file_path
stringlengths 11
79
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stringlengths 2
100
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list | end
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stringclasses 4
values | url
stringclasses 4
values | start
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|---|---|---|---|---|---|---|
Mathlib/Algebra/Algebra/Unitization.lean
|
Unitization.snd_zero
|
[] |
[
219,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/FieldTheory/Finite/Basic.lean
|
FiniteField.isSquare_iff
|
[
{
"state_after": "K : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\n⊢ IsSquare (Units.mk0 a ha) ↔ IsSquare a",
"state_before": "K : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\n⊢ IsSquare a ↔ a ^ (Fintype.card F / 2) = 1",
"tactic": "apply\n (iff_congr _ (by simp [Units.ext_iff])).mp (FiniteField.unit_isSquare_iff hF (Units.mk0 a ha))"
},
{
"state_after": "K : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\n⊢ (∃ r, a = ↑r * ↑r) ↔ ∃ r, a = r * r",
"state_before": "K : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\n⊢ IsSquare (Units.mk0 a ha) ↔ IsSquare a",
"tactic": "simp only [IsSquare, Units.ext_iff, Units.val_mk0, Units.val_mul]"
},
{
"state_after": "case mp\nK : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\n⊢ (∃ r, a = ↑r * ↑r) → ∃ r, a = r * r\n\ncase mpr\nK : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\n⊢ (∃ r, a = r * r) → ∃ r, a = ↑r * ↑r",
"state_before": "K : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\n⊢ (∃ r, a = ↑r * ↑r) ↔ ∃ r, a = r * r",
"tactic": "constructor"
},
{
"state_after": "no goals",
"state_before": "K : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\n⊢ Units.mk0 a ha ^ (Fintype.card F / 2) = 1 ↔ a ^ (Fintype.card F / 2) = 1",
"tactic": "simp [Units.ext_iff]"
},
{
"state_after": "case mp.intro\nK : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\ny : Fˣ\nhy : a = ↑y * ↑y\n⊢ ∃ r, a = r * r",
"state_before": "case mp\nK : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\n⊢ (∃ r, a = ↑r * ↑r) → ∃ r, a = r * r",
"tactic": "rintro ⟨y, hy⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro\nK : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\ny : Fˣ\nhy : a = ↑y * ↑y\n⊢ ∃ r, a = r * r",
"tactic": "exact ⟨y, hy⟩"
},
{
"state_after": "case mpr.intro\nK : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ny : F\nha : y * y ≠ 0\n⊢ ∃ r, y * y = ↑r * ↑r",
"state_before": "case mpr\nK : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\na : F\nha : a ≠ 0\n⊢ (∃ r, a = r * r) → ∃ r, a = ↑r * ↑r",
"tactic": "rintro ⟨y, rfl⟩"
},
{
"state_after": "case mpr.intro\nK : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ny : F\nha : y * y ≠ 0\nhy : y ≠ 0\n⊢ ∃ r, y * y = ↑r * ↑r",
"state_before": "case mpr.intro\nK : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ny : F\nha : y * y ≠ 0\n⊢ ∃ r, y * y = ↑r * ↑r",
"tactic": "have hy : y ≠ 0 := by rintro rfl; simp at ha"
},
{
"state_after": "case mpr.intro\nK : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ny : F\nha : y * y ≠ 0\nhy : y ≠ 0\n⊢ y * y = ↑(Units.mk0 y hy) * ↑(Units.mk0 y hy)",
"state_before": "case mpr.intro\nK : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ny : F\nha : y * y ≠ 0\nhy : y ≠ 0\n⊢ ∃ r, y * y = ↑r * ↑r",
"tactic": "refine' ⟨Units.mk0 y hy, _⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro\nK : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ny : F\nha : y * y ≠ 0\nhy : y ≠ 0\n⊢ y * y = ↑(Units.mk0 y hy) * ↑(Units.mk0 y hy)",
"tactic": "simp"
},
{
"state_after": "K : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\nha : 0 * 0 ≠ 0\n⊢ False",
"state_before": "K : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\ny : F\nha : y * y ≠ 0\n⊢ y ≠ 0",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "K : Type ?u.1283171\nR : Type ?u.1283174\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : Fintype F\nhF : ringChar F ≠ 2\nha : 0 * 0 ≠ 0\n⊢ False",
"tactic": "simp at ha"
}
] |
[
569,
38
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
560,
1
] |
Mathlib/Order/Filter/Basic.lean
|
Filter.eventuallyEq_of_left_inv_of_right_inv
|
[] |
[
2873,
95
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2870,
1
] |
Mathlib/Analysis/Convex/Star.lean
|
starConvex_zero_iff
|
[
{
"state_after": "case refine'_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.92819\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMulWithZero 𝕜 E\ns : Set E\nx : E\nx✝ : x ∈ s\nh : ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • 0 + b • x ∈ s\na : 𝕜\nha₀ : 0 ≤ a\nha₁ : a ≤ 1\n⊢ a • x ∈ s\n\ncase refine'_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.92819\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMulWithZero 𝕜 E\ns : Set E\nx : E\nx✝ : x ∈ s\nh : ∀ ⦃a : 𝕜⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a • 0 + b • x ∈ s",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.92819\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMulWithZero 𝕜 E\ns : Set E\n⊢ StarConvex 𝕜 0 s ↔ ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃a : 𝕜⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s",
"tactic": "refine'\n forall_congr' fun x => forall_congr' fun _ => ⟨fun h a ha₀ ha₁ => _, fun h a b ha hb hab => _⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.92819\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMulWithZero 𝕜 E\ns : Set E\nx : E\nx✝ : x ∈ s\nh : ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → a • 0 + b • x ∈ s\na : 𝕜\nha₀ : 0 ≤ a\nha₁ : a ≤ 1\n⊢ a • x ∈ s",
"tactic": "simpa only [sub_add_cancel, eq_self_iff_true, forall_true_left, zero_add, smul_zero] using\n h (sub_nonneg_of_le ha₁) ha₀"
},
{
"state_after": "case refine'_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.92819\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMulWithZero 𝕜 E\ns : Set E\nx : E\nx✝ : x ∈ s\nh : ∀ ⦃a : 𝕜⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ b • x ∈ s",
"state_before": "case refine'_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.92819\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMulWithZero 𝕜 E\ns : Set E\nx : E\nx✝ : x ∈ s\nh : ∀ ⦃a : 𝕜⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ a • 0 + b • x ∈ s",
"tactic": "rw [smul_zero, zero_add]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.92819\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMulWithZero 𝕜 E\ns : Set E\nx : E\nx✝ : x ∈ s\nh : ∀ ⦃a : 𝕜⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ b • x ∈ s",
"tactic": "exact h hb (by rw [← hab]; exact le_add_of_nonneg_left ha)"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.92819\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMulWithZero 𝕜 E\ns : Set E\nx : E\nx✝ : x ∈ s\nh : ∀ ⦃a : 𝕜⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ b ≤ a + b",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.92819\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMulWithZero 𝕜 E\ns : Set E\nx : E\nx✝ : x ∈ s\nh : ∀ ⦃a : 𝕜⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ b ≤ 1",
"tactic": "rw [← hab]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.92819\ninst✝² : OrderedRing 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMulWithZero 𝕜 E\ns : Set E\nx : E\nx✝ : x ∈ s\nh : ∀ ⦃a : 𝕜⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\n⊢ b ≤ a + b",
"tactic": "exact le_add_of_nonneg_left ha"
}
] |
[
326,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
319,
1
] |
Mathlib/Init/Algebra/Classes.lean
|
not_lt_of_lt
|
[] |
[
432,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
430,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.lift_spec_mul_add
|
[
{
"state_after": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_2\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\ng : R →+* P\nhg : ∀ (y : { x // x ∈ M }), IsUnit (↑g ↑y)\nz : S\nw w' v : (fun x => P) z\n⊢ ↑↑(RingHom.toMonoidWithZeroHom g)\n (Submonoid.LocalizationMap.sec (toLocalizationWithZeroMap M S).toLocalizationMap z).fst *\n w +\n ↑g ↑(Submonoid.LocalizationMap.sec (toLocalizationWithZeroMap M S).toLocalizationMap z).snd * w' =\n ↑g ↑(Submonoid.LocalizationMap.sec (toLocalizationWithZeroMap M S).toLocalizationMap z).snd * v ↔\n ↑g (Submonoid.LocalizationMap.sec (toLocalizationMap M S) z).fst * w +\n ↑g ↑(Submonoid.LocalizationMap.sec (toLocalizationMap M S) z).snd * w' =\n ↑g ↑(Submonoid.LocalizationMap.sec (toLocalizationMap M S) z).snd * v",
"state_before": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_2\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\ng : R →+* P\nhg : ∀ (y : { x // x ∈ M }), IsUnit (↑g ↑y)\nz : S\nw w' v : (fun x => P) z\n⊢ ↑(Submonoid.LocalizationWithZeroMap.lift (toLocalizationWithZeroMap M S) (RingHom.toMonoidWithZeroHom g) hg) z * w +\n w' =\n v ↔\n ↑g (Submonoid.LocalizationMap.sec (toLocalizationMap M S) z).fst * w +\n ↑g ↑(Submonoid.LocalizationMap.sec (toLocalizationMap M S) z).snd * w' =\n ↑g ↑(Submonoid.LocalizationMap.sec (toLocalizationMap M S) z).snd * v",
"tactic": "erw [mul_comm, ← mul_assoc, mul_add_inv_left hg, mul_comm]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁴ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝³ : CommSemiring S\ninst✝² : Algebra R S\nP : Type u_2\ninst✝¹ : CommSemiring P\ninst✝ : IsLocalization M S\ng : R →+* P\nhg : ∀ (y : { x // x ∈ M }), IsUnit (↑g ↑y)\nz : S\nw w' v : (fun x => P) z\n⊢ ↑↑(RingHom.toMonoidWithZeroHom g)\n (Submonoid.LocalizationMap.sec (toLocalizationWithZeroMap M S).toLocalizationMap z).fst *\n w +\n ↑g ↑(Submonoid.LocalizationMap.sec (toLocalizationWithZeroMap M S).toLocalizationMap z).snd * w' =\n ↑g ↑(Submonoid.LocalizationMap.sec (toLocalizationWithZeroMap M S).toLocalizationMap z).snd * v ↔\n ↑g (Submonoid.LocalizationMap.sec (toLocalizationMap M S) z).fst * w +\n ↑g ↑(Submonoid.LocalizationMap.sec (toLocalizationMap M S) z).snd * w' =\n ↑g ↑(Submonoid.LocalizationMap.sec (toLocalizationMap M S) z).snd * v",
"tactic": "rfl"
}
] |
[
473,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
468,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.sup_sdiff_left
|
[
{
"state_after": "case refine'_1\nF : Type ?u.198274\nα : Type u_2\nβ : Type ?u.198280\nγ : Type ?u.198283\nι : Type u_1\nκ : Type ?u.198289\ninst✝ : BooleanAlgebra α\ns✝ s : Finset ι\nf : ι → α\na : α\n⊢ (sup ∅ fun b => a \\ f b) = a \\ inf ∅ f\n\ncase refine'_2\nF : Type ?u.198274\nα : Type u_2\nβ : Type ?u.198280\nγ : Type ?u.198283\nι : Type u_1\nκ : Type ?u.198289\ninst✝ : BooleanAlgebra α\ns✝ s : Finset ι\nf : ι → α\na : α\nb : ι\nt : Finset ι\nx✝ : ¬b ∈ t\nh : (sup t fun b => a \\ f b) = a \\ inf t f\n⊢ (sup (cons b t x✝) fun b => a \\ f b) = a \\ inf (cons b t x✝) f",
"state_before": "F : Type ?u.198274\nα : Type u_2\nβ : Type ?u.198280\nγ : Type ?u.198283\nι : Type u_1\nκ : Type ?u.198289\ninst✝ : BooleanAlgebra α\ns✝ s : Finset ι\nf : ι → α\na : α\n⊢ (sup s fun b => a \\ f b) = a \\ inf s f",
"tactic": "refine' Finset.cons_induction_on s _ fun b t _ h => _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nF : Type ?u.198274\nα : Type u_2\nβ : Type ?u.198280\nγ : Type ?u.198283\nι : Type u_1\nκ : Type ?u.198289\ninst✝ : BooleanAlgebra α\ns✝ s : Finset ι\nf : ι → α\na : α\n⊢ (sup ∅ fun b => a \\ f b) = a \\ inf ∅ f",
"tactic": "rw [sup_empty, inf_empty, sdiff_top]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nF : Type ?u.198274\nα : Type u_2\nβ : Type ?u.198280\nγ : Type ?u.198283\nι : Type u_1\nκ : Type ?u.198289\ninst✝ : BooleanAlgebra α\ns✝ s : Finset ι\nf : ι → α\na : α\nb : ι\nt : Finset ι\nx✝ : ¬b ∈ t\nh : (sup t fun b => a \\ f b) = a \\ inf t f\n⊢ (sup (cons b t x✝) fun b => a \\ f b) = a \\ inf (cons b t x✝) f",
"tactic": "rw [sup_cons, inf_cons, h, sdiff_inf]"
}
] |
[
621,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
617,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean
|
Set.preimage_const_mul_Iic
|
[] |
[
612,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
611,
1
] |
Mathlib/RingTheory/IntegralClosure.lean
|
