhp✝ : p.fst ≠ 0\nhp : p.fst > 0\n⊢ ContinuousAt (fun x => exp (log x.fst * x.snd)) p",
"state_before": "case inr\np : ℝ × ℝ\nhp✝ : p.fst ≠ 0\nhp : p.fst > 0\n⊢ ContinuousAt (fun p => p.fst ^ p.snd) p",
"tactic": "rw [continuousAt_congr (rpow_eq_nhds_of_pos hp)]"
},
{
"state_after": "case inr\np : ℝ × ℝ\nhp✝ : p.fst ≠ 0\nhp : p.fst > 0\n⊢ ContinuousAt (fun x => log x.fst) p",
"state_before": "case inr\np : ℝ × ℝ\nhp✝ : p.fst ≠ 0\nhp : p.fst > 0\n⊢ ContinuousAt (fun x => exp (log x.fst * x.snd)) p",
"tactic": "refine' continuous_exp.continuousAt.comp (ContinuousAt.mul _ continuous_snd.continuousAt)"
},
{
"state_after": "case inr\np : ℝ × ℝ\nhp✝ : p.fst ≠ 0\nhp : p.fst > 0\n⊢ p.fst ≠ 0",
"state_before": "case inr\np : ℝ × ℝ\nhp✝ : p.fst ≠ 0\nhp : p.fst > 0\n⊢ ContinuousAt (fun x => log x.fst) p",
"tactic": "refine' (continuousAt_log _).comp continuous_fst.continuousAt"
},
{
"state_after": "no goals",
"state_before": "case inr\np : ℝ × ℝ\nhp✝ : p.fst ≠ 0\nhp : p.fst > 0\n⊢ p.fst ≠ 0",
"tactic": "exact hp.lt.ne.symm"
}
] |
[
237,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
222,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean
|
CategoryTheory.IsPullback.of_is_product'
|
[] |
[
262,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/Algebra/Algebra/Unitization.lean
|
Unitization.fst_zero
|
[] |
[
214,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Std/Data/Array/Init/Lemmas.lean
|
Array.appendList_data
|
[
{
"state_after": "α : Type u_1\narr : Array α\nl : List α\n⊢ (Array.appendList arr l).data = arr.data ++ l",
"state_before": "α : Type u_1\narr : Array α\nl : List α\n⊢ (arr ++ l).data = arr.data ++ l",
"tactic": "rw [← appendList_eq_append]"
},
{
"state_after": "α : Type u_1\narr : Array α\nl : List α\n⊢ (List.foldl (fun r v => push r v) arr l).data = arr.data ++ l",
"state_before": "α : Type u_1\narr : Array α\nl : List α\n⊢ (Array.appendList arr l).data = arr.data ++ l",
"tactic": "unfold Array.appendList"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\narr : Array α\nl : List α\n⊢ (List.foldl (fun r v => push r v) arr l).data = arr.data ++ l",
"tactic": "induction l generalizing arr <;> simp [*]"
}
] |
[
204,
44
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
201,
9
] |
Mathlib/Order/Filter/Pointwise.lean
|
Filter.pure_smul_pure
|
[] |
[
1031,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1030,
1
] |
Mathlib/Topology/Homeomorph.lean
|
Homeomorph.comap_nhds_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.69386\nδ : Type ?u.69389\ninst✝³ : TopologicalSpace α\ninst✝² : TopologicalSpace β\ninst✝¹ : TopologicalSpace γ\ninst✝ : TopologicalSpace δ\nh : α ≃ₜ β\ny : β\n⊢ comap (↑h) (𝓝 y) = 𝓝 (↑(Homeomorph.symm h) y)",
"tactic": "rw [h.nhds_eq_comap, h.apply_symm_apply]"
}
] |
[
389,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
388,
1
] |
Mathlib/Data/Nat/Basic.lean
|
Nat.one_le_div_iff
|
[
{
"state_after": "no goals",
"state_before": "m n k a b : ℕ\nhb : 0 < b\n⊢ 1 ≤ a / b ↔ b ≤ a",
"tactic": "rw [le_div_iff_mul_le hb, one_mul]"
}
] |
[
616,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
615,
1
] |
Mathlib/Combinatorics/SimpleGraph/Prod.lean
|
SimpleGraph.boxProd_connected
|
[] |
[
217,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
216,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Type.lean
|
Equiv.Perm.sign_of_cycleType
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\n⊢ prod (map (fun n => -(-1) ^ n) (cycleType f)) = (-1) ^ (sum (cycleType f) + ↑card (cycleType f))",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\n⊢ ↑sign f = (-1) ^ (sum (cycleType f) + ↑card (cycleType f))",
"tactic": "rw [sign_of_cycleType']"
},
{
"state_after": "case empty\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\n⊢ prod (map (fun n => -(-1) ^ n) 0) = (-1) ^ (sum 0 + ↑card 0)\n\ncase cons\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\na : ℕ\ns : Multiset ℕ\nihs : prod (map (fun n => -(-1) ^ n) s) = (-1) ^ (sum s + ↑card s)\n⊢ prod (map (fun n => -(-1) ^ n) (a ::ₘ s)) = (-1) ^ (sum (a ::ₘ s) + ↑card (a ::ₘ s))",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\n⊢ prod (map (fun n => -(-1) ^ n) (cycleType f)) = (-1) ^ (sum (cycleType f) + ↑card (cycleType f))",
"tactic": "induction' f.cycleType using Multiset.induction_on with a s ihs"
},
{
"state_after": "no goals",
"state_before": "case empty\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\n⊢ prod (map (fun n => -(-1) ^ n) 0) = (-1) ^ (sum 0 + ↑card 0)",
"tactic": "rfl"
},
{
"state_after": "case cons\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\na : ℕ\ns : Multiset ℕ\nihs : prod (map (fun n => -(-1) ^ n) s) = (-1) ^ (sum s + ↑card s)\n⊢ -(-1) ^ a * (-1) ^ (sum s + ↑card s) = (-1) ^ (a + sum s + (↑card s + 1))",
"state_before": "case cons\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\na : ℕ\ns : Multiset ℕ\nihs : prod (map (fun n => -(-1) ^ n) s) = (-1) ^ (sum s + ↑card s)\n⊢ prod (map (fun n => -(-1) ^ n) (a ::ₘ s)) = (-1) ^ (sum (a ::ₘ s) + ↑card (a ::ₘ s))",
"tactic": "rw [Multiset.map_cons, Multiset.prod_cons, Multiset.sum_cons, Multiset.card_cons, ihs]"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nf : Perm α\na : ℕ\ns : Multiset ℕ\nihs : prod (map (fun n => -(-1) ^ n) s) = (-1) ^ (sum s + ↑card s)\n⊢ -(-1) ^ a * (-1) ^ (sum s + ↑card s) = (-1) ^ (a + sum s + (↑card s + 1))",
"tactic": "simp only [pow_add, pow_one, mul_neg_one, neg_mul, mul_neg, mul_assoc, mul_one]"
}
] |
[
170,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Analysis/Convex/Basic.lean
|
convex_Icc
|
[] |
[
256,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
255,
1
] |
Mathlib/Topology/LocalHomeomorph.lean
|
LocalHomeomorph.right_inv
|
[] |
[
163,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
162,
1
] |
Mathlib/RingTheory/Valuation/Basic.lean
|
AddValuation.map_le_sum
|
[] |
[
701,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
699,
1
] |
Mathlib/Topology/Basic.lean
|
frontier_inter_subset
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns✝ s₁ s₂ t✝ : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\ns t : Set α\n⊢ closure (s ∩ t) ∩ (closure (sᶜ) ∪ closure (tᶜ)) ⊆\n closure s ∩ closure (sᶜ) ∩ closure t ∪ closure s ∩ (closure t ∩ closure (tᶜ))",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns✝ s₁ s₂ t✝ : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\ns t : Set α\n⊢ frontier (s ∩ t) ⊆ frontier s ∩ closure t ∪ closure s ∩ frontier t",
"tactic": "simp only [frontier_eq_closure_inter_closure, compl_inter, closure_union]"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns✝ s₁ s₂ t✝ : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\ns t : Set α\n⊢ closure s ∩ closure t ∩ (closure (sᶜ) ∪ closure (tᶜ)) =\n closure s ∩ closure (sᶜ) ∩ closure t ∪ closure s ∩ (closure t ∩ closure (tᶜ))",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns✝ s₁ s₂ t✝ : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\ns t : Set α\n⊢ closure (s ∩ t) ∩ (closure (sᶜ) ∪ closure (tᶜ)) ⊆\n closure s ∩ closure (sᶜ) ∩ closure t ∪ closure s ∩ (closure t ∩ closure (tᶜ))",
"tactic": "refine' (inter_subset_inter_left _ (closure_inter_subset_inter_closure s t)).trans_eq _"
},
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nι : Sort w\na : α\ns✝ s₁ s₂ t✝ : Set α\np p₁ p₂ : α → Prop\ninst✝ : TopologicalSpace α\ns t : Set α\n⊢ closure s ∩ closure t ∩ (closure (sᶜ) ∪ closure (tᶜ)) =\n closure s ∩ closure (sᶜ) ∩ closure t ∪ closure s ∩ (closure t ∩ closure (tᶜ))",
"tactic": "simp only [inter_distrib_left, inter_distrib_right, inter_assoc, inter_comm (closure t)]"
}
] |
[
748,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
744,
1
] |
Mathlib/Data/Polynomial/Basic.lean
|
Polynomial.sum_add'
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\na b : R\nm n : ℕ\ninst✝¹ : Semiring R\np✝ q : R[X]\nS : Type u_1\ninst✝ : AddCommMonoid S\np : R[X]\nf g : ℕ → R → S\n⊢ sum p (f + g) = sum p f + sum p g",
"tactic": "simp [sum_def, Finset.sum_add_distrib]"
}
] |
[
994,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
993,
1
] |
Mathlib/MeasureTheory/Decomposition/Lebesgue.lean
|
MeasureTheory.Measure.haveLebesgueDecomposition_of_finiteMeasure
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh :\n ∃ u,\n Monotone u ∧\n Filter.Tendsto u Filter.atTop (nhds (sSup (measurableLEEval ν μ))) ∧ ∀ (n : ℕ), u n ∈ measurableLEEval ν μ\n⊢ ∃ p, Measurable p.snd ∧ p.fst ⟂ₘ ν ∧ μ = p.fst + withDensity ν p.snd",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\n⊢ ∃ p, Measurable p.snd ∧ p.fst ⟂ₘ ν ∧ μ = p.fst + withDensity ν p.snd",
"tactic": "have h :=\n @exists_seq_tendsto_sSup _ _ _ _ _ (measurableLEEval ν μ)\n ⟨0, 0, zero_mem_measurableLE, by simp⟩ (OrderTop.bddAbove _)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\n⊢ ∃ p, Measurable p.snd ∧ p.fst ⟂ₘ ν ∧ μ = p.fst + withDensity ν p.snd",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nh :\n ∃ u,\n Monotone u ∧\n Filter.Tendsto u Filter.atTop (nhds (sSup (measurableLEEval ν μ))) ∧ ∀ (n : ℕ), u n ∈ measurableLEEval ν μ\n⊢ ∃ p, Measurable p.snd ∧ p.fst ⟂ₘ ν ∧ μ = p.fst + withDensity ν p.snd",
"tactic": "choose g _ hg₂ f hf₁ hf₂ using h"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\n⊢ ∃ p, Measurable p.snd ∧ p.fst ⟂ₘ ν ∧ μ = p.fst + withDensity ν p.snd",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\n⊢ ∃ p, Measurable p.snd ∧ p.fst ⟂ₘ ν ∧ μ = p.fst + withDensity ν p.snd",
"tactic": "set ξ := ⨆ (n) (k) (hk : k ≤ n), f k with hξ"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\n⊢ ∃ p, Measurable p.snd ∧ p.fst ⟂ₘ ν ∧ μ = p.fst + withDensity ν p.snd",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\n⊢ ∃ p, Measurable p.snd ∧ p.fst ⟂ₘ ν ∧ μ = p.fst + withDensity ν p.snd",
"tactic": "have hξm : Measurable ξ := by\n convert measurable_iSup fun n => (iSup_mem_measurableLE _ hf₁ n).1\n refine Option.ext fun x => ?_; simp [hξ]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\n⊢ ∃ p, Measurable p.snd ∧ p.fst ⟂ₘ ν ∧ μ = p.fst + withDensity ν p.snd",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\n⊢ ∃ p, Measurable p.snd ∧ p.fst ⟂ₘ ν ∧ μ = p.fst + withDensity ν p.snd",
"tactic": "set μ₁ := μ - ν.withDensity ξ with hμ₁"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\n⊢ ∃ p, Measurable p.snd ∧ p.fst ⟂ₘ ν ∧ μ = p.fst + withDensity ν p.snd",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\n⊢ ∃ p, Measurable p.snd ∧ p.fst ⟂ₘ ν ∧ μ = p.fst + withDensity ν p.snd",
"tactic": "have hle : ν.withDensity ξ ≤ μ := by\n intro B hB\n rw [hξ, withDensity_apply _ hB]\n simp_rw [iSup_apply]\n rw [lintegral_iSup (fun i => (iSup_mem_measurableLE _ hf₁ i).1) (iSup_monotone _)]\n exact iSup_le fun i => (iSup_mem_measurableLE _ hf₁ i).2 B hB"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\n⊢ ∃ p, Measurable p.snd ∧ p.fst ⟂ₘ ν ∧ μ = p.fst + withDensity ν p.snd",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\n⊢ ∃ p, Measurable p.snd ∧ p.fst ⟂ₘ ν ∧ μ = p.fst + withDensity ν p.snd",
"tactic": "have : IsFiniteMeasure (ν.withDensity ξ) := by\n refine' isFiniteMeasure_withDensity _\n have hle' := hle univ MeasurableSet.univ\n rw [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ] at hle'\n exact ne_top_of_le_ne_top (measure_ne_top _ _) hle'"
},
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\n⊢ (μ₁, ξ).fst ⟂ₘ ν\n\ncase refine'_2\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\n⊢ μ = (μ₁, ξ).fst + withDensity ν (μ₁, ξ).snd",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\n⊢ ∃ p, Measurable p.snd ∧ p.fst ⟂ₘ ν ∧ μ = p.fst + withDensity ν p.snd",
"tactic": "refine' ⟨⟨μ₁, ξ⟩, hξm, _, _⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\n⊢ (fun f => ∫⁻ (x : α), f x ∂ν) 0 = 0",
"tactic": "simp"
},
{
"state_after": "case refine_4\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nthis :\n Filter.Tendsto (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) Filter.atTop\n (nhds (∫⁻ (x : α), iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) x ∂ν))\n⊢ sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\n\ncase refine_1\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\n⊢ ∀ (n : ℕ), AEMeasurable ((fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) n)\n\ncase refine_2\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\n⊢ ∀ᵐ (x : α) ∂ν, Monotone fun n => (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) n x\n\ncase refine_3\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\n⊢ ∀ᵐ (x : α) ∂ν,\n Filter.Tendsto (fun n => (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) n x) Filter.atTop\n (nhds (iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) x))",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\n⊢ sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν",
"tactic": "have :=\n @lintegral_tendsto_of_tendsto_of_monotone _ _ ν (fun n => ⨆ (k) (_ : k ≤ n), f k)\n (⨆ (n) (k) (_ : k ≤ n), f k) ?_ ?_ ?_"
},
{
"state_after": "case refine_4\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nthis :\n Filter.Tendsto (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) Filter.atTop\n (nhds (∫⁻ (x : α), iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) x ∂ν))\n⊢ Filter.Tendsto (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) Filter.atTop\n (nhds (sSup (measurableLEEval ν μ)))",
