file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Logic/Function/Basic.lean
|
eq_mpr_bijective
|
[
{
"state_after": "case refl\nα : Sort u_1\n⊢ Bijective (Eq.mpr (_ : α = α))",
"state_before": "α β : Sort u_1\nh : α = β\n⊢ Bijective (Eq.mpr h)",
"tactic": "cases h"
},
{
"state_after": "no goals",
"state_before": "case refl\nα : Sort u_1\n⊢ Bijective (Eq.mpr (_ : α = α))",
"tactic": "refine ⟨fun _ _ ↦ id, fun x ↦ ⟨x, rfl⟩⟩"
}
] |
[
1009,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1007,
1
] |
Mathlib/RingTheory/WittVector/StructurePolynomial.lean
|
constantCoeff_wittStructureRat
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nR : Type ?u.1929960\nidx : Type u_1\ninst✝ : CommRing R\nhp : Fact (Nat.Prime p)\nΦ : MvPolynomial idx ℚ\nh : ↑constantCoeff Φ = 0\nn : ℕ\n⊢ ↑constantCoeff (wittStructureRat p Φ n) = 0",
"tactic": "simp only [wittStructureRat, eval₂Hom_zero'_apply, h, bind₁, map_aeval, constantCoeff_rename,\n constantCoeff_wittPolynomial, constantCoeff_comp_algebraMap, RingHom.id_apply,\n constantCoeff_xInTermsOfW]"
}
] |
[
370,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
366,
1
] |
Mathlib/Algebra/CharP/Basic.lean
|
CharP.pow_prime_pow_mul_eq_one_iff
|
[
{
"state_after": "case zero\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsReduced R\np m : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx : R\n⊢ x ^ (p ^ Nat.zero * m) = 1 ↔ x ^ m = 1\n\ncase succ\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsReduced R\np m : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx : R\nk : ℕ\nhk : x ^ (p ^ k * m) = 1 ↔ x ^ m = 1\n⊢ x ^ (p ^ Nat.succ k * m) = 1 ↔ x ^ m = 1",
"state_before": "R : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsReduced R\np k m : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx : R\n⊢ x ^ (p ^ k * m) = 1 ↔ x ^ m = 1",
"tactic": "induction' k with k hk"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsReduced R\np m : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx : R\n⊢ x ^ (p ^ Nat.zero * m) = 1 ↔ x ^ m = 1",
"tactic": "rw [pow_zero, one_mul]"
},
{
"state_after": "case succ.refine'_1\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsReduced R\np m : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx : R\nk : ℕ\nhk : x ^ (p ^ k * m) = 1 ↔ x ^ m = 1\nh : x ^ (p ^ Nat.succ k * m) = 1\n⊢ x ^ m = 1\n\ncase succ.refine'_2\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsReduced R\np m : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx : R\nk : ℕ\nhk : x ^ (p ^ k * m) = 1 ↔ x ^ m = 1\nh : x ^ m = 1\n⊢ x ^ (p ^ Nat.succ k * m) = 1",
"state_before": "case succ\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsReduced R\np m : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx : R\nk : ℕ\nhk : x ^ (p ^ k * m) = 1 ↔ x ^ m = 1\n⊢ x ^ (p ^ Nat.succ k * m) = 1 ↔ x ^ m = 1",
"tactic": "refine' ⟨fun h => _, fun h => _⟩"
},
{
"state_after": "case succ.refine'_1\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsReduced R\np m : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx : R\nk : ℕ\nhk : x ^ (p ^ k * m) = 1 ↔ x ^ m = 1\nh✝ : ↑(frobenius R p) (x ^ (p ^ k * m)) = 1\nh : ↑(frobenius R p) (x ^ (p ^ k * m)) = ↑(frobenius R p) 1\n⊢ x ^ m = 1",
"state_before": "case succ.refine'_1\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsReduced R\np m : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx : R\nk : ℕ\nhk : x ^ (p ^ k * m) = 1 ↔ x ^ m = 1\nh : x ^ (p ^ Nat.succ k * m) = 1\n⊢ x ^ m = 1",
"tactic": "rw [pow_succ, mul_assoc, pow_mul', ← frobenius_def, ← frobenius_one p] at h"
},
{
"state_after": "no goals",
"state_before": "case succ.refine'_1\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsReduced R\np m : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx : R\nk : ℕ\nhk : x ^ (p ^ k * m) = 1 ↔ x ^ m = 1\nh✝ : ↑(frobenius R p) (x ^ (p ^ k * m)) = 1\nh : ↑(frobenius R p) (x ^ (p ^ k * m)) = ↑(frobenius R p) 1\n⊢ x ^ m = 1",
"tactic": "exact hk.1 (frobenius_inj R p h)"
},
{
"state_after": "no goals",
"state_before": "case succ.refine'_2\nR : Type u_1\ninst✝³ : CommRing R\ninst✝² : IsReduced R\np m : ℕ\ninst✝¹ : Fact (Nat.Prime p)\ninst✝ : CharP R p\nx : R\nk : ℕ\nhk : x ^ (p ^ k * m) = 1 ↔ x ^ m = 1\nh : x ^ m = 1\n⊢ x ^ (p ^ Nat.succ k * m) = 1",
"tactic": "rw [pow_mul', h, one_pow]"
}
] |
[
504,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
497,
1
] |
Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean
|
Matrix.left_inv_eq_left_inv
|
[
{
"state_after": "no goals",
"state_before": "l : Type ?u.259825\nm : Type u\nn : Type u'\nα : Type v\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing α\nA B C : Matrix n n α\nh : B ⬝ A = 1\ng : C ⬝ A = 1\n⊢ B = C",
"tactic": "rw [← inv_eq_left_inv h, ← inv_eq_left_inv g]"
}
] |
[
463,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
462,
1
] |
Mathlib/GroupTheory/Subsemigroup/Operations.lean
|
Subsemigroup.comap_top
|
[] |
[
363,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
362,
1
] |
Mathlib/Topology/FiberBundle/Trivialization.lean
|
Pretrivialization.preimage_symm_proj_baseSet
|
[
{
"state_after": "ι : Type ?u.9079\nB : Type u_1\nF : Type u_2\nE : B → Type ?u.9090\nZ : Type u_3\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nproj : Z → B\ne : Pretrivialization F proj\nx✝ : Z\nx : B × F\nhx : x ∈ e.target\n⊢ x ∈ ↑(LocalEquiv.symm e.toLocalEquiv) ⁻¹' (proj ⁻¹' e.baseSet)",
"state_before": "ι : Type ?u.9079\nB : Type u_1\nF : Type u_2\nE : B → Type ?u.9090\nZ : Type u_3\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nproj : Z → B\ne : Pretrivialization F proj\nx : Z\n⊢ ↑(LocalEquiv.symm e.toLocalEquiv) ⁻¹' (proj ⁻¹' e.baseSet) ∩ e.target = e.target",
"tactic": "refine' inter_eq_right_iff_subset.mpr fun x hx => _"
},
{
"state_after": "ι : Type ?u.9079\nB : Type u_1\nF : Type u_2\nE : B → Type ?u.9090\nZ : Type u_3\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nproj : Z → B\ne : Pretrivialization F proj\nx✝ : Z\nx : B × F\nhx : x ∈ e.target\n⊢ x.fst ∈ e.baseSet",
"state_before": "ι : Type ?u.9079\nB : Type u_1\nF : Type u_2\nE : B → Type ?u.9090\nZ : Type u_3\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nproj : Z → B\ne : Pretrivialization F proj\nx✝ : Z\nx : B × F\nhx : x ∈ e.target\n⊢ x ∈ ↑(LocalEquiv.symm e.toLocalEquiv) ⁻¹' (proj ⁻¹' e.baseSet)",
"tactic": "simp only [mem_preimage, LocalEquiv.invFun_as_coe, e.proj_symm_apply hx]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.9079\nB : Type u_1\nF : Type u_2\nE : B → Type ?u.9090\nZ : Type u_3\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nproj : Z → B\ne : Pretrivialization F proj\nx✝ : Z\nx : B × F\nhx : x ∈ e.target\n⊢ x.fst ∈ e.baseSet",
"tactic": "exact e.mem_target.mp hx"
}
] |
[
188,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
184,
1
] |
src/lean/Init/SimpLemmas.lean
|
decide_eq_true_eq
|
[] |
[
139,
129
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
139,
9
] |
Mathlib/Algebra/Homology/Homotopy.lean
|
prevD_eq
|
[
{
"state_after": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nf : (i j : ι) → X C i ⟶ X D j\nj : ι\nw : ComplexShape.Rel c (ComplexShape.prev c j) j\n⊢ ↑(prevD j) f = f j (ComplexShape.prev c j) ≫ d D (ComplexShape.prev c j) j",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nf : (i j : ι) → X C i ⟶ X D j\nj j' : ι\nw : ComplexShape.Rel c j' j\n⊢ ↑(prevD j) f = f j j' ≫ d D j' j",
"tactic": "obtain rfl := c.prev_eq' w"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nV : Type u\ninst✝¹ : Category V\ninst✝ : Preadditive V\nc : ComplexShape ι\nC D E : HomologicalComplex V c\nf✝ g : C ⟶ D\nh k : D ⟶ E\ni : ι\nf : (i j : ι) → X C i ⟶ X D j\nj : ι\nw : ComplexShape.Rel c (ComplexShape.prev c j) j\n⊢ ↑(prevD j) f = f j (ComplexShape.prev c j) ≫ d D (ComplexShape.prev c j) j",
"tactic": "rfl"
}
] |
[
94,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
1
] |
Mathlib/RingTheory/Polynomial/Cyclotomic/Basic.lean
|
Polynomial.cyclotomic_three
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝ : Ring R\n⊢ cyclotomic 3 R = X ^ 2 + X + 1",
"tactic": "simp [cyclotomic_prime, sum_range_succ']"
}
] |
[
419,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
418,
1
] |
Mathlib/Topology/VectorBundle/Basic.lean
|
VectorBundleCore.continuous_proj
|
[] |
[
785,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
784,
1
] |
Mathlib/Topology/Order.lean
|
isClosed_induced_iff
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.19398\nt : TopologicalSpace β\ns : Set α\nf : α → β\nthis : TopologicalSpace α := TopologicalSpace.induced f t\n⊢ IsClosed s ↔ ∃ t_1, IsClosed t_1 ∧ f ⁻¹' t_1 = s",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.19398\nt : TopologicalSpace β\ns : Set α\nf : α → β\n⊢ IsClosed s ↔ ∃ t_1, IsClosed t_1 ∧ f ⁻¹' t_1 = s",
"tactic": "letI := t.induced f"
},
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.19398\nt : TopologicalSpace β\ns : Set α\nf : α → β\nthis : TopologicalSpace α := TopologicalSpace.induced f t\n⊢ (∃ t_1, IsOpen t_1 ∧ f ⁻¹' t_1 = sᶜ) ↔ ∃ t_1, IsOpen (t_1ᶜ) ∧ f ⁻¹' t_1 = s",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.19398\nt : TopologicalSpace β\ns : Set α\nf : α → β\nthis : TopologicalSpace α := TopologicalSpace.induced f t\n⊢ IsClosed s ↔ ∃ t_1, IsClosed t_1 ∧ f ⁻¹' t_1 = s",
"tactic": "simp only [← isOpen_compl_iff, isOpen_induced_iff]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.19398\nt : TopologicalSpace β\ns : Set α\nf : α → β\nthis : TopologicalSpace α := TopologicalSpace.induced f t\n⊢ (∃ t_1, IsOpen t_1 ∧ f ⁻¹' t_1 = sᶜ) ↔ ∃ t_1, IsOpen (t_1ᶜ) ∧ f ⁻¹' t_1 = s",
"tactic": "exact compl_surjective.exists.trans (by simp only [preimage_compl, compl_inj_iff])"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.19398\nt : TopologicalSpace β\ns : Set α\nf : α → β\nthis : TopologicalSpace α := TopologicalSpace.induced f t\n⊢ (∃ x, IsOpen (xᶜ) ∧ f ⁻¹' xᶜ = sᶜ) ↔ ∃ t_1, IsOpen (t_1ᶜ) ∧ f ⁻¹' t_1 = s",
"tactic": "simp only [preimage_compl, compl_inj_iff]"
}
] |
[
377,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
373,
1
] |
Mathlib/Data/List/ToFinsupp.lean
|
List.coe_toFinsupp
|
[] |
[
59,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
58,
1
] |
Std/Data/Nat/Gcd.lean
|
Nat.coprime.coprime_div_left
|
[
{
"state_after": "no goals",
"state_before": "m n a : Nat\ncmn : coprime m n\ndvd : a ∣ m\n⊢ coprime (m / a) n",
"tactic": "match eq_zero_or_pos a with\n| .inl h0 =>\n rw [h0] at dvd\n rw [Nat.eq_zero_of_zero_dvd dvd] at cmn ⊢\n simp; assumption\n| .inr hpos =>\n let ⟨k, hk⟩ := dvd\n rw [hk, Nat.mul_div_cancel_left _ hpos]\n rw [hk] at cmn\n exact cmn.coprime_mul_left"
},
{
"state_after": "m n a : Nat\ncmn : coprime m n\ndvd : 0 ∣ m\nh0 : a = 0\n⊢ coprime (m / a) n",
"state_before": "m n a : Nat\ncmn : coprime m n\ndvd : a ∣ m\nh0 : a = 0\n⊢ coprime (m / a) n",
"tactic": "rw [h0] at dvd"
},
{
"state_after": "m n a : Nat\ncmn : coprime 0 n\ndvd : 0 ∣ m\nh0 : a = 0\n⊢ coprime (0 / a) n",
"state_before": "m n a : Nat\ncmn : coprime m n\ndvd : 0 ∣ m\nh0 : a = 0\n⊢ coprime (m / a) n",
"tactic": "rw [Nat.eq_zero_of_zero_dvd dvd] at cmn ⊢"
},
{
"state_after": "m n a : Nat\ncmn : coprime 0 n\ndvd : 0 ∣ m\nh0 : a = 0\n⊢ coprime 0 n",
"state_before": "m n a : Nat\ncmn : coprime 0 n\ndvd : 0 ∣ m\nh0 : a = 0\n⊢ coprime (0 / a) n",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "m n a : Nat\ncmn : coprime 0 n\ndvd : 0 ∣ m\nh0 : a = 0\n⊢ coprime 0 n",
"tactic": "assumption"
},
{
"state_after": "m n a : Nat\ncmn : coprime m n\ndvd : a ∣ m\nhpos : a > 0\nk : Nat\nhk : m = a * k\n⊢ coprime (m / a) n",
"state_before": "m n a : Nat\ncmn : coprime m n\ndvd : a ∣ m\nhpos : a > 0\n⊢ coprime (m / a) n",
"tactic": "let ⟨k, hk⟩ := dvd"
},
{
"state_after": "m n a : Nat\ncmn : coprime m n\ndvd : a ∣ m\nhpos : a > 0\nk : Nat\nhk : m = a * k\n⊢ coprime k n",
"state_before": "m n a : Nat\ncmn : coprime m n\ndvd : a ∣ m\nhpos : a > 0\nk : Nat\nhk : m = a * k\n⊢ coprime (m / a) n",
"tactic": "rw [hk, Nat.mul_div_cancel_left _ hpos]"
},
{
"state_after": "m n a : Nat\ndvd : a ∣ m\nhpos : a > 0\nk : Nat\ncmn : coprime (a * k) n\nhk : m = a * k\n⊢ coprime k n",
"state_before": "m n a : Nat\ncmn : coprime m n\ndvd : a ∣ m\nhpos : a > 0\nk : Nat\nhk : m = a * k\n⊢ coprime k n",
"tactic": "rw [hk] at cmn"
},
{
"state_after": "no goals",
"state_before": "m n a : Nat\ndvd : a ∣ m\nhpos : a > 0\nk : Nat\ncmn : coprime (a * k) n\nhk : m = a * k\n⊢ coprime k n",
"tactic": "exact cmn.coprime_mul_left"
}
] |
[
328,
31
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
318,
1
] |
Mathlib/Analysis/Analytic/Basic.lean
|
FormalMultilinearSeries.le_radius_of_summable_norm
|
[] |
[
276,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
274,
1
] |
Mathlib/Logic/Encodable/Basic.lean
|
Encodable.encode_none
|
[] |
[
169,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/LinearAlgebra/Matrix/Symmetric.lean
|
Matrix.IsSymm.map
|
[] |
[
88,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
Mathlib/Init/Algebra/Order.lean
|
not_le
|
[] |
[
378,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
377,
1
] |
Mathlib/Order/Monotone/Basic.lean
|
Subsingleton.antitone'
|
[] |
[
495,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
494,
1
] |
Mathlib/CategoryTheory/Subobject/Lattice.lean
|
CategoryTheory.Subobject.finset_inf_factors
|
[
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nI : Type u_1\nA B : C\ns : Finset I\nP : I → Subobject B\nf : A ⟶ B\n⊢ Factors (Finset.inf s P) f ↔ ∀ (i : I), i ∈ s → Factors (P i) f",
"tactic": "classical\ninduction' s using Finset.induction_on with _ _ _ ih\n. simp [top_factors]\n. simp [ih]"
},
{
"state_after": "case empty\nC : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nI : Type u_1\nA B : C\nP : I → Subobject B\nf : A ⟶ B\n⊢ Factors (Finset.inf ∅ P) f ↔ ∀ (i : I), i ∈ ∅ → Factors (P i) f\n\ncase insert\nC : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nI : Type u_1\nA B : C\nP : I → Subobject B\nf : A ⟶ B\na✝¹ : I\ns✝ : Finset I\na✝ : ¬a✝¹ ∈ s✝\nih : Factors (Finset.inf s✝ P) f ↔ ∀ (i : I), i ∈ s✝ → Factors (P i) f\n⊢ Factors (Finset.inf (insert a✝¹ s✝) P) f ↔ ∀ (i : I), i ∈ insert a✝¹ s✝ → Factors (P i) f",
"state_before": "C : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nI : Type u_1\nA B : C\ns : Finset I\nP : I → Subobject B\nf : A ⟶ B\n⊢ Factors (Finset.inf s P) f ↔ ∀ (i : I), i ∈ s → Factors (P i) f",
"tactic": "induction' s using Finset.induction_on with _ _ _ ih"
},
{
"state_after": "case insert\nC : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nI : Type u_1\nA B : C\nP : I → Subobject B\nf : A ⟶ B\na✝¹ : I\ns✝ : Finset I\na✝ : ¬a✝¹ ∈ s✝\nih : Factors (Finset.inf s✝ P) f ↔ ∀ (i : I), i ∈ s✝ → Factors (P i) f\n⊢ Factors (Finset.inf (insert a✝¹ s✝) P) f ↔ ∀ (i : I), i ∈ insert a✝¹ s✝ → Factors (P i) f",
"state_before": "case empty\nC : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nI : Type u_1\nA B : C\nP : I → Subobject B\nf : A ⟶ B\n⊢ Factors (Finset.inf ∅ P) f ↔ ∀ (i : I), i ∈ ∅ → Factors (P i) f\n\ncase insert\nC : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nI : Type u_1\nA B : C\nP : I → Subobject B\nf : A ⟶ B\na✝¹ : I\ns✝ : Finset I\na✝ : ¬a✝¹ ∈ s✝\nih : Factors (Finset.inf s✝ P) f ↔ ∀ (i : I), i ∈ s✝ → Factors (P i) f\n⊢ Factors (Finset.inf (insert a✝¹ s✝) P) f ↔ ∀ (i : I), i ∈ insert a✝¹ s✝ → Factors (P i) f",