isIntegral_zero
|
[] |
[
481,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
480,
1
] |
Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean
|
GromovHausdorff.maxVar_bound
|
[
{
"state_after": "no goals",
"state_before": "X : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ diam univ = diam (range inl ∪ range inr)",
"tactic": "rw [range_inl_union_range_inr]"
},
{
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"tactic": "rw [isometry_inl.diam_range, isometry_inr.diam_range]"
},
{
"state_after": "no goals",
"state_before": "X : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ diam univ + dist (inl default) (inr default) + diam univ =\n diam univ + (dist default default + 1 + dist default default) + diam univ\n\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ Type ?u.17724\n\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ Dist ?m.18397\n\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ Inhabited ?m.18397\n\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ Inhabited ?m.18397\n\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ Type ?u.19088\n\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ Dist ?m.19761\n\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ Inhabited ?m.19761\n\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ Inhabited ?m.19761\n\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ Type ?u.17724\n\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ Dist ?m.18397\n\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ Inhabited ?m.18397\n\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ Inhabited ?m.18397\n\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ Type ?u.19088\n\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ Dist ?m.19761\n\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ Inhabited ?m.19761\n\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ Inhabited ?m.19761",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "X : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ diam univ + (dist default default + 1 + dist default default) + diam univ = 1 * diam univ + 1 + 1 * diam univ",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "X : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : GromovHausdorff.ProdSpaceFun X Y\nx y z t : X ⊕ Y\n⊢ 1 * diam univ + 1 + 1 * diam univ ≤ 2 * diam univ + 1 + 2 * diam univ",
"tactic": "gcongr <;> norm_num"
}
] |
[
121,
88
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
108,
9
] |
Mathlib/Data/List/AList.lean
|
AList.insertRec_insert
|
[
{
"state_after": "case mk\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\nC : AList β → Sort u_1\nH0 : C ∅\nIH : (a : α) → (b : β a) → (l : AList β) → ¬a ∈ l → C l → C (insert a b l)\nc : Sigma β\nl : List (Sigma β)\nhl : NodupKeys l\nh : ¬c.fst ∈ { entries := l, nodupKeys := hl }\n⊢ insertRec H0 IH (insert c.fst c.snd { entries := l, nodupKeys := hl }) =\n IH c.fst c.snd { entries := l, nodupKeys := hl } h (insertRec H0 IH { entries := l, nodupKeys := hl })",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\nC : AList β → Sort u_1\nH0 : C ∅\nIH : (a : α) → (b : β a) → (l : AList β) → ¬a ∈ l → C l → C (insert a b l)\nc : Sigma β\nl : AList β\nh : ¬c.fst ∈ l\n⊢ insertRec H0 IH (insert c.fst c.snd l) = IH c.fst c.snd l h (insertRec H0 IH l)",
"tactic": "cases' l with l hl"
},
{
"state_after": "case mk\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\nC : AList β → Sort u_1\nH0 : C ∅\nIH : (a : α) → (b : β a) → (l : AList β) → ¬a ∈ l → C l → C (insert a b l)\nc : Sigma β\nl : List (Sigma β)\nhl : NodupKeys l\nh : ¬c.fst ∈ { entries := l, nodupKeys := hl }\n⊢ HEq (insertRec H0 IH { entries := c :: l, nodupKeys := (_ : NodupKeys (c :: l)) })\n (IH c.fst c.snd { entries := l, nodupKeys := hl } h (insertRec H0 IH { entries := l, nodupKeys := hl }))",
"state_before": "case mk\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\nC : AList β → Sort u_1\nH0 : C ∅\nIH : (a : α) → (b : β a) → (l : AList β) → ¬a ∈ l → C l → C (insert a b l)\nc : Sigma β\nl : List (Sigma β)\nhl : NodupKeys l\nh : ¬c.fst ∈ { entries := l, nodupKeys := hl }\n⊢ insertRec H0 IH (insert c.fst c.snd { entries := l, nodupKeys := hl }) =\n IH c.fst c.snd { entries := l, nodupKeys := hl } h (insertRec H0 IH { entries := l, nodupKeys := hl })",
"tactic": "suffices HEq (@insertRec α β _ C H0 IH ⟨c :: l, nodupKeys_cons.2 ⟨h, hl⟩⟩)\n (IH c.1 c.2 ⟨l, hl⟩ h (@insertRec α β _ C H0 IH ⟨l, hl⟩)) by\n cases c\n apply eq_of_heq\n convert this <;> rw [insert_of_neg h]"
},
{
"state_after": "case mk\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\nC : AList β → Sort u_1\nH0 : C ∅\nIH : (a : α) → (b : β a) → (l : AList β) → ¬a ∈ l → C l → C (insert a b l)\nc : Sigma β\nl : List (Sigma β)\nhl : NodupKeys l\nh : ¬c.fst ∈ { entries := l, nodupKeys := hl }\n⊢ HEq\n (Eq.mpr\n (_ :\n C { entries := c :: l, nodupKeys := (_ : NodupKeys (c :: l)) } =\n C (insert c.fst c.snd { entries := l, nodupKeys := (_ : NodupKeys l) }))\n (IH c.fst c.snd { entries := l, nodupKeys := (_ : NodupKeys l) }\n (_ : ¬c.fst ∈ List.keys { entries := l, nodupKeys := (_ : NodupKeys l) }.entries)\n (insertRec H0 IH { entries := l, nodupKeys := (_ : NodupKeys l) })))\n (IH c.fst c.snd { entries := l, nodupKeys := hl } h (insertRec H0 IH { entries := l, nodupKeys := hl }))",
"state_before": "case mk\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\nC : AList β → Sort u_1\nH0 : C ∅\nIH : (a : α) → (b : β a) → (l : AList β) → ¬a ∈ l → C l → C (insert a b l)\nc : Sigma β\nl : List (Sigma β)\nhl : NodupKeys l\nh : ¬c.fst ∈ { entries := l, nodupKeys := hl }\n⊢ HEq (insertRec H0 IH { entries := c :: l, nodupKeys := (_ : NodupKeys (c :: l)) })\n (IH c.fst c.snd { entries := l, nodupKeys := hl } h (insertRec H0 IH { entries := l, nodupKeys := hl }))",
"tactic": "rw [insertRec]"
},
{
"state_after": "no goals",
"state_before": "case mk\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\nC : AList β → Sort u_1\nH0 : C ∅\nIH : (a : α) → (b : β a) → (l : AList β) → ¬a ∈ l → C l → C (insert a b l)\nc : Sigma β\nl : List (Sigma β)\nhl : NodupKeys l\nh : ¬c.fst ∈ { entries := l, nodupKeys := hl }\n⊢ HEq\n (Eq.mpr\n (_ :\n C { entries := c :: l, nodupKeys := (_ : NodupKeys (c :: l)) } =\n C (insert c.fst c.snd { entries := l, nodupKeys := (_ : NodupKeys l) }))\n (IH c.fst c.snd { entries := l, nodupKeys := (_ : NodupKeys l) }\n (_ : ¬c.fst ∈ List.keys { entries := l, nodupKeys := (_ : NodupKeys l) }.entries)\n (insertRec H0 IH { entries := l, nodupKeys := (_ : NodupKeys l) })))\n (IH c.fst c.snd { entries := l, nodupKeys := hl } h (insertRec H0 IH { entries := l, nodupKeys := hl }))",
"tactic": "apply cast_heq"
},
{
"state_after": "case mk\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\nC : AList β → Sort u_1\nH0 : C ∅\nIH : (a : α) → (b : β a) → (l : AList β) → ¬a ∈ l → C l → C (insert a b l)\nl : List (Sigma β)\nhl : NodupKeys l\nfst✝ : α\nsnd✝ : β fst✝\nh : ¬{ fst := fst✝, snd := snd✝ }.fst ∈ { entries := l, nodupKeys := hl }\nthis :\n HEq\n (insertRec H0 IH\n { entries := { fst := fst✝, snd := snd✝ } :: l,\n nodupKeys := (_ : NodupKeys ({ fst := fst✝, snd := snd✝ } :: l)) })\n (IH { fst := fst✝, snd := snd✝ }.fst { fst := fst✝, snd := snd✝ }.snd { entries := l, nodupKeys := hl } h\n (insertRec H0 IH { entries := l, nodupKeys := hl }))\n⊢ insertRec H0 IH\n (insert { fst := fst✝, snd := snd✝ }.fst { fst := fst✝, snd := snd✝ }.snd { entries := l, nodupKeys := hl }) =\n IH { fst := fst✝, snd := snd✝ }.fst { fst := fst✝, snd := snd✝ }.snd { entries := l, nodupKeys := hl } h\n (insertRec H0 IH { entries := l, nodupKeys := hl })",
"state_before": "α : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\nC : AList β → Sort u_1\nH0 : C ∅\nIH : (a : α) → (b : β a) → (l : AList β) → ¬a ∈ l → C l → C (insert a b l)\nc : Sigma β\nl : List (Sigma β)\nhl : NodupKeys l\nh : ¬c.fst ∈ { entries := l, nodupKeys := hl }\nthis :\n HEq (insertRec H0 IH { entries := c :: l, nodupKeys := (_ : NodupKeys (c :: l)) })\n (IH c.fst c.snd { entries := l, nodupKeys := hl } h (insertRec H0 IH { entries := l, nodupKeys := hl }))\n⊢ insertRec H0 IH (insert c.fst c.snd { entries := l, nodupKeys := hl }) =\n IH c.fst c.snd { entries := l, nodupKeys := hl } h (insertRec H0 IH { entries := l, nodupKeys := hl })",
"tactic": "cases c"
},
{
"state_after": "case mk.h\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\nC : AList β → Sort u_1\nH0 : C ∅\nIH : (a : α) → (b : β a) → (l : AList β) → ¬a ∈ l → C l → C (insert a b l)\nl : List (Sigma β)\nhl : NodupKeys l\nfst✝ : α\nsnd✝ : β fst✝\nh : ¬{ fst := fst✝, snd := snd✝ }.fst ∈ { entries := l, nodupKeys := hl }\nthis :\n HEq\n (insertRec H0 IH\n { entries := { fst := fst✝, snd := snd✝ } :: l,\n nodupKeys := (_ : NodupKeys ({ fst := fst✝, snd := snd✝ } :: l)) })\n (IH { fst := fst✝, snd := snd✝ }.fst { fst := fst✝, snd := snd✝ }.snd { entries := l, nodupKeys := hl } h\n (insertRec H0 IH { entries := l, nodupKeys := hl }))\n⊢ HEq\n (insertRec H0 IH\n (insert { fst := fst✝, snd := snd✝ }.fst { fst := fst✝, snd := snd✝ }.snd { entries := l, nodupKeys := hl }))\n (IH { fst := fst✝, snd := snd✝ }.fst { fst := fst✝, snd := snd✝ }.snd { entries := l, nodupKeys := hl } h\n (insertRec H0 IH { entries := l, nodupKeys := hl }))",
"state_before": "case mk\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\nC : AList β → Sort u_1\nH0 : C ∅\nIH : (a : α) → (b : β a) → (l : AList β) → ¬a ∈ l → C l → C (insert a b l)\nl : List (Sigma β)\nhl : NodupKeys l\nfst✝ : α\nsnd✝ : β fst✝\nh : ¬{ fst := fst✝, snd := snd✝ }.fst ∈ { entries := l, nodupKeys := hl }\nthis :\n HEq\n (insertRec H0 IH\n { entries := { fst := fst✝, snd := snd✝ } :: l,\n nodupKeys := (_ : NodupKeys ({ fst := fst✝, snd := snd✝ } :: l)) })\n (IH { fst := fst✝, snd := snd✝ }.fst { fst := fst✝, snd := snd✝ }.snd { entries := l, nodupKeys := hl } h\n (insertRec H0 IH { entries := l, nodupKeys := hl }))\n⊢ insertRec H0 IH\n (insert { fst := fst✝, snd := snd✝ }.fst { fst := fst✝, snd := snd✝ }.snd { entries := l, nodupKeys := hl }) =\n IH { fst := fst✝, snd := snd✝ }.fst { fst := fst✝, snd := snd✝ }.snd { entries := l, nodupKeys := hl } h\n (insertRec H0 IH { entries := l, nodupKeys := hl })",
"tactic": "apply eq_of_heq"
},
{
"state_after": "no goals",
"state_before": "case mk.h\nα : Type u\nβ : α → Type v\ninst✝ : DecidableEq α\nC : AList β → Sort u_1\nH0 : C ∅\nIH : (a : α) → (b : β a) → (l : AList β) → ¬a ∈ l → C l → C (insert a b l)\nl : List (Sigma β)\nhl : NodupKeys l\nfst✝ : α\nsnd✝ : β fst✝\nh : ¬{ fst := fst✝, snd := snd✝ }.fst ∈ { entries := l, nodupKeys := hl }\nthis :\n HEq\n (insertRec H0 IH\n { entries := { fst := fst✝, snd := snd✝ } :: l,\n nodupKeys := (_ : NodupKeys ({ fst := fst✝, snd := snd✝ } :: l)) })\n (IH { fst := fst✝, snd := snd✝ }.fst { fst := fst✝, snd := snd✝ }.snd { entries := l, nodupKeys := hl } h\n (insertRec H0 IH { entries := l, nodupKeys := hl }))\n⊢ HEq\n (insertRec H0 IH\n (insert { fst := fst✝, snd := snd✝ }.fst { fst := fst✝, snd := snd✝ }.snd { entries := l, nodupKeys := hl }))\n (IH { fst := fst✝, snd := snd✝ }.fst { fst := fst✝, snd := snd✝ }.snd { entries := l, nodupKeys := hl } h\n (insertRec H0 IH { entries := l, nodupKeys := hl }))",
"tactic": "convert this <;> rw [insert_of_neg h]"
}
] |
[
387,
17
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
376,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.iUnion_ofWithBot
|
[
{
"state_after": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nboxes : Finset (WithBot (Box ι))\nle_of_mem : ∀ (J : WithBot (Box ι)), J ∈ boxes → J ≤ ↑I\npairwise_disjoint : Set.Pairwise (↑boxes) Disjoint\n⊢ (⋃ (J : Box ι) (_ : ↑J ∈ boxes), ↑J) = ⋃ (J : WithBot (Box ι)) (_ : J ∈ boxes), ↑J",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nboxes : Finset (WithBot (Box ι))\nle_of_mem : ∀ (J : WithBot (Box ι)), J ∈ boxes → J ≤ ↑I\npairwise_disjoint : Set.Pairwise (↑boxes) Disjoint\n⊢ Prepartition.iUnion (ofWithBot boxes le_of_mem pairwise_disjoint) = ⋃ (J : WithBot (Box ι)) (_ : J ∈ boxes), ↑J",