"state_before": "case refine_4\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nthis :\n Filter.Tendsto (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) Filter.atTop\n (nhds (∫⁻ (x : α), iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) x ∂ν))\n⊢ sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν",
"tactic": "refine' tendsto_nhds_unique _ this"
},
{
"state_after": "case refine_4.refine'_1\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nthis :\n Filter.Tendsto (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) Filter.atTop\n (nhds (∫⁻ (x : α), iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) x ∂ν))\n⊢ g ≤ fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν\n\ncase refine_4.refine'_2\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nthis :\n Filter.Tendsto (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) Filter.atTop\n (nhds (∫⁻ (x : α), iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) x ∂ν))\n⊢ (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) ≤ fun x => sSup (measurableLEEval ν μ)",
"state_before": "case refine_4\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nthis :\n Filter.Tendsto (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) Filter.atTop\n (nhds (∫⁻ (x : α), iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) x ∂ν))\n⊢ Filter.Tendsto (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) Filter.atTop\n (nhds (sSup (measurableLEEval ν μ)))",
"tactic": "refine' tendsto_of_tendsto_of_tendsto_of_le_of_le hg₂ tendsto_const_nhds _ _"
},
{
"state_after": "case refine_4.refine'_1\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nthis :\n Filter.Tendsto (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) Filter.atTop\n (nhds (∫⁻ (x : α), iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) x ∂ν))\nn : ℕ\n⊢ g n ≤ (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) n",
"state_before": "case refine_4.refine'_1\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nthis :\n Filter.Tendsto (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) Filter.atTop\n (nhds (∫⁻ (x : α), iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) x ∂ν))\n⊢ g ≤ fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν",
"tactic": "intro n"
},
{
"state_after": "case refine_4.refine'_1\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nthis :\n Filter.Tendsto (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) Filter.atTop\n (nhds (∫⁻ (x : α), iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) x ∂ν))\nn : ℕ\n⊢ (fun f => ∫⁻ (x : α), f x ∂ν) (f n) ≤ (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) n",
"state_before": "case refine_4.refine'_1\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nthis :\n Filter.Tendsto (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) Filter.atTop\n (nhds (∫⁻ (x : α), iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) x ∂ν))\nn : ℕ\n⊢ g n ≤ (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) n",
"tactic": "rw [← hf₂ n]"
},
{
"state_after": "case refine_4.refine'_1.hfg\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nthis :\n Filter.Tendsto (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) Filter.atTop\n (nhds (∫⁻ (x : α), iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) x ∂ν))\nn : ℕ\n⊢ (fun a => f n a) ≤ fun a => iSup (fun k => ⨆ (_ : k ≤ n), f k) a",
"state_before": "case refine_4.refine'_1\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nthis :\n Filter.Tendsto (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) Filter.atTop\n (nhds (∫⁻ (x : α), iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) x ∂ν))\nn : ℕ\n⊢ (fun f => ∫⁻ (x : α), f x ∂ν) (f n) ≤ (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) n",
"tactic": "apply lintegral_mono"
},
{
"state_after": "no goals",
"state_before": "case refine_4.refine'_1.hfg\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nthis :\n Filter.Tendsto (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) Filter.atTop\n (nhds (∫⁻ (x : α), iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) x ∂ν))\nn : ℕ\n⊢ (fun a => f n a) ≤ fun a => iSup (fun k => ⨆ (_ : k ≤ n), f k) a",
"tactic": "simp only [iSup_apply, iSup_le_le f n n le_rfl]"
},
{
"state_after": "case refine_4.refine'_2\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nthis :\n Filter.Tendsto (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) Filter.atTop\n (nhds (∫⁻ (x : α), iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) x ∂ν))\nn : ℕ\n⊢ (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) n ≤ (fun x => sSup (measurableLEEval ν μ)) n",
"state_before": "case refine_4.refine'_2\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nthis :\n Filter.Tendsto (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) Filter.atTop\n (nhds (∫⁻ (x : α), iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) x ∂ν))\n⊢ (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) ≤ fun x => sSup (measurableLEEval ν μ)",
"tactic": "intro n"
},
{
"state_after": "no goals",
"state_before": "case refine_4.refine'_2\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nthis :\n Filter.Tendsto (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) Filter.atTop\n (nhds (∫⁻ (x : α), iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) x ∂ν))\nn : ℕ\n⊢ (fun n => ∫⁻ (x : α), iSup (fun k => ⨆ (_ : k ≤ n), f k) x ∂ν) n ≤ (fun x => sSup (measurableLEEval ν μ)) n",
"tactic": "exact le_sSup ⟨⨆ (k : ℕ) (_ : k ≤ n), f k, iSup_mem_measurableLE' _ hf₁ _, rfl⟩"
},
{
"state_after": "case refine_1\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nn : ℕ\n⊢ AEMeasurable ((fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) n)",
"state_before": "case refine_1\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\n⊢ ∀ (n : ℕ), AEMeasurable ((fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) n)",
"tactic": "intro n"
},
{
"state_after": "case refine_1\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nn : ℕ\n⊢ Measurable ((fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) n)",
"state_before": "case refine_1\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nn : ℕ\n⊢ AEMeasurable ((fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) n)",
"tactic": "refine' Measurable.aemeasurable _"
},
{
"state_after": "case h.e'_5.h\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nn : ℕ\nx✝ : α\n⊢ (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) n x✝ = ⨆ (k : ℕ) (_ : k ≤ n), f k x✝",
"state_before": "case refine_1\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nn : ℕ\n⊢ Measurable ((fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) n)",
"tactic": "convert (iSup_mem_measurableLE _ hf₁ n).1"
},
{
"state_after": "case h.e'_5.h\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nn : ℕ\nx✝ : α\nx : ℝ≥0\n⊢ x ∈ (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) n x✝ ↔ x ∈ ⨆ (k : ℕ) (_ : k ≤ n), f k x✝",
"state_before": "case h.e'_5.h\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nn : ℕ\nx✝ : α\n⊢ (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) n x✝ = ⨆ (k : ℕ) (_ : k ≤ n), f k x✝",
"tactic": "refine Option.ext fun x => ?_"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nn : ℕ\nx✝ : α\nx : ℝ≥0\n⊢ x ∈ (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) n x✝ ↔ x ∈ ⨆ (k : ℕ) (_ : k ≤ n), f k x✝",
"tactic": "simp"
},
{
"state_after": "case refine_2\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\na : α\n⊢ Monotone fun n => (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) n a",
"state_before": "case refine_2\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\n⊢ ∀ᵐ (x : α) ∂ν, Monotone fun n => (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) n x",
"tactic": "refine' Filter.eventually_of_forall fun a => _"
},
{
"state_after": "no goals",
"state_before": "case refine_2\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\na : α\n⊢ Monotone fun n => (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) n a",
"tactic": "simp [iSup_monotone' f _]"
},
{
"state_after": "case refine_3\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\na : α\n⊢ Filter.Tendsto (fun n => (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) n a) Filter.atTop\n (nhds (iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) a))",
"state_before": "case refine_3\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\n⊢ ∀ᵐ (x : α) ∂ν,\n Filter.Tendsto (fun n => (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) n x) Filter.atTop\n (nhds (iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) x))",
"tactic": "refine' Filter.eventually_of_forall fun a => _"
},
{
"state_after": "no goals",
"state_before": "case refine_3\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\na : α\n⊢ Filter.Tendsto (fun n => (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) n a) Filter.atTop\n (nhds (iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) a))",
"tactic": "simp [tendsto_atTop_iSup (iSup_monotone' f a)]"
},
{
"state_after": "case h.e'_5.h\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nx✝ : α\n⊢ ξ x✝ = ⨆ (i : ℕ) (k : ℕ) (_ : k ≤ i), f k x✝",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\n⊢ Measurable ξ",
"tactic": "convert measurable_iSup fun n => (iSup_mem_measurableLE _ hf₁ n).1"
},
{
"state_after": "case h.e'_5.h\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nx✝ : α\nx : ℝ≥0\n⊢ x ∈ ξ x✝ ↔ x ∈ ⨆ (i : ℕ) (k : ℕ) (_ : k ≤ i), f k x✝",
"state_before": "case h.e'_5.h\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nx✝ : α\n⊢ ξ x✝ = ⨆ (i : ℕ) (k : ℕ) (_ : k ≤ i), f k x✝",
"tactic": "refine Option.ext fun x => ?_"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nx✝ : α\nx : ℝ≥0\n⊢ x ∈ ξ x✝ ↔ x ∈ ⨆ (i : ℕ) (k : ℕ) (_ : k ≤ i), f k x✝",
"tactic": "simp [hξ]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nB : Set α\nhB : MeasurableSet B\n⊢ ↑↑(withDensity ν ξ) B ≤ ↑↑μ B",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\n⊢ withDensity ν ξ ≤ μ",
"tactic": "intro B hB"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nB : Set α\nhB : MeasurableSet B\n⊢ (∫⁻ (a : α) in B, iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) a ∂ν) ≤ ↑↑μ B",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nB : Set α\nhB : MeasurableSet B\n⊢ ↑↑(withDensity ν ξ) B ≤ ↑↑μ B",
"tactic": "rw [hξ, withDensity_apply _ hB]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nB : Set α\nhB : MeasurableSet B\n⊢ (∫⁻ (a : α) in B, ⨆ (i : ℕ) (i_1 : ℕ) (_ : i_1 ≤ i), f i_1 a ∂ν) ≤ ↑↑μ B",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nB : Set α\nhB : MeasurableSet B\n⊢ (∫⁻ (a : α) in B, iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) a ∂ν) ≤ ↑↑μ B",
"tactic": "simp_rw [iSup_apply]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nB : Set α\nhB : MeasurableSet B\n⊢ (⨆ (n : ℕ), ∫⁻ (a : α) in B, ⨆ (k : ℕ) (_ : k ≤ n), f k a ∂ν) ≤ ↑↑μ B",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nB : Set α\nhB : MeasurableSet B\n⊢ (∫⁻ (a : α) in B, ⨆ (i : ℕ) (i_1 : ℕ) (_ : i_1 ≤ i), f i_1 a ∂ν) ≤ ↑↑μ B",
"tactic": "rw [lintegral_iSup (fun i => (iSup_mem_measurableLE _ hf₁ i).1) (iSup_monotone _)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nB : Set α\nhB : MeasurableSet B\n⊢ (⨆ (n : ℕ), ∫⁻ (a : α) in B, ⨆ (k : ℕ) (_ : k ≤ n), f k a ∂ν) ≤ ↑↑μ B",
"tactic": "exact iSup_le fun i => (iSup_mem_measurableLE _ hf₁ i).2 B hB"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\n⊢ (∫⁻ (a : α), ξ a ∂ν) ≠ ⊤",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\n⊢ IsFiniteMeasure (withDensity ν ξ)",
"tactic": "refine' isFiniteMeasure_withDensity _"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nhle' : ↑↑(withDensity ν ξ) univ ≤ ↑↑μ univ\n⊢ (∫⁻ (a : α), ξ a ∂ν) ≠ ⊤",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\n⊢ (∫⁻ (a : α), ξ a ∂ν) ≠ ⊤",
"tactic": "have hle' := hle univ MeasurableSet.univ"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nhle' : (∫⁻ (a : α), ξ a ∂ν) ≤ ↑↑μ univ\n⊢ (∫⁻ (a : α), ξ a ∂ν) ≠ ⊤",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nhle' : ↑↑(withDensity ν ξ) univ ≤ ↑↑μ univ\n⊢ (∫⁻ (a : α), ξ a ∂ν) ≠ ⊤",
"tactic": "rw [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ] at hle'"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nhle' : (∫⁻ (a : α), ξ a ∂ν) ≤ ↑↑μ univ\n⊢ (∫⁻ (a : α), ξ a ∂ν) ≠ ⊤",
"tactic": "exact ne_top_of_le_ne_top (measure_ne_top _ _) hle'"
},
{
"state_after": "case refine'_1\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\n⊢ False",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\n⊢ (μ₁, ξ).fst ⟂ₘ ν",
"tactic": "by_contra h"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ : VectorMeasure.restrict 0 E ≤ VectorMeasure.restrict (toSignedMeasure μ₁ - toSignedMeasure (ε • ν)) E\n⊢ False",
"state_before": "case refine'_1\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\n⊢ False",
"tactic": "obtain ⟨ε, hε₁, E, hE₁, hE₂, hE₃⟩ := exists_positive_of_not_mutuallySingular μ₁ ν h"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\n⊢ False",