"tactic": ". simp [top_factors]"
},
{
"state_after": "no goals",
"state_before": "case insert\nC : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nI : Type u_1\nA B : C\nP : I → Subobject B\nf : A ⟶ B\na✝¹ : I\ns✝ : Finset I\na✝ : ¬a✝¹ ∈ s✝\nih : Factors (Finset.inf s✝ P) f ↔ ∀ (i : I), i ∈ s✝ → Factors (P i) f\n⊢ Factors (Finset.inf (insert a✝¹ s✝) P) f ↔ ∀ (i : I), i ∈ insert a✝¹ s✝ → Factors (P i) f",
"tactic": ". simp [ih]"
},
{
"state_after": "no goals",
"state_before": "case empty\nC : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nI : Type u_1\nA B : C\nP : I → Subobject B\nf : A ⟶ B\n⊢ Factors (Finset.inf ∅ P) f ↔ ∀ (i : I), i ∈ ∅ → Factors (P i) f",
"tactic": "simp [top_factors]"
},
{
"state_after": "no goals",
"state_before": "case insert\nC : Type u₁\ninst✝² : Category C\nX Y Z : C\nD : Type u₂\ninst✝¹ : Category D\ninst✝ : HasPullbacks C\nI : Type u_1\nA B : C\nP : I → Subobject B\nf : A ⟶ B\na✝¹ : I\ns✝ : Finset I\na✝ : ¬a✝¹ ∈ s✝\nih : Factors (Finset.inf s✝ P) f ↔ ∀ (i : I), i ∈ s✝ → Factors (P i) f\n⊢ Factors (Finset.inf (insert a✝¹ s✝) P) f ↔ ∀ (i : I), i ∈ insert a✝¹ s✝ → Factors (P i) f",
"tactic": "simp [ih]"
}
] |
[
442,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
437,
1
] |
Mathlib/MeasureTheory/Integral/VitaliCaratheodory.lean
|
MeasureTheory.SimpleFunc.exists_upperSemicontinuous_le_lintegral_le
|
[
{
"state_after": "case h_ind\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\n⊢ ∃ g,\n (∀ (x : α), g x ≤ ↑(piecewise s hs (const α c) (const α 0)) x) ∧\n UpperSemicontinuous g ∧\n (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\n\ncase h_add\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\n⊢ ∃ g,\n (∀ (x : α), g x ≤ ↑(f₁ + f₂) x) ∧\n UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\n⊢ ∃ g, (∀ (x : α), g x ≤ ↑f x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"tactic": "induction' f using MeasureTheory.SimpleFunc.induction with c s hs f₁ f₂ _ h₁ h₂ generalizing ε"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : c = 0\n⊢ ∃ g,\n (∀ (x : α), g x ≤ ↑(piecewise s hs (const α c) (const α 0)) x) ∧\n UpperSemicontinuous g ∧\n (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\n\ncase neg\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\n⊢ ∃ g,\n (∀ (x : α), g x ≤ ↑(piecewise s hs (const α c) (const α 0)) x) ∧\n UpperSemicontinuous g ∧\n (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"state_before": "case h_ind\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\n⊢ ∃ g,\n (∀ (x : α), g x ≤ ↑(piecewise s hs (const α c) (const α 0)) x) ∧\n UpperSemicontinuous g ∧\n (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"tactic": "by_cases hc : c = 0"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\n⊢ ∃ g,\n (∀ (x : α), g x ≤ ↑(piecewise s hs (const α c) (const α 0)) x) ∧\n UpperSemicontinuous g ∧\n (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"state_before": "case neg\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\n⊢ ∃ g,\n (∀ (x : α), g x ≤ ↑(piecewise s hs (const α c) (const α 0)) x) ∧\n UpperSemicontinuous g ∧\n (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"tactic": "have μs_lt_top : μ s < ∞ := by\n classical\n simpa only [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const, or_false_iff,\n lintegral_const, ENNReal.coe_indicator, Set.univ_inter, ENNReal.coe_ne_top,\n Measure.restrict_apply MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,\n Function.const_apply, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,\n SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise,\n false_and_iff] using int_f"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\nthis : 0 < ε / ↑c\n⊢ ∃ g,\n (∀ (x : α), g x ≤ ↑(piecewise s hs (const α c) (const α 0)) x) ∧\n UpperSemicontinuous g ∧\n (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"state_before": "case neg\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\n⊢ ∃ g,\n (∀ (x : α), g x ≤ ↑(piecewise s hs (const α c) (const α 0)) x) ∧\n UpperSemicontinuous g ∧\n (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"tactic": "have : (0 : ℝ≥0∞) < ε / c := ENNReal.div_pos_iff.2 ⟨ε0, ENNReal.coe_ne_top⟩"
},
{
"state_after": "case neg.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\nthis : 0 < ε / ↑c\nF : Set α\nFs : F ⊆ s\nF_closed : IsClosed F\nμF : ↑↑μ s < ↑↑μ F + ε / ↑c\n⊢ ∃ g,\n (∀ (x : α), g x ≤ ↑(piecewise s hs (const α c) (const α 0)) x) ∧\n UpperSemicontinuous g ∧\n (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"state_before": "case neg\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\nthis : 0 < ε / ↑c\n⊢ ∃ g,\n (∀ (x : α), g x ≤ ↑(piecewise s hs (const α c) (const α 0)) x) ∧\n UpperSemicontinuous g ∧\n (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"tactic": "obtain ⟨F, Fs, F_closed, μF⟩ : ∃ (F : _), F ⊆ s ∧ IsClosed F ∧ μ s < μ F + ε / c :=\n hs.exists_isClosed_lt_add μs_lt_top.ne this.ne'"
},
{
"state_after": "case neg.intro.intro.intro.refine'_1\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\nthis : 0 < ε / ↑c\nF : Set α\nFs : F ⊆ s\nF_closed : IsClosed F\nμF : ↑↑μ s < ↑↑μ F + ε / ↑c\nx : α\n⊢ Set.indicator F (fun x => c) x ≤ ↑(piecewise s hs (const α c) (const α 0)) x\n\ncase neg.intro.intro.intro.refine'_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\nthis : 0 < ε / ↑c\nF : Set α\nFs : F ⊆ s\nF_closed : IsClosed F\nμF : ↑↑μ s < ↑↑μ F + ε / ↑c\n⊢ (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≤\n (∫⁻ (x : α), ↑(Set.indicator F (fun x => c) x) ∂μ) + ε",
"state_before": "case neg.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\nthis : 0 < ε / ↑c\nF : Set α\nFs : F ⊆ s\nF_closed : IsClosed F\nμF : ↑↑μ s < ↑↑μ F + ε / ↑c\n⊢ ∃ g,\n (∀ (x : α), g x ≤ ↑(piecewise s hs (const α c) (const α 0)) x) ∧\n UpperSemicontinuous g ∧\n (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"tactic": "refine'\n ⟨Set.indicator F fun _ => c, fun x => _, F_closed.upperSemicontinuous_indicator (zero_le _),\n _⟩"
},
{
"state_after": "case pos.refine'_1\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : c = 0\n⊢ ∀ (x : α), (fun x => 0) x ≤ ↑(piecewise s hs (const α c) (const α 0)) x\n\ncase pos.refine'_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : c = 0\n⊢ (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≤ (∫⁻ (x : α), ↑((fun x => 0) x) ∂μ) + ε",
"state_before": "case pos\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : c = 0\n⊢ ∃ g,\n (∀ (x : α), g x ≤ ↑(piecewise s hs (const α c) (const α 0)) x) ∧\n UpperSemicontinuous g ∧\n (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"tactic": "refine' ⟨fun _ => 0, _, upperSemicontinuous_const, _⟩"
},
{
"state_after": "no goals",
"state_before": "case pos.refine'_1\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : c = 0\n⊢ ∀ (x : α), (fun x => 0) x ≤ ↑(piecewise s hs (const α c) (const α 0)) x",
"tactic": "classical\nsimp only [hc, Set.indicator_zero', Pi.zero_apply, SimpleFunc.const_zero, imp_true_iff,\n eq_self_iff_true, SimpleFunc.coe_zero, Set.piecewise_eq_indicator,\n SimpleFunc.coe_piecewise, le_zero_iff]"
},
{
"state_after": "no goals",
"state_before": "case pos.refine'_1\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : c = 0\n⊢ ∀ (x : α), (fun x => 0) x ≤ ↑(piecewise s hs (const α c) (const α 0)) x",
"tactic": "simp only [hc, Set.indicator_zero', Pi.zero_apply, SimpleFunc.const_zero, imp_true_iff,\n eq_self_iff_true, SimpleFunc.coe_zero, Set.piecewise_eq_indicator,\n SimpleFunc.coe_piecewise, le_zero_iff]"
},
{
"state_after": "no goals",
"state_before": "case pos.refine'_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : c = 0\n⊢ (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≤ (∫⁻ (x : α), ↑((fun x => 0) x) ∂μ) + ε",
"tactic": "classical\nsimp only [hc, Set.indicator_zero', lintegral_const, MulZeroClass.zero_mul, Pi.zero_apply,\n SimpleFunc.const_zero, zero_add, zero_le', SimpleFunc.coe_zero,\n Set.piecewise_eq_indicator, ENNReal.coe_zero, SimpleFunc.coe_piecewise, zero_le]"
},
{
"state_after": "no goals",
"state_before": "case pos.refine'_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : c = 0\n⊢ (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≤ (∫⁻ (x : α), ↑((fun x => 0) x) ∂μ) + ε",
"tactic": "simp only [hc, Set.indicator_zero', lintegral_const, MulZeroClass.zero_mul, Pi.zero_apply,\n SimpleFunc.const_zero, zero_add, zero_le', SimpleFunc.coe_zero,\n Set.piecewise_eq_indicator, ENNReal.coe_zero, SimpleFunc.coe_piecewise, zero_le]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\n⊢ ↑↑μ s < ⊤",
"tactic": "classical\nsimpa only [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const, or_false_iff,\n lintegral_const, ENNReal.coe_indicator, Set.univ_inter, ENNReal.coe_ne_top,\n Measure.restrict_apply MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,\n Function.const_apply, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,\n SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise,\n false_and_iff] using int_f"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\n⊢ ↑↑μ s < ⊤",
"tactic": "simpa only [hs, hc, lt_top_iff_ne_top, true_and_iff, SimpleFunc.coe_const, or_false_iff,\n lintegral_const, ENNReal.coe_indicator, Set.univ_inter, ENNReal.coe_ne_top,\n Measure.restrict_apply MeasurableSet.univ, ENNReal.mul_eq_top, SimpleFunc.const_zero,\n Function.const_apply, lintegral_indicator, ENNReal.coe_eq_zero, Ne.def, not_false_iff,\n SimpleFunc.coe_zero, Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise,\n false_and_iff] using int_f"
},
{
"state_after": "case neg.intro.intro.intro.refine'_1\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\nthis : 0 < ε / ↑c\nF : Set α\nFs : F ⊆ s\nF_closed : IsClosed F\nμF : ↑↑μ s < ↑↑μ F + ε / ↑c\nx : α\n⊢ Set.indicator F (fun x => c) x ≤ Set.piecewise s (Function.const α c) 0 x",
"state_before": "case neg.intro.intro.intro.refine'_1\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\nthis : 0 < ε / ↑c\nF : Set α\nFs : F ⊆ s\nF_closed : IsClosed F\nμF : ↑↑μ s < ↑↑μ F + ε / ↑c\nx : α\n⊢ Set.indicator F (fun x => c) x ≤ ↑(piecewise s hs (const α c) (const α 0)) x",
"tactic": "simp only [SimpleFunc.coe_const, SimpleFunc.const_zero, SimpleFunc.coe_zero,\n Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise]"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro.refine'_1\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\nthis : 0 < ε / ↑c\nF : Set α\nFs : F ⊆ s\nF_closed : IsClosed F\nμF : ↑↑μ s < ↑↑μ F + ε / ↑c\nx : α\n⊢ Set.indicator F (fun x => c) x ≤ Set.piecewise s (Function.const α c) 0 x",
"tactic": "exact Set.indicator_le_indicator_of_subset Fs (fun x => zero_le _) _"
},
{
"state_after": "case neg.intro.intro.intro.refine'_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\nthis : 0 < ε / ↑c\nF : Set α\nFs : F ⊆ s\nF_closed : IsClosed F\nμF : ↑↑μ s < ↑↑μ F + ε / ↑c\n⊢ ↑c * ↑↑μ s ≤ ↑c * ↑↑μ F + ε",
"state_before": "case neg.intro.intro.intro.refine'_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\nthis : 0 < ε / ↑c\nF : Set α\nFs : F ⊆ s\nF_closed : IsClosed F\nμF : ↑↑μ s < ↑↑μ F + ε / ↑c\n⊢ (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≤\n (∫⁻ (x : α), ↑(Set.indicator F (fun x => c) x) ∂μ) + ε",
"tactic": "suffices (c : ℝ≥0∞) * μ s ≤ c * μ F + ε by\n classical\n simpa only [hs, F_closed.measurableSet, SimpleFunc.coe_const, Function.const_apply,\n lintegral_const, ENNReal.coe_indicator, Set.univ_inter, MeasurableSet.univ,\n SimpleFunc.const_zero, lintegral_indicator, SimpleFunc.coe_zero,\n Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, Measure.restrict_apply]"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro.refine'_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\nthis : 0 < ε / ↑c\nF : Set α\nFs : F ⊆ s\nF_closed : IsClosed F\nμF : ↑↑μ s < ↑↑μ F + ε / ↑c\n⊢ ↑c * ↑↑μ s ≤ ↑c * ↑↑μ F + ε",
"tactic": "calc\n (c : ℝ≥0∞) * μ s ≤ c * (μ F + ε / c) := mul_le_mul_left' μF.le _\n _ = c * μ F + ε := by\n simp_rw [mul_add]\n rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]\n simpa using hc"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\nthis✝ : 0 < ε / ↑c\nF : Set α\nFs : F ⊆ s\nF_closed : IsClosed F\nμF : ↑↑μ s < ↑↑μ F + ε / ↑c\nthis : ↑c * ↑↑μ s ≤ ↑c * ↑↑μ F + ε\n⊢ (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≤\n (∫⁻ (x : α), ↑(Set.indicator F (fun x => c) x) ∂μ) + ε",
"tactic": "classical\nsimpa only [hs, F_closed.measurableSet, SimpleFunc.coe_const, Function.const_apply,\n lintegral_const, ENNReal.coe_indicator, Set.univ_inter, MeasurableSet.univ,\n SimpleFunc.const_zero, lintegral_indicator, SimpleFunc.coe_zero,\n Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, Measure.restrict_apply]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\nthis✝ : 0 < ε / ↑c\nF : Set α\nFs : F ⊆ s\nF_closed : IsClosed F\nμF : ↑↑μ s < ↑↑μ F + ε / ↑c\nthis : ↑c * ↑↑μ s ≤ ↑c * ↑↑μ F + ε\n⊢ (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≤\n (∫⁻ (x : α), ↑(Set.indicator F (fun x => c) x) ∂μ) + ε",
"tactic": "simpa only [hs, F_closed.measurableSet, SimpleFunc.coe_const, Function.const_apply,\n lintegral_const, ENNReal.coe_indicator, Set.univ_inter, MeasurableSet.univ,\n SimpleFunc.const_zero, lintegral_indicator, SimpleFunc.coe_zero,\n Set.piecewise_eq_indicator, SimpleFunc.coe_piecewise, Measure.restrict_apply]"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\nthis : 0 < ε / ↑c\nF : Set α\nFs : F ⊆ s\nF_closed : IsClosed F\nμF : ↑↑μ s < ↑↑μ F + ε / ↑c\n⊢ ↑c * ↑↑μ F + ↑c * (ε / ↑c) = ↑c * ↑↑μ F + ε",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\nthis : 0 < ε / ↑c\nF : Set α\nFs : F ⊆ s\nF_closed : IsClosed F\nμF : ↑↑μ s < ↑↑μ F + ε / ↑c\n⊢ ↑c * (↑↑μ F + ε / ↑c) = ↑c * ↑↑μ F + ε",
"tactic": "simp_rw [mul_add]"
},
{
"state_after": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\nthis : 0 < ε / ↑c\nF : Set α\nFs : F ⊆ s\nF_closed : IsClosed F\nμF : ↑↑μ s < ↑↑μ F + ε / ↑c\n⊢ ↑c ≠ 0",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\nthis : 0 < ε / ↑c\nF : Set α\nFs : F ⊆ s\nF_closed : IsClosed F\nμF : ↑↑μ s < ↑↑μ F + ε / ↑c\n⊢ ↑c * ↑↑μ F + ↑c * (ε / ↑c) = ↑c * ↑↑μ F + ε",
"tactic": "rw [ENNReal.mul_div_cancel' _ ENNReal.coe_ne_top]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nint_f : (∫⁻ (x : α), ↑(↑(piecewise s hs (const α c) (const α 0)) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nhc : ¬c = 0\nμs_lt_top : ↑↑μ s < ⊤\nthis : 0 < ε / ↑c\nF : Set α\nFs : F ⊆ s\nF_closed : IsClosed F\nμF : ↑↑μ s < ↑↑μ F + ε / ↑c\n⊢ ↑c ≠ 0",
"tactic": "simpa using hc"
},
{
"state_after": "case h_add\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nA : ((∫⁻ (x : α), ↑(↑f₁ x) ∂μ) + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤\n⊢ ∃ g,\n (∀ (x : α), g x ≤ ↑(f₁ + f₂) x) ∧\n UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"state_before": "case h_add\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\n⊢ ∃ g,\n (∀ (x : α), g x ≤ ↑(f₁ + f₂) x) ∧\n UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"tactic": "have A : ((∫⁻ x : α, f₁ x ∂μ) + ∫⁻ x : α, f₂ x ∂μ) ≠ ⊤ := by\n rwa [← lintegral_add_left f₁.measurable.coe_nnreal_ennreal]"
},
{