"tactic": "suffices (⋃ (J : Box ι) (_ : ↑J ∈ boxes), ↑J) = ⋃ J ∈ boxes, (J : Set (ι → ℝ)) by\n simpa [ofWithBot, Prepartition.iUnion]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nboxes : Finset (WithBot (Box ι))\nle_of_mem : ∀ (J : WithBot (Box ι)), J ∈ boxes → J ≤ ↑I\npairwise_disjoint : Set.Pairwise (↑boxes) Disjoint\n⊢ (⋃ (J : Box ι) (_ : ↑J ∈ boxes), ↑J) = ⋃ (J : WithBot (Box ι)) (_ : J ∈ boxes), ↑J",
"tactic": "simp only [← Box.biUnion_coe_eq_coe, @iUnion_comm _ _ (Box ι), @iUnion_comm _ _ (@Eq _ _ _),\n iUnion_iUnion_eq_right]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁ π₂ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nboxes : Finset (WithBot (Box ι))\nle_of_mem : ∀ (J : WithBot (Box ι)), J ∈ boxes → J ≤ ↑I\npairwise_disjoint : Set.Pairwise (↑boxes) Disjoint\nthis : (⋃ (J : Box ι) (_ : ↑J ∈ boxes), ↑J) = ⋃ (J : WithBot (Box ι)) (_ : J ∈ boxes), ↑J\n⊢ Prepartition.iUnion (ofWithBot boxes le_of_mem pairwise_disjoint) = ⋃ (J : WithBot (Box ι)) (_ : J ∈ boxes), ↑J",
"tactic": "simpa [ofWithBot, Prepartition.iUnion]"
}
] |
[
431,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
424,
1
] |
Mathlib/Order/Filter/Germ.lean
|
Filter.const_eventuallyEq
|
[] |
[
67,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
66,
1
] |
Mathlib/Data/Int/Cast/Lemmas.lean
|
Int.cast_natAbs
|
[
{
"state_after": "case ofNat\nF : Type ?u.33869\nι : Type ?u.33872\nα : Type u_1\nβ : Type ?u.33878\ninst✝ : LinearOrderedRing α\na b : ℤ\na✝ : ℕ\n⊢ ↑(natAbs (ofNat a✝)) = ↑(abs (ofNat a✝))\n\ncase negSucc\nF : Type ?u.33869\nι : Type ?u.33872\nα : Type u_1\nβ : Type ?u.33878\ninst✝ : LinearOrderedRing α\na b : ℤ\na✝ : ℕ\n⊢ ↑(natAbs -[a✝+1]) = ↑(abs -[a✝+1])",
"state_before": "F : Type ?u.33869\nι : Type ?u.33872\nα : Type u_1\nβ : Type ?u.33878\ninst✝ : LinearOrderedRing α\na b n : ℤ\n⊢ ↑(natAbs n) = ↑(abs n)",
"tactic": "cases n"
},
{
"state_after": "no goals",
"state_before": "case ofNat\nF : Type ?u.33869\nι : Type ?u.33872\nα : Type u_1\nβ : Type ?u.33878\ninst✝ : LinearOrderedRing α\na b : ℤ\na✝ : ℕ\n⊢ ↑(natAbs (ofNat a✝)) = ↑(abs (ofNat a✝))",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "case negSucc\nF : Type ?u.33869\nι : Type ?u.33872\nα : Type u_1\nβ : Type ?u.33878\ninst✝ : LinearOrderedRing α\na b : ℤ\na✝ : ℕ\n⊢ ↑(natAbs -[a✝+1]) = ↑(abs -[a✝+1])",
"tactic": "rw [abs_eq_natAbs, natAbs_negSucc, cast_succ, cast_ofNat, cast_succ]"
}
] |
[
210,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
207,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean
|
Subalgebra.coe_star
|
[] |
[
368,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
367,
1
] |
Mathlib/Algebra/FreeAlgebra.lean
|
FreeAlgebra.liftAux_eq
|
[
{
"state_after": "R : Type u_1\ninst✝² : CommSemiring R\nX : Type u_2\nA : Type u_3\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : X → A\n⊢ FreeAlgebra.liftAux R f =\n ↑{ toFun := FreeAlgebra.liftAux R, invFun := fun F => ↑F ∘ ι R,\n left_inv := (_ : ∀ (f : X → A), (fun F => ↑F ∘ ι R) (FreeAlgebra.liftAux R f) = f),\n right_inv := (_ : ∀ (F : FreeAlgebra R X →ₐ[R] A), FreeAlgebra.liftAux R ((fun F => ↑F ∘ ι R) F) = F) }\n f",
"state_before": "R : Type u_1\ninst✝² : CommSemiring R\nX : Type u_2\nA : Type u_3\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : X → A\n⊢ FreeAlgebra.liftAux R f = ↑(lift R) f",
"tactic": "rw [lift]"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝² : CommSemiring R\nX : Type u_2\nA : Type u_3\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nf : X → A\n⊢ FreeAlgebra.liftAux R f =\n ↑{ toFun := FreeAlgebra.liftAux R, invFun := fun F => ↑F ∘ ι R,\n left_inv := (_ : ∀ (f : X → A), (fun F => ↑F ∘ ι R) (FreeAlgebra.liftAux R f) = f),\n right_inv := (_ : ∀ (F : FreeAlgebra R X →ₐ[R] A), FreeAlgebra.liftAux R ((fun F => ↑F ∘ ι R) F) = F) }\n f",
"tactic": "rfl"
}
] |
[
328,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
326,
1
] |
Mathlib/Data/Finset/Fold.lean
|
Finset.fold_insert
|
[
{
"state_after": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.4116\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ninst✝ : DecidableEq α\nh : ¬a ∈ s\n⊢ Multiset.fold op b (Multiset.map f (insert a s).val) = op (f a) (Multiset.fold op b (Multiset.map f s.val))",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.4116\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ninst✝ : DecidableEq α\nh : ¬a ∈ s\n⊢ fold op b f (insert a s) = op (f a) (fold op b f s)",
"tactic": "unfold fold"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type ?u.4116\nop : β → β → β\nhc : IsCommutative β op\nha : IsAssociative β op\nf : α → β\nb : β\ns : Finset α\na : α\ninst✝ : DecidableEq α\nh : ¬a ∈ s\n⊢ Multiset.fold op b (Multiset.map f (insert a s).val) = op (f a) (Multiset.fold op b (Multiset.map f s.val))",
"tactic": "rw [insert_val, ndinsert_of_not_mem h, Multiset.map_cons, fold_cons_left]"
}
] |
[
58,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/RingTheory/NonUnitalSubsemiring/Basic.lean
|
NonUnitalSubsemiring.centralizer_le
|
[] |
[
541,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
539,
1
] |
Mathlib/Order/RelClasses.lean
|
wellFoundedLT_dual_iff
|
[] |
[
374,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
373,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean
|
InnerProductGeometry.angle_zero_right
|
[
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y x : V\n⊢ arccos (inner x 0 / (‖x‖ * ‖0‖)) = π / 2",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y x : V\n⊢ angle x 0 = π / 2",
"tactic": "unfold angle"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y x : V\n⊢ arccos (inner x 0 / (‖x‖ * ‖0‖)) = π / 2",
"tactic": "rw [inner_zero_right, zero_div, Real.arccos_zero]"
}
] |
[
129,
52
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
127,
1
] |
src/lean/Init/Data/Nat/Linear.lean
|
Nat.Linear.Poly.denote_le_cancel_eq
|
[] |
[
506,
86
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
505,
1
] |
Mathlib/RingTheory/Localization/Basic.lean
|
IsLocalization.isLocalization_iff_of_base_ringEquiv
|
[
{
"state_after": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_2\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n⊢ IsLocalization M S ↔ IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_2\ninst✝ : CommSemiring P\nh : R ≃+* P\n⊢ IsLocalization M S ↔ IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S",
"tactic": "letI : Algebra P S := ((algebraMap R S).comp h.symm.toRingHom).toAlgebra"
},
{
"state_after": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_2\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n⊢ IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S → IsLocalization M S",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_2\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n⊢ IsLocalization M S ↔ IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S",
"tactic": "refine' ⟨fun _ => isLocalization_of_base_ringEquiv M S h, _⟩"
},
{
"state_after": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_2\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n⊢ IsLocalization M S",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_2\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\n⊢ IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S → IsLocalization M S",
"tactic": "intro H"
},
{
"state_after": "case h.e'_3\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_2\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n⊢ M = Submonoid.map (RingEquiv.toMonoidHom (RingEquiv.symm h)) (Submonoid.map (RingEquiv.toMonoidHom h) M)\n\ncase h.e'_6\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_2\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n⊢ inst✝¹ = RingHom.toAlgebra (RingHom.comp (algebraMap P S) (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.symm h))))",
"state_before": "R : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_2\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n⊢ IsLocalization M S",
"tactic": "convert isLocalization_of_base_ringEquiv (Submonoid.map (RingEquiv.toMonoidHom h) M) S h.symm"
},
{
"state_after": "case h.e'_6\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_2\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n⊢ inst✝¹ =\n RingHom.toAlgebra\n (RingHom.comp (algebraMap R S)\n (RingHom.comp (RingEquiv.toRingHom (RingEquiv.symm h))\n (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.symm h)))))",
"state_before": "case h.e'_6\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_2\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n⊢ inst✝¹ = RingHom.toAlgebra (RingHom.comp (algebraMap P S) (RingEquiv.toRingHom (RingEquiv.symm (RingEquiv.symm h))))",
"tactic": "rw [RingHom.algebraMap_toAlgebra, RingHom.comp_assoc]"
},
{
"state_after": "case h.e'_6\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_2\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n⊢ inst✝¹ = RingHom.toAlgebra (algebraMap R S)",
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"tactic": "simp only [RingHom.comp_id, RingEquiv.symm_symm, RingEquiv.symm_toRingHom_comp_toRingHom]"
},
{
"state_after": "case h.e'_6.h\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_2\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n⊢ ∀ (r : R), ↑(algebraMap R S) r = ↑(algebraMap R S) r",
"state_before": "case h.e'_6\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_2\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n⊢ inst✝¹ = RingHom.toAlgebra (algebraMap R S)",
"tactic": "apply Algebra.algebra_ext"
},
{
"state_after": "case h.e'_6.h\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_2\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\nr : R\n⊢ ↑(algebraMap R S) r = ↑(algebraMap R S) r",
"state_before": "case h.e'_6.h\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_2\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n⊢ ∀ (r : R), ↑(algebraMap R S) r = ↑(algebraMap R S) r",
"tactic": "intro r"
},
{
"state_after": "no goals",
"state_before": "case h.e'_6.h\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_2\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\nr : R\n⊢ ↑(algebraMap R S) r = ↑(algebraMap R S) r",
"tactic": "rw [RingHom.algebraMap_toAlgebra]"
},
{
"state_after": "case h.e'_3.hf\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_2\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n⊢ Injective ↑(MulEquiv.toMonoidHom (MulEquiv.symm (RingEquiv.toMulEquiv (RingEquiv.symm h))))",
"state_before": "case h.e'_3\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_2\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n⊢ M = Submonoid.map (RingEquiv.toMonoidHom (RingEquiv.symm h)) (Submonoid.map (RingEquiv.toMonoidHom h) M)",
"tactic": "erw [Submonoid.map_equiv_eq_comap_symm, Submonoid.comap_map_eq_of_injective]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3.hf\nR : Type u_1\ninst✝³ : CommSemiring R\nM : Submonoid R\nS : Type u_3\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\nP : Type u_2\ninst✝ : CommSemiring P\nh : R ≃+* P\nthis : Algebra P S := RingHom.toAlgebra (RingHom.comp (algebraMap R S) (RingEquiv.toRingHom (RingEquiv.symm h)))\nH : IsLocalization (Submonoid.map (RingEquiv.toMonoidHom h) M) S\n⊢ Injective ↑(MulEquiv.toMonoidHom (MulEquiv.symm (RingEquiv.toMulEquiv (RingEquiv.symm h))))",
"tactic": "exact h.toEquiv.injective"
}
] |
[
818,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
804,
1
] |
Std/Data/PairingHeap.lean
|
Std.PairingHeapImp.Heap.noSibling_merge
|
[
{
"state_after": "α : Type u_1\nle : α → α → Bool\ns₁ s₂ : Heap α\n⊢ NoSibling\n (match s₁, s₂ with\n | nil, nil => nil\n | nil, node a₂ c₂ sibling => node a₂ c₂ nil\n | node a₁ c₁ sibling, nil => node a₁ c₁ nil\n | node a₁ c₁ sibling, node a₂ c₂ sibling_1 =>\n if le a₁ a₂ = true then node a₁ (node a₂ c₂ c₁) nil else node a₂ (node a₁ c₁ c₂) nil)",
"state_before": "α : Type u_1\nle : α → α → Bool\ns₁ s₂ : Heap α\n⊢ NoSibling (merge le s₁ s₂)",