"state_before": "case refine'_1.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ : VectorMeasure.restrict 0 E ≤ VectorMeasure.restrict (toSignedMeasure μ₁ - toSignedMeasure (ε • ν)) E\n⊢ False",
"tactic": "simp_rw [hμ₁] at hE₃"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\n⊢ False",
"state_before": "case refine'_1.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\n⊢ False",
"tactic": "have hξle : ∀ A, MeasurableSet A → (∫⁻ a in A, ξ a ∂ν) ≤ μ A := by\n intro A hA; rw [hξ]\n simp_rw [iSup_apply]\n rw [lintegral_iSup (fun n => (iSup_mem_measurableLE _ hf₁ n).1) (iSup_monotone _)]\n exact iSup_le fun n => (iSup_mem_measurableLE _ hf₁ n).2 A hA"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nhξε : (ξ + indicator E fun x => ↑ε) ∈ measurableLE ν μ\n⊢ False",
"state_before": "case refine'_1.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\n⊢ False",
"tactic": "have hξε : (ξ + E.indicator fun _ => (ε : ℝ≥0∞)) ∈ measurableLE ν μ := by\n refine' ⟨Measurable.add hξm (Measurable.indicator measurable_const hE₁), fun A hA => _⟩\n have :\n (∫⁻ a in A, (ξ + E.indicator fun _ => (ε : ℝ≥0∞)) a ∂ν) =\n (∫⁻ a in A ∩ E, ε + ξ a ∂ν) + ∫⁻ a in A \\ E, ξ a ∂ν := by\n simp only [lintegral_add_left measurable_const, lintegral_add_left hξm,\n set_lintegral_const, add_assoc, lintegral_inter_add_diff _ _ hE₁, Pi.add_apply,\n lintegral_indicator _ hE₁, restrict_apply hE₁]\n rw [inter_comm, add_comm]\n rw [this, ← measure_inter_add_diff A hE₁]\n exact add_le_add (hε₂ A hA) (hξle (A \\ E) (hA.diff hE₁))"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nhξε : (ξ + indicator E fun x => ↑ε) ∈ measurableLE ν μ\nthis : (∫⁻ (a : α), ξ a + indicator E (fun x => ↑ε) a ∂ν) ≤ sSup (measurableLEEval ν μ)\n⊢ False",
"state_before": "case refine'_1.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nhξε : (ξ + indicator E fun x => ↑ε) ∈ measurableLE ν μ\n⊢ False",
"tactic": "have : (∫⁻ a, ξ a + E.indicator (fun _ => (ε : ℝ≥0∞)) a ∂ν) ≤ sSup (measurableLEEval ν μ) :=\n le_sSup ⟨ξ + E.indicator fun _ => (ε : ℝ≥0∞), hξε, rfl⟩"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nhξε : (ξ + indicator E fun x => ↑ε) ∈ measurableLE ν μ\nthis : (∫⁻ (a : α), ξ a + indicator E (fun x => ↑ε) a ∂ν) ≤ sSup (measurableLEEval ν μ)\n⊢ sSup (measurableLEEval ν μ) < ∫⁻ (a : α), ξ a + indicator E (fun x => ↑ε) a ∂ν",
"state_before": "case refine'_1.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nhξε : (ξ + indicator E fun x => ↑ε) ∈ measurableLE ν μ\nthis : (∫⁻ (a : α), ξ a + indicator E (fun x => ↑ε) a ∂ν) ≤ sSup (measurableLEEval ν μ)\n⊢ False",
"tactic": "refine' not_lt.2 this _"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nhξε : (ξ + indicator E fun x => ↑ε) ∈ measurableLE ν μ\nthis : (∫⁻ (a : α), ξ a + indicator E (fun x => ↑ε) a ∂ν) ≤ sSup (measurableLEEval ν μ)\n⊢ (∫⁻ (a : α), ξ a ∂ν) < (∫⁻ (a : α), ξ a ∂ν) + ↑ε * ↑↑ν E",
"state_before": "case refine'_1.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nhξε : (ξ + indicator E fun x => ↑ε) ∈ measurableLE ν μ\nthis : (∫⁻ (a : α), ξ a + indicator E (fun x => ↑ε) a ∂ν) ≤ sSup (measurableLEEval ν μ)\n⊢ sSup (measurableLEEval ν μ) < ∫⁻ (a : α), ξ a + indicator E (fun x => ↑ε) a ∂ν",
"tactic": "rw [hξ₁, lintegral_add_left hξm, lintegral_indicator _ hE₁, set_lintegral_const]"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nhξε : (ξ + indicator E fun x => ↑ε) ∈ measurableLE ν μ\nthis : (∫⁻ (a : α), ξ a + indicator E (fun x => ↑ε) a ∂ν) ≤ sSup (measurableLEEval ν μ)\n⊢ (∫⁻ (a : α), ξ a ∂ν) ≠ ⊤",
"state_before": "case refine'_1.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nhξε : (ξ + indicator E fun x => ↑ε) ∈ measurableLE ν μ\nthis : (∫⁻ (a : α), ξ a + indicator E (fun x => ↑ε) a ∂ν) ≤ sSup (measurableLEEval ν μ)\n⊢ (∫⁻ (a : α), ξ a ∂ν) < (∫⁻ (a : α), ξ a ∂ν) + ↑ε * ↑↑ν E",
"tactic": "refine' ENNReal.lt_add_right _ (ENNReal.mul_pos_iff.2 ⟨ENNReal.coe_pos.2 hε₁, hE₂⟩).ne'"
},
{
"state_after": "case refine'_1.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝¹ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nhξε : (ξ + indicator E fun x => ↑ε) ∈ measurableLE ν μ\nthis✝ : (∫⁻ (a : α), ξ a + indicator E (fun x => ↑ε) a ∂ν) ≤ sSup (measurableLEEval ν μ)\nthis : ↑↑(withDensity ν ξ) univ ≠ ⊤\n⊢ (∫⁻ (a : α), ξ a ∂ν) ≠ ⊤",
"state_before": "case refine'_1.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nhξε : (ξ + indicator E fun x => ↑ε) ∈ measurableLE ν μ\nthis : (∫⁻ (a : α), ξ a + indicator E (fun x => ↑ε) a ∂ν) ≤ sSup (measurableLEEval ν μ)\n⊢ (∫⁻ (a : α), ξ a ∂ν) ≠ ⊤",
"tactic": "have := measure_ne_top (ν.withDensity ξ) univ"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro.intro.intro.intro.intro\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝¹ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nhξε : (ξ + indicator E fun x => ↑ε) ∈ measurableLE ν μ\nthis✝ : (∫⁻ (a : α), ξ a + indicator E (fun x => ↑ε) a ∂ν) ≤ sSup (measurableLEEval ν μ)\nthis : ↑↑(withDensity ν ξ) univ ≠ ⊤\n⊢ (∫⁻ (a : α), ξ a ∂ν) ≠ ⊤",
"tactic": "rwa [withDensity_apply _ MeasurableSet.univ, Measure.restrict_univ] at this"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nA : Set α\nhA : MeasurableSet A\n⊢ (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\n⊢ ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A",
"tactic": "intro A hA"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nA : Set α\nhA : MeasurableSet A\n⊢ (∫⁻ (a : α) in A, iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) a ∂ν) ≤ ↑↑μ A",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nA : Set α\nhA : MeasurableSet A\n⊢ (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A",
"tactic": "rw [hξ]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nA : Set α\nhA : MeasurableSet A\n⊢ (∫⁻ (a : α) in A, ⨆ (i : ℕ) (i_1 : ℕ) (_ : i_1 ≤ i), f i_1 a ∂ν) ≤ ↑↑μ A",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nA : Set α\nhA : MeasurableSet A\n⊢ (∫⁻ (a : α) in A, iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) a ∂ν) ≤ ↑↑μ A",
"tactic": "simp_rw [iSup_apply]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nA : Set α\nhA : MeasurableSet A\n⊢ (⨆ (n : ℕ), ∫⁻ (a : α) in A, ⨆ (k : ℕ) (_ : k ≤ n), f k a ∂ν) ≤ ↑↑μ A",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nA : Set α\nhA : MeasurableSet A\n⊢ (∫⁻ (a : α) in A, ⨆ (i : ℕ) (i_1 : ℕ) (_ : i_1 ≤ i), f i_1 a ∂ν) ≤ ↑↑μ A",
"tactic": "rw [lintegral_iSup (fun n => (iSup_mem_measurableLE _ hf₁ n).1) (iSup_monotone _)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nA : Set α\nhA : MeasurableSet A\n⊢ (⨆ (n : ℕ), ∫⁻ (a : α) in A, ⨆ (k : ℕ) (_ : k ≤ n), f k a ∂ν) ≤ ↑↑μ A",
"tactic": "exact iSup_le fun n => (iSup_mem_measurableLE _ hf₁ n).2 A hA"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nA : Set α\nhA : MeasurableSet A\n⊢ (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\n⊢ ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)",
"tactic": "intro A hA"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nA : Set α\nhA : MeasurableSet A\nthis : ↑0 (A ∩ E) ≤ ↑(toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) (A ∩ E)\n⊢ (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nA : Set α\nhA : MeasurableSet A\n⊢ (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)",
"tactic": "have := subset_le_of_restrict_le_restrict _ _ hE₁ hE₃ (inter_subset_right A E)"
},
{
"state_after": "case ha\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nA : Set α\nhA : MeasurableSet A\nthis : ENNReal.toReal (↑↑(ε • ν) (A ∩ E) + ↑↑(withDensity ν ξ) (A ∩ E)) ≤ ENNReal.toReal (↑↑μ (A ∩ E))\n⊢ ↑↑(ε • ν) (A ∩ E) + ↑↑(withDensity ν ξ) (A ∩ E) ≠ ⊤\n\ncase hb\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nA : Set α\nhA : MeasurableSet A\nthis : ENNReal.toReal (↑↑(ε • ν) (A ∩ E) + ↑↑(withDensity ν ξ) (A ∩ E)) ≤ ENNReal.toReal (↑↑μ (A ∩ E))\n⊢ ↑↑μ (A ∩ E) ≠ ⊤\n\ncase ha\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nA : Set α\nhA : MeasurableSet A\nthis : ENNReal.toReal (↑↑(ε • ν) (A ∩ E)) + ENNReal.toReal (↑↑(withDensity ν ξ) (A ∩ E)) ≤ ENNReal.toReal (↑↑μ (A ∩ E))\n⊢ ↑↑(ε • ν) (A ∩ E) ≠ ⊤\n\ncase hb\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nA : Set α\nhA : MeasurableSet A\nthis : ENNReal.toReal (↑↑(ε • ν) (A ∩ E)) + ENNReal.toReal (↑↑(withDensity ν ξ) (A ∩ E)) ≤ ENNReal.toReal (↑↑μ (A ∩ E))\n⊢ ↑↑(withDensity ν ξ) (A ∩ E) ≠ ⊤\n\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nA : Set α\nhA : MeasurableSet A\nthis : 0 ≤ ENNReal.toReal (↑↑μ (A ∩ E) - ↑↑(withDensity ν ξ) (A ∩ E)) - ENNReal.toReal (↑↑(ε • ν) (A ∩ E))\n⊢ ↑↑(withDensity ν ξ) (A ∩ E) ≤ ↑↑μ (A ∩ E)",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nA : Set α\nhA : MeasurableSet A\nthis : ↑0 (A ∩ E) ≤ ↑(toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) (A ∩ E)\n⊢ (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)",
"tactic": "rwa [zero_apply, toSignedMeasure_sub_apply (hA.inter hE₁),\n Measure.sub_apply (hA.inter hE₁) hle,\n ENNReal.toReal_sub_of_le _ (ne_of_lt (measure_lt_top _ _)), sub_nonneg, le_sub_iff_add_le,\n ← ENNReal.toReal_add, ENNReal.toReal_le_toReal, Measure.coe_smul, Pi.smul_apply,\n withDensity_apply _ (hA.inter hE₁), show ε • ν (A ∩ E) = (ε : ℝ≥0∞) * ν (A ∩ E) by rfl,\n ← set_lintegral_const, ← lintegral_add_left measurable_const] at this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nA : Set α\nhA : MeasurableSet A\nthis : (ε • ↑↑ν (A ∩ E) + ∫⁻ (a : α) in A ∩ E, ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\n⊢ ε • ↑↑ν (A ∩ E) = ↑ε * ↑↑ν (A ∩ E)",
"tactic": "rfl"
},
{
"state_after": "case ha\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nA : Set α\nhA : MeasurableSet A\nthis : ENNReal.toReal (↑↑(ε • ν) (A ∩ E) + ↑↑(withDensity ν ξ) (A ∩ E)) ≤ ENNReal.toReal (↑↑μ (A ∩ E))\n⊢ ¬↑↑(ε • ν) (A ∩ E) = ⊤ ∧ ¬↑↑(withDensity ν ξ) (A ∩ E) = ⊤",
"state_before": "case ha\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nA : Set α\nhA : MeasurableSet A\nthis : ENNReal.toReal (↑↑(ε • ν) (A ∩ E) + ↑↑(withDensity ν ξ) (A ∩ E)) ≤ ENNReal.toReal (↑↑μ (A ∩ E))\n⊢ ↑↑(ε • ν) (A ∩ E) + ↑↑(withDensity ν ξ) (A ∩ E) ≠ ⊤",
"tactic": "rw [Ne.def, ENNReal.add_eq_top, not_or]"
},
{
"state_after": "no goals",
"state_before": "case ha\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nA : Set α\nhA : MeasurableSet A\nthis : ENNReal.toReal (↑↑(ε • ν) (A ∩ E) + ↑↑(withDensity ν ξ) (A ∩ E)) ≤ ENNReal.toReal (↑↑μ (A ∩ E))\n⊢ ¬↑↑(ε • ν) (A ∩ E) = ⊤ ∧ ¬↑↑(withDensity ν ξ) (A ∩ E) = ⊤",
"tactic": "exact ⟨ne_of_lt (measure_lt_top _ _), ne_of_lt (measure_lt_top _ _)⟩"
},
{
"state_after": "no goals",
"state_before": "case hb\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nA : Set α\nhA : MeasurableSet A\nthis : ENNReal.toReal (↑↑(ε • ν) (A ∩ E) + ↑↑(withDensity ν ξ) (A ∩ E)) ≤ ENNReal.toReal (↑↑μ (A ∩ E))\n⊢ ↑↑μ (A ∩ E) ≠ ⊤",
"tactic": "exact ne_of_lt (measure_lt_top _ _)"
},
{
"state_after": "no goals",
"state_before": "case ha\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nA : Set α\nhA : MeasurableSet A\nthis : ENNReal.toReal (↑↑(ε • ν) (A ∩ E)) + ENNReal.toReal (↑↑(withDensity ν ξ) (A ∩ E)) ≤ ENNReal.toReal (↑↑μ (A ∩ E))\n⊢ ↑↑(ε • ν) (A ∩ E) ≠ ⊤",
"tactic": "exact ne_of_lt (measure_lt_top _ _)"
},
{
"state_after": "no goals",
"state_before": "case hb\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nA : Set α\nhA : MeasurableSet A\nthis : ENNReal.toReal (↑↑(ε • ν) (A ∩ E)) + ENNReal.toReal (↑↑(withDensity ν ξ) (A ∩ E)) ≤ ENNReal.toReal (↑↑μ (A ∩ E))\n⊢ ↑↑(withDensity ν ξ) (A ∩ E) ≠ ⊤",
"tactic": "exact ne_of_lt (measure_lt_top _ _)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nA : Set α\nhA : MeasurableSet A\nthis : 0 ≤ ENNReal.toReal (↑↑μ (A ∩ E) - ↑↑(withDensity ν ξ) (A ∩ E)) - ENNReal.toReal (↑↑(ε • ν) (A ∩ E))\n⊢ (∫⁻ (a : α) in A ∩ E, ξ a ∂ν) ≤ ↑↑μ (A ∩ E)",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nA : Set α\nhA : MeasurableSet A\nthis : 0 ≤ ENNReal.toReal (↑↑μ (A ∩ E) - ↑↑(withDensity ν ξ) (A ∩ E)) - ENNReal.toReal (↑↑(ε • ν) (A ∩ E))\n⊢ ↑↑(withDensity ν ξ) (A ∩ E) ≤ ↑↑μ (A ∩ E)",
"tactic": "rw [withDensity_apply _ (hA.inter hE₁)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nA : Set α\nhA : MeasurableSet A\nthis : 0 ≤ ENNReal.toReal (↑↑μ (A ∩ E) - ↑↑(withDensity ν ξ) (A ∩ E)) - ENNReal.toReal (↑↑(ε • ν) (A ∩ E))\n⊢ (∫⁻ (a : α) in A ∩ E, ξ a ∂ν) ≤ ↑↑μ (A ∩ E)",
"tactic": "exact hξle (A ∩ E) (hA.inter hE₁)"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nA : Set α\nhA : MeasurableSet A\n⊢ (∫⁻ (x : α) in A, (ξ + indicator E fun x => ↑ε) x ∂ν) ≤ ↑↑μ A",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\n⊢ (ξ + indicator E fun x => ↑ε) ∈ measurableLE ν μ",
"tactic": "refine' ⟨Measurable.add hξm (Measurable.indicator measurable_const hE₁), fun A hA => _⟩"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nA : Set α\nhA : MeasurableSet A\nthis :\n (∫⁻ (a : α) in A, (ξ + indicator E fun x => ↑ε) a ∂ν) =\n (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) + ∫⁻ (a : α) in A \\ E, ξ a ∂ν\n⊢ (∫⁻ (x : α) in A, (ξ + indicator E fun x => ↑ε) x ∂ν) ≤ ↑↑μ A",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nA : Set α\nhA : MeasurableSet A\n⊢ (∫⁻ (x : α) in A, (ξ + indicator E fun x => ↑ε) x ∂ν) ≤ ↑↑μ A",