"state_after": "case h_add.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nA : ((∫⁻ (x : α), ↑(↑f₁ x) ∂μ) + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤\ng₁ : α → ℝ≥0\nf₁_le_g₁ : ∀ (x : α), g₁ x ≤ ↑f₁ x\ng₁cont : UpperSemicontinuous g₁\ng₁int : (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2\n⊢ ∃ g,\n (∀ (x : α), g x ≤ ↑(f₁ + f₂) x) ∧\n UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"state_before": "case h_add\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nA : ((∫⁻ (x : α), ↑(↑f₁ x) ∂μ) + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤\n⊢ ∃ g,\n (∀ (x : α), g x ≤ ↑(f₁ + f₂) x) ∧\n UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"tactic": "rcases h₁ (ENNReal.add_ne_top.1 A).1 (ENNReal.half_pos ε0).ne' with\n ⟨g₁, f₁_le_g₁, g₁cont, g₁int⟩"
},
{
"state_after": "case h_add.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nA : ((∫⁻ (x : α), ↑(↑f₁ x) ∂μ) + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤\ng₁ : α → ℝ≥0\nf₁_le_g₁ : ∀ (x : α), g₁ x ≤ ↑f₁ x\ng₁cont : UpperSemicontinuous g₁\ng₁int : (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2\ng₂ : α → ℝ≥0\nf₂_le_g₂ : ∀ (x : α), g₂ x ≤ ↑f₂ x\ng₂cont : UpperSemicontinuous g₂\ng₂int : (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₂ x) ∂μ) + ε / 2\n⊢ ∃ g,\n (∀ (x : α), g x ≤ ↑(f₁ + f₂) x) ∧\n UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"state_before": "case h_add.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nA : ((∫⁻ (x : α), ↑(↑f₁ x) ∂μ) + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤\ng₁ : α → ℝ≥0\nf₁_le_g₁ : ∀ (x : α), g₁ x ≤ ↑f₁ x\ng₁cont : UpperSemicontinuous g₁\ng₁int : (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2\n⊢ ∃ g,\n (∀ (x : α), g x ≤ ↑(f₁ + f₂) x) ∧\n UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"tactic": "rcases h₂ (ENNReal.add_ne_top.1 A).2 (ENNReal.half_pos ε0).ne' with\n ⟨g₂, f₂_le_g₂, g₂cont, g₂int⟩"
},
{
"state_after": "case h_add.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nA : ((∫⁻ (x : α), ↑(↑f₁ x) ∂μ) + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤\ng₁ : α → ℝ≥0\nf₁_le_g₁ : ∀ (x : α), g₁ x ≤ ↑f₁ x\ng₁cont : UpperSemicontinuous g₁\ng₁int : (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2\ng₂ : α → ℝ≥0\nf₂_le_g₂ : ∀ (x : α), g₂ x ≤ ↑f₂ x\ng₂cont : UpperSemicontinuous g₂\ng₂int : (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₂ x) ∂μ) + ε / 2\n⊢ (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≤ (∫⁻ (x : α), ↑((fun x => g₁ x + g₂ x) x) ∂μ) + ε",
"state_before": "case h_add.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nA : ((∫⁻ (x : α), ↑(↑f₁ x) ∂μ) + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤\ng₁ : α → ℝ≥0\nf₁_le_g₁ : ∀ (x : α), g₁ x ≤ ↑f₁ x\ng₁cont : UpperSemicontinuous g₁\ng₁int : (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2\ng₂ : α → ℝ≥0\nf₂_le_g₂ : ∀ (x : α), g₂ x ≤ ↑f₂ x\ng₂cont : UpperSemicontinuous g₂\ng₂int : (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₂ x) ∂μ) + ε / 2\n⊢ ∃ g,\n (∀ (x : α), g x ≤ ↑(f₁ + f₂) x) ∧\n UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε",
"tactic": "refine'\n ⟨fun x => g₁ x + g₂ x, fun x => add_le_add (f₁_le_g₁ x) (f₂_le_g₂ x), g₁cont.add g₂cont, _⟩"
},
{
"state_after": "case h_add.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nA : ((∫⁻ (x : α), ↑(↑f₁ x) ∂μ) + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤\ng₁ : α → ℝ≥0\nf₁_le_g₁ : ∀ (x : α), g₁ x ≤ ↑f₁ x\ng₁cont : UpperSemicontinuous g₁\ng₁int : (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2\ng₂ : α → ℝ≥0\nf₂_le_g₂ : ∀ (x : α), g₂ x ≤ ↑f₂ x\ng₂cont : UpperSemicontinuous g₂\ng₂int : (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₂ x) ∂μ) + ε / 2\n⊢ (∫⁻ (x : α), ↑(↑f₁ x) + ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₁ x) + ↑(g₂ x) ∂μ) + ε",
"state_before": "case h_add.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nA : ((∫⁻ (x : α), ↑(↑f₁ x) ∂μ) + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤\ng₁ : α → ℝ≥0\nf₁_le_g₁ : ∀ (x : α), g₁ x ≤ ↑f₁ x\ng₁cont : UpperSemicontinuous g₁\ng₁int : (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2\ng₂ : α → ℝ≥0\nf₂_le_g₂ : ∀ (x : α), g₂ x ≤ ↑f₂ x\ng₂cont : UpperSemicontinuous g₂\ng₂int : (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₂ x) ∂μ) + ε / 2\n⊢ (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≤ (∫⁻ (x : α), ↑((fun x => g₁ x + g₂ x) x) ∂μ) + ε",
"tactic": "simp only [SimpleFunc.coe_add, ENNReal.coe_add, Pi.add_apply]"
},
{
"state_after": "case h_add.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nA : ((∫⁻ (x : α), ↑(↑f₁ x) ∂μ) + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤\ng₁ : α → ℝ≥0\nf₁_le_g₁ : ∀ (x : α), g₁ x ≤ ↑f₁ x\ng₁cont : UpperSemicontinuous g₁\ng₁int : (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2\ng₂ : α → ℝ≥0\nf₂_le_g₂ : ∀ (x : α), g₂ x ≤ ↑f₂ x\ng₂cont : UpperSemicontinuous g₂\ng₂int : (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₂ x) ∂μ) + ε / 2\n⊢ ((∫⁻ (a : α), ↑(↑f₁ a) ∂μ) + ∫⁻ (a : α), ↑(↑f₂ a) ∂μ) ≤ ((∫⁻ (a : α), ↑(g₁ a) ∂μ) + ∫⁻ (a : α), ↑(g₂ a) ∂μ) + ε",
"state_before": "case h_add.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nA : ((∫⁻ (x : α), ↑(↑f₁ x) ∂μ) + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤\ng₁ : α → ℝ≥0\nf₁_le_g₁ : ∀ (x : α), g₁ x ≤ ↑f₁ x\ng₁cont : UpperSemicontinuous g₁\ng₁int : (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2\ng₂ : α → ℝ≥0\nf₂_le_g₂ : ∀ (x : α), g₂ x ≤ ↑f₂ x\ng₂cont : UpperSemicontinuous g₂\ng₂int : (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₂ x) ∂μ) + ε / 2\n⊢ (∫⁻ (x : α), ↑(↑f₁ x) + ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₁ x) + ↑(g₂ x) ∂μ) + ε",
"tactic": "rw [lintegral_add_left f₁.measurable.coe_nnreal_ennreal,\n lintegral_add_left g₁cont.measurable.coe_nnreal_ennreal]"
},
{
"state_after": "case h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nA : ((∫⁻ (x : α), ↑(↑f₁ x) ∂μ) + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤\ng₁ : α → ℝ≥0\nf₁_le_g₁ : ∀ (x : α), g₁ x ≤ ↑f₁ x\ng₁cont : UpperSemicontinuous g₁\ng₁int : (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2\ng₂ : α → ℝ≥0\nf₂_le_g₂ : ∀ (x : α), g₂ x ≤ ↑f₂ x\ng₂cont : UpperSemicontinuous g₂\ng₂int : (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₂ x) ∂μ) + ε / 2\n⊢ ((∫⁻ (a : α), ↑(g₁ a) ∂μ) + ∫⁻ (a : α), ↑(g₂ a) ∂μ) + ε =\n (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2 + ((∫⁻ (x : α), ↑(g₂ x) ∂μ) + ε / 2)",
"state_before": "case h_add.intro.intro.intro.intro.intro.intro\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nA : ((∫⁻ (x : α), ↑(↑f₁ x) ∂μ) + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤\ng₁ : α → ℝ≥0\nf₁_le_g₁ : ∀ (x : α), g₁ x ≤ ↑f₁ x\ng₁cont : UpperSemicontinuous g₁\ng₁int : (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2\ng₂ : α → ℝ≥0\nf₂_le_g₂ : ∀ (x : α), g₂ x ≤ ↑f₂ x\ng₂cont : UpperSemicontinuous g₂\ng₂int : (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₂ x) ∂μ) + ε / 2\n⊢ ((∫⁻ (a : α), ↑(↑f₁ a) ∂μ) + ∫⁻ (a : α), ↑(↑f₂ a) ∂μ) ≤ ((∫⁻ (a : α), ↑(g₁ a) ∂μ) + ∫⁻ (a : α), ↑(g₂ a) ∂μ) + ε",
"tactic": "convert add_le_add g₁int g₂int using 1"
},
{
"state_after": "case h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nA : ((∫⁻ (x : α), ↑(↑f₁ x) ∂μ) + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤\ng₁ : α → ℝ≥0\nf₁_le_g₁ : ∀ (x : α), g₁ x ≤ ↑f₁ x\ng₁cont : UpperSemicontinuous g₁\ng₁int : (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2\ng₂ : α → ℝ≥0\nf₂_le_g₂ : ∀ (x : α), g₂ x ≤ ↑f₂ x\ng₂cont : UpperSemicontinuous g₂\ng₂int : (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₂ x) ∂μ) + ε / 2\n⊢ ((∫⁻ (a : α), ↑(g₁ a) ∂μ) + ∫⁻ (a : α), ↑(g₂ a) ∂μ) + ε =\n (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2 + ((∫⁻ (x : α), ↑(g₂ x) ∂μ) + ε / 2)",
"state_before": "case h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nA : ((∫⁻ (x : α), ↑(↑f₁ x) ∂μ) + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤\ng₁ : α → ℝ≥0\nf₁_le_g₁ : ∀ (x : α), g₁ x ≤ ↑f₁ x\ng₁cont : UpperSemicontinuous g₁\ng₁int : (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2\ng₂ : α → ℝ≥0\nf₂_le_g₂ : ∀ (x : α), g₂ x ≤ ↑f₂ x\ng₂cont : UpperSemicontinuous g₂\ng₂int : (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₂ x) ∂μ) + ε / 2\n⊢ ((∫⁻ (a : α), ↑(g₁ a) ∂μ) + ∫⁻ (a : α), ↑(g₂ a) ∂μ) + ε =\n (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2 + ((∫⁻ (x : α), ↑(g₂ x) ∂μ) + ε / 2)",
"tactic": "simp only"
},
{
"state_after": "case h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nA : ((∫⁻ (x : α), ↑(↑f₁ x) ∂μ) + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤\ng₁ : α → ℝ≥0\nf₁_le_g₁ : ∀ (x : α), g₁ x ≤ ↑f₁ x\ng₁cont : UpperSemicontinuous g₁\ng₁int : (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2\ng₂ : α → ℝ≥0\nf₂_le_g₂ : ∀ (x : α), g₂ x ≤ ↑f₂ x\ng₂cont : UpperSemicontinuous g₂\ng₂int : (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₂ x) ∂μ) + ε / 2\n⊢ ((∫⁻ (a : α), ↑(g₁ a) ∂μ) + ∫⁻ (a : α), ↑(g₂ a) ∂μ) + (ε / 2 + ε / 2) =\n (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2 + ((∫⁻ (x : α), ↑(g₂ x) ∂μ) + ε / 2)",
"state_before": "case h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nA : ((∫⁻ (x : α), ↑(↑f₁ x) ∂μ) + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤\ng₁ : α → ℝ≥0\nf₁_le_g₁ : ∀ (x : α), g₁ x ≤ ↑f₁ x\ng₁cont : UpperSemicontinuous g₁\ng₁int : (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2\ng₂ : α → ℝ≥0\nf₂_le_g₂ : ∀ (x : α), g₂ x ≤ ↑f₂ x\ng₂cont : UpperSemicontinuous g₂\ng₂int : (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₂ x) ∂μ) + ε / 2\n⊢ ((∫⁻ (a : α), ↑(g₁ a) ∂μ) + ∫⁻ (a : α), ↑(g₂ a) ∂μ) + ε =\n (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2 + ((∫⁻ (x : α), ↑(g₂ x) ∂μ) + ε / 2)",
"tactic": "conv_lhs => rw [← ENNReal.add_halves ε]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\nA : ((∫⁻ (x : α), ↑(↑f₁ x) ∂μ) + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤\ng₁ : α → ℝ≥0\nf₁_le_g₁ : ∀ (x : α), g₁ x ≤ ↑f₁ x\ng₁cont : UpperSemicontinuous g₁\ng₁int : (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2\ng₂ : α → ℝ≥0\nf₂_le_g₂ : ∀ (x : α), g₂ x ≤ ↑f₂ x\ng₂cont : UpperSemicontinuous g₂\ng₂int : (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g₂ x) ∂μ) + ε / 2\n⊢ ((∫⁻ (a : α), ↑(g₁ a) ∂μ) + ∫⁻ (a : α), ↑(g₂ a) ∂μ) + (ε / 2 + ε / 2) =\n (∫⁻ (x : α), ↑(g₁ x) ∂μ) + ε / 2 + ((∫⁻ (x : α), ↑(g₂ x) ∂μ) + ε / 2)",
"tactic": "abel"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : WeaklyRegular μ\nf : α →ₛ ℝ≥0\nint_f✝ : (∫⁻ (x : α), ↑(↑f x) ∂μ) ≠ ⊤\nε✝ : ℝ≥0∞\nε0✝ : ε✝ ≠ 0\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ↑f₁) (Function.support ↑f₂)\nh₁ :\n (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₁ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₁ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nh₂ :\n (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 →\n ∃ g, (∀ (x : α), g x ≤ ↑f₂ x) ∧ UpperSemicontinuous g ∧ (∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≤ (∫⁻ (x : α), ↑(g x) ∂μ) + ε\nint_f : (∫⁻ (x : α), ↑(↑(f₁ + f₂) x) ∂μ) ≠ ⊤\nε : ℝ≥0∞\nε0 : ε ≠ 0\n⊢ ((∫⁻ (x : α), ↑(↑f₁ x) ∂μ) + ∫⁻ (x : α), ↑(↑f₂ x) ∂μ) ≠ ⊤",
"tactic": "rwa [← lintegral_add_left f₁.measurable.coe_nnreal_ennreal]"
}
] |
[
388,
9
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
330,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean
|
pow_eq_pow_mod
|
[
{
"state_after": "α : Type ?u.27674\nM✝ : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝² : Monoid M✝\ninst✝¹ : AddMonoid A\nM : Type u_1\ninst✝ : Monoid M\nx : M\nm n : ℕ\nh : x ^ n = 1\nt : x ^ m = x ^ (n * (m / n) + m % n)\n⊢ x ^ m = x ^ (m % n)",
"state_before": "α : Type ?u.27674\nM✝ : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝² : Monoid M✝\ninst✝¹ : AddMonoid A\nM : Type u_1\ninst✝ : Monoid M\nx : M\nm n : ℕ\nh : x ^ n = 1\n⊢ x ^ m = x ^ (m % n)",
"tactic": "have t : x ^ m = x ^ (n * (m / n) + m % n) :=\n congr_arg (fun a => x ^ a) ((Nat.add_comm _ _).trans (Nat.mod_add_div _ _)).symm"
},
{
"state_after": "α : Type ?u.27674\nM✝ : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝² : Monoid M✝\ninst✝¹ : AddMonoid A\nM : Type u_1\ninst✝ : Monoid M\nx : M\nm n : ℕ\nh : x ^ n = 1\nt : x ^ m = x ^ (n * (m / n) + m % n)\n⊢ x ^ m = x ^ (m % n)",
"state_before": "α : Type ?u.27674\nM✝ : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝² : Monoid M✝\ninst✝¹ : AddMonoid A\nM : Type u_1\ninst✝ : Monoid M\nx : M\nm n : ℕ\nh : x ^ n = 1\nt : x ^ m = x ^ (n * (m / n) + m % n)\n⊢ x ^ m = x ^ (m % n)",
"tactic": "dsimp at t"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.27674\nM✝ : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝² : Monoid M✝\ninst✝¹ : AddMonoid A\nM : Type u_1\ninst✝ : Monoid M\nx : M\nm n : ℕ\nh : x ^ n = 1\nt : x ^ m = x ^ (n * (m / n) + m % n)\n⊢ x ^ m = x ^ (m % n)",
"tactic": "rw [t, pow_add, pow_mul, h, one_pow, one_mul]"
}
] |
[
183,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/LinearAlgebra/SymplecticGroup.lean
|
SymplecticGroup.symplectic_det
|
[
{
"state_after": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ∈ symplecticGroup l R\n⊢ ∃ b, det A * b = 1",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ∈ symplecticGroup l R\n⊢ IsUnit (det A)",
"tactic": "rw [isUnit_iff_exists_inv]"
},
{
"state_after": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ∈ symplecticGroup l R\n⊢ det A * det A = 1",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ∈ symplecticGroup l R\n⊢ ∃ b, det A * b = 1",
"tactic": "use A.det"
},
{
"state_after": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ∈ symplecticGroup l R\n⊢ det (J l R) * (det A * det A) = det (J l R) * 1",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ∈ symplecticGroup l R\n⊢ det A * det A = 1",
"tactic": "refine' (isUnit_det_J l R).mul_left_cancel _"
},
{
"state_after": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ∈ symplecticGroup l R\n⊢ det (J l R) * (det A * det A) = det (J l R)",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ∈ symplecticGroup l R\n⊢ det (J l R) * (det A * det A) = det (J l R) * 1",
"tactic": "rw [mul_one]"
},
{
"state_after": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ⬝ J l R ⬝ Aᵀ = J l R\n⊢ det (J l R) * (det A * det A) = det (J l R)",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ∈ symplecticGroup l R\n⊢ det (J l R) * (det A * det A) = det (J l R)",
"tactic": "rw [mem_iff] at hA"
},
{
"state_after": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : det (A ⬝ J l R ⬝ Aᵀ) = det (J l R)\n⊢ det (J l R) * (det A * det A) = det (J l R)",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : A ⬝ J l R ⬝ Aᵀ = J l R\n⊢ det (J l R) * (det A * det A) = det (J l R)",
"tactic": "apply_fun det at hA"
},
{
"state_after": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : det A * det (J l R) * det A = det (J l R)\n⊢ det (J l R) * (det A * det A) = det (J l R)",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : det (A ⬝ J l R ⬝ Aᵀ) = det (J l R)\n⊢ det (J l R) * (det A * det A) = det (J l R)",
"tactic": "simp only [det_mul, det_transpose] at hA"
},
{
"state_after": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : det (J l R) * (det A * det A) = det (J l R)\n⊢ det (J l R) * (det A * det A) = det (J l R)",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : det A * det (J l R) * det A = det (J l R)\n⊢ det (J l R) * (det A * det A) = det (J l R)",
"tactic": "rw [mul_comm A.det, mul_assoc] at hA"
},
{
"state_after": "no goals",