"tactic": "unfold merge"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nle : α → α → Bool\ns₁ s₂ : Heap α\n⊢ NoSibling\n (match s₁, s₂ with\n | nil, nil => nil\n | nil, node a₂ c₂ sibling => node a₂ c₂ nil\n | node a₁ c₁ sibling, nil => node a₁ c₁ nil\n | node a₁ c₁ sibling, node a₂ c₂ sibling_1 =>\n if le a₁ a₂ = true then node a₁ (node a₂ c₂ c₁) nil else node a₂ (node a₁ c₁ c₂) nil)",
"tactic": "(split <;> try split) <;> constructor"
},
{
"state_after": "case h_1\nα : Type u_1\nle : α → α → Bool\nx✝¹ x✝ : Heap α\n⊢ NoSibling nil\n\ncase h_2\nα : Type u_1\nle : α → α → Bool\nx✝¹ x✝ : Heap α\na₂✝ : α\nc₂✝ sibling✝ : Heap α\n⊢ NoSibling (node a₂✝ c₂✝ nil)\n\ncase h_3\nα : Type u_1\nle : α → α → Bool\nx✝¹ x✝ : Heap α\na₁✝ : α\nc₁✝ sibling✝ : Heap α\n⊢ NoSibling (node a₁✝ c₁✝ nil)\n\ncase h_4.inl\nα : Type u_1\nle : α → α → Bool\nx✝¹ x✝ : Heap α\na₁✝ : α\nc₁✝ sibling✝¹ : Heap α\na₂✝ : α\nc₂✝ sibling✝ : Heap α\nh✝ : le a₁✝ a₂✝ = true\n⊢ NoSibling (node a₁✝ (node a₂✝ c₂✝ c₁✝) nil)\n\ncase h_4.inr\nα : Type u_1\nle : α → α → Bool\nx✝¹ x✝ : Heap α\na₁✝ : α\nc₁✝ sibling✝¹ : Heap α\na₂✝ : α\nc₂✝ sibling✝ : Heap α\nh✝ : ¬le a₁✝ a₂✝ = true\n⊢ NoSibling (node a₂✝ (node a₁✝ c₁✝ c₂✝) nil)",
"state_before": "α : Type u_1\nle : α → α → Bool\ns₁ s₂ : Heap α\n⊢ NoSibling\n (match s₁, s₂ with\n | nil, nil => nil\n | nil, node a₂ c₂ sibling => node a₂ c₂ nil\n | node a₁ c₁ sibling, nil => node a₁ c₁ nil\n | node a₁ c₁ sibling, node a₂ c₂ sibling_1 =>\n if le a₁ a₂ = true then node a₁ (node a₂ c₂ c₁) nil else node a₂ (node a₁ c₁ c₂) nil)",
"tactic": "split <;> try split"
},
{
"state_after": "case h_4.inl\nα : Type u_1\nle : α → α → Bool\nx✝¹ x✝ : Heap α\na₁✝ : α\nc₁✝ sibling✝¹ : Heap α\na₂✝ : α\nc₂✝ sibling✝ : Heap α\nh✝ : le a₁✝ a₂✝ = true\n⊢ NoSibling (node a₁✝ (node a₂✝ c₂✝ c₁✝) nil)\n\ncase h_4.inr\nα : Type u_1\nle : α → α → Bool\nx✝¹ x✝ : Heap α\na₁✝ : α\nc₁✝ sibling✝¹ : Heap α\na₂✝ : α\nc₂✝ sibling✝ : Heap α\nh✝ : ¬le a₁✝ a₂✝ = true\n⊢ NoSibling (node a₂✝ (node a₁✝ c₁✝ c₂✝) nil)",
"state_before": "case h_4\nα : Type u_1\nle : α → α → Bool\nx✝¹ x✝ : Heap α\na₁✝ : α\nc₁✝ sibling✝¹ : Heap α\na₂✝ : α\nc₂✝ sibling✝ : Heap α\n⊢ NoSibling (if le a₁✝ a₂✝ = true then node a₁✝ (node a₂✝ c₂✝ c₁✝) nil else node a₂✝ (node a₁✝ c₁✝ c₂✝) nil)",
"tactic": "split"
}
] |
[
94,
40
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
91,
1
] |
Mathlib/RingTheory/NonZeroDivisors.lean
|
prod_zero_iff_exists_zero
|
[
{
"state_after": "case mp\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\ns : Multiset M₁\n⊢ Multiset.prod s = 0 → ∃ r x, r = 0\n\ncase mpr\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\ns : Multiset M₁\n⊢ (∃ r x, r = 0) → Multiset.prod s = 0",
"state_before": "M : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\ns : Multiset M₁\n⊢ Multiset.prod s = 0 ↔ ∃ r x, r = 0",
"tactic": "constructor"
},
{
"state_after": "case mpr\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\ns : Multiset M₁\n⊢ (∃ r x, r = 0) → Multiset.prod s = 0\n\ncase mp\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\ns : Multiset M₁\n⊢ Multiset.prod s = 0 → ∃ r x, r = 0",
"state_before": "case mp\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\ns : Multiset M₁\n⊢ Multiset.prod s = 0 → ∃ r x, r = 0\n\ncase mpr\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\ns : Multiset M₁\n⊢ (∃ r x, r = 0) → Multiset.prod s = 0",
"tactic": "swap"
},
{
"state_after": "case mp.empty\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\n⊢ Multiset.prod 0 = 0 → ∃ r x, r = 0\n\ncase mp.cons\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\na : M₁\ns : Multiset M₁\nih : Multiset.prod s = 0 → ∃ r x, r = 0\n⊢ Multiset.prod (a ::ₘ s) = 0 → ∃ r x, r = 0",
"state_before": "case mp\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\ns : Multiset M₁\n⊢ Multiset.prod s = 0 → ∃ r x, r = 0",
"tactic": "induction' s using Multiset.induction_on with a s ih"
},
{
"state_after": "case mpr.intro.intro\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\ns : Multiset M₁\nhrs : 0 ∈ s\n⊢ Multiset.prod s = 0",
"state_before": "case mpr\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\ns : Multiset M₁\n⊢ (∃ r x, r = 0) → Multiset.prod s = 0",
"tactic": "rintro ⟨r, hrs, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\ns : Multiset M₁\nhrs : 0 ∈ s\n⊢ Multiset.prod s = 0",
"tactic": "exact Multiset.prod_eq_zero hrs"
},
{
"state_after": "case mp.empty\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\nhabs : Multiset.prod 0 = 0\n⊢ ∃ r x, r = 0",
"state_before": "case mp.empty\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\n⊢ Multiset.prod 0 = 0 → ∃ r x, r = 0",
"tactic": "intro habs"
},
{
"state_after": "no goals",
"state_before": "case mp.empty\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\nhabs : Multiset.prod 0 = 0\n⊢ ∃ r x, r = 0",
"tactic": "simp at habs"
},
{
"state_after": "case mp.cons\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\na : M₁\ns : Multiset M₁\nih : Multiset.prod s = 0 → ∃ r x, r = 0\n⊢ a * Multiset.prod s = 0 → ∃ r x, r = 0",
"state_before": "case mp.cons\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\na : M₁\ns : Multiset M₁\nih : Multiset.prod s = 0 → ∃ r x, r = 0\n⊢ Multiset.prod (a ::ₘ s) = 0 → ∃ r x, r = 0",
"tactic": "rw [Multiset.prod_cons]"
},
{
"state_after": "case mp.cons\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\na : M₁\ns : Multiset M₁\nih : Multiset.prod s = 0 → ∃ r x, r = 0\nhprod : a * Multiset.prod s = 0\n⊢ ∃ r x, r = 0",
"state_before": "case mp.cons\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\na : M₁\ns : Multiset M₁\nih : Multiset.prod s = 0 → ∃ r x, r = 0\n⊢ a * Multiset.prod s = 0 → ∃ r x, r = 0",
"tactic": "intro hprod"
},
{
"state_after": "case mp.cons\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\na : M₁\ns : Multiset M₁\nih : Multiset.prod s = 0 → ∃ r x, r = 0\nhprod : a = 0 ∨ Multiset.prod s = 0\n⊢ ∃ r x, r = 0",
"state_before": "case mp.cons\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\na : M₁\ns : Multiset M₁\nih : Multiset.prod s = 0 → ∃ r x, r = 0\nhprod : a * Multiset.prod s = 0\n⊢ ∃ r x, r = 0",
"tactic": "replace hprod := eq_zero_or_eq_zero_of_mul_eq_zero hprod"
},
{
"state_after": "case mp.cons.inl\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\na : M₁\ns : Multiset M₁\nih : Multiset.prod s = 0 → ∃ r x, r = 0\nha : a = 0\n⊢ ∃ r x, r = 0\n\ncase mp.cons.inr\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\na : M₁\ns : Multiset M₁\nih : Multiset.prod s = 0 → ∃ r x, r = 0\nhb : Multiset.prod s = 0\n⊢ ∃ r x, r = 0",
"state_before": "case mp.cons\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\na : M₁\ns : Multiset M₁\nih : Multiset.prod s = 0 → ∃ r x, r = 0\nhprod : a = 0 ∨ Multiset.prod s = 0\n⊢ ∃ r x, r = 0",
"tactic": "cases' hprod with ha hb"
},
{
"state_after": "no goals",
"state_before": "case mp.cons.inl\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\na : M₁\ns : Multiset M₁\nih : Multiset.prod s = 0 → ∃ r x, r = 0\nha : a = 0\n⊢ ∃ r x, r = 0",
"tactic": "exact ⟨a, Multiset.mem_cons_self a s, ha⟩"
},
{
"state_after": "M : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\na : M₁\ns : Multiset M₁\nih : Multiset.prod s = 0 → ∃ r x, r = 0\nhb : Multiset.prod s = 0\n⊢ ∀ (a_1 : M₁), (∃ x, a_1 = 0) → ∃ x, a_1 = 0",
"state_before": "case mp.cons.inr\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\na : M₁\ns : Multiset M₁\nih : Multiset.prod s = 0 → ∃ r x, r = 0\nhb : Multiset.prod s = 0\n⊢ ∃ r x, r = 0",
"tactic": "apply (ih hb).imp _"
},
{
"state_after": "case intro\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\na : M₁\ns : Multiset M₁\nih : Multiset.prod s = 0 → ∃ r x, r = 0\nhb : Multiset.prod s = 0\nb : M₁\nhb₁ : b ∈ s\nhb₂ : b = 0\n⊢ ∃ x, b = 0",
"state_before": "M : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\na : M₁\ns : Multiset M₁\nih : Multiset.prod s = 0 → ∃ r x, r = 0\nhb : Multiset.prod s = 0\n⊢ ∀ (a_1 : M₁), (∃ x, a_1 = 0) → ∃ x, a_1 = 0",
"tactic": "rintro b ⟨hb₁, hb₂⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nM : Type ?u.75307\nM' : Type ?u.75310\nM₁ : Type u_1\nR : Type ?u.75316\nR' : Type ?u.75319\nF : Type ?u.75322\ninst✝⁶ : MonoidWithZero M\ninst✝⁵ : MonoidWithZero M'\ninst✝⁴ : CommMonoidWithZero M₁\ninst✝³ : Ring R\ninst✝² : CommRing R'\ninst✝¹ : NoZeroDivisors M₁\ninst✝ : Nontrivial M₁\na : M₁\ns : Multiset M₁\nih : Multiset.prod s = 0 → ∃ r x, r = 0\nhb : Multiset.prod s = 0\nb : M₁\nhb₁ : b ∈ s\nhb₂ : b = 0\n⊢ ∃ x, b = 0",
"tactic": "exact ⟨Multiset.mem_cons_of_mem hb₁, hb₂⟩"
}
] |
[
180,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
165,
1
] |
Mathlib/Order/SymmDiff.lean
|
symmDiff_sdiff_eq_sup
|
[
{
"state_after": "ι : Type ?u.32022\nα : Type u_1\nβ : Type ?u.32028\nπ : ι → Type ?u.32033\ninst✝ : GeneralizedCoheytingAlgebra α\na b c d : α\n⊢ a \\ (b \\ a) ⊔ b \\ a = a ⊔ b",
"state_before": "ι : Type ?u.32022\nα : Type u_1\nβ : Type ?u.32028\nπ : ι → Type ?u.32033\ninst✝ : GeneralizedCoheytingAlgebra α\na b c d : α\n⊢ a ∆ (b \\ a) = a ⊔ b",
"tactic": "rw [symmDiff, sdiff_idem]"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.32022\nα : Type u_1\nβ : Type ?u.32028\nπ : ι → Type ?u.32033\ninst✝ : GeneralizedCoheytingAlgebra α\na b c d : α\n⊢ a \\ (b \\ a) ⊔ b \\ a = a ⊔ b",
"tactic": "exact\n le_antisymm (sup_le_sup sdiff_le sdiff_le)\n (sup_le le_sdiff_sup <| le_sdiff_sup.trans <| sup_le le_sup_right le_sdiff_sup)"
}
] |
[
184,
86
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
180,
1
] |
Mathlib/Data/Subtype.lean
|
Subtype.forall'
|
[] |
[
53,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
52,
11
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
|
CategoryTheory.Limits.BinaryCofan.mk_inr
|
[] |
[
336,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
335,
1
] |
Mathlib/Topology/Filter.lean
|
Filter.sInter_nhds
|
[] |
[
152,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Data/Polynomial/AlgebraMap.lean
|
Polynomial.coeff_zero_eq_aeval_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\nA : Type z\nA' : Type ?u.1112049\nB' : Type ?u.1112052\na b : R\nn : ℕ\ninst✝⁶ : CommSemiring A'\ninst✝⁵ : Semiring B'\ninst✝⁴ : CommSemiring R\np✝ q : R[X]\ninst✝³ : Semiring A\ninst✝² : Algebra R A\nB : Type ?u.1112265\ninst✝¹ : Semiring B\ninst✝ : Algebra R B\nx : A\np : R[X]\n⊢ coeff p 0 = ↑(aeval 0) p",
"tactic": "simp [coeff_zero_eq_eval_zero]"
}
] |
[
321,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Mathlib/MeasureTheory/Constructions/BorelSpace/Basic.lean
|
measurableSet_interior
|
[] |
[
311,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
310,
1
] |
Mathlib/Algebra/Order/Hom/Monoid.lean
|
monotone_iff_map_nonneg
|
[
{
"state_after": "F : Type u_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.37806\nδ : Type ?u.37809\ninst✝² : OrderedAddCommGroup α\ninst✝¹ : OrderedAddCommMonoid β\ninst✝ : AddMonoidHomClass F α β\nf : F\nh : Monotone ↑f\na : α\n⊢ 0 ≤ a → ↑f 0 ≤ ↑f a",
"state_before": "F : Type u_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.37806\nδ : Type ?u.37809\ninst✝² : OrderedAddCommGroup α\ninst✝¹ : OrderedAddCommMonoid β\ninst✝ : AddMonoidHomClass F α β\nf : F\nh : Monotone ↑f\na : α\n⊢ 0 ≤ a → 0 ≤ ↑f a",
"tactic": "rw [← map_zero f]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.37806\nδ : Type ?u.37809\ninst✝² : OrderedAddCommGroup α\ninst✝¹ : OrderedAddCommMonoid β\ninst✝ : AddMonoidHomClass F α β\nf : F\nh : Monotone ↑f\na : α\n⊢ 0 ≤ a → ↑f 0 ≤ ↑f a",
"tactic": "apply h"
},
{
"state_after": "F : Type u_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.37806\nδ : Type ?u.37809\ninst✝² : OrderedAddCommGroup α\ninst✝¹ : OrderedAddCommMonoid β\ninst✝ : AddMonoidHomClass F α β\nf : F\nh : ∀ (a : α), 0 ≤ a → 0 ≤ ↑f a\na b : α\nhl : a ≤ b\n⊢ ↑f a ≤ ↑f (b - a) + ↑f a",
"state_before": "F : Type u_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.37806\nδ : Type ?u.37809\ninst✝² : OrderedAddCommGroup α\ninst✝¹ : OrderedAddCommMonoid β\ninst✝ : AddMonoidHomClass F α β\nf : F\nh : ∀ (a : α), 0 ≤ a → 0 ≤ ↑f a\na b : α\nhl : a ≤ b\n⊢ ↑f a ≤ ↑f b",
"tactic": "rw [← sub_add_cancel b a, map_add f]"