"tactic": "have :\n (∫⁻ a in A, (ξ + E.indicator fun _ => (ε : ℝ≥0∞)) a ∂ν) =\n (∫⁻ a in A ∩ E, ε + ξ a ∂ν) + ∫⁻ a in A \\ E, ξ a ∂ν := by\n simp only [lintegral_add_left measurable_const, lintegral_add_left hξm,\n set_lintegral_const, add_assoc, lintegral_inter_add_diff _ _ hE₁, Pi.add_apply,\n lintegral_indicator _ hE₁, restrict_apply hE₁]\n rw [inter_comm, add_comm]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nA : Set α\nhA : MeasurableSet A\nthis :\n (∫⁻ (a : α) in A, (ξ + indicator E fun x => ↑ε) a ∂ν) =\n (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) + ∫⁻ (a : α) in A \\ E, ξ a ∂ν\n⊢ ((∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) + ∫⁻ (a : α) in A \\ E, ξ a ∂ν) ≤ ↑↑μ (A ∩ E) + ↑↑μ (A \\ E)",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nA : Set α\nhA : MeasurableSet A\nthis :\n (∫⁻ (a : α) in A, (ξ + indicator E fun x => ↑ε) a ∂ν) =\n (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) + ∫⁻ (a : α) in A \\ E, ξ a ∂ν\n⊢ (∫⁻ (x : α) in A, (ξ + indicator E fun x => ↑ε) x ∂ν) ≤ ↑↑μ A",
"tactic": "rw [this, ← measure_inter_add_diff A hE₁]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis✝ : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nA : Set α\nhA : MeasurableSet A\nthis :\n (∫⁻ (a : α) in A, (ξ + indicator E fun x => ↑ε) a ∂ν) =\n (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) + ∫⁻ (a : α) in A \\ E, ξ a ∂ν\n⊢ ((∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) + ∫⁻ (a : α) in A \\ E, ξ a ∂ν) ≤ ↑↑μ (A ∩ E) + ↑↑μ (A \\ E)",
"tactic": "exact add_le_add (hε₂ A hA) (hξle (A \\ E) (hA.diff hE₁))"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nA : Set α\nhA : MeasurableSet A\n⊢ (∫⁻ (a : α) in A, iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) a ∂ν) + ↑ε * ↑↑ν (E ∩ A) =\n ↑ε * ↑↑ν (A ∩ E) + ∫⁻ (a : α) in A, iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) a ∂ν",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nA : Set α\nhA : MeasurableSet A\n⊢ (∫⁻ (a : α) in A, (ξ + indicator E fun x => ↑ε) a ∂ν) =\n (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) + ∫⁻ (a : α) in A \\ E, ξ a ∂ν",
"tactic": "simp only [lintegral_add_left measurable_const, lintegral_add_left hξm,\n set_lintegral_const, add_assoc, lintegral_inter_add_diff _ _ hE₁, Pi.add_apply,\n lintegral_indicator _ hE₁, restrict_apply hE₁]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nh : ¬(μ₁, ξ).fst ⟂ₘ ν\nε : ℝ≥0\nhε₁ : 0 < ε\nE : Set α\nhE₁ : MeasurableSet E\nhE₂ : 0 < ↑↑ν E\nhE₃ :\n VectorMeasure.restrict 0 E ≤\n VectorMeasure.restrict (toSignedMeasure (μ - withDensity ν ξ) - toSignedMeasure (ε • ν)) E\nhξle : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A, ξ a ∂ν) ≤ ↑↑μ A\nhε₂ : ∀ (A : Set α), MeasurableSet A → (∫⁻ (a : α) in A ∩ E, ↑ε + ξ a ∂ν) ≤ ↑↑μ (A ∩ E)\nA : Set α\nhA : MeasurableSet A\n⊢ (∫⁻ (a : α) in A, iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) a ∂ν) + ↑ε * ↑↑ν (E ∩ A) =\n ↑ε * ↑↑ν (A ∩ E) + ∫⁻ (a : α) in A, iSup (fun n => ⨆ (k : ℕ) (_ : k ≤ n), f k) a ∂ν",
"tactic": "rw [inter_comm, add_comm]"
},
{
"state_after": "case refine'_2\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\n⊢ μ = (μ - withDensity ν ξ, ξ).fst + withDensity ν (μ - withDensity ν ξ, ξ).snd",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\n⊢ μ = (μ₁, ξ).fst + withDensity ν (μ₁, ξ).snd",
"tactic": "rw [hμ₁]"
},
{
"state_after": "case refine'_2.h\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nA : Set α\nhA : MeasurableSet A\n⊢ ↑↑μ A = ↑↑((μ - withDensity ν ξ, ξ).fst + withDensity ν (μ - withDensity ν ξ, ξ).snd) A",
"state_before": "case refine'_2\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\n⊢ μ = (μ - withDensity ν ξ, ξ).fst + withDensity ν (μ - withDensity ν ξ, ξ).snd",
"tactic": "ext1 A hA"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.h\nα : Type u_1\nβ : Type ?u.139824\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\ng : ℕ → ℝ≥0∞\nh✝ : Monotone g\nhg₂ : Filter.Tendsto g Filter.atTop (nhds (sSup (measurableLEEval ν μ)))\nf : ℕ → α → ℝ≥0∞\nhf₁ : ∀ (n : ℕ), f n ∈ measurableLE ν μ\nhf₂ : ∀ (n : ℕ), (fun f => ∫⁻ (x : α), f x ∂ν) (f n) = g n\nξ : α → ℝ≥0∞ := ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ : ξ = ⨆ (n : ℕ) (k : ℕ) (_ : k ≤ n), f k\nhξ₁ : sSup (measurableLEEval ν μ) = ∫⁻ (a : α), ξ a ∂ν\nhξm : Measurable ξ\nμ₁ : Measure α := μ - withDensity ν ξ\nhμ₁ : μ₁ = μ - withDensity ν ξ\nhle : withDensity ν ξ ≤ μ\nthis : IsFiniteMeasure (withDensity ν ξ)\nA : Set α\nhA : MeasurableSet A\n⊢ ↑↑μ A = ↑↑((μ - withDensity ν ξ, ξ).fst + withDensity ν (μ - withDensity ν ξ, ξ).snd) A",
"tactic": "rw [Measure.coe_add, Pi.add_apply, Measure.sub_apply hA hle, add_comm,\n add_tsub_cancel_of_le (hle A hA)]"
}
] |
[
655,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
556,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpace.lean
|
MeasureTheory.Measure.QuasiMeasurePreserving.mono
|
[] |
[
2471,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2469,
1
] |
Mathlib/Tactic/IntervalCases.lean
|
Int.le_sub_one_of_not_le
|
[] |
[
239,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/Topology/Connected.lean
|
IsIrreducible.isConnected
|
[] |
[
81,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
80,
1
] |
Mathlib/Analysis/Analytic/Linear.lean
|
ContinuousLinearMap.fpowerSeriesBilinear_radius
|
[] |
[
97,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
95,
1
] |
Mathlib/Order/Basic.lean
|
le_of_eq_of_le
|
[] |
[
179,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/LinearAlgebra/Alternating.lean
|
AlternatingMap.curryLeft_compLinearMap
|
[
{
"state_after": "case refine'_1\nR : Type ?u.1350987\ninst✝¹⁹ : Semiring R\nM : Type ?u.1350993\ninst✝¹⁸ : AddCommMonoid M\ninst✝¹⁷ : Module R M\nN : Type ?u.1351025\ninst✝¹⁶ : AddCommMonoid N\ninst✝¹⁵ : Module R N\nP : Type ?u.1351055\ninst✝¹⁴ : AddCommMonoid P\ninst✝¹³ : Module R P\nM' : Type ?u.1351085\ninst✝¹² : AddCommGroup M'\ninst✝¹¹ : Module R M'\nN' : Type ?u.1351473\ninst✝¹⁰ : AddCommGroup N'\ninst✝⁹ : Module R N'\nι : Type ?u.1351861\nι' : Type ?u.1351864\nι'' : Type ?u.1351867\nR' : Type u_1\nM'' : Type u_3\nM₂'' : Type u_2\nN'' : Type u_4\nN₂'' : Type ?u.1351882\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommMonoid M''\ninst✝⁶ : AddCommMonoid M₂''\ninst✝⁵ : AddCommMonoid N''\ninst✝⁴ : AddCommMonoid N₂''\ninst✝³ : Module R' M''\ninst✝² : Module R' M₂''\ninst✝¹ : Module R' N''\ninst✝ : Module R' N₂''\nn : ℕ\ng : M₂'' →ₗ[R'] M''\nf : AlternatingMap R' M'' N'' (Fin (Nat.succ n))\nm : M₂''\nv : Fin n → M₂''\n⊢ ↑((fun x => g) 0) (Matrix.vecCons m v 0) = Matrix.vecCons (↑g m) (fun i => ↑((fun x => g) i) (v i)) 0\n\ncase refine'_2\nR : Type ?u.1350987\ninst✝¹⁹ : Semiring R\nM : Type ?u.1350993\ninst✝¹⁸ : AddCommMonoid M\ninst✝¹⁷ : Module R M\nN : Type ?u.1351025\ninst✝¹⁶ : AddCommMonoid N\ninst✝¹⁵ : Module R N\nP : Type ?u.1351055\ninst✝¹⁴ : AddCommMonoid P\ninst✝¹³ : Module R P\nM' : Type ?u.1351085\ninst✝¹² : AddCommGroup M'\ninst✝¹¹ : Module R M'\nN' : Type ?u.1351473\ninst✝¹⁰ : AddCommGroup N'\ninst✝⁹ : Module R N'\nι : Type ?u.1351861\nι' : Type ?u.1351864\nι'' : Type ?u.1351867\nR' : Type u_1\nM'' : Type u_3\nM₂'' : Type u_2\nN'' : Type u_4\nN₂'' : Type ?u.1351882\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommMonoid M''\ninst✝⁶ : AddCommMonoid M₂''\ninst✝⁵ : AddCommMonoid N''\ninst✝⁴ : AddCommMonoid N₂''\ninst✝³ : Module R' M''\ninst✝² : Module R' M₂''\ninst✝¹ : Module R' N''\ninst✝ : Module R' N₂''\nn : ℕ\ng : M₂'' →ₗ[R'] M''\nf : AlternatingMap R' M'' N'' (Fin (Nat.succ n))\nm : M₂''\nv : Fin n → M₂''\n⊢ ∀ (i : Fin n),\n ↑((fun x => g) (Fin.succ i)) (Matrix.vecCons m v (Fin.succ i)) =\n Matrix.vecCons (↑g m) (fun i => ↑((fun x => g) i) (v i)) (Fin.succ i)",
"state_before": "R : Type ?u.1350987\ninst✝¹⁹ : Semiring R\nM : Type ?u.1350993\ninst✝¹⁸ : AddCommMonoid M\ninst✝¹⁷ : Module R M\nN : Type ?u.1351025\ninst✝¹⁶ : AddCommMonoid N\ninst✝¹⁵ : Module R N\nP : Type ?u.1351055\ninst✝¹⁴ : AddCommMonoid P\ninst✝¹³ : Module R P\nM' : Type ?u.1351085\ninst✝¹² : AddCommGroup M'\ninst✝¹¹ : Module R M'\nN' : Type ?u.1351473\ninst✝¹⁰ : AddCommGroup N'\ninst✝⁹ : Module R N'\nι : Type ?u.1351861\nι' : Type ?u.1351864\nι'' : Type ?u.1351867\nR' : Type u_1\nM'' : Type u_3\nM₂'' : Type u_2\nN'' : Type u_4\nN₂'' : Type ?u.1351882\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommMonoid M''\ninst✝⁶ : AddCommMonoid M₂''\ninst✝⁵ : AddCommMonoid N''\ninst✝⁴ : AddCommMonoid N₂''\ninst✝³ : Module R' M''\ninst✝² : Module R' M₂''\ninst✝¹ : Module R' N''\ninst✝ : Module R' N₂''\nn : ℕ\ng : M₂'' →ₗ[R'] M''\nf : AlternatingMap R' M'' N'' (Fin (Nat.succ n))\nm : M₂''\nv : Fin n → M₂''\n⊢ ∀ (x : Fin (Nat.succ n)),\n ↑((fun x => g) x) (Matrix.vecCons m v x) = Matrix.vecCons (↑g m) (fun i => ↑((fun x => g) i) (v i)) x",
"tactic": "refine' Fin.cases _ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nR : Type ?u.1350987\ninst✝¹⁹ : Semiring R\nM : Type ?u.1350993\ninst✝¹⁸ : AddCommMonoid M\ninst✝¹⁷ : Module R M\nN : Type ?u.1351025\ninst✝¹⁶ : AddCommMonoid N\ninst✝¹⁵ : Module R N\nP : Type ?u.1351055\ninst✝¹⁴ : AddCommMonoid P\ninst✝¹³ : Module R P\nM' : Type ?u.1351085\ninst✝¹² : AddCommGroup M'\ninst✝¹¹ : Module R M'\nN' : Type ?u.1351473\ninst✝¹⁰ : AddCommGroup N'\ninst✝⁹ : Module R N'\nι : Type ?u.1351861\nι' : Type ?u.1351864\nι'' : Type ?u.1351867\nR' : Type u_1\nM'' : Type u_3\nM₂'' : Type u_2\nN'' : Type u_4\nN₂'' : Type ?u.1351882\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommMonoid M''\ninst✝⁶ : AddCommMonoid M₂''\ninst✝⁵ : AddCommMonoid N''\ninst✝⁴ : AddCommMonoid N₂''\ninst✝³ : Module R' M''\ninst✝² : Module R' M₂''\ninst✝¹ : Module R' N''\ninst✝ : Module R' N₂''\nn : ℕ\ng : M₂'' →ₗ[R'] M''\nf : AlternatingMap R' M'' N'' (Fin (Nat.succ n))\nm : M₂''\nv : Fin n → M₂''\n⊢ ↑((fun x => g) 0) (Matrix.vecCons m v 0) = Matrix.vecCons (↑g m) (fun i => ↑((fun x => g) i) (v i)) 0",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nR : Type ?u.1350987\ninst✝¹⁹ : Semiring R\nM : Type ?u.1350993\ninst✝¹⁸ : AddCommMonoid M\ninst✝¹⁷ : Module R M\nN : Type ?u.1351025\ninst✝¹⁶ : AddCommMonoid N\ninst✝¹⁵ : Module R N\nP : Type ?u.1351055\ninst✝¹⁴ : AddCommMonoid P\ninst✝¹³ : Module R P\nM' : Type ?u.1351085\ninst✝¹² : AddCommGroup M'\ninst✝¹¹ : Module R M'\nN' : Type ?u.1351473\ninst✝¹⁰ : AddCommGroup N'\ninst✝⁹ : Module R N'\nι : Type ?u.1351861\nι' : Type ?u.1351864\nι'' : Type ?u.1351867\nR' : Type u_1\nM'' : Type u_3\nM₂'' : Type u_2\nN'' : Type u_4\nN₂'' : Type ?u.1351882\ninst✝⁸ : CommSemiring R'\ninst✝⁷ : AddCommMonoid M''\ninst✝⁶ : AddCommMonoid M₂''\ninst✝⁵ : AddCommMonoid N''\ninst✝⁴ : AddCommMonoid N₂''\ninst✝³ : Module R' M''\ninst✝² : Module R' M₂''\ninst✝¹ : Module R' N''\ninst✝ : Module R' N₂''\nn : ℕ\ng : M₂'' →ₗ[R'] M''\nf : AlternatingMap R' M'' N'' (Fin (Nat.succ n))\nm : M₂''\nv : Fin n → M₂''\n⊢ ∀ (i : Fin n),\n ↑((fun x => g) (Fin.succ i)) (Matrix.vecCons m v (Fin.succ i)) =\n Matrix.vecCons (↑g m) (fun i => ↑((fun x => g) i) (v i)) (Fin.succ i)",
"tactic": "simp"
}
] |
[
1309,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1301,
1
] |
src/lean/Init/Data/Nat/Div.lean
|
Nat.div_lt_self
|
[
{
"state_after": "n k : Nat\nhLtN : 0 < n\nhLtK : 1 < k\n⊢ ite (0 < k ∧ k ≤ n) ((n - k) / k + 1) 0 < n",
"state_before": "n k : Nat\nhLtN : 0 < n\nhLtK : 1 < k\n⊢ n / k < n",
"tactic": "rw [div_eq]"
},
{
"state_after": "no goals",
"state_before": "n k : Nat\nhLtN : 0 < n\nhLtK : 1 < k\n⊢ ite (0 < k ∧ k ≤ n) ((n - k) / k + 1) 0 < n",
"tactic": "cases (inferInstance : Decidable (0 < k ∧ k ≤ n)) with\n| isFalse h => simp [hLtN, h]\n| isTrue h =>\n suffices (n - k) / k + 1 < n by simp [h, this]\n have ⟨_, hKN⟩ := h\n have : (n - k) / k ≤ n - k := div_le_self (n - k) k\n have := Nat.add_le_of_le_sub hKN this\n exact Nat.lt_of_lt_of_le (Nat.add_lt_add_left hLtK _) this"
},
{
"state_after": "no goals",
"state_before": "case isFalse\nn k : Nat\nhLtN : 0 < n\nhLtK : 1 < k\nh : ¬(0 < k ∧ k ≤ n)\n⊢ ite (0 < k ∧ k ≤ n) ((n - k) / k + 1) 0 < n",
"tactic": "simp [hLtN, h]"
},
{
"state_after": "case isTrue\nn k : Nat\nhLtN : 0 < n\nhLtK : 1 < k\nh : 0 < k ∧ k ≤ n\n⊢ (n - k) / k + 1 < n",
"state_before": "case isTrue\nn k : Nat\nhLtN : 0 < n\nhLtK : 1 < k\nh : 0 < k ∧ k ≤ n\n⊢ ite (0 < k ∧ k ≤ n) ((n - k) / k + 1) 0 < n",
"tactic": "suffices (n - k) / k + 1 < n by simp [h, this]"
},
{
"state_after": "case isTrue\nn k : Nat\nhLtN : 0 < n\nhLtK : 1 < k\nh : 0 < k ∧ k ≤ n\nleft✝ : 0 < k\nhKN : k ≤ n\n⊢ (n - k) / k + 1 < n",
"state_before": "case isTrue\nn k : Nat\nhLtN : 0 < n\nhLtK : 1 < k\nh : 0 < k ∧ k ≤ n\n⊢ (n - k) / k + 1 < n",
"tactic": "have ⟨_, hKN⟩ := h"
},
{
"state_after": "case isTrue\nn k : Nat\nhLtN : 0 < n\nhLtK : 1 < k\nh : 0 < k ∧ k ≤ n\nleft✝ : 0 < k\nhKN : k ≤ n\nthis : (n - k) / k ≤ n - k\n⊢ (n - k) / k + 1 < n",
"state_before": "case isTrue\nn k : Nat\nhLtN : 0 < n\nhLtK : 1 < k\nh : 0 < k ∧ k ≤ n\nleft✝ : 0 < k\nhKN : k ≤ n\n⊢ (n - k) / k + 1 < n",
"tactic": "have : (n - k) / k ≤ n - k := div_le_self (n - k) k"
},
{
"state_after": "case isTrue\nn k : Nat\nhLtN : 0 < n\nhLtK : 1 < k\nh : 0 < k ∧ k ≤ n\nleft✝ : 0 < k\nhKN : k ≤ n\nthis✝ : (n - k) / k ≤ n - k\nthis : (n - k) / k + k ≤ n\n⊢ (n - k) / k + 1 < n",
"state_before": "case isTrue\nn k : Nat\nhLtN : 0 < n\nhLtK : 1 < k\nh : 0 < k ∧ k ≤ n\nleft✝ : 0 < k\nhKN : k ≤ n\nthis : (n - k) / k ≤ n - k\n⊢ (n - k) / k + 1 < n",