"state_before": "l : Type u_1\nR : Type u_2\ninst✝² : DecidableEq l\ninst✝¹ : Fintype l\ninst✝ : CommRing R\nA : Matrix (l ⊕ l) (l ⊕ l) R\nhA : det (J l R) * (det A * det A) = det (J l R)\n⊢ det (J l R) * (det A * det A) = det (J l R)",
"tactic": "exact hA"
}
] |
[
154,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
145,
1
] |
Mathlib/Order/Atoms.lean
|
IsCompl.isCoatom_iff_isAtom
|
[] |
[
853,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
852,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.tan_eq_of_two_nsmul_eq
|
[
{
"state_after": "θ ψ : Angle\nh : θ = ψ ∨ θ = ψ + ↑π\n⊢ tan θ = tan ψ",
"state_before": "θ ψ : Angle\nh : 2 • θ = 2 • ψ\n⊢ tan θ = tan ψ",
"tactic": "rw [two_nsmul_eq_iff] at h"
},
{
"state_after": "case inl\nθ : Angle\n⊢ tan θ = tan θ\n\ncase inr\nψ : Angle\n⊢ tan (ψ + ↑π) = tan ψ",
"state_before": "θ ψ : Angle\nh : θ = ψ ∨ θ = ψ + ↑π\n⊢ tan θ = tan ψ",
"tactic": "rcases h with (rfl | rfl)"
},
{
"state_after": "no goals",
"state_before": "case inl\nθ : Angle\n⊢ tan θ = tan θ",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case inr\nψ : Angle\n⊢ tan (ψ + ↑π) = tan ψ",
"tactic": "exact tan_add_pi _"
}
] |
[
822,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
818,
1
] |
Mathlib/Combinatorics/Partition.lean
|
Nat.Partition.count_ofSums_zero
|
[] |
[
131,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/Order/Hom/CompleteLattice.lean
|
sInfHom.coe_copy
|
[] |
[
425,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
424,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Submodule.mem_map
|
[] |
[
713,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
712,
1
] |
Mathlib/Data/Set/Semiring.lean
|
SetSemiring.mul_def
|
[] |
[
154,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
153,
1
] |
Mathlib/Topology/Spectral/Hom.lean
|
SpectralMap.id_comp
|
[] |
[
213,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
212,
1
] |
Mathlib/Logic/Embedding/Basic.lean
|
Equiv.embeddingCongr_apply_trans
|
[
{
"state_after": "case h\nα₁ : Sort u_1\nβ₁ : Sort u_2\nγ₁ : Sort u_3\nα₂ : Sort u_4\nβ₂ : Sort u_5\nγ₂ : Sort u_6\nea : α₁ ≃ α₂\neb : β₁ ≃ β₂\nec : γ₁ ≃ γ₂\nf : α₁ ↪ β₁\ng : β₁ ↪ γ₁\nx✝ : α₂\n⊢ ↑(↑(embeddingCongr ea ec) (Embedding.trans f g)) x✝ =\n ↑(Embedding.trans (↑(embeddingCongr ea eb) f) (↑(embeddingCongr eb ec) g)) x✝",
"state_before": "α₁ : Sort u_1\nβ₁ : Sort u_2\nγ₁ : Sort u_3\nα₂ : Sort u_4\nβ₂ : Sort u_5\nγ₂ : Sort u_6\nea : α₁ ≃ α₂\neb : β₁ ≃ β₂\nec : γ₁ ≃ γ₂\nf : α₁ ↪ β₁\ng : β₁ ↪ γ₁\n⊢ ↑(embeddingCongr ea ec) (Embedding.trans f g) =\n Embedding.trans (↑(embeddingCongr ea eb) f) (↑(embeddingCongr eb ec) g)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nα₁ : Sort u_1\nβ₁ : Sort u_2\nγ₁ : Sort u_3\nα₂ : Sort u_4\nβ₂ : Sort u_5\nγ₂ : Sort u_6\nea : α₁ ≃ α₂\neb : β₁ ≃ β₂\nec : γ₁ ≃ γ₂\nf : α₁ ↪ β₁\ng : β₁ ↪ γ₁\nx✝ : α₂\n⊢ ↑(↑(embeddingCongr ea ec) (Embedding.trans f g)) x✝ =\n ↑(Embedding.trans (↑(embeddingCongr ea eb) f) (↑(embeddingCongr eb ec) g)) x✝",
"tactic": "simp"
}
] |
[
452,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
447,
1
] |
Mathlib/Data/Nat/Cast/Defs.lean
|
Nat.cast_eq_ofNat
|
[] |
[
62,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Order/RelIso/Basic.lean
|
RelIso.symm_apply_apply
|
[] |
[
791,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
790,
1
] |
Mathlib/Analysis/InnerProductSpace/Orthogonal.lean
|
Submodule.mem_orthogonal'
|
[
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\nE : Type u_1\nF : Type ?u.19423\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nK : Submodule 𝕜 E\nv : E\n⊢ v ∈ Kᗮ ↔ ∀ (u : E), u ∈ K → inner v u = 0",
"tactic": "simp_rw [mem_orthogonal, inner_eq_zero_symm]"
}
] |
[
63,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
62,
1
] |
Mathlib/Algebra/Order/Hom/Ring.lean
|
OrderRingIso.trans_apply
|
[] |
[
495,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
494,
1
] |
Mathlib/MeasureTheory/Integral/IntervalIntegral.lean
|
intervalIntegral.integral_div
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.13479101\n𝕜✝ : Type ?u.13479104\nE : Type ?u.13479107\nF : Type ?u.13479110\nA : Type ?u.13479113\ninst✝³ : NormedAddCommGroup E\ninst✝² : CompleteSpace E\ninst✝¹ : NormedSpace ℝ E\na b : ℝ\nf✝ g : ℝ → E\nμ : MeasureTheory.Measure ℝ\n𝕜 : Type u_1\ninst✝ : IsROrC 𝕜\nr : 𝕜\nf : ℝ → 𝕜\n⊢ (∫ (x : ℝ) in a..b, f x / r ∂μ) = (∫ (x : ℝ) in a..b, f x ∂μ) / r",
"tactic": "simpa only [div_eq_mul_inv] using integral_mul_const r⁻¹ f"
}
] |
[
631,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
629,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.pairwise_zero
|
[] |
[
3003,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3002,
1
] |
Mathlib/Topology/List.lean
|
Vector.tendsto_insertNth
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.65317\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nn : ℕ\ni : Fin (n + 1)\na : α\nl : List α\nhl : List.length l = n\n⊢ Tendsto (fun x => ↑(insertNth x.fst i x.snd)) (𝓝 a ×ˢ 𝓝 { val := l, property := hl })\n (𝓝\n ↑{ val := List.insertNth (↑i) a ↑{ val := l, property := hl },\n property := (_ : List.length (List.insertNth (↑i) a ↑{ val := l, property := hl }) = n + 1) })",
"state_before": "α : Type u_1\nβ : Type ?u.65317\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nn : ℕ\ni : Fin (n + 1)\na : α\nl : List α\nhl : List.length l = n\n⊢ Tendsto (fun p => insertNth p.fst i p.snd) (𝓝 a ×ˢ 𝓝 { val := l, property := hl })\n (𝓝 (insertNth a i { val := l, property := hl }))",
"tactic": "rw [insertNth, tendsto_subtype_rng]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.65317\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nn : ℕ\ni : Fin (n + 1)\na : α\nl : List α\nhl : List.length l = n\n⊢ Tendsto (fun x => List.insertNth (↑i) x.fst ↑x.snd) (𝓝 a ×ˢ 𝓝 { val := l, property := hl })\n (𝓝 (List.insertNth (↑i) a l))",
"state_before": "α : Type u_1\nβ : Type ?u.65317\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nn : ℕ\ni : Fin (n + 1)\na : α\nl : List α\nhl : List.length l = n\n⊢ Tendsto (fun x => ↑(insertNth x.fst i x.snd)) (𝓝 a ×ˢ 𝓝 { val := l, property := hl })\n (𝓝\n ↑{ val := List.insertNth (↑i) a ↑{ val := l, property := hl },\n property := (_ : List.length (List.insertNth (↑i) a ↑{ val := l, property := hl }) = n + 1) })",
"tactic": "simp [insertNth_val]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.65317\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nn : ℕ\ni : Fin (n + 1)\na : α\nl : List α\nhl : List.length l = n\n⊢ Tendsto (fun x => List.insertNth (↑i) x.fst ↑x.snd) (𝓝 a ×ˢ 𝓝 { val := l, property := hl })\n (𝓝 (List.insertNth (↑i) a l))",
"tactic": "exact List.tendsto_insertNth tendsto_fst (Tendsto.comp continuousAt_subtype_val tendsto_snd : _)"
}
] |
[
204,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
198,
1
] |
Mathlib/Algebra/GroupWithZero/Basic.lean
|
div_zero
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.23833\nM₀ : Type ?u.23836\nG₀ : Type u_1\nM₀' : Type ?u.23842\nG₀' : Type ?u.23845\nF : Type ?u.23848\nF' : Type ?u.23851\ninst✝ : GroupWithZero G₀\na✝ b c a : G₀\n⊢ a / 0 = 0",
"tactic": "rw [div_eq_mul_inv, inv_zero, mul_zero]"
}
] |
[
339,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
339,
1
] |
Mathlib/Data/Stream/Init.lean
|
Stream'.nth_succ_iterate'
|
[] |
[
274,
64
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
273,
1
] |
Mathlib/Data/Nat/ModEq.lean
|
Nat.mod_mul_left_mod
|
[] |
[
407,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
406,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Matrix.lean
|
AffineBasis.toMatrix_vecMul_coords
|
[
{
"state_after": "case h\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : Ring k\ninst✝² : Module k V\nb : AffineBasis ι k P\nι' : Type ?u.37207\ninst✝¹ : Fintype ι'\ninst✝ : Fintype ι\nb₂ : AffineBasis ι k P\nx : P\nj : ι\n⊢ vecMul (↑(coords b₂) x) (toMatrix b ↑b₂) j = ↑(coords b) x j",
"state_before": "ι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : Ring k\ninst✝² : Module k V\nb : AffineBasis ι k P\nι' : Type ?u.37207\ninst✝¹ : Fintype ι'\ninst✝ : Fintype ι\nb₂ : AffineBasis ι k P\nx : P\n⊢ vecMul (↑(coords b₂) x) (toMatrix b ↑b₂) = ↑(coords b) x",
"tactic": "ext j"
},
{
"state_after": "case h\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : Ring k\ninst✝² : Module k V\nb : AffineBasis ι k P\nι' : Type ?u.37207\ninst✝¹ : Fintype ι'\ninst✝ : Fintype ι\nb₂ : AffineBasis ι k P\nx : P\nj : ι\n⊢ vecMul (↑(coords b₂) x) (toMatrix b ↑b₂) j = ↑(coord b j) x",
"state_before": "case h\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : Ring k\ninst✝² : Module k V\nb : AffineBasis ι k P\nι' : Type ?u.37207\ninst✝¹ : Fintype ι'\ninst✝ : Fintype ι\nb₂ : AffineBasis ι k P\nx : P\nj : ι\n⊢ vecMul (↑(coords b₂) x) (toMatrix b ↑b₂) j = ↑(coords b) x j",
"tactic": "change _ = b.coord j x"
},
{
"state_after": "case h\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : Ring k\ninst✝² : Module k V\nb : AffineBasis ι k P\nι' : Type ?u.37207\ninst✝¹ : Fintype ι'\ninst✝ : Fintype ι\nb₂ : AffineBasis ι k P\nx : P\nj : ι\n⊢ vecMul (↑(coords b₂) x) (toMatrix b ↑b₂) j =\n ↑(coord b j) (↑(Finset.affineCombination k Finset.univ ↑b₂) fun i => ↑(coord b₂ i) x)",
"state_before": "case h\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : Ring k\ninst✝² : Module k V\nb : AffineBasis ι k P\nι' : Type ?u.37207\ninst✝¹ : Fintype ι'\ninst✝ : Fintype ι\nb₂ : AffineBasis ι k P\nx : P\nj : ι\n⊢ vecMul (↑(coords b₂) x) (toMatrix b ↑b₂) j = ↑(coord b j) x",
"tactic": "conv_rhs => rw [← b₂.affineCombination_coord_eq_self x]"
},
{
"state_after": "case h\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : Ring k\ninst✝² : Module k V\nb : AffineBasis ι k P\nι' : Type ?u.37207\ninst✝¹ : Fintype ι'\ninst✝ : Fintype ι\nb₂ : AffineBasis ι k P\nx : P\nj : ι\n⊢ vecMul (↑(coords b₂) x) (toMatrix b ↑b₂) j =\n ↑(Finset.affineCombination ((fun a => k) x) Finset.univ (↑(coord b j) ∘ ↑b₂)) fun i => ↑(coord b₂ i) x",
"state_before": "case h\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : Ring k\ninst✝² : Module k V\nb : AffineBasis ι k P\nι' : Type ?u.37207\ninst✝¹ : Fintype ι'\ninst✝ : Fintype ι\nb₂ : AffineBasis ι k P\nx : P\nj : ι\n⊢ vecMul (↑(coords b₂) x) (toMatrix b ↑b₂) j =\n ↑(coord b j) (↑(Finset.affineCombination k Finset.univ ↑b₂) fun i => ↑(coord b₂ i) x)",
"tactic": "rw [Finset.map_affineCombination _ _ _ (b₂.sum_coord_apply_eq_one x)]"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u₁\nk : Type u₂\nV : Type u₃\nP : Type u₄\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : AffineSpace V P\ninst✝³ : Ring k\ninst✝² : Module k V\nb : AffineBasis ι k P\nι' : Type ?u.37207\ninst✝¹ : Fintype ι'\ninst✝ : Fintype ι\nb₂ : AffineBasis ι k P\nx : P\nj : ι\n⊢ vecMul (↑(coords b₂) x) (toMatrix b ↑b₂) j =\n ↑(Finset.affineCombination ((fun a => k) x) Finset.univ (↑(coord b j) ∘ ↑b₂)) fun i => ↑(coord b₂ i) x",
"tactic": "simp [Matrix.vecMul, Matrix.dotProduct, toMatrix_apply, coords]"
}
] |
[
118,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
113,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.top_prod_top
|
[] |
[
1753,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1752,
1
] |
Mathlib/Data/List/Basic.lean
|
List.reduceOption_nil
|
[] |
[
3430,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
3429,
1
] |
Mathlib/RingTheory/WittVector/MulP.lean
|
WittVector.mulN_isPoly
|
[
{
"state_after": "case h\np : ℕ\nR✝ : Type ?u.50758\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R✝\nn : ℕ\nR : Type u_1\n_Rcr : CommRing R\nx : 𝕎 R\nk : ℕ\n⊢ coeff (x * ↑n) k = ↑(aeval x.coeff) (wittMulN p n k)",
"state_before": "p : ℕ\nR✝ : Type ?u.50758\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R✝\nn : ℕ\nR : Type u_1\n_Rcr : CommRing R\nx : 𝕎 R\n⊢ (x * ↑n).coeff = fun n_1 => ↑(aeval x.coeff) (wittMulN p n n_1)",
"tactic": "funext k"
},
{
"state_after": "no goals",
"state_before": "case h\np : ℕ\nR✝ : Type ?u.50758\nhp : Fact (Nat.Prime p)\ninst✝ : CommRing R✝\nn : ℕ\nR : Type u_1\n_Rcr : CommRing R\nx : 𝕎 R\nk : ℕ\n⊢ coeff (x * ↑n) k = ↑(aeval x.coeff) (wittMulN p n k)",
"tactic": "exact mulN_coeff n x k"
}
] |
[
72,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
71,
1
] |
Mathlib/Order/SuccPred/Basic.lean
|
Order.wcovby_succ
|
[] |
[
246,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/Data/Fintype/Prod.lean
|
Set.toFinset_prod
|
[
{
"state_after": "case a\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.32\ns✝ t✝ s : Set α\nt : Set β\ninst✝² : Fintype ↑s\ninst✝¹ : Fintype ↑t\ninst✝ : Fintype ↑(s ×ˢ t)\na✝ : α × β\n⊢ a✝ ∈ toFinset (s ×ˢ t) ↔ a✝ ∈ toFinset s ×ˢ toFinset t",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.32\ns✝ t✝ s : Set α\nt : Set β\ninst✝² : Fintype ↑s\ninst✝¹ : Fintype ↑t\ninst✝ : Fintype ↑(s ×ˢ t)\n⊢ toFinset (s ×ˢ t) = toFinset s ×ˢ toFinset t",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.32\ns✝ t✝ s : Set α\nt : Set β\ninst✝² : Fintype ↑s\ninst✝¹ : Fintype ↑t\ninst✝ : Fintype ↑(s ×ˢ t)\na✝ : α × β\n⊢ a✝ ∈ toFinset (s ×ˢ t) ↔ a✝ ∈ toFinset s ×ˢ toFinset t",
"tactic": "simp"
}
] |
[
37,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
34,
1
] |
Mathlib/CategoryTheory/Types.lean
|
CategoryTheory.types_comp_apply
|
[] |
[
80,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
79,
1
] |
Mathlib/RingTheory/EuclideanDomain.lean
|
right_div_gcd_ne_zero
|
[
{
"state_after": "case intro\nR : Type u_1\ninst✝¹ : EuclideanDomain R\ninst✝ : GCDMonoid R\np q : R\nhq : q ≠ 0\nr : R\nhr : q = GCDMonoid.gcd p q * r\n⊢ q / GCDMonoid.gcd p q ≠ 0",
"state_before": "R : Type u_1\ninst✝¹ : EuclideanDomain R\ninst✝ : GCDMonoid R\np q : R\nhq : q ≠ 0\n⊢ q / GCDMonoid.gcd p q ≠ 0",
"tactic": "obtain ⟨r, hr⟩ := GCDMonoid.gcd_dvd_right p q"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝¹ : EuclideanDomain R\ninst✝ : GCDMonoid R\np q : R\nhq : q ≠ 0\nr : R\nhr : q = GCDMonoid.gcd p q * r\npq0 : GCDMonoid.gcd p q ≠ 0\nr0 : r ≠ 0\n⊢ q / GCDMonoid.gcd p q ≠ 0",
"state_before": "case intro\nR : Type u_1\ninst✝¹ : EuclideanDomain R\ninst✝ : GCDMonoid R\np q : R\nhq : q ≠ 0\nr : R\nhr : q = GCDMonoid.gcd p q * r\n⊢ q / GCDMonoid.gcd p q ≠ 0",
"tactic": "obtain ⟨pq0, r0⟩ : GCDMonoid.gcd p q ≠ 0 ∧ r ≠ 0 := mul_ne_zero_iff.mp (hr ▸ hq)"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝¹ : EuclideanDomain R\ninst✝ : GCDMonoid R\np q : R\nhq : q ≠ 0\nr : R\nhr : q = GCDMonoid.gcd p q * r\npq0 : GCDMonoid.gcd p q ≠ 0\nr0 : r ≠ 0\n⊢ GCDMonoid.gcd p q * r / GCDMonoid.gcd p q ≠ 0",
"state_before": "case intro.intro\nR : Type u_1\ninst✝¹ : EuclideanDomain R\ninst✝ : GCDMonoid R\np q : R\nhq : q ≠ 0\nr : R\nhr : q = GCDMonoid.gcd p q * r\npq0 : GCDMonoid.gcd p q ≠ 0\nr0 : r ≠ 0\n⊢ q / GCDMonoid.gcd p q ≠ 0",
"tactic": "nth_rw 1 [hr]"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝¹ : EuclideanDomain R\ninst✝ : GCDMonoid R\np q : R\nhq : q ≠ 0\nr : R\nhr : q = GCDMonoid.gcd p q * r\npq0 : GCDMonoid.gcd p q ≠ 0\nr0 : r ≠ 0\n⊢ r ≠ 0",
"state_before": "case intro.intro\nR : Type u_1\ninst✝¹ : EuclideanDomain R\ninst✝ : GCDMonoid R\np q : R\nhq : q ≠ 0\nr : R\nhr : q = GCDMonoid.gcd p q * r\npq0 : GCDMonoid.gcd p q ≠ 0\nr0 : r ≠ 0\n⊢ GCDMonoid.gcd p q * r / GCDMonoid.gcd p q ≠ 0",