},
{
"state_after": "no goals",
"state_before": "F : Type u_3\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.37806\nδ : Type ?u.37809\ninst✝² : OrderedAddCommGroup α\ninst✝¹ : OrderedAddCommMonoid β\ninst✝ : AddMonoidHomClass F α β\nf : F\nh : ∀ (a : α), 0 ≤ a → 0 ≤ ↑f a\na b : α\nhl : a ≤ b\n⊢ ↑f a ≤ ↑f (b - a) + ↑f a",
"tactic": "exact le_add_of_nonneg_left (h _ <| sub_nonneg.2 hl)"
}
] |
[
243,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
238,
1
] |
Mathlib/Analysis/NormedSpace/Exponential.lean
|
expSeries_div_summable_of_mem_ball
|
[] |
[
347,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
345,
1
] |
Mathlib/Topology/MetricSpace/GromovHausdorffRealized.lean
|
GromovHausdorff.HD_lipschitz_aux3
|
[] |
[
409,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
407,
9
] |
Mathlib/Topology/SubsetProperties.lean
|
isPreirreducible_singleton
|
[] |
[
1712,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1711,
1
] |
Std/Logic.lean
|
Decidable.or_iff_not_and_not
|
[
{
"state_after": "no goals",
"state_before": "a b : Prop\ninst✝¹ : Decidable a\ninst✝ : Decidable b\n⊢ a ∨ b ↔ ¬(¬a ∧ ¬b)",
"tactic": "rw [← not_or, not_not]"
}
] |
[
606,
25
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
605,
1
] |
Mathlib/Analysis/Calculus/MeanValue.lean
|
StrictMonoOn.strictConvexOn_of_deriv
|
[
{
"state_after": "E : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\n⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)",
"state_before": "E : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\n⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)",
"tactic": "have hxzD : Icc x z ⊆ D := hD.ordConnected.out hx hz"
},
{
"state_after": "E : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\nhxyD : Icc x y ⊆ D\n⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)",
"state_before": "E : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\n⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)",
"tactic": "have hxyD : Icc x y ⊆ D := (Icc_subset_Icc_right hyz.le).trans hxzD"
},
{
"state_after": "E : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\nhxyD : Icc x y ⊆ D\nhxyD' : Ioo x y ⊆ interior D\n⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)",
"state_before": "E : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\nhxyD : Icc x y ⊆ D\n⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)",
"tactic": "have hxyD' : Ioo x y ⊆ interior D :=\n subset_sUnion_of_mem ⟨isOpen_Ioo, Ioo_subset_Icc_self.trans hxyD⟩"
},
{
"state_after": "E : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\nhxyD : Icc x y ⊆ D\nhxyD' : Ioo x y ⊆ interior D\nhyzD : Icc y z ⊆ D\n⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)",
"state_before": "E : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\nhxyD : Icc x y ⊆ D\nhxyD' : Ioo x y ⊆ interior D\n⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)",
"tactic": "have hyzD : Icc y z ⊆ D := (Icc_subset_Icc_left hxy.le).trans hxzD"
},
{
"state_after": "E : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\nhxyD : Icc x y ⊆ D\nhxyD' : Ioo x y ⊆ interior D\nhyzD : Icc y z ⊆ D\nhyzD' : Ioo y z ⊆ interior D\n⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)",
"state_before": "E : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\nhxyD : Icc x y ⊆ D\nhxyD' : Ioo x y ⊆ interior D\nhyzD : Icc y z ⊆ D\n⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)",
"tactic": "have hyzD' : Ioo y z ⊆ interior D :=\n subset_sUnion_of_mem ⟨isOpen_Ioo, Ioo_subset_Icc_self.trans hyzD⟩"
},
{
"state_after": "E : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\nhxyD : Icc x y ⊆ D\nhxyD' : Ioo x y ⊆ interior D\nhyzD : Icc y z ⊆ D\nhyzD' : Ioo y z ⊆ interior D\n⊢ ∃ a, a ∈ Ioo x y ∧ (f y - f x) / (y - x) < deriv f a\n\ncase intro.intro.intro\nE : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\nhxyD : Icc x y ⊆ D\nhxyD' : Ioo x y ⊆ interior D\nhyzD : Icc y z ⊆ D\nhyzD' : Ioo y z ⊆ interior D\na : ℝ\nha : (f y - f x) / (y - x) < deriv f a\nhxa : x < a\nhay : a < y\n⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)",
"state_before": "E : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\nhxyD : Icc x y ⊆ D\nhxyD' : Ioo x y ⊆ interior D\nhyzD : Icc y z ⊆ D\nhyzD' : Ioo y z ⊆ interior D\n⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)",
"tactic": "obtain ⟨a, ⟨hxa, hay⟩, ha⟩ : ∃ a ∈ Ioo x y, (f y - f x) / (y - x) < deriv f a"
},
{
"state_after": "E : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\nhxyD : Icc x y ⊆ D\nhxyD' : Ioo x y ⊆ interior D\nhyzD : Icc y z ⊆ D\nhyzD' : Ioo y z ⊆ interior D\na : ℝ\nha : (f y - f x) / (y - x) < deriv f a\nhxa : x < a\nhay : a < y\n⊢ ∃ b, b ∈ Ioo y z ∧ deriv f b < (f z - f y) / (z - y)\n\ncase intro.intro.intro.intro.intro.intro\nE : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\nhxyD : Icc x y ⊆ D\nhxyD' : Ioo x y ⊆ interior D\nhyzD : Icc y z ⊆ D\nhyzD' : Ioo y z ⊆ interior D\na : ℝ\nha : (f y - f x) / (y - x) < deriv f a\nhxa : x < a\nhay : a < y\nb : ℝ\nhb : deriv f b < (f z - f y) / (z - y)\nhyb : y < b\nhbz : b < z\n⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)",
"state_before": "case intro.intro.intro\nE : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\nhxyD : Icc x y ⊆ D\nhxyD' : Ioo x y ⊆ interior D\nhyzD : Icc y z ⊆ D\nhyzD' : Ioo y z ⊆ interior D\na : ℝ\nha : (f y - f x) / (y - x) < deriv f a\nhxa : x < a\nhay : a < y\n⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)",
"tactic": "obtain ⟨b, ⟨hyb, hbz⟩, hb⟩ : ∃ b ∈ Ioo y z, deriv f b < (f z - f y) / (z - y)"
},
{
"state_after": "E : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\nhxyD : Icc x y ⊆ D\nhxyD' : Ioo x y ⊆ interior D\nhyzD : Icc y z ⊆ D\nhyzD' : Ioo y z ⊆ interior D\na : ℝ\nha : (f y - f x) / (y - x) < deriv f a\nhxa : x < a\nhay : a < y\nb : ℝ\nhb : deriv f b < (f z - f y) / (z - y)\nhyb : y < b\nhbz : b < z\n⊢ deriv f a < deriv f b",
"state_before": "case intro.intro.intro.intro.intro.intro\nE : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\nhxyD : Icc x y ⊆ D\nhxyD' : Ioo x y ⊆ interior D\nhyzD : Icc y z ⊆ D\nhyzD' : Ioo y z ⊆ interior D\na : ℝ\nha : (f y - f x) / (y - x) < deriv f a\nhxa : x < a\nhay : a < y\nb : ℝ\nhb : deriv f b < (f z - f y) / (z - y)\nhyb : y < b\nhbz : b < z\n⊢ (f y - f x) / (y - x) < (f z - f y) / (z - y)",
"tactic": "apply ha.trans (lt_trans _ hb)"
},
{
"state_after": "no goals",
"state_before": "E : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\nhxyD : Icc x y ⊆ D\nhxyD' : Ioo x y ⊆ interior D\nhyzD : Icc y z ⊆ D\nhyzD' : Ioo y z ⊆ interior D\na : ℝ\nha : (f y - f x) / (y - x) < deriv f a\nhxa : x < a\nhay : a < y\nb : ℝ\nhb : deriv f b < (f z - f y) / (z - y)\nhyb : y < b\nhbz : b < z\n⊢ deriv f a < deriv f b",
"tactic": "exact hf' (hxyD' ⟨hxa, hay⟩) (hyzD' ⟨hyb, hbz⟩) (hay.trans hyb)"
},
{
"state_after": "no goals",
"state_before": "E : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\nhxyD : Icc x y ⊆ D\nhxyD' : Ioo x y ⊆ interior D\nhyzD : Icc y z ⊆ D\nhyzD' : Ioo y z ⊆ interior D\n⊢ ∃ a, a ∈ Ioo x y ∧ (f y - f x) / (y - x) < deriv f a",
"tactic": "exact StrictMonoOn.exists_slope_lt_deriv (hf.mono hxyD) hxy (hf'.mono hxyD')"
},
{
"state_after": "no goals",
"state_before": "E : Type ?u.445897\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nF : Type ?u.445993\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nD : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : StrictMonoOn (deriv f) (interior D)\nx y z : ℝ\nhx : x ∈ D\nhz : z ∈ D\nhxy : x < y\nhyz : y < z\nhxzD : Icc x z ⊆ D\nhxyD : Icc x y ⊆ D\nhxyD' : Ioo x y ⊆ interior D\nhyzD : Icc y z ⊆ D\nhyzD' : Ioo y z ⊆ interior D\na : ℝ\nha : (f y - f x) / (y - x) < deriv f a\nhxa : x < a\nhay : a < y\n⊢ ∃ b, b ∈ Ioo y z ∧ deriv f b < (f z - f y) / (z - y)",
"tactic": "exact StrictMonoOn.exists_deriv_lt_slope (hf.mono hyzD) hyz (hf'.mono hyzD')"
}
] |
[
1122,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1104,
1
] |
src/lean/Init/SimpLemmas.lean
|
false_iff
|
[] |
[
96,
94
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
96,
9
] |
Mathlib/Data/Set/Lattice.lean
|
Set.iUnion_prod_of_monotone
|
[
{
"state_after": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort ?u.237166\nι' : Sort ?u.237169\nι₂ : Sort ?u.237172\nκ : ι → Sort ?u.237177\nκ₁ : ι → Sort ?u.237182\nκ₂ : ι → Sort ?u.237187\nκ' : ι' → Sort ?u.237192\ninst✝ : SemilatticeSup α\ns : α → Set β\nt : α → Set γ\nhs : Monotone s\nht : Monotone t\nz : β\nw : γ\n⊢ ((z, w) ∈ ⋃ (x : α), s x ×ˢ t x) ↔ (z, w) ∈ (⋃ (x : α), s x) ×ˢ ⋃ (x : α), t x",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort ?u.237166\nι' : Sort ?u.237169\nι₂ : Sort ?u.237172\nκ : ι → Sort ?u.237177\nκ₁ : ι → Sort ?u.237182\nκ₂ : ι → Sort ?u.237187\nκ' : ι' → Sort ?u.237192\ninst✝ : SemilatticeSup α\ns : α → Set β\nt : α → Set γ\nhs : Monotone s\nht : Monotone t\n⊢ (⋃ (x : α), s x ×ˢ t x) = (⋃ (x : α), s x) ×ˢ ⋃ (x : α), t x",
"tactic": "ext ⟨z, w⟩"
},
{
"state_after": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort ?u.237166\nι' : Sort ?u.237169\nι₂ : Sort ?u.237172\nκ : ι → Sort ?u.237177\nκ₁ : ι → Sort ?u.237182\nκ₂ : ι → Sort ?u.237187\nκ' : ι' → Sort ?u.237192\ninst✝ : SemilatticeSup α\ns : α → Set β\nt : α → Set γ\nhs : Monotone s\nht : Monotone t\nz : β\nw : γ\n⊢ (∀ (x : α), z ∈ s x → w ∈ t x → (∃ i, z ∈ s i) ∧ ∃ i, w ∈ t i) ∧\n ∀ (x : α), z ∈ s x → ∀ (x : α), w ∈ t x → ∃ i, z ∈ s i ∧ w ∈ t i",
"state_before": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort ?u.237166\nι' : Sort ?u.237169\nι₂ : Sort ?u.237172\nκ : ι → Sort ?u.237177\nκ₁ : ι → Sort ?u.237182\nκ₂ : ι → Sort ?u.237187\nκ' : ι' → Sort ?u.237192\ninst✝ : SemilatticeSup α\ns : α → Set β\nt : α → Set γ\nhs : Monotone s\nht : Monotone t\nz : β\nw : γ\n⊢ ((z, w) ∈ ⋃ (x : α), s x ×ˢ t x) ↔ (z, w) ∈ (⋃ (x : α), s x) ×ˢ ⋃ (x : α), t x",
"tactic": "simp only [mem_prod, mem_iUnion, exists_imp, and_imp, iff_def]"
},
{
"state_after": "case h.mk.left\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort ?u.237166\nι' : Sort ?u.237169\nι₂ : Sort ?u.237172\nκ : ι → Sort ?u.237177\nκ₁ : ι → Sort ?u.237182\nκ₂ : ι → Sort ?u.237187\nκ' : ι' → Sort ?u.237192\ninst✝ : SemilatticeSup α\ns : α → Set β\nt : α → Set γ\nhs : Monotone s\nht : Monotone t\nz : β\nw : γ\n⊢ ∀ (x : α), z ∈ s x → w ∈ t x → (∃ i, z ∈ s i) ∧ ∃ i, w ∈ t i\n\ncase h.mk.right\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort ?u.237166\nι' : Sort ?u.237169\nι₂ : Sort ?u.237172\nκ : ι → Sort ?u.237177\nκ₁ : ι → Sort ?u.237182\nκ₂ : ι → Sort ?u.237187\nκ' : ι' → Sort ?u.237192\ninst✝ : SemilatticeSup α\ns : α → Set β\nt : α → Set γ\nhs : Monotone s\nht : Monotone t\nz : β\nw : γ\n⊢ ∀ (x : α), z ∈ s x → ∀ (x : α), w ∈ t x → ∃ i, z ∈ s i ∧ w ∈ t i",
"state_before": "case h.mk\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort ?u.237166\nι' : Sort ?u.237169\nι₂ : Sort ?u.237172\nκ : ι → Sort ?u.237177\nκ₁ : ι → Sort ?u.237182\nκ₂ : ι → Sort ?u.237187\nκ' : ι' → Sort ?u.237192\ninst✝ : SemilatticeSup α\ns : α → Set β\nt : α → Set γ\nhs : Monotone s\nht : Monotone t\nz : β\nw : γ\n⊢ (∀ (x : α), z ∈ s x → w ∈ t x → (∃ i, z ∈ s i) ∧ ∃ i, w ∈ t i) ∧\n ∀ (x : α), z ∈ s x → ∀ (x : α), w ∈ t x → ∃ i, z ∈ s i ∧ w ∈ t i",
"tactic": "constructor"
},
{
"state_after": "case h.mk.left\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort ?u.237166\nι' : Sort ?u.237169\nι₂ : Sort ?u.237172\nκ : ι → Sort ?u.237177\nκ₁ : ι → Sort ?u.237182\nκ₂ : ι → Sort ?u.237187\nκ' : ι' → Sort ?u.237192\ninst✝ : SemilatticeSup α\ns : α → Set β\nt : α → Set γ\nhs : Monotone s\nht : Monotone t\nz : β\nw : γ\nx : α\nhz : z ∈ s x\nhw : w ∈ t x\n⊢ (∃ i, z ∈ s i) ∧ ∃ i, w ∈ t i",
"state_before": "case h.mk.left\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort ?u.237166\nι' : Sort ?u.237169\nι₂ : Sort ?u.237172\nκ : ι → Sort ?u.237177\nκ₁ : ι → Sort ?u.237182\nκ₂ : ι → Sort ?u.237187\nκ' : ι' → Sort ?u.237192\ninst✝ : SemilatticeSup α\ns : α → Set β\nt : α → Set γ\nhs : Monotone s\nht : Monotone t\nz : β\nw : γ\n⊢ ∀ (x : α), z ∈ s x → w ∈ t x → (∃ i, z ∈ s i) ∧ ∃ i, w ∈ t i",