"tactic": "have := Nat.add_le_of_le_sub hKN this"
},
{
"state_after": "no goals",
"state_before": "case isTrue\nn k : Nat\nhLtN : 0 < n\nhLtK : 1 < k\nh : 0 < k ∧ k ≤ n\nleft✝ : 0 < k\nhKN : k ≤ n\nthis✝ : (n - k) / k ≤ n - k\nthis : (n - k) / k + k ≤ n\n⊢ (n - k) / k + 1 < n",
"tactic": "exact Nat.lt_of_lt_of_le (Nat.add_lt_add_left hLtK _) this"
},
{
"state_after": "no goals",
"state_before": "n k : Nat\nhLtN : 0 < n\nhLtK : 1 < k\nh : 0 < k ∧ k ≤ n\nthis : (n - k) / k + 1 < n\n⊢ ite (0 < k ∧ k ≤ n) ((n - k) / k + 1) 0 < n",
"tactic": "simp [h, this]"
}
] |
[
65,
63
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
56,
1
] |
Mathlib/Algebra/ContinuedFractions/Computation/Translations.lean
|
GeneralizedContinuedFraction.IntFractPair.stream_succ_of_int
|
[
{
"state_after": "case zero\nK : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\na : ℤ\n⊢ IntFractPair.stream (↑a) (Nat.zero + 1) = none\n\ncase succ\nK : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn✝ : ℕ\na : ℤ\nn : ℕ\nih : IntFractPair.stream (↑a) (n + 1) = none\n⊢ IntFractPair.stream (↑a) (Nat.succ n + 1) = none",
"state_before": "K : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn✝ : ℕ\na : ℤ\nn : ℕ\n⊢ IntFractPair.stream (↑a) (n + 1) = none",
"tactic": "induction' n with n ih"
},
{
"state_after": "case zero\nK : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\na : ℤ\n⊢ (IntFractPair.of ↑a).fr = 0",
"state_before": "case zero\nK : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\na : ℤ\n⊢ IntFractPair.stream (↑a) (Nat.zero + 1) = none",
"tactic": "refine' IntFractPair.stream_eq_none_of_fr_eq_zero (IntFractPair.stream_zero (a : K)) _"
},
{
"state_after": "no goals",
"state_before": "case zero\nK : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn : ℕ\na : ℤ\n⊢ (IntFractPair.of ↑a).fr = 0",
"tactic": "simp only [IntFractPair.of, Int.fract_intCast]"
},
{
"state_after": "no goals",
"state_before": "case succ\nK : Type u_1\ninst✝¹ : LinearOrderedField K\ninst✝ : FloorRing K\nv : K\nn✝ : ℕ\na : ℤ\nn : ℕ\nih : IntFractPair.stream (↑a) (n + 1) = none\n⊢ IntFractPair.stream (↑a) (Nat.succ n + 1) = none",
"tactic": "exact IntFractPair.succ_nth_stream_eq_none_iff.mpr (Or.inl ih)"
}
] |
[
112,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
1
] |
Mathlib/RingTheory/Localization/FractionRing.lean
|
IsFractionRing.coe_inj
|
[] |
[
97,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
96,
1
] |
Mathlib/Order/WellFoundedSet.lean
|
Set.IsWf.min_union
|
[
{
"state_after": "ι : Type ?u.131877\nα : Type u_1\nβ : Type ?u.131883\ninst✝ : LinearOrder α\ns t : Set α\na : α\nhs : IsWf s\nhsn : Set.Nonempty s\nht : IsWf t\nhtn : Set.Nonempty t\n⊢ Min.min (min hs hsn) (min ht htn) ≤ min (_ : IsWf (s ∪ t)) (_ : Set.Nonempty (s ∪ t))",
"state_before": "ι : Type ?u.131877\nα : Type u_1\nβ : Type ?u.131883\ninst✝ : LinearOrder α\ns t : Set α\na : α\nhs : IsWf s\nhsn : Set.Nonempty s\nht : IsWf t\nhtn : Set.Nonempty t\n⊢ min (_ : IsWf (s ∪ t)) (_ : Set.Nonempty (s ∪ t)) = Min.min (min hs hsn) (min ht htn)",
"tactic": "refine' le_antisymm (le_min (IsWf.min_le_min_of_subset (subset_union_left _ _))\n (IsWf.min_le_min_of_subset (subset_union_right _ _))) _"
},
{
"state_after": "ι : Type ?u.131877\nα : Type u_1\nβ : Type ?u.131883\ninst✝ : LinearOrder α\ns t : Set α\na : α\nhs : IsWf s\nhsn : Set.Nonempty s\nht : IsWf t\nhtn : Set.Nonempty t\n⊢ min hs hsn ≤ min (_ : IsWf (s ∪ t)) (_ : Set.Nonempty (s ∪ t)) ∨\n min ht htn ≤ min (_ : IsWf (s ∪ t)) (_ : Set.Nonempty (s ∪ t))",
"state_before": "ι : Type ?u.131877\nα : Type u_1\nβ : Type ?u.131883\ninst✝ : LinearOrder α\ns t : Set α\na : α\nhs : IsWf s\nhsn : Set.Nonempty s\nht : IsWf t\nhtn : Set.Nonempty t\n⊢ Min.min (min hs hsn) (min ht htn) ≤ min (_ : IsWf (s ∪ t)) (_ : Set.Nonempty (s ∪ t))",
"tactic": "rw [min_le_iff]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.131877\nα : Type u_1\nβ : Type ?u.131883\ninst✝ : LinearOrder α\ns t : Set α\na : α\nhs : IsWf s\nhsn : Set.Nonempty s\nht : IsWf t\nhtn : Set.Nonempty t\n⊢ min hs hsn ≤ min (_ : IsWf (s ∪ t)) (_ : Set.Nonempty (s ∪ t)) ∨\n min ht htn ≤ min (_ : IsWf (s ∪ t)) (_ : Set.Nonempty (s ∪ t))",
"tactic": "exact ((mem_union _ _ _).1 ((hs.union ht).min_mem (union_nonempty.2 (.inl hsn)))).imp\n (hs.min_le _) (ht.min_le _)"
}
] |
[
662,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
655,
1
] |
Mathlib/LinearAlgebra/LinearIndependent.lean
|
LinearIndependent.disjoint_span_image
|
[
{
"state_after": "ι : Type u'\nι' : Type ?u.335804\nR : Type u_1\nK : Type ?u.335810\nM : Type u_2\nM' : Type ?u.335816\nM'' : Type ?u.335819\nV : Type u\nV' : Type ?u.335824\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns t : Set ι\nhs : Disjoint s t\n⊢ ∀ (x : M),\n (∃ l, l ∈ Finsupp.supported R R s ∧ ↑(Finsupp.total ι M R v) l = x) →\n (∃ l, l ∈ Finsupp.supported R R t ∧ ↑(Finsupp.total ι M R v) l = x) → x = 0",
"state_before": "ι : Type u'\nι' : Type ?u.335804\nR : Type u_1\nK : Type ?u.335810\nM : Type u_2\nM' : Type ?u.335816\nM'' : Type ?u.335819\nV : Type u\nV' : Type ?u.335824\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns t : Set ι\nhs : Disjoint s t\n⊢ Disjoint (span R (v '' s)) (span R (v '' t))",
"tactic": "simp only [disjoint_def, Finsupp.mem_span_image_iff_total]"
},
{
"state_after": "case intro.intro.intro.intro\nι : Type u'\nι' : Type ?u.335804\nR : Type u_1\nK : Type ?u.335810\nM : Type u_2\nM' : Type ?u.335816\nM'' : Type ?u.335819\nV : Type u\nV' : Type ?u.335824\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns t : Set ι\nhs : Disjoint s t\nl₁ : ι →₀ R\nhl₁ : l₁ ∈ Finsupp.supported R R s\nl₂ : ι →₀ R\nhl₂ : l₂ ∈ Finsupp.supported R R t\nH : ↑(Finsupp.total ι M R v) l₂ = ↑(Finsupp.total ι M R v) l₁\n⊢ ↑(Finsupp.total ι M R v) l₁ = 0",
"state_before": "ι : Type u'\nι' : Type ?u.335804\nR : Type u_1\nK : Type ?u.335810\nM : Type u_2\nM' : Type ?u.335816\nM'' : Type ?u.335819\nV : Type u\nV' : Type ?u.335824\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns t : Set ι\nhs : Disjoint s t\n⊢ ∀ (x : M),\n (∃ l, l ∈ Finsupp.supported R R s ∧ ↑(Finsupp.total ι M R v) l = x) →\n (∃ l, l ∈ Finsupp.supported R R t ∧ ↑(Finsupp.total ι M R v) l = x) → x = 0",
"tactic": "rintro _ ⟨l₁, hl₁, rfl⟩ ⟨l₂, hl₂, H⟩"
},
{
"state_after": "case intro.intro.intro.intro\nι : Type u'\nι' : Type ?u.335804\nR : Type u_1\nK : Type ?u.335810\nM : Type u_2\nM' : Type ?u.335816\nM'' : Type ?u.335819\nV : Type u\nV' : Type ?u.335824\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns t : Set ι\nhs : Disjoint s t\nl₁ : ι →₀ R\nhl₁ : l₁ ∈ Finsupp.supported R R s\nl₂ : ι →₀ R\nhl₂ : l₂ ∈ Finsupp.supported R R t\nH : l₂ = l₁\n⊢ ↑(Finsupp.total ι M R v) l₁ = 0",
"state_before": "case intro.intro.intro.intro\nι : Type u'\nι' : Type ?u.335804\nR : Type u_1\nK : Type ?u.335810\nM : Type u_2\nM' : Type ?u.335816\nM'' : Type ?u.335819\nV : Type u\nV' : Type ?u.335824\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns t : Set ι\nhs : Disjoint s t\nl₁ : ι →₀ R\nhl₁ : l₁ ∈ Finsupp.supported R R s\nl₂ : ι →₀ R\nhl₂ : l₂ ∈ Finsupp.supported R R t\nH : ↑(Finsupp.total ι M R v) l₂ = ↑(Finsupp.total ι M R v) l₁\n⊢ ↑(Finsupp.total ι M R v) l₁ = 0",
"tactic": "rw [hv.injective_total.eq_iff] at H"
},
{
"state_after": "case intro.intro.intro.intro\nι : Type u'\nι' : Type ?u.335804\nR : Type u_1\nK : Type ?u.335810\nM : Type u_2\nM' : Type ?u.335816\nM'' : Type ?u.335819\nV : Type u\nV' : Type ?u.335824\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns t : Set ι\nhs : Disjoint s t\nl₁ : ι →₀ R\nhl₁ : l₁ ∈ Finsupp.supported R R s\nhl₂ : l₁ ∈ Finsupp.supported R R t\n⊢ ↑(Finsupp.total ι M R v) l₁ = 0",
"state_before": "case intro.intro.intro.intro\nι : Type u'\nι' : Type ?u.335804\nR : Type u_1\nK : Type ?u.335810\nM : Type u_2\nM' : Type ?u.335816\nM'' : Type ?u.335819\nV : Type u\nV' : Type ?u.335824\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns t : Set ι\nhs : Disjoint s t\nl₁ : ι →₀ R\nhl₁ : l₁ ∈ Finsupp.supported R R s\nl₂ : ι →₀ R\nhl₂ : l₂ ∈ Finsupp.supported R R t\nH : l₂ = l₁\n⊢ ↑(Finsupp.total ι M R v) l₁ = 0",
"tactic": "subst l₂"
},
{
"state_after": "case intro.intro.intro.intro\nι : Type u'\nι' : Type ?u.335804\nR : Type u_1\nK : Type ?u.335810\nM : Type u_2\nM' : Type ?u.335816\nM'' : Type ?u.335819\nV : Type u\nV' : Type ?u.335824\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns t : Set ι\nhs : Disjoint s t\nl₁ : ι →₀ R\nhl₁ : l₁ ∈ Finsupp.supported R R s\nhl₂ : l₁ ∈ Finsupp.supported R R t\nthis : l₁ = 0\n⊢ ↑(Finsupp.total ι M R v) l₁ = 0",
"state_before": "case intro.intro.intro.intro\nι : Type u'\nι' : Type ?u.335804\nR : Type u_1\nK : Type ?u.335810\nM : Type u_2\nM' : Type ?u.335816\nM'' : Type ?u.335819\nV : Type u\nV' : Type ?u.335824\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns t : Set ι\nhs : Disjoint s t\nl₁ : ι →₀ R\nhl₁ : l₁ ∈ Finsupp.supported R R s\nhl₂ : l₁ ∈ Finsupp.supported R R t\n⊢ ↑(Finsupp.total ι M R v) l₁ = 0",
"tactic": "have : l₁ = 0 := Submodule.disjoint_def.mp (Finsupp.disjoint_supported_supported hs) _ hl₁ hl₂"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro\nι : Type u'\nι' : Type ?u.335804\nR : Type u_1\nK : Type ?u.335810\nM : Type u_2\nM' : Type ?u.335816\nM'' : Type ?u.335819\nV : Type u\nV' : Type ?u.335824\nv : ι → M\ninst✝⁶ : Ring R\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : AddCommGroup M'\ninst✝³ : AddCommGroup M''\ninst✝² : Module R M\ninst✝¹ : Module R M'\ninst✝ : Module R M''\na b : R\nx y : M\nhv : LinearIndependent R v\ns t : Set ι\nhs : Disjoint s t\nl₁ : ι →₀ R\nhl₁ : l₁ ∈ Finsupp.supported R R s\nhl₂ : l₁ ∈ Finsupp.supported R R t\nthis : l₁ = 0\n⊢ ↑(Finsupp.total ι M R v) l₁ = 0",
"tactic": "simp [this]"
}
] |
[
617,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
611,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
RingHom.is_integral_map
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nS : Type u_2\nA : Type ?u.6854\ninst✝³ : CommRing R\ninst✝² : Ring A\ninst✝¹ : Ring S\nf : R →+* S\ninst✝ : Algebra R A\nx : R\n⊢ eval₂ f (↑f x) (X - ↑C x) = 0",
"tactic": "simp"
}
] |
[
78,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
77,
1
] |
Mathlib/Data/Finset/Basic.lean
|
Finset.mem_of_mem_insert_of_ne
|
[] |
[
1073,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1072,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.isLeast_min'
|
[] |
[
1337,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1336,
1
] |
Mathlib/Data/Finset/LocallyFinite.lean
|
Finset.Ioo_eq_empty_iff
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.2143\nα : Type u_1\ninst✝² : Preorder α\ninst✝¹ : LocallyFiniteOrder α\na a₁ a₂ b b₁ b₂ c x : α\ninst✝ : DenselyOrdered α\n⊢ Ioo a b = ∅ ↔ ¬a < b",
"tactic": "rw [← coe_eq_empty, coe_Ioo, Set.Ioo_eq_empty_iff]"
}
] |
[
85,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
84,
1
] |
Mathlib/Dynamics/PeriodicPts.lean
|
Function.mem_ptsOfPeriod
|
[] |
[
201,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
200,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.lf_of_le_moveLeft
|
[] |
[
486,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
485,
1
] |
Mathlib/Analysis/NormedSpace/Dual.lean
|
NormedSpace.polar_closure
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\n⊢ IsClosed\n ((LinearMap.polar (LinearMap.flip (LinearMap.flip (dualPairing 𝕜 E))) ∘ ↑OrderDual.ofDual)\n (LinearMap.polar (LinearMap.flip (dualPairing 𝕜 E)) s))",
"tactic": "simpa [LinearMap.flip_flip] using\n (isClosed_polar _ _).preimage (inclusionInDoubleDual 𝕜 E).continuous"
}
] |
[
218,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
213,
1
] |
Mathlib/Analysis/Convex/Segment.lean
|
openSegment_subset_segment
|
[] |
[
92,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Affine.lean
|
EuclideanGeometry.angle_ne_zero_of_not_collinear
|
[] |
[
459,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
457,
1
] |
Mathlib/Algebra/Symmetrized.lean
|
SymAlg.unsym_injective
|
[] |
[
111,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
110,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Type.lean
|
Equiv.Perm.two_dvd_card_support
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Fintype α\ninst✝ : DecidableEq α\nσ : Perm α\nhσ : σ ^ 2 = 1\nn : ℕ\nhn : n ∈ cycleType σ\n⊢ 2 ∣ n",
"tactic": "rw [le_antisymm\n (Nat.le_of_dvd zero_lt_two <|\n (dvd_of_mem_cycleType hn).trans <| orderOf_dvd_of_pow_eq_one hσ)\n (two_le_of_mem_cycleType hn)]"
}
] |
[
203,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
197,
1
] |
Mathlib/Order/Antisymmetrization.lean
|
AntisymmRel.symm
|
[] |
[
59,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Mathlib/CategoryTheory/Subobject/Lattice.lean
|
CategoryTheory.Subobject.sSup_le
|
[
{
"state_after": "case f\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\n⊢ underlying.obj (sSup s) ⟶ underlying.obj f\n\ncase w\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\n⊢ ?f ≫ arrow f = arrow (sSup s)",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\n⊢ sSup s ≤ f",
"tactic": "fapply le_of_comm"
},
{
"state_after": "no goals",