"tactic": "rw [mul_comm, EuclideanDomain.mul_div_cancel _ pq0]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u_1\ninst✝¹ : EuclideanDomain R\ninst✝ : GCDMonoid R\np q : R\nhq : q ≠ 0\nr : R\nhr : q = GCDMonoid.gcd p q * r\npq0 : GCDMonoid.gcd p q ≠ 0\nr0 : r ≠ 0\n⊢ r ≠ 0",
"tactic": "exact r0"
}
] |
[
60,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean
|
LocalHomeomorph.continuousAt_extend
|
[] |
[
833,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
832,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.sum_repr
|
[] |
[
933,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
932,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.mk_finsupp_of_fintype
|
[
{
"state_after": "no goals",
"state_before": "α✝ β✝ α β : Type u\ninst✝¹ : Fintype α\ninst✝ : Zero β\n⊢ (#α →₀ β) = (#β) ^ Fintype.card α",
"tactic": "simp"
}
] |
[
1333,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1332,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.sinh_sub_cosh
|
[
{
"state_after": "no goals",
"state_before": "x y : ℂ\n⊢ sinh x - cosh x = -exp (-x)",
"tactic": "rw [← neg_sub, cosh_sub_sinh]"
}
] |
[
751,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
751,
1
] |
Mathlib/Data/List/Indexes.lean
|
List.foldrIdx_eq_foldrIdxSpec
|
[
{
"state_after": "case nil\nα : Type u\nβ : Type v\nf : ℕ → α → β → β\nb : β\nstart : ℕ\n⊢ foldrIdx f b [] start = foldrIdxSpec f b [] start\n\ncase cons\nα : Type u\nβ : Type v\nf : ℕ → α → β → β\nb : β\nhead✝ : α\ntail✝ : List α\ntail_ih✝ : ∀ (start : ℕ), foldrIdx f b tail✝ start = foldrIdxSpec f b tail✝ start\nstart : ℕ\n⊢ foldrIdx f b (head✝ :: tail✝) start = foldrIdxSpec f b (head✝ :: tail✝) start",
"state_before": "α : Type u\nβ : Type v\nf : ℕ → α → β → β\nb : β\nas : List α\nstart : ℕ\n⊢ foldrIdx f b as start = foldrIdxSpec f b as start",
"tactic": "induction as generalizing start"
},
{
"state_after": "no goals",
"state_before": "case nil\nα : Type u\nβ : Type v\nf : ℕ → α → β → β\nb : β\nstart : ℕ\n⊢ foldrIdx f b [] start = foldrIdxSpec f b [] start",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u\nβ : Type v\nf : ℕ → α → β → β\nb : β\nhead✝ : α\ntail✝ : List α\ntail_ih✝ : ∀ (start : ℕ), foldrIdx f b tail✝ start = foldrIdxSpec f b tail✝ start\nstart : ℕ\n⊢ foldrIdx f b (head✝ :: tail✝) start = foldrIdxSpec f b (head✝ :: tail✝) start",
"tactic": "simp only [foldrIdx, foldrIdxSpec_cons, *]"
}
] |
[
231,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
227,
1
] |
Mathlib/Data/Real/NNReal.lean
|
Real.toNNReal_prod_of_nonneg
|
[
{
"state_after": "α : Type u_1\ns : Finset α\nf : α → ℝ\nhf : ∀ (a : α), a ∈ s → 0 ≤ f a\n⊢ ∏ i in s, f i = ∏ a in s, ↑(Real.toNNReal (f a))",
"state_before": "α : Type u_1\ns : Finset α\nf : α → ℝ\nhf : ∀ (a : α), a ∈ s → 0 ≤ f a\n⊢ Real.toNNReal (∏ a in s, f a) = ∏ a in s, Real.toNNReal (f a)",
"tactic": "rw [← NNReal.coe_eq, NNReal.coe_prod, Real.coe_toNNReal _ (Finset.prod_nonneg hf)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns : Finset α\nf : α → ℝ\nhf : ∀ (a : α), a ∈ s → 0 ≤ f a\n⊢ ∏ i in s, f i = ∏ a in s, ↑(Real.toNNReal (f a))",
"tactic": "exact Finset.prod_congr rfl fun x hxs => by rw [Real.coe_toNNReal _ (hf x hxs)]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ns : Finset α\nf : α → ℝ\nhf : ∀ (a : α), a ∈ s → 0 ≤ f a\nx : α\nhxs : x ∈ s\n⊢ f x = ↑(Real.toNNReal (f x))",
"tactic": "rw [Real.coe_toNNReal _ (hf x hxs)]"
}
] |
[
349,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
345,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.Walk.reverse_map
|
[
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\nf : G →g G'\nf' : G' →g G''\nu v u' v' : V\np : Walk G u v\n⊢ reverse (Walk.map f p) = Walk.map f (reverse p)",
"tactic": "induction p <;> simp [map_append, *]"
}
] |
[
1507,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1507,
1
] |
Std/Data/String/Lemmas.lean
|
String.set_of_valid
|
[] |
[
247,
66
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
245,
1
] |
Mathlib/MeasureTheory/Integral/Layercake.lean
|
MeasureTheory.lintegral_comp_eq_lintegral_meas_le_mul_of_measurable
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\n⊢ (∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) ∂μ) =\n ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\n⊢ (∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) ∂μ) =\n ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)",
"tactic": "have integrand_eq :\n ∀ ω, ENNReal.ofReal (∫ t in (0)..f ω, g t) = ∫⁻ t in Ioc 0 (f ω), ENNReal.ofReal (g t) := by\n intro ω\n have g_ae_nn : 0 ≤ᵐ[volume.restrict (Ioc 0 (f ω))] g := by\n filter_upwards [self_mem_ae_restrict (measurableSet_Ioc : MeasurableSet (Ioc 0 (f ω)))] with x\n hx using g_nn x hx.1\n rw [← ofReal_integral_eq_lintegral_ofReal (g_intble' (f ω) (f_nn ω)).1 g_ae_nn]\n congr\n exact intervalIntegral.integral_of_le (f_nn ω)"
},
{
"state_after": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\n⊢ (∫⁻ (ω : α), ∫⁻ (a : ℝ), indicator (Ioc 0 (f ω)) (fun t => ENNReal.ofReal (g t)) a ∂μ) =\n ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\n⊢ (∫⁻ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) ∂μ) =\n ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)",
"tactic": "simp_rw [integrand_eq, ← lintegral_indicator (fun t => ENNReal.ofReal (g t)) measurableSet_Ioc]"
},
{
"state_after": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\n⊢ (∫⁻ (ω : α), ∫⁻ (a : ℝ), indicator (Ioc 0 (f ω)) (fun t => ENNReal.ofReal (g t)) a ∂μ) =\n ∫⁻ (a : ℝ), indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) a",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\n⊢ (∫⁻ (ω : α), ∫⁻ (a : ℝ), indicator (Ioc 0 (f ω)) (fun t => ENNReal.ofReal (g t)) a ∂μ) =\n ∫⁻ (t : ℝ) in Ioi 0, ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)",
"tactic": "rw [← lintegral_indicator _ measurableSet_Ioi]"
},
{
"state_after": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\n⊢ (∫⁻ (y : ℝ), ∫⁻ (x : α), indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y ∂μ) =\n ∫⁻ (a : ℝ), indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) a\n\ncase hf\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\n⊢ AEMeasurable (Function.uncurry fun ω a => indicator (Ioc 0 (f ω)) (fun t => ENNReal.ofReal (g t)) a)",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\n⊢ (∫⁻ (ω : α), ∫⁻ (a : ℝ), indicator (Ioc 0 (f ω)) (fun t => ENNReal.ofReal (g t)) a ∂μ) =\n ∫⁻ (a : ℝ), indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) a",
"tactic": "rw [lintegral_lintegral_swap]"
},
{
"state_after": "case hf\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\naux₂ :\n (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd)\n⊢ AEMeasurable (indicator {p | p.snd ∈ Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd))",
"state_before": "case hf\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\naux₂ :\n (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd)\n⊢ AEMeasurable (Function.uncurry fun ω a => indicator (Ioc 0 (f ω)) (fun t => ENNReal.ofReal (g t)) a)",
"tactic": "rw [aux₂]"
},
{
"state_after": "case hf\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\naux₂ :\n (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd)\nmble : MeasurableSet {p | p.fst ∈ univ ∧ p.snd ∈ Ioc (OfNat.ofNat 0 p.fst) (f p.fst)}\n⊢ AEMeasurable (indicator {p | p.snd ∈ Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd))",
"state_before": "case hf\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\naux₂ :\n (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd)\n⊢ AEMeasurable (indicator {p | p.snd ∈ Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd))",
"tactic": "have mble := measurableSet_region_between_oc measurable_zero f_mble MeasurableSet.univ"
},
{
"state_after": "case hf\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\naux₂ :\n (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd)\nmble : MeasurableSet {p | p.snd ∈ Ioc 0 (f p.fst)}\n⊢ AEMeasurable (indicator {p | p.snd ∈ Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd))",
"state_before": "case hf\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\naux₂ :\n (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd)\nmble : MeasurableSet {p | p.fst ∈ univ ∧ p.snd ∈ Ioc (OfNat.ofNat 0 p.fst) (f p.fst)}\n⊢ AEMeasurable (indicator {p | p.snd ∈ Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd))",
"tactic": "simp_rw [mem_univ, Pi.zero_apply, true_and_iff] at mble"
},
{
"state_after": "no goals",
"state_before": "case hf\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\naux₂ :\n (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd)\nmble : MeasurableSet {p | p.snd ∈ Ioc 0 (f p.fst)}\n⊢ AEMeasurable (indicator {p | p.snd ∈ Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd))",
"tactic": "exact (ENNReal.measurable_ofReal.comp (g_mble.comp measurable_snd)).aemeasurable.indicator mble"
},
{
"state_after": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\nt : ℝ\nht : 0 ≤ t\n⊢ IntervalIntegrable g volume 0 t",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\n⊢ ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t",
"tactic": "intro t ht"
},
{
"state_after": "case inl\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\nt : ℝ\nht : 0 ≤ t\nh : 0 = t\n⊢ IntervalIntegrable g volume 0 t\n\ncase inr\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\nt : ℝ\nht : 0 ≤ t\nh : 0 < t\n⊢ IntervalIntegrable g volume 0 t",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\nt : ℝ\nht : 0 ≤ t\n⊢ IntervalIntegrable g volume 0 t",
"tactic": "cases' eq_or_lt_of_le ht with h h"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\nt : ℝ\nht : 0 ≤ t\nh : 0 = t\n⊢ IntervalIntegrable g volume 0 t",
"tactic": "simp [← h]"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\nt : ℝ\nht : 0 ≤ t\nh : 0 < t\n⊢ IntervalIntegrable g volume 0 t",
"tactic": "exact g_intble t h"
},
{
"state_after": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nω : α\n⊢ ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\n⊢ ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)",
"tactic": "intro ω"
},
{
"state_after": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nω : α\ng_ae_nn : 0 ≤ᵐ[Measure.restrict volume (Ioc 0 (f ω))] g\n⊢ ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nω : α\n⊢ ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)",
"tactic": "have g_ae_nn : 0 ≤ᵐ[volume.restrict (Ioc 0 (f ω))] g := by\n filter_upwards [self_mem_ae_restrict (measurableSet_Ioc : MeasurableSet (Ioc 0 (f ω)))] with x\n hx using g_nn x hx.1"
},
{
"state_after": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nω : α\ng_ae_nn : 0 ≤ᵐ[Measure.restrict volume (Ioc 0 (f ω))] g\n⊢ ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ENNReal.ofReal (∫ (x : ℝ) in Ioc 0 (f ω), g x)",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nω : α\ng_ae_nn : 0 ≤ᵐ[Measure.restrict volume (Ioc 0 (f ω))] g\n⊢ ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)",
"tactic": "rw [← ofReal_integral_eq_lintegral_ofReal (g_intble' (f ω) (f_nn ω)).1 g_ae_nn]"
},
{
"state_after": "case e_r\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nω : α\ng_ae_nn : 0 ≤ᵐ[Measure.restrict volume (Ioc 0 (f ω))] g\n⊢ (∫ (t : ℝ) in 0 ..f ω, g t) = ∫ (x : ℝ) in Ioc 0 (f ω), g x",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nω : α\ng_ae_nn : 0 ≤ᵐ[Measure.restrict volume (Ioc 0 (f ω))] g\n⊢ ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ENNReal.ofReal (∫ (x : ℝ) in Ioc 0 (f ω), g x)",
"tactic": "congr"
},
{
"state_after": "no goals",
"state_before": "case e_r\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nω : α\ng_ae_nn : 0 ≤ᵐ[Measure.restrict volume (Ioc 0 (f ω))] g\n⊢ (∫ (t : ℝ) in 0 ..f ω, g t) = ∫ (x : ℝ) in Ioc 0 (f ω), g x",
"tactic": "exact intervalIntegral.integral_of_le (f_nn ω)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nω : α\n⊢ 0 ≤ᵐ[Measure.restrict volume (Ioc 0 (f ω))] g",
"tactic": "filter_upwards [self_mem_ae_restrict (measurableSet_Ioc : MeasurableSet (Ioc 0 (f ω)))] with x\n hx using g_nn x hx.1"
},
{
"state_after": "case h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\n⊢ (fun y => ∫⁻ (x : α), indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y ∂μ) = fun a =>\n indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) a",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\n⊢ (∫⁻ (y : ℝ), ∫⁻ (x : α), indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y ∂μ) =\n ∫⁻ (a : ℝ), indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) a",
"tactic": "apply congr_arg"
},
{
"state_after": "case h.h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\n⊢ (∫⁻ (x : α), indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s ∂μ) =\n indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) s",
"state_before": "case h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\n⊢ (fun y => ∫⁻ (x : α), indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y ∂μ) = fun a =>\n indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) a",
"tactic": "funext s"
},
{
"state_after": "case h.h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ (∫⁻ (x : α), ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) ∂μ) =\n indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) s",
"state_before": "case h.h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ (∫⁻ (x : α), indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s ∂μ) =\n indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) s",
"tactic": "simp_rw [aux₁]"
},
{
"state_after": "case h.h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ (ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * ∫⁻ (a : α), indicator (Ici s) (fun x => 1) (f a) ∂μ) =\n indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) s\n\ncase h.h.hr\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s ≠ ⊤",
"state_before": "case h.h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ (∫⁻ (x : α), ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x) ∂μ) =\n indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) s",
"tactic": "rw [lintegral_const_mul']"
},
{
"state_after": "case h.h.hr\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s ≠ ⊤\n\ncase h.h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ (ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * ∫⁻ (a : α), indicator (Ici s) (fun x => 1) (f a) ∂μ) =\n indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) s",
"state_before": "case h.h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ (ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * ∫⁻ (a : α), indicator (Ici s) (fun x => 1) (f a) ∂μ) =\n indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) s\n\ncase h.h.hr\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s ≠ ⊤",
"tactic": "swap"
},
{
"state_after": "case h.h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ (ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * ∫⁻ (a : α), indicator {a | s ≤ f a} (fun x => 1) a ∂μ) =\n indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) s",
"state_before": "case h.h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ (ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * ∫⁻ (a : α), indicator (Ici s) (fun x => 1) (f a) ∂μ) =\n indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) s",
"tactic": "simp_rw [show\n (fun a => (Ici s).indicator (fun _ : ℝ => (1 : ℝ≥0∞)) (f a)) = fun a =>\n {a : α | s ≤ f a}.indicator (fun _ => 1) a\n by funext a; by_cases s ≤ f a <;> simp [h]]"