"tactic": "intro x hz hw"
},
{
"state_after": "no goals",
"state_before": "case h.mk.left\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort ?u.237166\nι' : Sort ?u.237169\nι₂ : Sort ?u.237172\nκ : ι → Sort ?u.237177\nκ₁ : ι → Sort ?u.237182\nκ₂ : ι → Sort ?u.237187\nκ' : ι' → Sort ?u.237192\ninst✝ : SemilatticeSup α\ns : α → Set β\nt : α → Set γ\nhs : Monotone s\nht : Monotone t\nz : β\nw : γ\nx : α\nhz : z ∈ s x\nhw : w ∈ t x\n⊢ (∃ i, z ∈ s i) ∧ ∃ i, w ∈ t i",
"tactic": "exact ⟨⟨x, hz⟩, x, hw⟩"
},
{
"state_after": "case h.mk.right\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort ?u.237166\nι' : Sort ?u.237169\nι₂ : Sort ?u.237172\nκ : ι → Sort ?u.237177\nκ₁ : ι → Sort ?u.237182\nκ₂ : ι → Sort ?u.237187\nκ' : ι' → Sort ?u.237192\ninst✝ : SemilatticeSup α\ns : α → Set β\nt : α → Set γ\nhs : Monotone s\nht : Monotone t\nz : β\nw : γ\nx : α\nhz : z ∈ s x\nx' : α\nhw : w ∈ t x'\n⊢ ∃ i, z ∈ s i ∧ w ∈ t i",
"state_before": "case h.mk.right\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort ?u.237166\nι' : Sort ?u.237169\nι₂ : Sort ?u.237172\nκ : ι → Sort ?u.237177\nκ₁ : ι → Sort ?u.237182\nκ₂ : ι → Sort ?u.237187\nκ' : ι' → Sort ?u.237192\ninst✝ : SemilatticeSup α\ns : α → Set β\nt : α → Set γ\nhs : Monotone s\nht : Monotone t\nz : β\nw : γ\n⊢ ∀ (x : α), z ∈ s x → ∀ (x : α), w ∈ t x → ∃ i, z ∈ s i ∧ w ∈ t i",
"tactic": "intro x hz x' hw"
},
{
"state_after": "no goals",
"state_before": "case h.mk.right\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Sort ?u.237166\nι' : Sort ?u.237169\nι₂ : Sort ?u.237172\nκ : ι → Sort ?u.237177\nκ₁ : ι → Sort ?u.237182\nκ₂ : ι → Sort ?u.237187\nκ' : ι' → Sort ?u.237192\ninst✝ : SemilatticeSup α\ns : α → Set β\nt : α → Set γ\nhs : Monotone s\nht : Monotone t\nz : β\nw : γ\nx : α\nhz : z ∈ s x\nx' : α\nhw : w ∈ t x'\n⊢ ∃ i, z ∈ s i ∧ w ∈ t i",
"tactic": "exact ⟨x ⊔ x', hs le_sup_left hz, ht le_sup_right hw⟩"
}
] |
[
1817,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1811,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.measure_union_ne_top
|
[] |
[
321,
48
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
320,
1
] |
Mathlib/Analysis/InnerProductSpace/Basic.lean
|
real_inner_div_norm_mul_norm_eq_neg_one_of_ne_zero_of_neg_mul
|
[
{
"state_after": "𝕜 : Type ?u.3160254\nE : Type ?u.3160257\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx : F\nr : ℝ\nhx : x ≠ 0\nhr : r < 0\n⊢ r * (‖x‖ * ‖x‖) ≠ 0",
"state_before": "𝕜 : Type ?u.3160254\nE : Type ?u.3160257\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx : F\nr : ℝ\nhx : x ≠ 0\nhr : r < 0\n⊢ inner x (r • x) / (‖x‖ * ‖r • x‖) = -1",
"tactic": "rw [real_inner_smul_self_right, norm_smul, Real.norm_eq_abs, ← mul_assoc ‖x‖, mul_comm _ (|r|),\n mul_assoc, abs_of_neg hr, neg_mul, div_neg_eq_neg_div, div_self]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type ?u.3160254\nE : Type ?u.3160257\nF : Type u_1\ninst✝⁴ : IsROrC 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\ndec_E : DecidableEq E\nx : F\nr : ℝ\nhx : x ≠ 0\nhr : r < 0\n⊢ r * (‖x‖ * ‖x‖) ≠ 0",
"tactic": "exact mul_ne_zero hr.ne (mul_self_ne_zero.2 (norm_ne_zero_iff.2 hx))"
}
] |
[
1586,
71
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1582,
1
] |
Mathlib/SetTheory/Cardinal/Basic.lean
|
Cardinal.cast_toNat_of_aleph0_le
|
[
{
"state_after": "no goals",
"state_before": "α β : Type u\nc : Cardinal\nh : ℵ₀ ≤ c\n⊢ ↑(↑toNat c) = 0",
"tactic": "rw [toNat_apply_of_aleph0_le h, Nat.cast_zero]"
}
] |
[
1690,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1689,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.add_le_add_iff_left
|
[] |
[
774,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
773,
11
] |
Mathlib/Analysis/Convex/Between.lean
|
wbtw_smul_vadd_smul_vadd_of_nonneg_of_le
|
[
{
"state_after": "R : Type u_1\nV : Type u_2\nV' : Type ?u.518717\nP : Type u_3\nP' : Type ?u.518723\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx : P\nv : V\nr₁ r₂ : R\nhr₁ : 0 ≤ r₁\nhr₂ : r₁ ≤ r₂\n⊢ ↑(lineMap x (r₂ • v +ᵥ x)) (r₁ / r₂) = r₁ • v +ᵥ x",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.518717\nP : Type u_3\nP' : Type ?u.518723\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx : P\nv : V\nr₁ r₂ : R\nhr₁ : 0 ≤ r₁\nhr₂ : r₁ ≤ r₂\n⊢ Wbtw R x (r₁ • v +ᵥ x) (r₂ • v +ᵥ x)",
"tactic": "refine' ⟨r₁ / r₂, ⟨div_nonneg hr₁ (hr₁.trans hr₂), div_le_one_of_le hr₂ (hr₁.trans hr₂)⟩, _⟩"
},
{
"state_after": "case pos\nR : Type u_1\nV : Type u_2\nV' : Type ?u.518717\nP : Type u_3\nP' : Type ?u.518723\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx : P\nv : V\nr₁ r₂ : R\nhr₁ : 0 ≤ r₁\nhr₂ : r₁ ≤ r₂\nh : r₁ = 0\n⊢ ↑(lineMap x (r₂ • v +ᵥ x)) (r₁ / r₂) = r₁ • v +ᵥ x\n\ncase neg\nR : Type u_1\nV : Type u_2\nV' : Type ?u.518717\nP : Type u_3\nP' : Type ?u.518723\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx : P\nv : V\nr₁ r₂ : R\nhr₁ : 0 ≤ r₁\nhr₂ : r₁ ≤ r₂\nh : ¬r₁ = 0\n⊢ ↑(lineMap x (r₂ • v +ᵥ x)) (r₁ / r₂) = r₁ • v +ᵥ x",
"state_before": "R : Type u_1\nV : Type u_2\nV' : Type ?u.518717\nP : Type u_3\nP' : Type ?u.518723\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx : P\nv : V\nr₁ r₂ : R\nhr₁ : 0 ≤ r₁\nhr₂ : r₁ ≤ r₂\n⊢ ↑(lineMap x (r₂ • v +ᵥ x)) (r₁ / r₂) = r₁ • v +ᵥ x",
"tactic": "by_cases h : r₁ = 0"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u_1\nV : Type u_2\nV' : Type ?u.518717\nP : Type u_3\nP' : Type ?u.518723\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx : P\nv : V\nr₁ r₂ : R\nhr₁ : 0 ≤ r₁\nhr₂ : r₁ ≤ r₂\nh : ¬r₁ = 0\n⊢ ↑(lineMap x (r₂ • v +ᵥ x)) (r₁ / r₂) = r₁ • v +ᵥ x",
"tactic": "simp [lineMap_apply, smul_smul, ((hr₁.lt_of_ne' h).trans_le hr₂).ne.symm]"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u_1\nV : Type u_2\nV' : Type ?u.518717\nP : Type u_3\nP' : Type ?u.518723\ninst✝³ : LinearOrderedField R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx : P\nv : V\nr₁ r₂ : R\nhr₁ : 0 ≤ r₁\nhr₂ : r₁ ≤ r₂\nh : r₁ = 0\n⊢ ↑(lineMap x (r₂ • v +ᵥ x)) (r₁ / r₂) = r₁ • v +ᵥ x",
"tactic": "simp [h]"
}
] |
[
771,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
767,
1
] |
Mathlib/MeasureTheory/Integral/SetIntegral.lean
|
MeasureTheory.integral_Icc_eq_integral_Ioc'
|
[] |
[
612,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
610,
1
] |
Mathlib/Computability/Halting.lean
|
Nat.Partrec'.to_part
|
[
{
"state_after": "case prim\nn : ℕ\nf : Vector ℕ n →. ℕ\nn✝ : ℕ\nf✝ : Vector ℕ n✝ → ℕ\na✝ : Primrec' f✝\n⊢ _root_.Partrec ↑f✝\n\ncase comp\nn : ℕ\nf : Vector ℕ n →. ℕ\nm✝ n✝ : ℕ\nf✝ : Vector ℕ n✝ →. ℕ\ng✝ : Fin n✝ → Vector ℕ m✝ →. ℕ\na✝¹ : Partrec' f✝\na✝ : ∀ (i : Fin n✝), Partrec' (g✝ i)\na_ih✝¹ : _root_.Partrec f✝\na_ih✝ : ∀ (i : Fin n✝), _root_.Partrec (g✝ i)\n⊢ _root_.Partrec fun v => (mOfFn fun i => g✝ i v) >>= f✝\n\ncase rfind\nn : ℕ\nf : Vector ℕ n →. ℕ\nn✝ : ℕ\nf✝ : Vector ℕ (n✝ + 1) → ℕ\na✝ : Partrec' ↑f✝\na_ih✝ : _root_.Partrec ↑f✝\n⊢ _root_.Partrec fun v => Nat.rfind fun n => Part.some (decide (f✝ (n ::ᵥ v) = 0))",
"state_before": "n : ℕ\nf : Vector ℕ n →. ℕ\npf : Partrec' f\n⊢ _root_.Partrec f",
"tactic": "induction pf"
},
{
"state_after": "case comp\nn : ℕ\nf : Vector ℕ n →. ℕ\nm✝ n✝ : ℕ\nf✝ : Vector ℕ n✝ →. ℕ\ng✝ : Fin n✝ → Vector ℕ m✝ →. ℕ\na✝¹ : Partrec' f✝\na✝ : ∀ (i : Fin n✝), Partrec' (g✝ i)\na_ih✝¹ : _root_.Partrec f✝\na_ih✝ : ∀ (i : Fin n✝), _root_.Partrec (g✝ i)\n⊢ _root_.Partrec fun v => (mOfFn fun i => g✝ i v) >>= f✝\n\ncase rfind\nn : ℕ\nf : Vector ℕ n →. ℕ\nn✝ : ℕ\nf✝ : Vector ℕ (n✝ + 1) → ℕ\na✝ : Partrec' ↑f✝\na_ih✝ : _root_.Partrec ↑f✝\n⊢ _root_.Partrec fun v => Nat.rfind fun n => Part.some (decide (f✝ (n ::ᵥ v) = 0))",
"state_before": "case prim\nn : ℕ\nf : Vector ℕ n →. ℕ\nn✝ : ℕ\nf✝ : Vector ℕ n✝ → ℕ\na✝ : Primrec' f✝\n⊢ _root_.Partrec ↑f✝\n\ncase comp\nn : ℕ\nf : Vector ℕ n →. ℕ\nm✝ n✝ : ℕ\nf✝ : Vector ℕ n✝ →. ℕ\ng✝ : Fin n✝ → Vector ℕ m✝ →. ℕ\na✝¹ : Partrec' f✝\na✝ : ∀ (i : Fin n✝), Partrec' (g✝ i)\na_ih✝¹ : _root_.Partrec f✝\na_ih✝ : ∀ (i : Fin n✝), _root_.Partrec (g✝ i)\n⊢ _root_.Partrec fun v => (mOfFn fun i => g✝ i v) >>= f✝\n\ncase rfind\nn : ℕ\nf : Vector ℕ n →. ℕ\nn✝ : ℕ\nf✝ : Vector ℕ (n✝ + 1) → ℕ\na✝ : Partrec' ↑f✝\na_ih✝ : _root_.Partrec ↑f✝\n⊢ _root_.Partrec fun v => Nat.rfind fun n => Part.some (decide (f✝ (n ::ᵥ v) = 0))",
"tactic": "case prim n f hf => exact hf.to_prim.to_comp"
},
{
"state_after": "case rfind\nn : ℕ\nf : Vector ℕ n →. ℕ\nn✝ : ℕ\nf✝ : Vector ℕ (n✝ + 1) → ℕ\na✝ : Partrec' ↑f✝\na_ih✝ : _root_.Partrec ↑f✝\n⊢ _root_.Partrec fun v => Nat.rfind fun n => Part.some (decide (f✝ (n ::ᵥ v) = 0))",
"state_before": "case comp\nn : ℕ\nf : Vector ℕ n →. ℕ\nm✝ n✝ : ℕ\nf✝ : Vector ℕ n✝ →. ℕ\ng✝ : Fin n✝ → Vector ℕ m✝ →. ℕ\na✝¹ : Partrec' f✝\na✝ : ∀ (i : Fin n✝), Partrec' (g✝ i)\na_ih✝¹ : _root_.Partrec f✝\na_ih✝ : ∀ (i : Fin n✝), _root_.Partrec (g✝ i)\n⊢ _root_.Partrec fun v => (mOfFn fun i => g✝ i v) >>= f✝\n\ncase rfind\nn : ℕ\nf : Vector ℕ n →. ℕ\nn✝ : ℕ\nf✝ : Vector ℕ (n✝ + 1) → ℕ\na✝ : Partrec' ↑f✝\na_ih✝ : _root_.Partrec ↑f✝\n⊢ _root_.Partrec fun v => Nat.rfind fun n => Part.some (decide (f✝ (n ::ᵥ v) = 0))",
"tactic": "case comp m n f g _ _ hf hg => exact (Partrec.vector_mOfFn fun i => hg i).bind (hf.comp snd)"
},
{
"state_after": "no goals",
"state_before": "case rfind\nn : ℕ\nf : Vector ℕ n →. ℕ\nn✝ : ℕ\nf✝ : Vector ℕ (n✝ + 1) → ℕ\na✝ : Partrec' ↑f✝\na_ih✝ : _root_.Partrec ↑f✝\n⊢ _root_.Partrec fun v => Nat.rfind fun n => Part.some (decide (f✝ (n ::ᵥ v) = 0))",
"tactic": "case rfind n f _ hf =>\n have := hf.comp (vector_cons.comp snd fst)\n have :=\n ((Primrec.eq.comp _root_.Primrec.id (_root_.Primrec.const 0)).to_comp.comp\n this).to₂.partrec₂\n exact _root_.Partrec.rfind this"
},
{
"state_after": "no goals",
"state_before": "n✝ : ℕ\nf✝ : Vector ℕ n✝ →. ℕ\nn : ℕ\nf : Vector ℕ n → ℕ\nhf : Primrec' f\n⊢ _root_.Partrec ↑f",
"tactic": "exact hf.to_prim.to_comp"
},
{
"state_after": "no goals",
"state_before": "n✝ : ℕ\nf✝ : Vector ℕ n✝ →. ℕ\nm n : ℕ\nf : Vector ℕ n →. ℕ\ng : Fin n → Vector ℕ m →. ℕ\na✝¹ : Partrec' f\na✝ : ∀ (i : Fin n), Partrec' (g i)\nhf : _root_.Partrec f\nhg : ∀ (i : Fin n), _root_.Partrec (g i)\n⊢ _root_.Partrec fun v => (mOfFn fun i => g i v) >>= f",
"tactic": "exact (Partrec.vector_mOfFn fun i => hg i).bind (hf.comp snd)"
},
{
"state_after": "n✝ : ℕ\nf✝ : Vector ℕ n✝ →. ℕ\nn : ℕ\nf : Vector ℕ (n + 1) → ℕ\na✝ : Partrec' ↑f\nhf : _root_.Partrec ↑f\nthis : _root_.Partrec fun a => ↑f (a.snd ::ᵥ a.fst)\n⊢ _root_.Partrec fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::ᵥ v) = 0))",
"state_before": "n✝ : ℕ\nf✝ : Vector ℕ n✝ →. ℕ\nn : ℕ\nf : Vector ℕ (n + 1) → ℕ\na✝ : Partrec' ↑f\nhf : _root_.Partrec ↑f\n⊢ _root_.Partrec fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::ᵥ v) = 0))",
"tactic": "have := hf.comp (vector_cons.comp snd fst)"
},
{
"state_after": "n✝ : ℕ\nf✝ : Vector ℕ n✝ →. ℕ\nn : ℕ\nf : Vector ℕ (n + 1) → ℕ\na✝ : Partrec' ↑f\nhf : _root_.Partrec ↑f\nthis✝ : _root_.Partrec fun a => ↑f (a.snd ::ᵥ a.fst)\nthis : Partrec₂ fun a => ↑fun b => decide ((fun a => id a = 0) (f ((a, b).snd ::ᵥ (a, b).fst)))\n⊢ _root_.Partrec fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::ᵥ v) = 0))",
"state_before": "n✝ : ℕ\nf✝ : Vector ℕ n✝ →. ℕ\nn : ℕ\nf : Vector ℕ (n + 1) → ℕ\na✝ : Partrec' ↑f\nhf : _root_.Partrec ↑f\nthis : _root_.Partrec fun a => ↑f (a.snd ::ᵥ a.fst)\n⊢ _root_.Partrec fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::ᵥ v) = 0))",
"tactic": "have :=\n ((Primrec.eq.comp _root_.Primrec.id (_root_.Primrec.const 0)).to_comp.comp\n this).to₂.partrec₂"
},
{
"state_after": "no goals",
"state_before": "n✝ : ℕ\nf✝ : Vector ℕ n✝ →. ℕ\nn : ℕ\nf : Vector ℕ (n + 1) → ℕ\na✝ : Partrec' ↑f\nhf : _root_.Partrec ↑f\nthis✝ : _root_.Partrec fun a => ↑f (a.snd ::ᵥ a.fst)\nthis : Partrec₂ fun a => ↑fun b => decide ((fun a => id a = 0) (f ((a, b).snd ::ᵥ (a, b).fst)))\n⊢ _root_.Partrec fun v => Nat.rfind fun n_1 => Part.some (decide (f (n_1 ::ᵥ v) = 0))",
"tactic": "exact _root_.Partrec.rfind this"
}
] |
[
306,
36
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
297,
1
] |
Mathlib/Analysis/NormedSpace/ContinuousAffineMap.lean