"state_before": "case w\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\n⊢ ((underlyingIso (image.ι (smallCoproductDesc s))).hom ≫\n image.lift\n (MonoFactorisation.mk (underlying.obj f) (arrow f)\n (Sigma.desc fun b =>\n Subtype.casesOn b fun g m =>\n underlying.map (homOfLE (_ : ↑(equivShrink (Subobject A)).symm ↑{ val := g, property := m } ≤ f))))) ≫\n arrow f =\n arrow (sSup s)",
"tactic": ". dsimp [sSup]\n rw [assoc, image.lift_fac, underlyingIso_hom_comp_eq_mk]"
},
{
"state_after": "case f.refine'_1\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\n⊢ (∐ fun j => underlying.obj (↑(equivShrink (Subobject A)).symm ↑j)) ⟶ underlying.obj f\n\ncase f.refine'_2\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\n⊢ ?f.refine'_1 ≫ arrow f = smallCoproductDesc s",
"state_before": "case f\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\n⊢ underlying.obj (sSup s) ⟶ underlying.obj f",
"tactic": "refine'(underlyingIso _).hom ≫ image.lift ⟨_, f.arrow, _, _⟩"
},
{
"state_after": "case f.refine'_1\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\n⊢ (b : ↑(↑(equivShrink (Subobject A)) '' s)) → underlying.obj (↑(equivShrink (Subobject A)).symm ↑b) ⟶ underlying.obj f",
"state_before": "case f.refine'_1\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\n⊢ (∐ fun j => underlying.obj (↑(equivShrink (Subobject A)).symm ↑j)) ⟶ underlying.obj f",
"tactic": "refine' Sigma.desc _"
},
{
"state_after": "case f.refine'_1.mk\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\ng : Shrink (Subobject A)\nm : g ∈ ↑(equivShrink (Subobject A)) '' s\n⊢ underlying.obj (↑(equivShrink (Subobject A)).symm ↑{ val := g, property := m }) ⟶ underlying.obj f",
"state_before": "case f.refine'_1\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\n⊢ (b : ↑(↑(equivShrink (Subobject A)) '' s)) → underlying.obj (↑(equivShrink (Subobject A)).symm ↑b) ⟶ underlying.obj f",
"tactic": "rintro ⟨g, m⟩"
},
{
"state_after": "case f.refine'_1.mk\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\ng : Shrink (Subobject A)\nm : g ∈ ↑(equivShrink (Subobject A)) '' s\n⊢ ↑(equivShrink (Subobject A)).symm ↑{ val := g, property := m } ∈ s",
"state_before": "case f.refine'_1.mk\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\ng : Shrink (Subobject A)\nm : g ∈ ↑(equivShrink (Subobject A)) '' s\n⊢ underlying.obj (↑(equivShrink (Subobject A)).symm ↑{ val := g, property := m }) ⟶ underlying.obj f",
"tactic": "refine' underlying.map (homOfLE (k _ _))"
},
{
"state_after": "no goals",
"state_before": "case f.refine'_1.mk\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\ng : Shrink (Subobject A)\nm : g ∈ ↑(equivShrink (Subobject A)) '' s\n⊢ ↑(equivShrink (Subobject A)).symm ↑{ val := g, property := m } ∈ s",
"tactic": "simpa [symm_apply_mem_iff_mem_image] using m"
},
{
"state_after": "case f.refine'_2.h\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\nb✝ : ↑(↑(equivShrink (Subobject A)) '' s)\n⊢ Sigma.ι (fun j => underlying.obj (↑(equivShrink (Subobject A)).symm ↑j)) b✝ ≫\n (Sigma.desc fun b =>\n Subtype.casesOn b fun g m =>\n underlying.map (homOfLE (_ : ↑(equivShrink (Subobject A)).symm ↑{ val := g, property := m } ≤ f))) ≫\n arrow f =\n Sigma.ι (fun j => underlying.obj (↑(equivShrink (Subobject A)).symm ↑j)) b✝ ≫ smallCoproductDesc s",
"state_before": "case f.refine'_2\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\n⊢ (Sigma.desc fun b =>\n Subtype.casesOn b fun g m =>\n underlying.map (homOfLE (_ : ↑(equivShrink (Subobject A)).symm ↑{ val := g, property := m } ≤ f))) ≫\n arrow f =\n smallCoproductDesc s",
"tactic": "ext"
},
{
"state_after": "case f.refine'_2.h\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\nb✝ : ↑(↑(equivShrink (Subobject A)) '' s)\n⊢ Sigma.ι (fun j => underlying.obj (↑(equivShrink (Subobject A)).symm ↑j)) b✝ ≫\n (Sigma.desc fun b => underlying.map (homOfLE (_ : ↑(equivShrink (Subobject A)).symm ↑b ≤ f))) ≫ arrow f =\n Sigma.ι (fun j => underlying.obj (↑(equivShrink (Subobject A)).symm ↑j)) b✝ ≫\n Sigma.desc fun j => arrow (↑(equivShrink (Subobject A)).symm ↑j)",
"state_before": "case f.refine'_2.h\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\nb✝ : ↑(↑(equivShrink (Subobject A)) '' s)\n⊢ Sigma.ι (fun j => underlying.obj (↑(equivShrink (Subobject A)).symm ↑j)) b✝ ≫\n (Sigma.desc fun b =>\n Subtype.casesOn b fun g m =>\n underlying.map (homOfLE (_ : ↑(equivShrink (Subobject A)).symm ↑{ val := g, property := m } ≤ f))) ≫\n arrow f =\n Sigma.ι (fun j => underlying.obj (↑(equivShrink (Subobject A)).symm ↑j)) b✝ ≫ smallCoproductDesc s",
"tactic": "dsimp [smallCoproductDesc]"
},
{
"state_after": "no goals",
"state_before": "case f.refine'_2.h\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\nb✝ : ↑(↑(equivShrink (Subobject A)) '' s)\n⊢ Sigma.ι (fun j => underlying.obj (↑(equivShrink (Subobject A)).symm ↑j)) b✝ ≫\n (Sigma.desc fun b => underlying.map (homOfLE (_ : ↑(equivShrink (Subobject A)).symm ↑b ≤ f))) ≫ arrow f =\n Sigma.ι (fun j => underlying.obj (↑(equivShrink (Subobject A)).symm ↑j)) b✝ ≫\n Sigma.desc fun j => arrow (↑(equivShrink (Subobject A)).symm ↑j)",
"tactic": "simp"
},
{
"state_after": "case w\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\n⊢ ((underlyingIso (image.ι (smallCoproductDesc s))).hom ≫\n image.lift\n (MonoFactorisation.mk (underlying.obj f) (arrow f)\n (Sigma.desc fun b => underlying.map (homOfLE (_ : ↑(equivShrink (Subobject A)).symm ↑b ≤ f))))) ≫\n arrow f =\n arrow (mk (image.ι (smallCoproductDesc s)))",
"state_before": "case w\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\n⊢ ((underlyingIso (image.ι (smallCoproductDesc s))).hom ≫\n image.lift\n (MonoFactorisation.mk (underlying.obj f) (arrow f)\n (Sigma.desc fun b =>\n Subtype.casesOn b fun g m =>\n underlying.map (homOfLE (_ : ↑(equivShrink (Subobject A)).symm ↑{ val := g, property := m } ≤ f))))) ≫\n arrow f =\n arrow (sSup s)",
"tactic": "dsimp [sSup]"
},
{
"state_after": "no goals",
"state_before": "case w\nC : Type u₁\ninst✝⁴ : Category C\nX Y Z : C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasCoproducts C\ninst✝ : HasImages C\nA : C\ns : Set (Subobject A)\nf : Subobject A\nk : ∀ (g : Subobject A), g ∈ s → g ≤ f\n⊢ ((underlyingIso (image.ι (smallCoproductDesc s))).hom ≫\n image.lift\n (MonoFactorisation.mk (underlying.obj f) (arrow f)\n (Sigma.desc fun b => underlying.map (homOfLE (_ : ↑(equivShrink (Subobject A)).symm ↑b ≤ f))))) ≫\n arrow f =\n arrow (mk (image.ι (smallCoproductDesc s)))",
"tactic": "rw [assoc, image.lift_fac, underlyingIso_hom_comp_eq_mk]"
}
] |
[
715,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
703,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
FormalMultilinearSeries.norm_changeOriginSeriesTerm
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_3\nE : Type u_1\nF : Type u_2\nG : Type ?u.1191203\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 E F\nx y : E\nr R : ℝ≥0\nk l : ℕ\ns : Finset (Fin (k + l))\nhs : Finset.card s = l\n⊢ ‖changeOriginSeriesTerm p k l s hs‖ = ‖p (k + l)‖",
"tactic": "simp only [changeOriginSeriesTerm, LinearIsometryEquiv.norm_map]"
}
] |
[
1116,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1114,
1
] |
Mathlib/Computability/Primrec.lean
|
Nat.Primrec.casesOn1
|
[
{
"state_after": "no goals",
"state_before": "f : ℕ → ℕ\nm : ℕ\nhf : Nat.Primrec f\n⊢ ∀ (n : ℕ), Nat.rec m (fun y IH => f (unpair (Nat.pair y IH)).fst) n = Nat.casesOn n m f",
"tactic": "simp"
}
] |
[
112,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
111,
1
] |
Mathlib/Topology/Algebra/Polynomial.lean
|
Polynomial.tendsto_abv_eval₂_atTop
|
[
{
"state_after": "R : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\np : R[X]\nhd : 0 < degree p\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\n⊢ ↑f (leadingCoeff p) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) p)) l atTop",
"state_before": "R : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\np : R[X]\nhd : 0 < degree p\nhf : ↑f (leadingCoeff p) ≠ 0\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\n⊢ Tendsto (fun x => abv (eval₂ f (z x) p)) l atTop",
"tactic": "revert hf"
},
{
"state_after": "case refine'_1\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\n⊢ ∀ {a : R}, a ≠ 0 → ↑f (leadingCoeff (↑C a * X)) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) (↑C a * X))) l atTop\n\ncase refine'_2\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\n⊢ ∀ {p : R[X]},\n 0 < degree p →\n (↑f (leadingCoeff p) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) p)) l atTop) →\n ↑f (leadingCoeff (p * X)) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) (p * X))) l atTop\n\ncase refine'_3\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\n⊢ ∀ {p : R[X]} {a : R},\n 0 < degree p →\n (↑f (leadingCoeff p) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) p)) l atTop) →\n ↑f (leadingCoeff (p + ↑C a)) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) (p + ↑C a))) l atTop",
"state_before": "R : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\np : R[X]\nhd : 0 < degree p\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\n⊢ ↑f (leadingCoeff p) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) p)) l atTop",
"tactic": "refine' degree_pos_induction_on p hd _ _ _ <;> clear hd p"
},
{
"state_after": "case refine'_1\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\na✝ : R\nhc : ↑f (leadingCoeff (↑C a✝ * X)) ≠ 0\n⊢ Tendsto (fun x => abv (eval₂ f (z x) (↑C a✝ * X))) l atTop",
"state_before": "case refine'_1\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\n⊢ ∀ {a : R}, a ≠ 0 → ↑f (leadingCoeff (↑C a * X)) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) (↑C a * X))) l atTop",
"tactic": "rintro _ - hc"
},
{
"state_after": "case refine'_1\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\na✝ : R\nhc : ↑f a✝ ≠ 0\n⊢ Tendsto (fun x => abv (eval₂ f (z x) (↑C a✝ * X))) l atTop",
"state_before": "case refine'_1\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\na✝ : R\nhc : ↑f (leadingCoeff (↑C a✝ * X)) ≠ 0\n⊢ Tendsto (fun x => abv (eval₂ f (z x) (↑C a✝ * X))) l atTop",
"tactic": "rw [leadingCoeff_mul_X, leadingCoeff_C] at hc"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\na✝ : R\nhc : ↑f a✝ ≠ 0\n⊢ Tendsto (fun x => abv (eval₂ f (z x) (↑C a✝ * X))) l atTop",
"tactic": "simpa [abv_mul abv] using hz.const_mul_atTop ((abv_pos abv).2 hc)"
},
{
"state_after": "case refine'_2\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\np✝ : R[X]\na✝ : 0 < degree p✝\nihp : ↑f (leadingCoeff p✝) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) p✝)) l atTop\nhf : ↑f (leadingCoeff (p✝ * X)) ≠ 0\n⊢ Tendsto (fun x => abv (eval₂ f (z x) (p✝ * X))) l atTop",
"state_before": "case refine'_2\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\n⊢ ∀ {p : R[X]},\n 0 < degree p →\n (↑f (leadingCoeff p) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) p)) l atTop) →\n ↑f (leadingCoeff (p * X)) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) (p * X))) l atTop",
"tactic": "intro _ _ ihp hf"
},
{
"state_after": "case refine'_2\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\np✝ : R[X]\na✝ : 0 < degree p✝\nihp : ↑f (leadingCoeff p✝) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) p✝)) l atTop\nhf : ↑f (leadingCoeff p✝) ≠ 0\n⊢ Tendsto (fun x => abv (eval₂ f (z x) (p✝ * X))) l atTop",
"state_before": "case refine'_2\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\np✝ : R[X]\na✝ : 0 < degree p✝\nihp : ↑f (leadingCoeff p✝) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) p✝)) l atTop\nhf : ↑f (leadingCoeff (p✝ * X)) ≠ 0\n⊢ Tendsto (fun x => abv (eval₂ f (z x) (p✝ * X))) l atTop",
"tactic": "rw [leadingCoeff_mul_X] at hf"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\np✝ : R[X]\na✝ : 0 < degree p✝\nihp : ↑f (leadingCoeff p✝) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) p✝)) l atTop\nhf : ↑f (leadingCoeff p✝) ≠ 0\n⊢ Tendsto (fun x => abv (eval₂ f (z x) (p✝ * X))) l atTop",
"tactic": "simpa [abv_mul abv] using (ihp hf).atTop_mul_atTop hz"
},
{
"state_after": "case refine'_3\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\np✝ : R[X]\na : R\nhd : 0 < degree p✝\nihp : ↑f (leadingCoeff p✝) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) p✝)) l atTop\nhf : ↑f (leadingCoeff (p✝ + ↑C a)) ≠ 0\n⊢ Tendsto (fun x => abv (eval₂ f (z x) (p✝ + ↑C a))) l atTop",
"state_before": "case refine'_3\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\n⊢ ∀ {p : R[X]} {a : R},\n 0 < degree p →\n (↑f (leadingCoeff p) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) p)) l atTop) →\n ↑f (leadingCoeff (p + ↑C a)) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) (p + ↑C a))) l atTop",
"tactic": "intro _ a hd ihp hf"
},
{
"state_after": "case refine'_3\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\np✝ : R[X]\na : R\nhd : 0 < degree p✝\nihp : ↑f (leadingCoeff p✝) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) p✝)) l atTop\nhf : ↑f (leadingCoeff p✝) ≠ 0\n⊢ Tendsto (fun x => abv (eval₂ f (z x) (p✝ + ↑C a))) l atTop",
"state_before": "case refine'_3\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\np✝ : R[X]\na : R\nhd : 0 < degree p✝\nihp : ↑f (leadingCoeff p✝) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) p✝)) l atTop\nhf : ↑f (leadingCoeff (p✝ + ↑C a)) ≠ 0\n⊢ Tendsto (fun x => abv (eval₂ f (z x) (p✝ + ↑C a))) l atTop",