},
{
"state_after": "case h.h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ (ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * ∫⁻ (a : α) in {a | s ≤ f a}, 1 ∂μ) =\n indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) s\n\ncase h.h.hs\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ MeasurableSet {a | s ≤ f a}",
"state_before": "case h.h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ (ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * ∫⁻ (a : α), indicator {a | s ≤ f a} (fun x => 1) a ∂μ) =\n indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) s",
"tactic": "rw [lintegral_indicator]"
},
{
"state_after": "case h.h.hs\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ MeasurableSet {a | s ≤ f a}\n\ncase h.h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ (ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * ∫⁻ (a : α) in {a | s ≤ f a}, 1 ∂μ) =\n indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) s",
"state_before": "case h.h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ (ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * ∫⁻ (a : α) in {a | s ≤ f a}, 1 ∂μ) =\n indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) s\n\ncase h.h.hs\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ MeasurableSet {a | s ≤ f a}",
"tactic": "swap"
},
{
"state_after": "case h.h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ ENNReal.ofReal (g s) * indicator (Ioi 0) (fun _x => 1 * ↑↑μ {a | s ≤ f a}) s =\n indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a}) s * ENNReal.ofReal (g s)",
"state_before": "case h.h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ (ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * ∫⁻ (a : α) in {a | s ≤ f a}, 1 ∂μ) =\n indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a} * ENNReal.ofReal (g t)) s",
"tactic": "rw [lintegral_one, Measure.restrict_apply MeasurableSet.univ, univ_inter, indicator_mul_left,\n mul_assoc,\n show\n (Ioi 0).indicator (fun _x : ℝ => (1 : ℝ≥0∞)) s * μ {a : α | s ≤ f a} =\n (Ioi 0).indicator (fun _x : ℝ => 1 * μ {a : α | s ≤ f a}) s\n by by_cases 0 < s <;> simp [h]]"
},
{
"state_after": "case h.h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ ENNReal.ofReal (g s) * indicator (Ioi 0) (fun _x => ↑↑μ {a | s ≤ f a}) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a}) s",
"state_before": "case h.h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ ENNReal.ofReal (g s) * indicator (Ioi 0) (fun _x => 1 * ↑↑μ {a | s ≤ f a}) s =\n indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a}) s * ENNReal.ofReal (g s)",
"tactic": "simp_rw [mul_comm _ (ENNReal.ofReal _), one_mul]"
},
{
"state_after": "no goals",
"state_before": "case h.h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ ENNReal.ofReal (g s) * indicator (Ioi 0) (fun _x => ↑↑μ {a | s ≤ f a}) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun t => ↑↑μ {a | t ≤ f a}) s",
"tactic": "rfl"
},
{
"state_after": "case h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\na : α\n⊢ indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\n⊢ (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)",
"tactic": "funext a"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\na : α\nh : s ∈ Ioc 0 (f a)\n⊢ indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)\n\ncase neg\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\na : α\nh : ¬s ∈ Ioc 0 (f a)\n⊢ indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)",
"state_before": "case h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\na : α\n⊢ indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)",
"tactic": "by_cases s ∈ Ioc (0 : ℝ) (f a)"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\na : α\nh : s ∈ Ioc 0 (f a)\n⊢ indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)",
"tactic": "simp only [h, show s ∈ Ioi (0 : ℝ) from h.1, show f a ∈ Ici s from h.2, indicator_of_mem,\n mul_one]"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\na : α\nh h_copy : ¬s ∈ Ioc 0 (f a)\n⊢ indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\na : α\nh : ¬s ∈ Ioc 0 (f a)\n⊢ indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)",
"tactic": "have h_copy := h"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\na : α\nh_copy : ¬s ∈ Ioc 0 (f a)\nh : 0 < s → f a < s\n⊢ indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\na : α\nh h_copy : ¬s ∈ Ioc 0 (f a)\n⊢ indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)",
"tactic": "simp only [mem_Ioc, not_and, not_le] at h"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\na : α\nh_copy : ¬s ∈ Ioc 0 (f a)\nh : 0 < s → f a < s\nh' : 0 < s\n⊢ indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)\n\ncase neg\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\na : α\nh_copy : ¬s ∈ Ioc 0 (f a)\nh : 0 < s → f a < s\nh' : ¬0 < s\n⊢ indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\na : α\nh_copy : ¬s ∈ Ioc 0 (f a)\nh : 0 < s → f a < s\n⊢ indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)",
"tactic": "by_cases h' : 0 < s"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\na : α\nh_copy : ¬s ∈ Ioc 0 (f a)\nh : 0 < s → f a < s\nh' : 0 < s\n⊢ indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)",
"tactic": "simp only [h_copy, h h', indicator_of_not_mem, not_false_iff, mem_Ici, not_le,\n MulZeroClass.mul_zero]"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\na : α\nh_copy : ¬s ∈ Ioc 0 (f a)\nh : 0 < s → f a < s\nh' : ¬0 < s\nthis : ¬s ∈ Ioi 0\n⊢ indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\na : α\nh_copy : ¬s ∈ Ioc 0 (f a)\nh : 0 < s → f a < s\nh' : ¬0 < s\n⊢ indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)",
"tactic": "have : s ∉ Ioi (0 : ℝ) := h'"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\na : α\nh_copy : ¬s ∈ Ioc 0 (f a)\nh : 0 < s → f a < s\nh' : ¬0 < s\nthis : ¬s ∈ Ioi 0\n⊢ indicator (Ioc 0 (f a)) (fun t => ENNReal.ofReal (g t)) s =\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f a)",
"tactic": "simp only [this, h', indicator_of_not_mem, not_false_iff, MulZeroClass.mul_zero,\n MulZeroClass.zero_mul, mem_Ioc, false_and_iff]"
},
{
"state_after": "case h.h.hr\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ indicator (Ioi 0) (fun x => 1) s ≠ ⊤",
"state_before": "case h.h.hr\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s ≠ ⊤",
"tactic": "apply ENNReal.mul_ne_top ENNReal.ofReal_ne_top"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\nh : ¬0 < s\n⊢ indicator (Ioi 0) (fun x => 1) s ≠ ⊤",
"tactic": "simp [indicator_apply, h]"
},
{
"state_after": "case h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\na : α\n⊢ indicator (Ici s) (fun x => 1) (f a) = indicator {a | s ≤ f a} (fun x => 1) a",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ (fun a => indicator (Ici s) (fun x => 1) (f a)) = fun a => indicator {a | s ≤ f a} (fun x => 1) a",
"tactic": "funext a"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\na : α\n⊢ indicator (Ici s) (fun x => 1) (f a) = indicator {a | s ≤ f a} (fun x => 1) a",
"tactic": "by_cases s ≤ f a <;> simp [h]"
},
{
"state_after": "no goals",
"state_before": "case h.h.hs\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ MeasurableSet {a | s ≤ f a}",
"tactic": "exact f_mble measurableSet_Ici"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns✝ : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\ns : ℝ\naux₁ :\n (fun x => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) s) = fun x =>\n ENNReal.ofReal (g s) * indicator (Ioi 0) (fun x => 1) s * indicator (Ici s) (fun x => 1) (f x)\n⊢ indicator (Ioi 0) (fun _x => 1) s * ↑↑μ {a | s ≤ f a} = indicator (Ioi 0) (fun _x => 1 * ↑↑μ {a | s ≤ f a}) s",
"tactic": "by_cases 0 < s <;> simp [h]"
},
{
"state_after": "case h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\np : α × ℝ\n⊢ Function.uncurry (fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) p =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) p",
"state_before": "α : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\n⊢ (Function.uncurry fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} fun p => ENNReal.ofReal (g p.snd)",
"tactic": "funext p"
},
{
"state_after": "case h.mk\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\np_fst : α\np_snd : ℝ\n⊢ Function.uncurry (fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) (p_fst, p_snd) =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) (p_fst, p_snd)",
"state_before": "case h\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\np : α × ℝ\n⊢ Function.uncurry (fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) p =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) p",
"tactic": "cases p with | mk p_fst p_snd => ?_"
},
{
"state_after": "case h.mk\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\np_fst : α\np_snd : ℝ\n⊢ indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) (p_fst, p_snd)",
"state_before": "case h.mk\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\np_fst : α\np_snd : ℝ\n⊢ Function.uncurry (fun x y => indicator (Ioc 0 (f x)) (fun t => ENNReal.ofReal (g t)) y) (p_fst, p_snd) =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) (p_fst, p_snd)",
"tactic": "rw [Function.uncurry_apply_pair]"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\np_fst : α\np_snd : ℝ\nh : p_snd ∈ Ioc 0 (f p_fst)\n⊢ indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) (p_fst, p_snd)\n\ncase neg\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\np_fst : α\np_snd : ℝ\nh : ¬p_snd ∈ Ioc 0 (f p_fst)\n⊢ indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) (p_fst, p_snd)",
"state_before": "case h.mk\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\np_fst : α\np_snd : ℝ\n⊢ indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) (p_fst, p_snd)",
"tactic": "by_cases p_snd ∈ Ioc 0 (f p_fst)"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\np_fst : α\np_snd : ℝ\nh : p_snd ∈ Ioc 0 (f p_fst)\nh' : (p_fst, p_snd) ∈ {p | p.snd ∈ Ioc 0 (f p.fst)}\n⊢ indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) (p_fst, p_snd)",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\np_fst : α\np_snd : ℝ\nh : p_snd ∈ Ioc 0 (f p_fst)\n⊢ indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) (p_fst, p_snd)",
"tactic": "have h' : (p_fst, p_snd) ∈ {p : α × ℝ | p.snd ∈ Ioc 0 (f p.fst)} := h"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\np_fst : α\np_snd : ℝ\nh : p_snd ∈ Ioc 0 (f p_fst)\nh' : (p_fst, p_snd) ∈ {p | p.snd ∈ Ioc 0 (f p.fst)}\n⊢ indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) (p_fst, p_snd)",
"tactic": "rw [Set.indicator_of_mem h', Set.indicator_of_mem h]"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\np_fst : α\np_snd : ℝ\nh : ¬p_snd ∈ Ioc 0 (f p_fst)\nh' : ¬(p_fst, p_snd) ∈ {p | p.snd ∈ Ioc 0 (f p.fst)}\n⊢ indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) (p_fst, p_snd)",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\np_fst : α\np_snd : ℝ\nh : ¬p_snd ∈ Ioc 0 (f p_fst)\n⊢ indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) (p_fst, p_snd)",
"tactic": "have h' : (p_fst, p_snd) ∉ {p : α × ℝ | p.snd ∈ Ioc 0 (f p.fst)} := h"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\ns : Set α\nμ : Measure α\ninst✝ : SigmaFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ (t : ℝ), t > 0 → IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ (t : ℝ), t > 0 → 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume 0 t\nintegrand_eq : ∀ (ω : α), ENNReal.ofReal (∫ (t : ℝ) in 0 ..f ω, g t) = ∫⁻ (t : ℝ) in Ioc 0 (f ω), ENNReal.ofReal (g t)\np_fst : α\np_snd : ℝ\nh : ¬p_snd ∈ Ioc 0 (f p_fst)\nh' : ¬(p_fst, p_snd) ∈ {p | p.snd ∈ Ioc 0 (f p.fst)}\n⊢ indicator (Ioc 0 (f p_fst)) (fun t => ENNReal.ofReal (g t)) p_snd =\n indicator {p | p.snd ∈ Ioc 0 (f p.fst)} (fun p => ENNReal.ofReal (g p.snd)) (p_fst, p_snd)",
"tactic": "rw [Set.indicator_of_not_mem h', Set.indicator_of_not_mem h]"
}
] |
[
162,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
83,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.filter_eq_iff
|
[] |
[
130,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
129,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.mul_mem
|
[] |
[
318,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
317,
11
] |
Mathlib/Algebra/BigOperators/Fin.lean
|
Fin.prod_univ_zero
|
[] |
[
64,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
63,
1
] |
Mathlib/Algebra/Ring/Divisibility.lean
|
dvd_sub_left
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.10227\ninst✝ : NonUnitalRing α\na b c : α\nh : a ∣ c\n⊢ a ∣ b - c ↔ a ∣ b",
"tactic": "simpa only [← sub_eq_add_neg] using dvd_add_left ((dvd_neg (α := α)).2 h)"
}
] |
[
108,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/Algebra/Periodic.lean
|
Function.Antiperiodic.neg
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.156251\nf g : α → β\nc c₁ c₂ x : α\ninst✝¹ : AddGroup α\ninst✝ : InvolutiveNeg β\nh : Antiperiodic f c\n⊢ Antiperiodic f (-c)",
"tactic": "simpa only [sub_eq_add_neg, Antiperiodic] using h.sub_eq"
}
] |
[
416,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
415,
11
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.toNat_surjective
|
[] |
[
1729,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1728,
1
] |
Mathlib/Order/Filter/Bases.lean
|
Filter.countable_biInf_eq_iInf_seq
|
[] |
[
1031,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1028,
1
] |
Mathlib/AlgebraicGeometry/StructureSheaf.lean
|
AlgebraicGeometry.StructureSheaf.stalkSpecializes_stalk_to_fiber
|
[
{
"state_after": "R✝ : Type u\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : CommRing R\nx y : PrimeSpectrum R\nh : x ⤳ y\n⊢ stalkSpecializes (Sheaf.presheaf (structureSheaf R)) h ≫ (stalkIso R x).hom =\n (stalkIso R y).hom ≫\n let_fun this := PrimeSpectrum.localizationMapOfSpecializes h;\n this",
"state_before": "R✝ : Type u\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : CommRing R\nx y : PrimeSpectrum R\nh : x ⤳ y\n⊢ stalkSpecializes (Sheaf.presheaf (structureSheaf R)) h ≫ stalkToFiberRingHom R x =\n stalkToFiberRingHom R y ≫\n let_fun this := PrimeSpectrum.localizationMapOfSpecializes h;\n this",
"tactic": "change _ ≫ (StructureSheaf.stalkIso R x).hom = (StructureSheaf.stalkIso R y).hom ≫ _"
},
{
"state_after": "R✝ : Type u\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : CommRing R\nx y : PrimeSpectrum R\nh : x ⤳ y\n⊢ (stalkIso R y).inv ≫ stalkSpecializes (Sheaf.presheaf (structureSheaf R)) h =\n (let_fun this := PrimeSpectrum.localizationMapOfSpecializes h;\n this) ≫\n (stalkIso R x).inv",
"state_before": "R✝ : Type u\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : CommRing R\nx y : PrimeSpectrum R\nh : x ⤳ y\n⊢ stalkSpecializes (Sheaf.presheaf (structureSheaf R)) h ≫ (stalkIso R x).hom =\n (stalkIso R y).hom ≫\n let_fun this := PrimeSpectrum.localizationMapOfSpecializes h;\n this",
"tactic": "rw [← Iso.eq_comp_inv, Category.assoc, ← Iso.inv_comp_eq]"
},
{
"state_after": "no goals",
"state_before": "R✝ : Type u\ninst✝¹ : CommRing R✝\nR : Type u_1\ninst✝ : CommRing R\nx y : PrimeSpectrum R\nh : x ⤳ y\n⊢ (stalkIso R y).inv ≫ stalkSpecializes (Sheaf.presheaf (structureSheaf R)) h =\n (let_fun this := PrimeSpectrum.localizationMapOfSpecializes h;\n this) ≫\n (stalkIso R x).inv",