|
ContinuousAffineMap.neg_contLinear
|
[] |
[
148,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
147,
1
] |
Mathlib/GroupTheory/SpecificGroups/Dihedral.lean
|
DihedralGroup.card
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\ninst✝ : NeZero n\n⊢ Fintype.card (DihedralGroup n) = 2 * n",
"tactic": "rw [← Fintype.card_eq.mpr ⟨fintypeHelper⟩, Fintype.card_sum, ZMod.card, two_mul]"
}
] |
[
126,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/RingTheory/Polynomial/RationalRoot.lean
|
den_dvd_of_is_root
|
[
{
"state_after": "A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\n⊢ ↑(den A r) ∣ leadingCoeff p * num A r ^ natDegree p",
"state_before": "A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\n⊢ ↑(den A r) ∣ leadingCoeff p",
"tactic": "suffices (den A r : A) ∣ p.leadingCoeff * num A r ^ p.natDegree by\n refine'\n dvd_of_dvd_mul_left_of_no_prime_factors (mem_nonZeroDivisors_iff_ne_zero.mp (den A r).2) _\n this\n intro q dvd_den dvd_num_pow hq\n apply hq.not_unit\n exact num_den_reduced A r (hq.dvd_of_dvd_pow dvd_num_pow) dvd_den"
},
{
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"tactic": "rw [← coeff_scaleRoots_natDegree]"
},
{
"state_after": "A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\n⊢ ∀ (j : ℕ), j ≠ natDegree p → ↑(den A r) ∣ coeff (scaleRoots p ↑(den A r)) j * num A r ^ j",
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"tactic": "apply dvd_term_of_isRoot_of_dvd_terms _ (num_isRoot_scaleRoots_of_aeval_eq_zero hr)"
},
{
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"tactic": "intro j hj"
},
{
"state_after": "case pos\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nj : ℕ\nhj : j ≠ natDegree p\nh : j < natDegree p\n⊢ ↑(den A r) ∣ coeff (scaleRoots p ↑(den A r)) j * num A r ^ j\n\ncase neg\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nj : ℕ\nhj : j ≠ natDegree p\nh : ¬j < natDegree p\n⊢ ↑(den A r) ∣ coeff (scaleRoots p ↑(den A r)) j * num A r ^ j",
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"tactic": "by_cases h : j < p.natDegree"
},
{
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"tactic": "rw [← natDegree_scaleRoots p (den A r)] at *"
},
{
"state_after": "case neg\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nj : ℕ\nhj : j ≠ natDegree (scaleRoots p ↑(den A r))\nh : ¬j < natDegree (scaleRoots p ↑(den A r))\n⊢ ↑(den A r) ∣ 0",
"state_before": "case neg\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nj : ℕ\nhj : j ≠ natDegree (scaleRoots p ↑(den A r))\nh : ¬j < natDegree (scaleRoots p ↑(den A r))\n⊢ ↑(den A r) ∣ coeff (scaleRoots p ↑(den A r)) j * num A r ^ j",
"tactic": "rw [coeff_eq_zero_of_natDegree_lt (lt_of_le_of_ne (le_of_not_gt h) hj.symm),\n MulZeroClass.zero_mul]"
},
{
"state_after": "no goals",
"state_before": "case neg\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nj : ℕ\nhj : j ≠ natDegree (scaleRoots p ↑(den A r))\nh : ¬j < natDegree (scaleRoots p ↑(den A r))\n⊢ ↑(den A r) ∣ 0",
"tactic": "exact dvd_zero _"
},
{
"state_after": "A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nthis : ↑(den A r) ∣ leadingCoeff p * num A r ^ natDegree p\n⊢ ∀ {d : A}, d ∣ ↑(den A r) → d ∣ num A r ^ natDegree p → ¬Prime d",
"state_before": "A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nthis : ↑(den A r) ∣ leadingCoeff p * num A r ^ natDegree p\n⊢ ↑(den A r) ∣ leadingCoeff p",
"tactic": "refine'\n dvd_of_dvd_mul_left_of_no_prime_factors (mem_nonZeroDivisors_iff_ne_zero.mp (den A r).2) _\n this"
},
{
"state_after": "A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nthis : ↑(den A r) ∣ leadingCoeff p * num A r ^ natDegree p\nq : A\ndvd_den : q ∣ ↑(den A r)\ndvd_num_pow : q ∣ num A r ^ natDegree p\nhq : Prime q\n⊢ False",
"state_before": "A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nthis : ↑(den A r) ∣ leadingCoeff p * num A r ^ natDegree p\n⊢ ∀ {d : A}, d ∣ ↑(den A r) → d ∣ num A r ^ natDegree p → ¬Prime d",
"tactic": "intro q dvd_den dvd_num_pow hq"
},
{
"state_after": "A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nthis : ↑(den A r) ∣ leadingCoeff p * num A r ^ natDegree p\nq : A\ndvd_den : q ∣ ↑(den A r)\ndvd_num_pow : q ∣ num A r ^ natDegree p\nhq : Prime q\n⊢ IsUnit q",
"state_before": "A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nthis : ↑(den A r) ∣ leadingCoeff p * num A r ^ natDegree p\nq : A\ndvd_den : q ∣ ↑(den A r)\ndvd_num_pow : q ∣ num A r ^ natDegree p\nhq : Prime q\n⊢ False",
"tactic": "apply hq.not_unit"
},
{
"state_after": "no goals",
"state_before": "A : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nthis : ↑(den A r) ∣ leadingCoeff p * num A r ^ natDegree p\nq : A\ndvd_den : q ∣ ↑(den A r)\ndvd_num_pow : q ∣ num A r ^ natDegree p\nhq : Prime q\n⊢ IsUnit q",
"tactic": "exact num_den_reduced A r (hq.dvd_of_dvd_pow dvd_num_pow) dvd_den"
},
{
"state_after": "case pos\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nj : ℕ\nhj : j ≠ natDegree p\nh : j < natDegree p\n⊢ ↑(den A r) ∣ coeff p j * ↑(den A r) ^ (natDegree p - j) * num A r ^ j",
"state_before": "case pos\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nj : ℕ\nhj : j ≠ natDegree p\nh : j < natDegree p\n⊢ ↑(den A r) ∣ coeff (scaleRoots p ↑(den A r)) j * num A r ^ j",
"tactic": "rw [coeff_scaleRoots]"
},
{
"state_after": "case pos\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nj : ℕ\nhj : j ≠ natDegree p\nh : j < natDegree p\n⊢ ↑(den A r) ∣ ↑(den A r) ^ (natDegree p - j)",
"state_before": "case pos\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nj : ℕ\nhj : j ≠ natDegree p\nh : j < natDegree p\n⊢ ↑(den A r) ∣ coeff p j * ↑(den A r) ^ (natDegree p - j) * num A r ^ j",
"tactic": "refine' (dvd_mul_of_dvd_right _ _).mul_right _"
},
{
"state_after": "case h.e'_3\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nj : ℕ\nhj : j ≠ natDegree p\nh : j < natDegree p\n⊢ ↑(den A r) = ↑(den A r) ^ Nat.succ ?pos.convert_1✝\n\ncase pos.convert_1\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nj : ℕ\nhj : j ≠ natDegree p\nh : j < natDegree p\n⊢ ℕ\n\ncase pos.convert_4\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nj : ℕ\nhj : j ≠ natDegree p\nh : j < natDegree p\n⊢ j + ?pos.convert_1✝ < natDegree p",
"state_before": "case pos\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nj : ℕ\nhj : j ≠ natDegree p\nh : j < natDegree p\n⊢ ↑(den A r) ∣ ↑(den A r) ^ (natDegree p - j)",
"tactic": "convert pow_dvd_pow (den A r : A) (Nat.succ_le_iff.mpr (lt_tsub_iff_left.mpr _))"
},
{
"state_after": "no goals",
"state_before": "case pos.convert_4\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nj : ℕ\nhj : j ≠ natDegree p\nh : j < natDegree p\n⊢ j + 0 < natDegree p",
"tactic": "simpa using h"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nA : Type u_1\nK : Type u_2\ninst✝⁵ : CommRing A\ninst✝⁴ : IsDomain A\ninst✝³ : UniqueFactorizationMonoid A\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\np : A[X]\nr : K\nhr : ↑(aeval r) p = 0\nj : ℕ\nhj : j ≠ natDegree p\nh : j < natDegree p\n⊢ ↑(den A r) = ↑(den A r) ^ Nat.succ ?pos.convert_1✝",
"tactic": "exact (pow_one _).symm"
}
] |
[
118,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
97,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
List.stronglyMeasurable_prod
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.113287\nγ : Type ?u.113290\nι : Type ?u.113293\ninst✝³ : Countable ι\nf g : α → β\nM : Type u_2\ninst✝² : Monoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nm : MeasurableSpace α\nl : List (α → M)\nhl : ∀ (f : α → M), f ∈ l → StronglyMeasurable f\n⊢ StronglyMeasurable fun x => List.prod (List.map (fun f => f x) l)",
"tactic": "simpa only [← Pi.list_prod_apply] using l.stronglyMeasurable_prod' hl"
}
] |
[
556,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
553,
1
] |
Mathlib/Topology/Partial.lean
|
pcontinuous_iff'
|
[
{
"state_after": "case mp\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\n⊢ PContinuous f → ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\n\ncase mpr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\n⊢ (∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)) → PContinuous f",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\n⊢ PContinuous f ↔ ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)",
"tactic": "constructor"
},
{
"state_after": "case mpr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\n⊢ IsOpen (PFun.preimage f s)",
"state_before": "case mpr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\n⊢ (∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)) → PContinuous f",
"tactic": "intro hf s os"
},
{
"state_after": "case mpr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\n⊢ ∀ (a : α), a ∈ PFun.preimage f s → 𝓝 a ≤ 𝓟 (PFun.preimage f s)",
"state_before": "case mpr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\n⊢ IsOpen (PFun.preimage f s)",
"tactic": "rw [isOpen_iff_nhds]"
},
{
"state_after": "case mpr.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\nx : α\ny : β\nys : y ∈ s\nfxy : y ∈ f x\nt : Set α\n⊢ t ∈ 𝓟 (PFun.preimage f s) → t ∈ 𝓝 x",
"state_before": "case mpr\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\n⊢ ∀ (a : α), a ∈ PFun.preimage f s → 𝓝 a ≤ 𝓟 (PFun.preimage f s)",
"tactic": "rintro x ⟨y, ys, fxy⟩ t"
},
{
"state_after": "case mpr.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\nx : α\ny : β\nys : y ∈ s\nfxy : y ∈ f x\nt : Set α\n⊢ PFun.preimage f s ⊆ t → t ∈ 𝓝 x",
"state_before": "case mpr.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\nx : α\ny : β\nys : y ∈ s\nfxy : y ∈ f x\nt : Set α\n⊢ t ∈ 𝓟 (PFun.preimage f s) → t ∈ 𝓝 x",
"tactic": "rw [mem_principal]"
},
{
"state_after": "case mpr.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\nx : α\ny : β\nys : y ∈ s\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s ⊆ t\n⊢ t ∈ 𝓝 x",
"state_before": "case mpr.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\nx : α\ny : β\nys : y ∈ s\nfxy : y ∈ f x\nt : Set α\n⊢ PFun.preimage f s ⊆ t → t ∈ 𝓝 x",
"tactic": "intro (h : f.preimage s ⊆ t)"
},
{
"state_after": "case mpr.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\nx : α\ny : β\nys : y ∈ s\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s ⊆ t\n⊢ t ∈ 𝓝 x",
"state_before": "case mpr.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\nx : α\ny : β\nys : y ∈ s\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s ⊆ t\n⊢ t ∈ 𝓝 x",
"tactic": "change t ∈ 𝓝 x"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\nx : α\ny : β\nys : y ∈ s\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s ⊆ t\n⊢ PFun.preimage f s ∈ 𝓝 x",
"state_before": "case mpr.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\nx : α\ny : β\nys : y ∈ s\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s ⊆ t\n⊢ t ∈ 𝓝 x",
"tactic": "apply mem_of_superset _ h"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\nx : α\ny : β\nys : y ∈ s\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s ⊆ t\nh' : ∀ (s : Set β), s ∈ 𝓝 y → PFun.preimage f s ∈ 𝓝 x\n⊢ PFun.preimage f s ∈ 𝓝 x",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\nx : α\ny : β\nys : y ∈ s\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s ⊆ t\n⊢ PFun.preimage f s ∈ 𝓝 x",
"tactic": "have h' : ∀ s ∈ 𝓝 y, f.preimage s ∈ 𝓝 x := by\n intro s hs\n have : PTendsto' f (𝓝 x) (𝓝 y) := hf fxy\n rw [ptendsto'_def] at this\n exact this s hs"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\nx : α\ny : β\nys : y ∈ s\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s ⊆ t\nh' : ∀ (s : Set β), s ∈ 𝓝 y → PFun.preimage f s ∈ 𝓝 x\n⊢ PFun.preimage f s ∈ 𝓝 x",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\nx : α\ny : β\nys : y ∈ s\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s ⊆ t\nh' : ∀ (s : Set β), s ∈ 𝓝 y → PFun.preimage f s ∈ 𝓝 x\n⊢ PFun.preimage f s ∈ 𝓝 x",