"tactic": "rw [add_comm, leadingCoeff_add_of_degree_lt (degree_C_le.trans_lt hd)] at hf"
},
{
"state_after": "case refine'_3\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\np✝ : R[X]\na : R\nhd : 0 < degree p✝\nihp : ↑f (leadingCoeff p✝) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) p✝)) l atTop\nhf : ↑f (leadingCoeff p✝) ≠ 0\n⊢ Tendsto (fun x => abv (eval₂ f (z x) (p✝ + ↑C a)) + abv (-↑f a)) l atTop",
"state_before": "case refine'_3\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\np✝ : R[X]\na : R\nhd : 0 < degree p✝\nihp : ↑f (leadingCoeff p✝) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) p✝)) l atTop\nhf : ↑f (leadingCoeff p✝) ≠ 0\n⊢ Tendsto (fun x => abv (eval₂ f (z x) (p✝ + ↑C a))) l atTop",
"tactic": "refine' tendsto_atTop_of_add_const_right (abv (-f a)) _"
},
{
"state_after": "case refine'_3\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\np✝ : R[X]\na : R\nhd : 0 < degree p✝\nihp : ↑f (leadingCoeff p✝) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) p✝)) l atTop\nhf : ↑f (leadingCoeff p✝) ≠ 0\n⊢ Tendsto (fun x => abv (eval₂ f (z x) (p✝ + ↑C a) + -↑f a)) l atTop",
"state_before": "case refine'_3\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\np✝ : R[X]\na : R\nhd : 0 < degree p✝\nihp : ↑f (leadingCoeff p✝) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) p✝)) l atTop\nhf : ↑f (leadingCoeff p✝) ≠ 0\n⊢ Tendsto (fun x => abv (eval₂ f (z x) (p✝ + ↑C a)) + abv (-↑f a)) l atTop",
"tactic": "refine' tendsto_atTop_mono (fun _ => abv_add abv _ _) _"
},
{
"state_after": "no goals",
"state_before": "case refine'_3\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝³ : Semiring R\ninst✝² : Ring S\ninst✝¹ : LinearOrderedField k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\np✝ : R[X]\na : R\nhd : 0 < degree p✝\nihp : ↑f (leadingCoeff p✝) ≠ 0 → Tendsto (fun x => abv (eval₂ f (z x) p✝)) l atTop\nhf : ↑f (leadingCoeff p✝) ≠ 0\n⊢ Tendsto (fun x => abv (eval₂ f (z x) (p✝ + ↑C a) + -↑f a)) l atTop",
"tactic": "simpa using ihp hf"
}
] |
[
117,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
102,
1
] |
Mathlib/Topology/LocallyConstant/Basic.lean
|
IsLocallyConstant.isClopen_fiber
|
[] |
[
83,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.tan_sq_div_one_add_tan_sq
|
[
{
"state_after": "no goals",
"state_before": "x✝ y x : ℂ\nhx : cos x ≠ 0\n⊢ tan x ^ 2 / (1 + tan x ^ 2) = sin x ^ 2",
"tactic": "simp only [← tan_mul_cos hx, mul_pow, ← inv_one_add_tan_sq hx, div_eq_mul_inv, one_mul]"
}
] |
[
1059,
90
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1057,
1
] |
Mathlib/MeasureTheory/Group/Arithmetic.lean
|
measurable_const_smul_iff₀
|
[] |
[
761,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
759,
1
] |
Mathlib/SetTheory/Ordinal/NaturalOps.lean
|
Ordinal.nat_nadd
|
[
{
"state_after": "no goals",
"state_before": "a b c : Ordinal\nn : ℕ\n⊢ ↑n ♯ a = a + ↑n",
"tactic": "rw [nadd_comm, nadd_nat]"
}
] |
[
323,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
323,
1
] |
Mathlib/Order/GaloisConnection.lean
|
GaloisConnection.isGreatest_u
|
[] |
[
147,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
146,
1
] |
Mathlib/Data/Rat/NNRat.lean
|
NNRat.coe_nonneg
|
[] |
[
115,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
114,
1
] |
Mathlib/RingTheory/Ideal/Cotangent.lean
|
AlgHom.ker_ker_sqare_lift
|
[
{
"state_after": "case a\nR : Type u\nS : Type v\nS' : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : CommSemiring S'\ninst✝⁶ : Algebra S' R\ninst✝⁵ : Algebra S S'\ninst✝⁴ : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\n⊢ RingHom.ker ↑(AlgHom.kerSquareLift f) ≤ cotangentIdeal (RingHom.ker ↑f)\n\ncase a\nR : Type u\nS : Type v\nS' : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : CommSemiring S'\ninst✝⁶ : Algebra S' R\ninst✝⁵ : Algebra S S'\ninst✝⁴ : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\n⊢ cotangentIdeal (RingHom.ker ↑f) ≤ RingHom.ker ↑(AlgHom.kerSquareLift f)",
"state_before": "R : Type u\nS : Type v\nS' : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : CommSemiring S'\ninst✝⁶ : Algebra S' R\ninst✝⁵ : Algebra S S'\ninst✝⁴ : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\n⊢ RingHom.ker ↑(AlgHom.kerSquareLift f) = cotangentIdeal (RingHom.ker ↑f)",
"tactic": "apply le_antisymm"
},
{
"state_after": "case a\nR : Type u\nS : Type v\nS' : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : CommSemiring S'\ninst✝⁶ : Algebra S' R\ninst✝⁵ : Algebra S S'\ninst✝⁴ : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nx : A ⧸ RingHom.ker ↑f ^ 2\nhx : x ∈ RingHom.ker ↑(AlgHom.kerSquareLift f)\n⊢ x ∈ cotangentIdeal (RingHom.ker ↑f)",
"state_before": "case a\nR : Type u\nS : Type v\nS' : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : CommSemiring S'\ninst✝⁶ : Algebra S' R\ninst✝⁵ : Algebra S S'\ninst✝⁴ : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\n⊢ RingHom.ker ↑(AlgHom.kerSquareLift f) ≤ cotangentIdeal (RingHom.ker ↑f)",
"tactic": "intro x hx"
},
{
"state_after": "case a.intro\nR : Type u\nS : Type v\nS' : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : CommSemiring S'\ninst✝⁶ : Algebra S' R\ninst✝⁵ : Algebra S S'\ninst✝⁴ : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nx : A\nhx : ↑(Quotient.mk (RingHom.ker ↑f ^ 2)) x ∈ RingHom.ker ↑(AlgHom.kerSquareLift f)\n⊢ ↑(Quotient.mk (RingHom.ker ↑f ^ 2)) x ∈ cotangentIdeal (RingHom.ker ↑f)",
"state_before": "case a\nR : Type u\nS : Type v\nS' : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : CommSemiring S'\ninst✝⁶ : Algebra S' R\ninst✝⁵ : Algebra S S'\ninst✝⁴ : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nx : A ⧸ RingHom.ker ↑f ^ 2\nhx : x ∈ RingHom.ker ↑(AlgHom.kerSquareLift f)\n⊢ x ∈ cotangentIdeal (RingHom.ker ↑f)",
"tactic": "obtain ⟨x, rfl⟩ := Ideal.Quotient.mk_surjective x"
},
{
"state_after": "no goals",
"state_before": "case a.intro\nR : Type u\nS : Type v\nS' : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : CommSemiring S'\ninst✝⁶ : Algebra S' R\ninst✝⁵ : Algebra S S'\ninst✝⁴ : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nx : A\nhx : ↑(Quotient.mk (RingHom.ker ↑f ^ 2)) x ∈ RingHom.ker ↑(AlgHom.kerSquareLift f)\n⊢ ↑(Quotient.mk (RingHom.ker ↑f ^ 2)) x ∈ cotangentIdeal (RingHom.ker ↑f)",
"tactic": "exact ⟨x, hx, rfl⟩"
},
{
"state_after": "case a.intro.intro\nR : Type u\nS : Type v\nS' : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : CommSemiring S'\ninst✝⁶ : Algebra S' R\ninst✝⁵ : Algebra S S'\ninst✝⁴ : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nx : A\nhx : x ∈ ↑(RingHom.ker ↑f)\n⊢ ↑(RingHom.toSemilinearMap (Quotient.mk (RingHom.ker ↑f ^ 2))) x ∈ RingHom.ker ↑(AlgHom.kerSquareLift f)",
"state_before": "case a\nR : Type u\nS : Type v\nS' : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : CommSemiring S'\ninst✝⁶ : Algebra S' R\ninst✝⁵ : Algebra S S'\ninst✝⁴ : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\n⊢ cotangentIdeal (RingHom.ker ↑f) ≤ RingHom.ker ↑(AlgHom.kerSquareLift f)",
"tactic": "rintro _ ⟨x, hx, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case a.intro.intro\nR : Type u\nS : Type v\nS' : Type w\ninst✝¹⁰ : CommRing R\ninst✝⁹ : CommSemiring S\ninst✝⁸ : Algebra S R\ninst✝⁷ : CommSemiring S'\ninst✝⁶ : Algebra S' R\ninst✝⁵ : Algebra S S'\ninst✝⁴ : IsScalarTower S S' R\nI : Ideal R\nA : Type u_1\nB : Type u_2\ninst✝³ : CommRing A\ninst✝² : CommRing B\ninst✝¹ : Algebra R A\ninst✝ : Algebra R B\nf : A →ₐ[R] B\nx : A\nhx : x ∈ ↑(RingHom.ker ↑f)\n⊢ ↑(RingHom.toSemilinearMap (Quotient.mk (RingHom.ker ↑f ^ 2))) x ∈ RingHom.ker ↑(AlgHom.kerSquareLift f)",
"tactic": "exact hx"
}
] |
[
194,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
ne_one_of_mem_sphere
|
[
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.105197\n𝕜 : Type ?u.105200\nα : Type ?u.105203\nι : Type ?u.105206\nκ : Type ?u.105209\nE : Type u_1\nF : Type ?u.105215\nG : Type ?u.105218\ninst✝² : SeminormedGroup E\ninst✝¹ : SeminormedGroup F\ninst✝ : SeminormedGroup G\ns : Set E\na a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\nhr : r ≠ 0\nx : ↑(sphere 1 r)\n⊢ ‖↑x‖ ≠ 0",
"tactic": "rwa [norm_eq_of_mem_sphere' x]"
}
] |
[
714,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
713,
1
] |
Mathlib/Logic/Equiv/Defs.lean
|
Equiv.sigmaCongrRight_trans
|
[] |
[
739,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
736,
1
] |
Mathlib/Data/Option/Basic.lean
|
Option.pmap_eq_none_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.12556\nδ : Type ?u.12559\np : α → Prop\nf : (a : α) → p a → β\nx : Option α\nh : ∀ (a : α), a ∈ x → p a\n⊢ pmap f x h = none ↔ x = none",
"tactic": "cases x <;> simp"
}
] |
[
237,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
237,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/ZeroMorphisms.lean
|
CategoryTheory.Limits.image.ι_zero'
|
[
{
"state_after": "C : Type u\ninst✝⁵ : Category C\nD : Type u'\ninst✝⁴ : Category D\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasZeroObject C\ninst✝¹ : HasEqualizers C\nX Y : C\nf : X ⟶ Y\nh : f = 0\ninst✝ : HasImage f\n⊢ (eqToIso h).hom ≫ ι 0 = 0",
"state_before": "C : Type u\ninst✝⁵ : Category C\nD : Type u'\ninst✝⁴ : Category D\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasZeroObject C\ninst✝¹ : HasEqualizers C\nX Y : C\nf : X ⟶ Y\nh : f = 0\ninst✝ : HasImage f\n⊢ ι f = 0",
"tactic": "rw [image.eq_fac h]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝⁵ : Category C\nD : Type u'\ninst✝⁴ : Category D\ninst✝³ : HasZeroMorphisms C\ninst✝² : HasZeroObject C\ninst✝¹ : HasEqualizers C\nX Y : C\nf : X ⟶ Y\nh : f = 0\ninst✝ : HasImage f\n⊢ (eqToIso h).hom ≫ ι 0 = 0",
"tactic": "simp"
}
] |
[
627,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
624,
1
] |
Mathlib/CategoryTheory/IsConnected.lean
|
CategoryTheory.isPreconnected_of_equivalent
|
[] |
[
219,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
210,
1
] |
Mathlib/Logic/Equiv/Set.lean
|
Equiv.Set.sumCompl_apply_inl
|
[] |
[
322,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Mathlib/Data/Set/Finite.lean
|
Set.Finite.map
|
[] |
[
868,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
867,
1
] |
Mathlib/CategoryTheory/PathCategory.lean
|
CategoryTheory.composePath_toPath
|
[] |
[
175,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
175,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
Dense.borel_eq_generateFrom_Ico_mem_aux
|
[
{
"state_after": "α✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\n⊢ borel α = MeasurableSpace.generateFrom S",
"state_before": "α✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\n⊢ borel α = MeasurableSpace.generateFrom {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}",
"tactic": "set S : Set (Set α) := { S | ∃ l ∈ s, ∃ u ∈ s, l < u ∧ Ico l u = S }"
},
{
"state_after": "α✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\n⊢ borel α ≤ MeasurableSpace.generateFrom S",
"state_before": "α✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\n⊢ borel α = MeasurableSpace.generateFrom S",
"tactic": "refine' le_antisymm _ (generateFrom_Ico_mem_le_borel _ _)"
},
{
"state_after": "α✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\n⊢ borel α ≤ MeasurableSpace.generateFrom S",
"state_before": "α✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\n⊢ borel α ≤ MeasurableSpace.generateFrom S",
"tactic": "letI : MeasurableSpace α := generateFrom S"
},
{
"state_after": "α✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\n⊢ MeasurableSpace.generateFrom (range Iio) ≤ MeasurableSpace.generateFrom S",
"state_before": "α✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\n⊢ borel α ≤ MeasurableSpace.generateFrom S",
"tactic": "rw [borel_eq_generateFrom_Iio]"
},
{
"state_after": "α✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na✝ b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\na : α\n⊢ MeasurableSet (Iio a)",
"state_before": "α✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\n⊢ MeasurableSpace.generateFrom (range Iio) ≤ MeasurableSpace.generateFrom S",
"tactic": "refine' generateFrom_le (forall_range_iff.2 fun a => _)"
},
{
"state_after": "case intro.intro.intro.intro.intro\nα✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t✝ u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na✝ b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\na : α\nt : Set α\nhts : t ⊆ s\nhc : Set.Countable t\nhtd : Dense t\nhtb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t\n⊢ MeasurableSet (Iio a)",
"state_before": "α✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na✝ b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\na : α\n⊢ MeasurableSet (Iio a)",
"tactic": "rcases hd.exists_countable_dense_subset_bot_top with ⟨t, hts, hc, htd, htb, -⟩"
},
{
"state_after": "case pos\nα✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t✝ u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na✝ b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\na : α\nt : Set α\nhts : t ⊆ s\nhc : Set.Countable t\nhtd : Dense t\nhtb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t\nha : ∀ (b : α), b < a → Set.Nonempty (Ioo b a)\n⊢ MeasurableSet (Iio a)\n\ncase neg\nα✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t✝ u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na✝ b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\na : α\nt : Set α\nhts : t ⊆ s\nhc : Set.Countable t\nhtd : Dense t\nhtb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t\nha : ¬∀ (b : α), b < a → Set.Nonempty (Ioo b a)\n⊢ MeasurableSet (Iio a)",