"tactic": "exact localizationToStalk_stalkSpecializes h"
}
] |
[
1053,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1041,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.induction_succ
|
[
{
"state_after": "case mk\nn m : ℕ\nC : Fin (n + 1) → Sort u_1\nh0 : C 0\nhs : (i : Fin n) → C (↑castSucc i) → C (succ i)\nval✝ : ℕ\nisLt✝ : val✝ < n\n⊢ (fun i => induction h0 hs i) (succ { val := val✝, isLt := isLt✝ }) =\n hs { val := val✝, isLt := isLt✝ } (induction h0 hs (↑castSucc { val := val✝, isLt := isLt✝ }))",
"state_before": "n m : ℕ\nC : Fin (n + 1) → Sort u_1\nh0 : C 0\nhs : (i : Fin n) → C (↑castSucc i) → C (succ i)\ni : Fin n\n⊢ (fun i => induction h0 hs i) (succ i) = hs i (induction h0 hs (↑castSucc i))",
"tactic": "cases i"
},
{
"state_after": "no goals",
"state_before": "case mk\nn m : ℕ\nC : Fin (n + 1) → Sort u_1\nh0 : C 0\nhs : (i : Fin n) → C (↑castSucc i) → C (succ i)\nval✝ : ℕ\nisLt✝ : val✝ < n\n⊢ (fun i => induction h0 hs i) (succ { val := val✝, isLt := isLt✝ }) =\n hs { val := val✝, isLt := isLt✝ } (induction h0 hs (↑castSucc { val := val✝, isLt := isLt✝ }))",
"tactic": "rfl"
}
] |
[
1712,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1709,
1
] |
Mathlib/Analysis/Normed/Group/Basic.lean
|
norm_le_zero_iff'''
|
[
{
"state_after": "𝓕 : Type ?u.499976\n𝕜 : Type ?u.499979\nα : Type ?u.499982\nι : Type ?u.499985\nκ : Type ?u.499988\nE : Type u_1\nF : Type ?u.499994\nG : Type ?u.499997\ninst✝³ : SeminormedGroup E\ninst✝² : SeminormedGroup F\ninst✝¹ : SeminormedGroup G\ns : Set E\na✝ a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : T0Space E\na : E\nthis : NormedGroup E :=\n let src := inst✝³;\n NormedGroup.mk\n⊢ ‖a‖ ≤ 0 ↔ a = 1",
"state_before": "𝓕 : Type ?u.499976\n𝕜 : Type ?u.499979\nα : Type ?u.499982\nι : Type ?u.499985\nκ : Type ?u.499988\nE : Type u_1\nF : Type ?u.499994\nG : Type ?u.499997\ninst✝³ : SeminormedGroup E\ninst✝² : SeminormedGroup F\ninst✝¹ : SeminormedGroup G\ns : Set E\na✝ a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : T0Space E\na : E\n⊢ ‖a‖ ≤ 0 ↔ a = 1",
"tactic": "letI : NormedGroup E :=\n { ‹SeminormedGroup E› with toMetricSpace := MetricSpace.ofT0PseudoMetricSpace E }"
},
{
"state_after": "no goals",
"state_before": "𝓕 : Type ?u.499976\n𝕜 : Type ?u.499979\nα : Type ?u.499982\nι : Type ?u.499985\nκ : Type ?u.499988\nE : Type u_1\nF : Type ?u.499994\nG : Type ?u.499997\ninst✝³ : SeminormedGroup E\ninst✝² : SeminormedGroup F\ninst✝¹ : SeminormedGroup G\ns : Set E\na✝ a₁ a₂ b b₁ b₂ : E\nr r₁ r₂ : ℝ\ninst✝ : T0Space E\na : E\nthis : NormedGroup E :=\n let src := inst✝³;\n NormedGroup.mk\n⊢ ‖a‖ ≤ 0 ↔ a = 1",
"tactic": "rw [← dist_one_right, dist_le_zero]"
}
] |
[
1275,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1272,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean
|
MeasureTheory.extend_union
|
[
{
"state_after": "α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns₁ s₂ : Set α\nhd : Disjoint s₁ s₂\nh₁ : P s₁\nh₂ : P s₂\n⊢ ∑ b : Bool, extend m (bif b then s₁ else s₂) = extend m s₁ + extend m s₂",
"state_before": "α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns₁ s₂ : Set α\nhd : Disjoint s₁ s₂\nh₁ : P s₁\nh₂ : P s₂\n⊢ extend m (s₁ ∪ s₂) = extend m s₁ + extend m s₂",
"tactic": "rw [union_eq_iUnion,\n extend_iUnion P0 m0 PU mU (pairwise_disjoint_on_bool.2 hd) (Bool.forall_bool.2 ⟨h₂, h₁⟩),\n tsum_fintype]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nP : Set α → Prop\nm : (s : Set α) → P s → ℝ≥0∞\nP0 : P ∅\nm0 : m ∅ P0 = 0\nPU : ∀ ⦃f : ℕ → Set α⦄, (∀ (i : ℕ), P (f i)) → P (⋃ (i : ℕ), f i)\nmU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n Pairwise (Disjoint on f) → m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) = ∑' (i : ℕ), m (f i) (_ : P (f i))\nmsU :\n ∀ ⦃f : ℕ → Set α⦄ (hm : ∀ (i : ℕ), P (f i)),\n m (⋃ (i : ℕ), f i) (_ : P (⋃ (i : ℕ), f i)) ≤ ∑' (i : ℕ), m (f i) (_ : P (f i))\nm_mono : ∀ ⦃s₁ s₂ : Set α⦄ (hs₁ : P s₁) (hs₂ : P s₂), s₁ ⊆ s₂ → m s₁ hs₁ ≤ m s₂ hs₂\ns₁ s₂ : Set α\nhd : Disjoint s₁ s₂\nh₁ : P s₁\nh₂ : P s₂\n⊢ ∑ b : Bool, extend m (bif b then s₁ else s₂) = extend m s₁ + extend m s₂",
"tactic": "simp"
}
] |
[
1434,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1429,
1
] |
Mathlib/Analysis/Calculus/FDeriv/Mul.lean
|
DifferentiableOn.smul_const
|
[] |
[
255,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
254,
1
] |
Mathlib/Algebra/DirectSum/Decomposition.lean
|
DirectSum.Decomposition.decompose'_eq
|
[] |
[
103,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/GroupTheory/CommutingProbability.lean
|
card_comm_eq_card_conjClasses_mul_card
|
[
{
"state_after": "M : Type ?u.6837\ninst✝³ : Mul M\ninst✝² : Finite M\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : Finite G\nthis : Fintype G\n⊢ Nat.card { p // p.fst * p.snd = p.snd * p.fst } = Nat.card (ConjClasses G) * Nat.card G",
"state_before": "M : Type ?u.6837\ninst✝³ : Mul M\ninst✝² : Finite M\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : Finite G\n⊢ Nat.card { p // p.fst * p.snd = p.snd * p.fst } = Nat.card (ConjClasses G) * Nat.card G",
"tactic": "haveI := Fintype.ofFinite G"
},
{
"state_after": "M : Type ?u.6837\ninst✝³ : Mul M\ninst✝² : Finite M\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : Finite G\nthis : Fintype G\n⊢ card { p // p.fst * p.snd = p.snd * p.fst } = card (ConjClasses G) * card G",
"state_before": "M : Type ?u.6837\ninst✝³ : Mul M\ninst✝² : Finite M\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : Finite G\nthis : Fintype G\n⊢ Nat.card { p // p.fst * p.snd = p.snd * p.fst } = Nat.card (ConjClasses G) * Nat.card G",
"tactic": "simp only [Nat.card_eq_fintype_card]"
},
{
"state_after": "no goals",
"state_before": "M : Type ?u.6837\ninst✝³ : Mul M\ninst✝² : Finite M\nG : Type u_1\ninst✝¹ : Group G\ninst✝ : Finite G\nthis : Fintype G\n⊢ card { p // p.fst * p.snd = p.snd * p.fst } = card (ConjClasses G) * card G",
"tactic": "rw [card_congr (Equiv.subtypeProdEquivSigmaSubtype fun g h : G ↦ g * h = h * g), card_sigma,\n sum_equiv ConjAct.toConjAct.toEquiv (fun a ↦ card { b // a * b = b * a })\n (fun g ↦ card (MulAction.fixedBy (ConjAct G) G g))\n fun g ↦ card_congr' <| congr_arg _ <| funext fun h ↦ mul_inv_eq_iff_eq_mul.symm.to_eq,\n MulAction.sum_card_fixedBy_eq_card_orbits_mul_card_group, ConjAct.card,\n (Setoid.ext fun g h ↦ (Setoid.comm' _).trans isConj_iff.symm :\n MulAction.orbitRel (ConjAct G) G = IsConj.setoid G),\n @card_congr' (Quotient (IsConj.setoid G)) (ConjClasses G) _ _ rfl]"
}
] |
[
87,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
75,
1
] |
Mathlib/LinearAlgebra/AffineSpace/Basis.lean
|
AffineBasis.linear_eq_sumCoords
|
[] |
[
161,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
160,
1
] |
Mathlib/Algebra/Category/ModuleCat/EpiMono.lean
|
ModuleCat.range_eq_top_of_epi
|
[] |
[
38,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
37,
1
] |
Mathlib/Analysis/Normed/Field/Basic.lean
|
List.nnnorm_prod_le'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.112914\nγ : Type ?u.112917\nι : Type ?u.112920\ninst✝ : SeminormedRing α\nl : List α\nhl : l ≠ []\n⊢ prod (map norm l) = (fun a => ↑a) (prod (map nnnorm l))",
"tactic": "simp [NNReal.coe_list_prod, List.map_map]"
}
] |
[
324,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
323,
1
] |
Mathlib/Logic/Relation.lean
|
flip_eq_iff
|
[] |
[
93,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
92,
1
] |
Mathlib/Topology/Instances/EReal.lean
|
EReal.continuousAt_add_bot_coe
|
[
{
"state_after": "α : Type ?u.23308\ninst✝ : TopologicalSpace α\na : ℝ\n⊢ ∀ (x : ℝ), ∀ᶠ (a : EReal × EReal) in 𝓝 (⊥, ↑a), a.fst + a.snd < ↑x",
"state_before": "α : Type ?u.23308\ninst✝ : TopologicalSpace α\na : ℝ\n⊢ ContinuousAt (fun p => p.fst + p.snd) (⊥, ↑a)",
"tactic": "simp only [ContinuousAt, tendsto_nhds_bot_iff_real, bot_add]"
},
{
"state_after": "α : Type ?u.23308\ninst✝ : TopologicalSpace α\na r : ℝ\nx✝ : EReal × EReal\nh : x✝.fst < ↑(r - (a + 1)) ∧ x✝.snd < ↑(a + 1)\n⊢ x✝.fst + x✝.snd < ↑r",
"state_before": "α : Type ?u.23308\ninst✝ : TopologicalSpace α\na : ℝ\n⊢ ∀ (x : ℝ), ∀ᶠ (a : EReal × EReal) in 𝓝 (⊥, ↑a), a.fst + a.snd < ↑x",
"tactic": "refine fun r ↦ ((gt_mem_nhds (bot_lt_coe (r - (a + 1)))).prod_nhds\n (gt_mem_nhds <| EReal.coe_lt_coe_iff.2 <| lt_add_one _)).mono fun _ h ↦ ?_"
},
{
"state_after": "no goals",
"state_before": "α : Type ?u.23308\ninst✝ : TopologicalSpace α\na r : ℝ\nx✝ : EReal × EReal\nh : x✝.fst < ↑(r - (a + 1)) ∧ x✝.snd < ↑(a + 1)\n⊢ x✝.fst + x✝.snd < ↑r",
"tactic": "simpa only [← coe_add, sub_add_cancel] using add_lt_add h.1 h.2"
}
] |
[
210,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Combinatorics/Pigeonhole.lean
|
Finset.exists_sum_fiber_le_of_maps_to_of_sum_le_nsmul
|
[] |
[
180,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
178,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
StrictAnti.mul_antitone'
|
[] |
[
1539,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1537,
1
] |
Mathlib/CategoryTheory/Conj.lean
|
CategoryTheory.Iso.trans_conjAut
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝ : Category C\nX Y : C\nα : X ≅ Y\nZ : C\nβ : Y ≅ Z\nf : Aut X\n⊢ ↑(conjAut (α ≪≫ β)) f = ↑(conjAut β) (↑(conjAut α) f)",
"tactic": "simp only [conjAut_apply, Iso.trans_symm, Iso.trans_assoc]"
}
] |
[
144,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
142,
1
] |
Mathlib/Order/CompleteLattice.lean
|
iInf_insert
|
[] |
[
1456,
81
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1454,
1
] |
Mathlib/Order/Monotone/Monovary.lean
|
monovary_toDual_right
|
[] |
[
254,
14
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
253,
1
] |
Mathlib/Combinatorics/SetFamily/Shadow.lean
|
Finset.mem_upShadow_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Fintype α\n𝒜 : Finset (Finset α)\ns t : Finset α\na : α\nk r : ℕ\n⊢ s ∈ (∂⁺ ) 𝒜 ↔ ∃ t, t ∈ 𝒜 ∧ ∃ a x, insert a t = s",
"tactic": "simp_rw [upShadow, mem_sup, mem_image, exists_prop, mem_compl]"
}
] |
[
209,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean
|
Polynomial.degree_mul_X
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝¹ : Semiring R\ninst✝ : Nontrivial R\np q : R[X]\n⊢ degree (p * X) = degree p + 1",
"tactic": "simp [monic_X.degree_mul]"
}
] |
[
1280,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1280,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Real.one_lt_rpow
|
[
{
"state_after": "x✝ y z✝ x z : ℝ\nhx : 1 < x\nhz : 0 < z\n⊢ 1 ^ z < x ^ z",
"state_before": "x✝ y z✝ x z : ℝ\nhx : 1 < x\nhz : 0 < z\n⊢ 1 < x ^ z",
"tactic": "rw [← one_rpow z]"
},
{
"state_after": "no goals",
"state_before": "x✝ y z✝ x z : ℝ\nhx : 1 < x\nhz : 0 < z\n⊢ 1 ^ z < x ^ z",
"tactic": "exact rpow_lt_rpow zero_le_one hx hz"
}
] |
[
536,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
534,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.mem_normalizer_iff
|
[] |
[
2154,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2153,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean
|
Multiset.disjoint_list_sum_left
|
[
{
"state_after": "case nil\nι : Type ?u.892330\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf g : α → β\na : Multiset α\n⊢ Disjoint (List.sum []) a ↔ ∀ (b : Multiset α), b ∈ [] → Disjoint b a\n\ncase cons\nι : Type ?u.892330\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf g : α → β\na b : Multiset α\nbs : List (Multiset α)\nih : Disjoint (List.sum bs) a ↔ ∀ (b : Multiset α), b ∈ bs → Disjoint b a\n⊢ Disjoint (List.sum (b :: bs)) a ↔ ∀ (b_1 : Multiset α), b_1 ∈ b :: bs → Disjoint b_1 a",
"state_before": "ι : Type ?u.892330\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf g : α → β\na : Multiset α\nl : List (Multiset α)\n⊢ Disjoint (List.sum l) a ↔ ∀ (b : Multiset α), b ∈ l → Disjoint b a",
"tactic": "induction' l with b bs ih"
},
{
"state_after": "no goals",
"state_before": "case nil\nι : Type ?u.892330\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf g : α → β\na : Multiset α\n⊢ Disjoint (List.sum []) a ↔ ∀ (b : Multiset α), b ∈ [] → Disjoint b a",
"tactic": "simp only [zero_disjoint, List.not_mem_nil, IsEmpty.forall_iff, forall_const, List.sum_nil]"
},
{
"state_after": "case cons\nι : Type ?u.892330\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf g : α → β\na b : Multiset α\nbs : List (Multiset α)\nih : Disjoint (List.sum bs) a ↔ ∀ (b : Multiset α), b ∈ bs → Disjoint b a\n⊢ Disjoint b a ∧ Disjoint (List.sum bs) a ↔ Disjoint b a ∧ ∀ (a_1 : Multiset α), a_1 ∈ bs → Disjoint a_1 a",
"state_before": "case cons\nι : Type ?u.892330\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf g : α → β\na b : Multiset α\nbs : List (Multiset α)\nih : Disjoint (List.sum bs) a ↔ ∀ (b : Multiset α), b ∈ bs → Disjoint b a\n⊢ Disjoint (List.sum (b :: bs)) a ↔ ∀ (b_1 : Multiset α), b_1 ∈ b :: bs → Disjoint b_1 a",
"tactic": "simp_rw [List.sum_cons, disjoint_add_left, List.mem_cons, forall_eq_or_imp]"
},
{
"state_after": "no goals",
"state_before": "case cons\nι : Type ?u.892330\nβ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf g : α → β\na b : Multiset α\nbs : List (Multiset α)\nih : Disjoint (List.sum bs) a ↔ ∀ (b : Multiset α), b ∈ bs → Disjoint b a\n⊢ Disjoint b a ∧ Disjoint (List.sum bs) a ↔ Disjoint b a ∧ ∀ (a_1 : Multiset α), a_1 ∈ bs → Disjoint a_1 a",
"tactic": "simp [and_congr_left_iff, iff_self_iff, ih]"
}
] |
[
2059,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2054,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Angle.lean
|
Real.Angle.coe_toIocMod
|
[
{
"state_after": "θ ψ : ℝ\n⊢ ∃ k, toIocMod two_pi_pos ψ θ - θ = 2 * π * ↑k",
"state_before": "θ ψ : ℝ\n⊢ ↑(toIocMod two_pi_pos ψ θ) = ↑θ",
"tactic": "rw [angle_eq_iff_two_pi_dvd_sub]"
},
{
"state_after": "θ ψ : ℝ\n⊢ toIocMod two_pi_pos ψ θ - θ = 2 * π * ↑(-toIocDiv two_pi_pos ψ θ)",
"state_before": "θ ψ : ℝ\n⊢ ∃ k, toIocMod two_pi_pos ψ θ - θ = 2 * π * ↑k",
"tactic": "refine' ⟨-toIocDiv two_pi_pos ψ θ, _⟩"
},
{
"state_after": "no goals",
"state_before": "θ ψ : ℝ\n⊢ toIocMod two_pi_pos ψ θ - θ = 2 * π * ↑(-toIocDiv two_pi_pos ψ θ)",
"tactic": "rw [toIocMod_sub_self, zsmul_eq_mul, mul_comm]"
}
] |
[
518,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
515,
1
] |
Mathlib/LinearAlgebra/Matrix/ZPow.lean
|
Matrix.one_div_pow
|
[
{
"state_after": "no goals",
"state_before": "n' : Type u_1\ninst✝² : DecidableEq n'\ninst✝¹ : Fintype n'\nR : Type u_2\ninst✝ : CommRing R\nA : M\nn : ℕ\n⊢ (1 / A) ^ n = 1 / A ^ n",
"tactic": "simp only [one_div, inv_pow']"
}
] |
[
331,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
331,
1
] |
Mathlib/RingTheory/Derivation/ToSquareZero.lean
|
liftOfDerivationToSquareZero_mk_apply
|
[
{
"state_after": "R : Type u\nA : Type v\nB : Type w\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\nI : Ideal B\nhI : I ^ 2 = ⊥\ninst✝¹ : Algebra A B\ninst✝ : IsScalarTower R A B\nd : Derivation R A { x // x ∈ I }\nx : A\n⊢ ↑(Ideal.Quotient.mk I) (↑(algebraMap A B) x) = ↑(algebraMap A (B ⧸ I)) x",
"state_before": "R : Type u\nA : Type v\nB : Type w\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\nI : Ideal B\nhI : I ^ 2 = ⊥\ninst✝¹ : Algebra A B\ninst✝ : IsScalarTower R A B\nd : Derivation R A { x // x ∈ I }\nx : A\n⊢ ↑(Ideal.Quotient.mk I) (↑(liftOfDerivationToSquareZero I hI d) x) = ↑(algebraMap A (B ⧸ I)) x",