"state_before": "case intro.intro.intro.intro.intro\nα✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t✝ u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na✝ b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\na : α\nt : Set α\nhts : t ⊆ s\nhc : Set.Countable t\nhtd : Dense t\nhtb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t\n⊢ MeasurableSet (Iio a)",
"tactic": "by_cases ha : ∀ b < a, (Ioo b a).Nonempty"
},
{
"state_after": "case h.e'_3\nα✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t✝ u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na✝ b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\na : α\nt : Set α\nhts : t ⊆ s\nhc : Set.Countable t\nhtd : Dense t\nhtb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t\nha : ∀ (b : α), b < a → Set.Nonempty (Ioo b a)\n⊢ Iio a = ⋃ (l : α) (_ : l ∈ t) (u : α) (_ : u ∈ t) (_ : l < u) (_ : u ≤ a), Ico l u\n\ncase pos\nα✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t✝ u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na✝ b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\na : α\nt : Set α\nhts : t ⊆ s\nhc : Set.Countable t\nhtd : Dense t\nhtb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t\nha : ∀ (b : α), b < a → Set.Nonempty (Ioo b a)\n⊢ MeasurableSet (⋃ (l : α) (_ : l ∈ t) (u : α) (_ : u ∈ t) (_ : l < u) (_ : u ≤ a), Ico l u)",
"state_before": "case pos\nα✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t✝ u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na✝ b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\na : α\nt : Set α\nhts : t ⊆ s\nhc : Set.Countable t\nhtd : Dense t\nhtb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t\nha : ∀ (b : α), b < a → Set.Nonempty (Ioo b a)\n⊢ MeasurableSet (Iio a)",
"tactic": "convert_to MeasurableSet (⋃ (l ∈ t) (u ∈ t) (_ : l < u) (_ : u ≤ a), Ico l u)"
},
{
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"state_before": "case h.e'_3\nα✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t✝ u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na✝ b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\na : α\nt : Set α\nhts : t ⊆ s\nhc : Set.Countable t\nhtd : Dense t\nhtb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t\nha : ∀ (b : α), b < a → Set.Nonempty (Ioo b a)\n⊢ Iio a = ⋃ (l : α) (_ : l ∈ t) (u : α) (_ : u ∈ t) (_ : l < u) (_ : u ≤ a), Ico l u",
"tactic": "ext y"
},
{
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"state_before": "case h.e'_3.h\nα✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t✝ u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na✝ b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\na : α\nt : Set α\nhts : t ⊆ s\nhc : Set.Countable t\nhtd : Dense t\nhtb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t\nha : ∀ (b : α), b < a → Set.Nonempty (Ioo b a)\ny : α\n⊢ y ∈ Iio a ↔ y ∈ ⋃ (l : α) (_ : l ∈ t) (u : α) (_ : u ∈ t) (_ : l < u) (_ : u ≤ a), Ico l u",
"tactic": "simp only [mem_iUnion, mem_Iio, mem_Ico]"
},
{
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"tactic": "constructor"
},
{
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"tactic": "intro hy"
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{
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"tactic": "rcases htd.exists_le' (fun b hb => htb _ hb (hbot b hb)) y with ⟨l, hlt, hly⟩"
},
{
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"tactic": "rcases htd.exists_mem_open isOpen_Ioo (ha y hy) with ⟨u, hut, hyu, hua⟩"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.h.mp.intro.intro.intro.intro.intro\nα✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t✝ u✝ : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na✝ b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\na : α\nt : Set α\nhts : t ⊆ s\nhc : Set.Countable t\nhtd : Dense t\nhtb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t\nha : ∀ (b : α), b < a → Set.Nonempty (Ioo b a)\ny : α\nhy : y < a\nl : α\nhlt : l ∈ t\nhly : l ≤ y\nu : α\nhut : u ∈ t\nhyu : y < u\nhua : u < a\n⊢ ∃ i h i_1 h h h, i ≤ y ∧ y < i_1",
"tactic": "exact ⟨l, hlt, u, hut, hly.trans_lt hyu, hua.le, hly, hyu⟩"
},
{
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"tactic": "rintro ⟨l, -, u, -, -, hua, -, hyu⟩"
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"tactic": "exact hyu.trans_le hua"
},
{
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"tactic": "refine' MeasurableSet.biUnion hc fun a ha => MeasurableSet.biUnion hc fun b hb => _"
},
{
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"tactic": "refine' MeasurableSet.iUnion fun hab => MeasurableSet.iUnion fun _ => _"
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"tactic": "exact .basic _ ⟨a, hts ha, b, hts hb, hab, mem_singleton _⟩"
},
{
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"tactic": "replace ha : a ∈ s := hIoo ha.choose a ha.choose_spec.fst ha.choose_spec.snd"
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"tactic": "convert_to MeasurableSet (⋃ (l ∈ t) (_ : l < a), Ico l a)"
},
{
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"tactic": "simp only [← Ici_inter_Iio, ← iUnion_inter, inter_eq_right_iff_subset, subset_def, mem_iUnion,\n mem_Ici, mem_Iio]"
},
{
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"tactic": "intro x hx"
},
{
"state_after": "case h.e'_3.intro.intro\nα✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t✝ u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na✝ b x✝ : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\na : α\nt : Set α\nhts : t ⊆ s\nhc : Set.Countable t\nhtd : Dense t\nhtb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t\nha : a ∈ s\nx : α\nhx : x < a\nz : α\nhzt : z ∈ t\nhzx : z ≤ x\n⊢ ∃ i h h, i ≤ x",
"state_before": "case h.e'_3\nα✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t✝ u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na✝ b x✝ : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\na : α\nt : Set α\nhts : t ⊆ s\nhc : Set.Countable t\nhtd : Dense t\nhtb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t\nha : a ∈ s\nx : α\nhx : x < a\n⊢ ∃ i h h, i ≤ x",
"tactic": "rcases htd.exists_le' (fun b hb => htb _ hb (hbot b hb)) x with ⟨z, hzt, hzx⟩"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.intro.intro\nα✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t✝ u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na✝ b x✝ : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\na : α\nt : Set α\nhts : t ⊆ s\nhc : Set.Countable t\nhtd : Dense t\nhtb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t\nha : a ∈ s\nx : α\nhx : x < a\nz : α\nhzt : z ∈ t\nhzx : z ≤ x\n⊢ ∃ i h h, i ≤ x",
"tactic": "exact ⟨z, hzt, hzx.trans_lt hx, hzx⟩"
},
{
"state_after": "case neg\nα✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t✝ u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na✝ b x✝ : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\na : α\nt : Set α\nhts : t ⊆ s\nhc : Set.Countable t\nhtd : Dense t\nhtb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t\nha : a ∈ s\nx : α\nhx : x ∈ t\nhlt : x < a\n⊢ MeasurableSet (Ico x a)",
"state_before": "case neg\nα✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t✝ u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na✝ b x : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\na : α\nt : Set α\nhts : t ⊆ s\nhc : Set.Countable t\nhtd : Dense t\nhtb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t\nha : a ∈ s\n⊢ MeasurableSet (⋃ (l : α) (_ : l ∈ t) (_ : l < a), Ico l a)",
"tactic": "refine' .biUnion hc fun x hx => MeasurableSet.iUnion fun hlt => _"
},
{
"state_after": "no goals",
"state_before": "case neg\nα✝ : Type ?u.609263\nβ : Type ?u.609266\nγ : Type ?u.609269\nγ₂ : Type ?u.609272\nδ : Type ?u.609275\nι : Sort y\ns✝ t✝ u : Set α✝\ninst✝²⁰ : TopologicalSpace α✝\ninst✝¹⁹ : MeasurableSpace α✝\ninst✝¹⁸ : OpensMeasurableSpace α✝\ninst✝¹⁷ : TopologicalSpace β\ninst✝¹⁶ : MeasurableSpace β\ninst✝¹⁵ : OpensMeasurableSpace β\ninst✝¹⁴ : TopologicalSpace γ\ninst✝¹³ : MeasurableSpace γ\ninst✝¹² : BorelSpace γ\ninst✝¹¹ : TopologicalSpace γ₂\ninst✝¹⁰ : MeasurableSpace γ₂\ninst✝⁹ : BorelSpace γ₂\ninst✝⁸ : MeasurableSpace δ\nα' : Type ?u.609368\ninst✝⁷ : TopologicalSpace α'\ninst✝⁶ : MeasurableSpace α'\ninst✝⁵ : LinearOrder α✝\ninst✝⁴ : OrderClosedTopology α✝\na✝ b x✝ : α✝\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : SecondCountableTopology α\ns : Set α\nhd : Dense s\nhbot : ∀ (x : α), IsBot x → x ∈ s\nhIoo : ∀ (x y : α), x < y → Ioo x y = ∅ → y ∈ s\nS : Set (Set α) := {S | ∃ l, l ∈ s ∧ ∃ u, u ∈ s ∧ l < u ∧ Ico l u = S}\nthis : MeasurableSpace α := MeasurableSpace.generateFrom S\na : α\nt : Set α\nhts : t ⊆ s\nhc : Set.Countable t\nhtd : Dense t\nhtb : ∀ (x : α), IsBot x → x ∈ s → x ∈ t\nha : a ∈ s\nx : α\nhx : x ∈ t\nhlt : x < a\n⊢ MeasurableSet (Ico x a)",
"tactic": "exact .basic _ ⟨x, hts hx, a, ha, hlt, mem_singleton _⟩"
}
] |
[
612,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
578,
1
] |
Mathlib/NumberTheory/Padics/PadicVal.lean
|
dvd_iff_padicValNat_ne_zero
|
[] |
[
248,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/Dynamics/PeriodicPts.lean
|
Function.IsPeriodicPt.map
|
[] |
[
87,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
85,
11
] |
Mathlib/Algebra/Ring/Basic.lean
|
inv_neg'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Group α\ninst✝ : HasDistribNeg α\na : α\n⊢ (-a)⁻¹ = -a⁻¹",
"tactic": "rw [eq_comm, eq_inv_iff_mul_eq_one, neg_mul, mul_neg, neg_neg, mul_left_inv]"
}
] |
[
116,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/Algebra/GroupWithZero/Divisibility.lean
|
eq_of_forall_dvd
|
[] |
[
136,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
135,
1
] |
src/lean/Init/Data/Nat/Basic.lean
|
Nat.pred_zero
|
[] |
[
684,
6
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
683,
19
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
norm_prod_le_iff
|
[] |
[
2389,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2388,
1
] |
Mathlib/Data/Real/NNReal.lean
|
NNReal.toNNReal_coe_nat
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ ↑(Real.toNNReal ↑n) = ↑↑n",
"tactic": "simp [Real.coe_toNNReal]"
}
] |
[
405,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
404,
1
] |
Mathlib/CategoryTheory/Limits/HasLimits.lean
|
CategoryTheory.Limits.colimit.isoColimitCocone_ι_hom
|
[
{
"state_after": "J : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u\ninst✝¹ : Category C\nF✝ F : J ⥤ C\ninst✝ : HasColimit F\nt : ColimitCocone F\nj : J\n⊢ ι F j ≫ desc F t.cocone = t.cocone.ι.app j",
"state_before": "J : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u\ninst✝¹ : Category C\nF✝ F : J ⥤ C\ninst✝ : HasColimit F\nt : ColimitCocone F\nj : J\n⊢ ι F j ≫ (isoColimitCocone t).hom = t.cocone.ι.app j",
"tactic": "dsimp [colimit.isoColimitCocone, IsColimit.coconePointUniqueUpToIso]"
},
{
"state_after": "no goals",
"state_before": "J : Type u₁\ninst✝³ : Category J\nK : Type u₂\ninst✝² : Category K\nC : Type u\ninst✝¹ : Category C\nF✝ F : J ⥤ C\ninst✝ : HasColimit F\nt : ColimitCocone F\nj : J\n⊢ ι F j ≫ desc F t.cocone = t.cocone.ι.app j",
"tactic": "aesop_cat"
}
] |
[
826,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
823,
1
] |
Mathlib/Data/Set/Intervals/Instances.lean
|
Set.Ioc.coe_eq_one
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : StrictOrderedSemiring α\ninst✝ : Nontrivial α\nx : ↑(Ioc 0 1)\n⊢ x = 1 ↔ ↑x = 1",
"state_before": "α : Type u_1\ninst✝¹ : StrictOrderedSemiring α\ninst✝ : Nontrivial α\nx : ↑(Ioc 0 1)\n⊢ ↑x = 1 ↔ x = 1",
"tactic": "symm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : StrictOrderedSemiring α\ninst✝ : Nontrivial α\nx : ↑(Ioc 0 1)\n⊢ x = 1 ↔ ↑x = 1",
"tactic": "exact Subtype.ext_iff"
}
] |
[
269,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
267,
1
] |
Mathlib/NumberTheory/Padics/PadicVal.lean
|
zero_le_padicValRat_of_nat
|
[
{
"state_after": "no goals",
"state_before": "p n : ℕ\n⊢ 0 ≤ padicValRat p ↑n",
"tactic": "simp"
}
] |
[
214,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
214,
1
] |
Mathlib/Algebra/MonoidAlgebra/Basic.lean
|
AddMonoidAlgebra.algHom_ext'
|
[] |
[
1950,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1945,
1
] |
Mathlib/Order/Basic.lean
|
Prod.mk_le_mk
|
[] |
[
1198,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1197,
1
] |
Mathlib/ModelTheory/Satisfiability.lean
|
FirstOrder.Language.Theory.isSatisfiable_of_isSatisfiable_onTheory
|
[] |
[
93,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/Algebra/Periodic.lean
|
Function.Periodic.sub_nsmul_eq
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.99178\nf g : α → β\nc c₁ c₂ x : α\ninst✝ : AddGroup α\nh : Periodic f c\nn : ℕ\n⊢ f (x - n • c) = f x",
"tactic": "simpa only [sub_eq_add_neg] using h.neg_nsmul n x"
}
] |
[
218,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
217,
1
] |
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