"tactic": "rw [liftOfDerivationToSquareZero_apply, map_add, Ideal.Quotient.eq_zero_iff_mem.mpr (d x).prop,\n zero_add]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nA : Type v\nB : Type w\ninst✝⁶ : CommSemiring R\ninst✝⁵ : CommSemiring A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R A\ninst✝² : Algebra R B\nI : Ideal B\nhI : I ^ 2 = ⊥\ninst✝¹ : Algebra A B\ninst✝ : IsScalarTower R A B\nd : Derivation R A { x // x ∈ I }\nx : A\n⊢ ↑(Ideal.Quotient.mk I) (↑(algebraMap A B) x) = ↑(algebraMap A (B ⧸ I)) x",
"tactic": "rfl"
}
] |
[
111,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
107,
1
] |
Mathlib/Combinatorics/Quiver/SingleObj.lean
|
Quiver.SingleObj.pathEquivList_symm_nil
|
[] |
[
169,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
168,
1
] |
Mathlib/Algebra/Lie/Basic.lean
|
LieHom.map_neg
|
[] |
[
322,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
321,
1
] |
Mathlib/Data/Finset/NatAntidiagonal.lean
|
Finset.Nat.map_swap_antidiagonal
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ (map { toFun := Prod.swap, inj' := (_ : Function.Injective Prod.swap) } (antidiagonal n)).val = (antidiagonal n).val",
"tactic": "simp [antidiagonal, Multiset.Nat.map_swap_antidiagonal]"
}
] |
[
90,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
88,
1
] |
Mathlib/Algebra/CharZero/Defs.lean
|
Nat.cast_injective
|
[] |
[
68,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
Mathlib/GroupTheory/GroupAction/Basic.lean
|
MulAction.smul_mem_orbit_smul
|
[
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\ninst✝¹ : Group α\ninst✝ : MulAction α β\ng h : α\na : β\n⊢ g • a ∈ orbit α (h • a)",
"tactic": "simp only [orbit_smul, mem_orbit]"
}
] |
[
266,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
265,
1
] |
Mathlib/FieldTheory/Subfield.lean
|
Subfield.coe_zero
|
[] |
[
397,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
396,
1
] |
Mathlib/Combinatorics/Colex.lean
|
Colex.singleton_lt_iff_lt
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : LinearOrder α\nr s : α\n⊢ toColex {r} < toColex {s} ↔ r < s",
"tactic": "simp [lt_singleton_iff_mem_lt]"
}
] |
[
301,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
300,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.injOn_sin
|
[] |
[
611,
25
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
610,
1
] |
Mathlib/Data/Complex/Basic.lean
|
Complex.not_lt_iff
|
[
{
"state_after": "no goals",
"state_before": "z w : ℂ\n⊢ ¬z < w ↔ w.re ≤ z.re ∨ z.im ≠ w.im",
"tactic": "rw [lt_def, not_and_or, not_lt]"
}
] |
[
1180,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1179,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.stronglyMeasurable_uncurry_of_continuous_of_stronglyMeasurable
|
[
{
"state_after": "α✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\n⊢ StronglyMeasurable (uncurry u)",
"state_before": "α✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\n⊢ StronglyMeasurable (uncurry u)",
"tactic": "borelize β"
},
{
"state_after": "case intro\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\n⊢ StronglyMeasurable (uncurry u)",
"state_before": "α✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\n⊢ StronglyMeasurable (uncurry u)",
"tactic": "obtain ⟨t_sf, ht_sf⟩ :\n ∃ t : ℕ → SimpleFunc ι ι, ∀ j x, Tendsto (fun n => u (t n j) x) atTop (𝓝 <| u j x) := by\n have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id\n refine' ⟨h_str_meas.approx, fun j x => _⟩\n exact ((hu_cont x).tendsto j).comp (h_str_meas.tendsto_approx j)"
},
{
"state_after": "case intro\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\n⊢ StronglyMeasurable (uncurry u)",
"state_before": "case intro\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\n⊢ StronglyMeasurable (uncurry u)",
"tactic": "let U (n : ℕ) (p : ι × α) := u (t_sf n p.fst) p.snd"
},
{
"state_after": "case intro\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\n⊢ StronglyMeasurable (uncurry u)",
"state_before": "case intro\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\n⊢ StronglyMeasurable (uncurry u)",
"tactic": "have h_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd) := by\n rw [tendsto_pi_nhds]\n exact fun p => ht_sf p.fst p.snd"
},
{
"state_after": "case intro\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\n⊢ StronglyMeasurable (U n)",
"state_before": "case intro\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\n⊢ StronglyMeasurable (uncurry u)",
"tactic": "refine' stronglyMeasurable_of_tendsto _ (fun n => _) h_tendsto"
},
{
"state_after": "case intro\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nh_str_meas : StronglyMeasurable fun p => u (↑p.fst) p.snd\nthis :\n (fun p => u (↑(t_sf n) p.fst) p.snd) =\n (fun p => u (↑p.fst) p.snd) ∘ fun p =>\n ({ val := ↑(t_sf n) p.fst, property := (_ : ↑(t_sf n) p.fst ∈ SimpleFunc.range (t_sf n)) }, p.snd)\n⊢ StronglyMeasurable (U n)",
"state_before": "case intro\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nh_str_meas : StronglyMeasurable fun p => u (↑p.fst) p.snd\n⊢ StronglyMeasurable (U n)",
"tactic": "have :\n (fun p : ι × α => u (t_sf n p.fst) p.snd) =\n (fun p : ↥(t_sf n).range × α => u p.fst p.snd) ∘ fun p : ι × α =>\n (⟨t_sf n p.fst, SimpleFunc.mem_range_self _ _⟩, p.snd) :=\n rfl"
},
{
"state_after": "case intro\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nh_str_meas : StronglyMeasurable fun p => u (↑p.fst) p.snd\nthis :\n (fun p => u (↑(t_sf n) p.fst) p.snd) =\n (fun p => u (↑p.fst) p.snd) ∘ fun p =>\n ({ val := ↑(t_sf n) p.fst, property := (_ : ↑(t_sf n) p.fst ∈ SimpleFunc.range (t_sf n)) }, p.snd)\n⊢ StronglyMeasurable\n ((fun p => u (↑p.fst) p.snd) ∘ fun p =>\n ({ val := ↑(t_sf n) p.fst, property := (_ : ↑(t_sf n) p.fst ∈ SimpleFunc.range (t_sf n)) }, p.snd))",
"state_before": "case intro\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nh_str_meas : StronglyMeasurable fun p => u (↑p.fst) p.snd\nthis :\n (fun p => u (↑(t_sf n) p.fst) p.snd) =\n (fun p => u (↑p.fst) p.snd) ∘ fun p =>\n ({ val := ↑(t_sf n) p.fst, property := (_ : ↑(t_sf n) p.fst ∈ SimpleFunc.range (t_sf n)) }, p.snd)\n⊢ StronglyMeasurable (U n)",
"tactic": "simp_rw [this]"
},
{
"state_after": "case intro\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nh_str_meas : StronglyMeasurable fun p => u (↑p.fst) p.snd\nthis :\n (fun p => u (↑(t_sf n) p.fst) p.snd) =\n (fun p => u (↑p.fst) p.snd) ∘ fun p =>\n ({ val := ↑(t_sf n) p.fst, property := (_ : ↑(t_sf n) p.fst ∈ SimpleFunc.range (t_sf n)) }, p.snd)\n⊢ Measurable fun p => { val := ↑(t_sf n) p.fst, property := (_ : ↑(t_sf n) p.fst ∈ SimpleFunc.range (t_sf n)) }",
"state_before": "case intro\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nh_str_meas : StronglyMeasurable fun p => u (↑p.fst) p.snd\nthis :\n (fun p => u (↑(t_sf n) p.fst) p.snd) =\n (fun p => u (↑p.fst) p.snd) ∘ fun p =>\n ({ val := ↑(t_sf n) p.fst, property := (_ : ↑(t_sf n) p.fst ∈ SimpleFunc.range (t_sf n)) }, p.snd)\n⊢ StronglyMeasurable\n ((fun p => u (↑p.fst) p.snd) ∘ fun p =>\n ({ val := ↑(t_sf n) p.fst, property := (_ : ↑(t_sf n) p.fst ∈ SimpleFunc.range (t_sf n)) }, p.snd))",
"tactic": "refine' h_str_meas.comp_measurable (Measurable.prod_mk _ measurable_snd)"
},
{
"state_after": "no goals",
"state_before": "case intro\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nh_str_meas : StronglyMeasurable fun p => u (↑p.fst) p.snd\nthis :\n (fun p => u (↑(t_sf n) p.fst) p.snd) =\n (fun p => u (↑p.fst) p.snd) ∘ fun p =>\n ({ val := ↑(t_sf n) p.fst, property := (_ : ↑(t_sf n) p.fst ∈ SimpleFunc.range (t_sf n)) }, p.snd)\n⊢ Measurable fun p => { val := ↑(t_sf n) p.fst, property := (_ : ↑(t_sf n) p.fst ∈ SimpleFunc.range (t_sf n)) }",
"tactic": "exact ((t_sf n).measurable.comp measurable_fst).subtype_mk"
},
{
"state_after": "α✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nh_str_meas : StronglyMeasurable id\n⊢ ∃ t, ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t n) j) x) atTop (𝓝 (u j x))",
"state_before": "α✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\n⊢ ∃ t, ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t n) j) x) atTop (𝓝 (u j x))",
"tactic": "have h_str_meas : StronglyMeasurable (id : ι → ι) := stronglyMeasurable_id"
},
{
"state_after": "α✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nh_str_meas : StronglyMeasurable id\nj : ι\nx : α\n⊢ Tendsto (fun n => u (↑(StronglyMeasurable.approx h_str_meas n) j) x) atTop (𝓝 (u j x))",
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"tactic": "refine' ⟨h_str_meas.approx, fun j x => _⟩"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nh_str_meas : StronglyMeasurable id\nj : ι\nx : α\n⊢ Tendsto (fun n => u (↑(StronglyMeasurable.approx h_str_meas n) j) x) atTop (𝓝 (u j x))",
"tactic": "exact ((hu_cont x).tendsto j).comp (h_str_meas.tendsto_approx j)"
},
{
"state_after": "α✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\n⊢ ∀ (x : ι × α), Tendsto (fun i => U i x) atTop (𝓝 (u x.fst x.snd))",
"state_before": "α✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\n⊢ Tendsto U atTop (𝓝 fun p => u p.fst p.snd)",
"tactic": "rw [tendsto_pi_nhds]"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\n⊢ ∀ (x : ι × α), Tendsto (fun i => U i x) atTop (𝓝 (u x.fst x.snd))",
"tactic": "exact fun p => ht_sf p.fst p.snd"
},
{
"state_after": "case refine'_1\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\n⊢ Measurable fun p => u (↑p.fst) p.snd\n\ncase refine'_2\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\n⊢ IsSeparable (range fun p => u (↑p.fst) p.snd)",
"state_before": "α✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\n⊢ StronglyMeasurable fun p => u (↑p.fst) p.snd",
"tactic": "refine' stronglyMeasurable_iff_measurable_separable.2 ⟨_, _⟩"
},
{
"state_after": "case refine'_1\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nthis : (fun p => u (↑p.fst) p.snd) = (fun p => u (↑p.snd) p.fst) ∘ Prod.swap\n⊢ Measurable fun p => u (↑p.fst) p.snd",
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"tactic": "have :\n (fun p : ↥(t_sf n).range × α => u (↑p.fst) p.snd) =\n (fun p : α × (t_sf n).range => u (↑p.snd) p.fst) ∘ Prod.swap :=\n rfl"
},
{
"state_after": "case refine'_1\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nthis : (fun p => u (↑p.fst) p.snd) = (fun p => u (↑p.snd) p.fst) ∘ Prod.swap\n⊢ Measurable fun p => u (↑p.snd) p.fst",
"state_before": "case refine'_1\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nthis : (fun p => u (↑p.fst) p.snd) = (fun p => u (↑p.snd) p.fst) ∘ Prod.swap\n⊢ Measurable fun p => u (↑p.fst) p.snd",
"tactic": "rw [this, measurable_swap_iff]"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nthis : (fun p => u (↑p.fst) p.snd) = (fun p => u (↑p.snd) p.fst) ∘ Prod.swap\n⊢ Measurable fun p => u (↑p.snd) p.fst",
"tactic": "exact measurable_from_prod_countable fun j => (h j).measurable"
},
{
"state_after": "case refine'_2\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nthis : IsSeparable (⋃ (i : { x // x ∈ SimpleFunc.range (t_sf n) }), range (u ↑i))\n⊢ IsSeparable (range fun p => u (↑p.fst) p.snd)",
"state_before": "case refine'_2\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\n⊢ IsSeparable (range fun p => u (↑p.fst) p.snd)",
"tactic": "have : IsSeparable (⋃ i : (t_sf n).range, range (u i)) :=\n isSeparable_iUnion fun i => (h i).isSeparable_range"
},
{
"state_after": "case refine'_2\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nthis : IsSeparable (⋃ (i : { x // x ∈ SimpleFunc.range (t_sf n) }), range (u ↑i))\n⊢ (range fun p => u (↑p.fst) p.snd) ⊆ ⋃ (i : { x // x ∈ SimpleFunc.range (t_sf n) }), range (u ↑i)",
"state_before": "case refine'_2\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nthis : IsSeparable (⋃ (i : { x // x ∈ SimpleFunc.range (t_sf n) }), range (u ↑i))\n⊢ IsSeparable (range fun p => u (↑p.fst) p.snd)",
"tactic": "apply this.mono"
},
{
"state_after": "case refine'_2.intro.mk\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nthis : IsSeparable (⋃ (i : { x // x ∈ SimpleFunc.range (t_sf n) }), range (u ↑i))\ni : { x // x ∈ SimpleFunc.range (t_sf n) }\nx : α\n⊢ (fun p => u (↑p.fst) p.snd) (i, x) ∈ ⋃ (i : { x // x ∈ SimpleFunc.range (t_sf n) }), range (u ↑i)",
"state_before": "case refine'_2\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nthis : IsSeparable (⋃ (i : { x // x ∈ SimpleFunc.range (t_sf n) }), range (u ↑i))\n⊢ (range fun p => u (↑p.fst) p.snd) ⊆ ⋃ (i : { x // x ∈ SimpleFunc.range (t_sf n) }), range (u ↑i)",
"tactic": "rintro _ ⟨⟨i, x⟩, rfl⟩"
},
{
"state_after": "case refine'_2.intro.mk\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nthis : IsSeparable (⋃ (i : { x // x ∈ SimpleFunc.range (t_sf n) }), range (u ↑i))\ni : { x // x ∈ SimpleFunc.range (t_sf n) }\nx : α\n⊢ ∃ i_1 y, u (↑i_1) y = u (↑i) x",
"state_before": "case refine'_2.intro.mk\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nthis : IsSeparable (⋃ (i : { x // x ∈ SimpleFunc.range (t_sf n) }), range (u ↑i))\ni : { x // x ∈ SimpleFunc.range (t_sf n) }\nx : α\n⊢ (fun p => u (↑p.fst) p.snd) (i, x) ∈ ⋃ (i : { x // x ∈ SimpleFunc.range (t_sf n) }), range (u ↑i)",
"tactic": "simp only [mem_iUnion, mem_range]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.mk\nα✝ : Type ?u.560896\nβ✝ : Type ?u.560899\nγ : Type ?u.560902\nι✝ : Type ?u.560905\ninst✝⁸ : Countable ι✝\nα : Type u_1\nβ : Type u_2\nι : Type u_3\ninst✝⁷ : TopologicalSpace ι\ninst✝⁶ : MetrizableSpace ι\ninst✝⁵ : MeasurableSpace ι\ninst✝⁴ : SecondCountableTopology ι\ninst✝³ : OpensMeasurableSpace ι\ninst✝² : TopologicalSpace β\ninst✝¹ : PseudoMetrizableSpace β\ninst✝ : MeasurableSpace α\nu : ι → α → β\nhu_cont : ∀ (x : α), Continuous fun i => u i x\nh : ∀ (i : ι), StronglyMeasurable (u i)\nthis✝¹ : MeasurableSpace β := borel β\nthis✝ : BorelSpace β\nt_sf : ℕ → ι →ₛ ι\nht_sf : ∀ (j : ι) (x : α), Tendsto (fun n => u (↑(t_sf n) j) x) atTop (𝓝 (u j x))\nU : ℕ → ι × α → β := fun n p => u (↑(t_sf n) p.fst) p.snd\nh_tendsto : Tendsto U atTop (𝓝 fun p => u p.fst p.snd)\nn : ℕ\nthis : IsSeparable (⋃ (i : { x // x ∈ SimpleFunc.range (t_sf n) }), range (u ↑i))\ni : { x // x ∈ SimpleFunc.range (t_sf n) }\nx : α\n⊢ ∃ i_1 y, u (↑i_1) y = u (↑i) x",
"tactic": "exact ⟨i, x, rfl⟩"
}
] |
[
2048,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2011,
1
] |
Mathlib/Order/Monotone/Basic.lean
|
Antitone.ne_of_lt_of_lt_nat
|
[
{
"state_after": "ι : Type ?u.45443\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.45452\nπ : ι → Type ?u.45457\nr : α → α → Prop\ninst✝ : Preorder α\nf : ℕ → α\nhf : Antitone f\nn a : ℕ\nh1 : f (n + 1) < f a\nh2 : f a < f n\n⊢ False",
"state_before": "ι : Type ?u.45443\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.45452\nπ : ι → Type ?u.45457\nr : α → α → Prop\ninst✝ : Preorder α\nf : ℕ → α\nhf : Antitone f\nn : ℕ\nx : α\nh1 : f (n + 1) < x\nh2 : x < f n\na : ℕ\n⊢ f a ≠ x",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.45443\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.45452\nπ : ι → Type ?u.45457\nr : α → α → Prop\ninst✝ : Preorder α\nf : ℕ → α\nhf : Antitone f\nn a : ℕ\nh1 : f (n + 1) < f a\nh2 : f a < f n\n⊢ False",
"tactic": "exact (hf.reflect_lt h2).not_le (Nat.le_of_lt_succ <| hf.reflect_lt h1)"
}
] |
[
1112,
74
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1109,
1
] |
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