"tactic": "show f.preimage s ∈ 𝓝 x"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\nx : α\ny : β\nys : y ∈ s\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s ⊆ t\nh' : ∀ (s : Set β), s ∈ 𝓝 y → PFun.preimage f s ∈ 𝓝 x\n⊢ s ∈ 𝓝 y",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\nx : α\ny : β\nys : y ∈ s\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s ⊆ t\nh' : ∀ (s : Set β), s ∈ 𝓝 y → PFun.preimage f s ∈ 𝓝 x\n⊢ PFun.preimage f s ∈ 𝓝 x",
"tactic": "apply h'"
},
{
"state_after": "case a\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\nx : α\ny : β\nys : y ∈ s\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s ⊆ t\nh' : ∀ (s : Set β), s ∈ 𝓝 y → PFun.preimage f s ∈ 𝓝 x\n⊢ ∃ t, t ⊆ s ∧ IsOpen t ∧ y ∈ t",
"state_before": "case a\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\nx : α\ny : β\nys : y ∈ s\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s ⊆ t\nh' : ∀ (s : Set β), s ∈ 𝓝 y → PFun.preimage f s ∈ 𝓝 x\n⊢ s ∈ 𝓝 y",
"tactic": "rw [mem_nhds_iff]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\nx : α\ny : β\nys : y ∈ s\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s ⊆ t\nh' : ∀ (s : Set β), s ∈ 𝓝 y → PFun.preimage f s ∈ 𝓝 x\n⊢ ∃ t, t ⊆ s ∧ IsOpen t ∧ y ∈ t",
"tactic": "exact ⟨s, Set.Subset.refl _, os, ys⟩"
},
{
"state_after": "case mp\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nh : PContinuous f\nx : α\ny : β\nh' : y ∈ f x\n⊢ PTendsto' f (𝓝 x) (𝓝 y)",
"state_before": "case mp\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\n⊢ PContinuous f → ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)",
"tactic": "intro h x y h'"
},
{
"state_after": "case mp\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nh : PContinuous f\nx : α\ny : β\nh' : y ∈ f x\n⊢ ∀ (s : Set β), (∃ t, t ⊆ s ∧ IsOpen t ∧ y ∈ t) → ∃ t, t ⊆ PFun.preimage f s ∧ IsOpen t ∧ x ∈ t",
"state_before": "case mp\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nh : PContinuous f\nx : α\ny : β\nh' : y ∈ f x\n⊢ PTendsto' f (𝓝 x) (𝓝 y)",
"tactic": "simp only [ptendsto'_def, mem_nhds_iff]"
},
{
"state_after": "case mp.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nh : PContinuous f\nx : α\ny : β\nh' : y ∈ f x\ns t : Set β\ntsubs : t ⊆ s\nopent : IsOpen t\nyt : y ∈ t\n⊢ ∃ t, t ⊆ PFun.preimage f s ∧ IsOpen t ∧ x ∈ t",
"state_before": "case mp\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nh : PContinuous f\nx : α\ny : β\nh' : y ∈ f x\n⊢ ∀ (s : Set β), (∃ t, t ⊆ s ∧ IsOpen t ∧ y ∈ t) → ∃ t, t ⊆ PFun.preimage f s ∧ IsOpen t ∧ x ∈ t",
"tactic": "rintro s ⟨t, tsubs, opent, yt⟩"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro\nα : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nh : PContinuous f\nx : α\ny : β\nh' : y ∈ f x\ns t : Set β\ntsubs : t ⊆ s\nopent : IsOpen t\nyt : y ∈ t\n⊢ ∃ t, t ⊆ PFun.preimage f s ∧ IsOpen t ∧ x ∈ t",
"tactic": "exact ⟨f.preimage t, PFun.preimage_mono _ tsubs, h _ opent, ⟨y, yt, h'⟩⟩"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns✝ : Set β\nos : IsOpen s✝\nx : α\ny : β\nys : y ∈ s✝\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s✝ ⊆ t\ns : Set β\nhs : s ∈ 𝓝 y\n⊢ PFun.preimage f s ∈ 𝓝 x",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set β\nos : IsOpen s\nx : α\ny : β\nys : y ∈ s\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s ⊆ t\n⊢ ∀ (s : Set β), s ∈ 𝓝 y → PFun.preimage f s ∈ 𝓝 x",
"tactic": "intro s hs"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns✝ : Set β\nos : IsOpen s✝\nx : α\ny : β\nys : y ∈ s✝\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s✝ ⊆ t\ns : Set β\nhs : s ∈ 𝓝 y\nthis : PTendsto' f (𝓝 x) (𝓝 y)\n⊢ PFun.preimage f s ∈ 𝓝 x",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns✝ : Set β\nos : IsOpen s✝\nx : α\ny : β\nys : y ∈ s✝\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s✝ ⊆ t\ns : Set β\nhs : s ∈ 𝓝 y\n⊢ PFun.preimage f s ∈ 𝓝 x",
"tactic": "have : PTendsto' f (𝓝 x) (𝓝 y) := hf fxy"
},
{
"state_after": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns✝ : Set β\nos : IsOpen s✝\nx : α\ny : β\nys : y ∈ s✝\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s✝ ⊆ t\ns : Set β\nhs : s ∈ 𝓝 y\nthis : ∀ (s : Set β), s ∈ 𝓝 y → PFun.preimage f s ∈ 𝓝 x\n⊢ PFun.preimage f s ∈ 𝓝 x",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns✝ : Set β\nos : IsOpen s✝\nx : α\ny : β\nys : y ∈ s✝\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s✝ ⊆ t\ns : Set β\nhs : s ∈ 𝓝 y\nthis : PTendsto' f (𝓝 x) (𝓝 y)\n⊢ PFun.preimage f s ∈ 𝓝 x",
"tactic": "rw [ptendsto'_def] at this"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α →. β\nhf : ∀ {x : α} {y : β}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns✝ : Set β\nos : IsOpen s✝\nx : α\ny : β\nys : y ∈ s✝\nfxy : y ∈ f x\nt : Set α\nh : PFun.preimage f s✝ ⊆ t\ns : Set β\nhs : s ∈ 𝓝 y\nthis : ∀ (s : Set β), s ∈ 𝓝 y → PFun.preimage f s ∈ 𝓝 x\n⊢ PFun.preimage f s ∈ 𝓝 x",
"tactic": "exact this s hs"
}
] |
[
86,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean
|
Ordinal.bmex_le_blsub
|
[] |
[
2103,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2101,
1
] |
Mathlib/Data/Nat/Dist.lean
|
Nat.dist_eq_sub_of_le_right
|
[
{
"state_after": "n m : ℕ\nh : m ≤ n\n⊢ dist m n = n - m",
"state_before": "n m : ℕ\nh : m ≤ n\n⊢ dist n m = n - m",
"tactic": "rw [dist_comm]"
},
{
"state_after": "no goals",
"state_before": "n m : ℕ\nh : m ≤ n\n⊢ dist m n = n - m",
"tactic": "apply dist_eq_sub_of_le h"
}
] |
[
54,
47
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
53,
1
] |
Mathlib/Topology/Algebra/Module/Basic.lean
|
ContinuousLinearEquiv.trans_apply
|
[] |
[
2032,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2030,
1
] |
Mathlib/Algebra/Hom/Group.lean
|
MonoidWithZeroHom.coe_comp
|
[] |
[
1108,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1107,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Equalizers.lean
|
CategoryTheory.Limits.isIso_colimit_cocone_parallelPair_of_eq
|
[] |
[
1051,
78
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1049,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean
|
SimpleGraph.reachable_deleteEdges_iff_exists_cycle.aux
|
[
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nc : Walk G u u\nhc : Walk.IsTrail c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))\nhv : v ∈ Walk.support c\n⊢ False",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nc : Walk G u u\nhc : Walk.IsTrail c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))\n⊢ False",
"tactic": "have hv := c.fst_mem_support_of_mem_edges he"
},
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nc : Walk G u u\nhc : Walk.IsTrail c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))\nhv : v ∈ Walk.support c\npuw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw\n⊢ False",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nc : Walk G u u\nhc : Walk.IsTrail c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))\nhv : v ∈ Walk.support c\n⊢ False",
"tactic": "let puw := (c.takeUntil v hv).takeUntil w hw"
},
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nc : Walk G u u\nhc : Walk.IsTrail c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))\nhv : v ∈ Walk.support c\npuw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw\npwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw\n⊢ False",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nc : Walk G u u\nhc : Walk.IsTrail c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))\nhv : v ∈ Walk.support c\npuw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw\n⊢ False",
"tactic": "let pwv := (c.takeUntil v hv).dropUntil w hw"
},
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nc : Walk G u u\nhc : Walk.IsTrail c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))\nhv : v ∈ Walk.support c\npuw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw\npwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw\npvu : Walk G v u := Walk.dropUntil c v hv\n⊢ False",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nc : Walk G u u\nhc : Walk.IsTrail c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))\nhv : v ∈ Walk.support c\npuw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw\npwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw\n⊢ False",
"tactic": "let pvu := c.dropUntil v hv"
},
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nc : Walk G u u\nhc : Walk.IsTrail c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))\nhv : v ∈ Walk.support c\npuw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw\npwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw\npvu : Walk G v u := Walk.dropUntil c v hv\nthis : c = Walk.append (Walk.append puw pwv) pvu\n⊢ False",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nc : Walk G u u\nhc : Walk.IsTrail c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))\nhv : v ∈ Walk.support c\npuw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw\npwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw\npvu : Walk G v u := Walk.dropUntil c v hv\n⊢ False",
"tactic": "have : c = (puw.append pwv).append pvu := by simp"
},
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nc : Walk G u u\nhc : Walk.IsTrail c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))\nhv : v ∈ Walk.support c\npuw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw\npwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw\npvu : Walk G v u := Walk.dropUntil c v hv\nthis : c = Walk.append (Walk.append puw pwv) pvu\nhbq : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges (Walk.append pvu puw)\n⊢ False",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nc : Walk G u u\nhc : Walk.IsTrail c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))\nhv : v ∈ Walk.support c\npuw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw\npwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw\npvu : Walk G v u := Walk.dropUntil c v hv\nthis : c = Walk.append (Walk.append puw pwv) pvu\n⊢ False",
"tactic": "have hbq := hb (pvu.append puw)"
},
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nc : Walk G u u\nhc : Walk.IsTrail c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))\nhv : v ∈ Walk.support c\npuw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw\npwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw\npvu : Walk G v u := Walk.dropUntil c v hv\nthis : c = Walk.append (Walk.append puw pwv) pvu\nhbq : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges (Walk.append pvu puw)\nhpq' : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges (Walk.reverse pwv)\n⊢ False",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nc : Walk G u u\nhc : Walk.IsTrail c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))\nhv : v ∈ Walk.support c\npuw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw\npwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw\npvu : Walk G v u := Walk.dropUntil c v hv\nthis : c = Walk.append (Walk.append puw pwv) pvu\nhbq : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges (Walk.append pvu puw)\n⊢ False",
"tactic": "have hpq' := hb pwv.reverse"
},
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nc : Walk G u u\nhc : Walk.IsTrail c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))\nhv : v ∈ Walk.support c\npuw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw\npwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw\npvu : Walk G v u := Walk.dropUntil c v hv\nthis : c = Walk.append (Walk.append puw pwv) pvu\nhbq : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges (Walk.append pvu puw)\nhpq' : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges pwv\n⊢ False",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nc : Walk G u u\nhc : Walk.IsTrail c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))\nhv : v ∈ Walk.support c\npuw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw\npwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw\npvu : Walk G v u := Walk.dropUntil c v hv\nthis : c = Walk.append (Walk.append puw pwv) pvu\nhbq : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges (Walk.append pvu puw)\nhpq' : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges (Walk.reverse pwv)\n⊢ False",
"tactic": "rw [Walk.edges_reverse, List.mem_reverse'] at hpq'"
},
{
"state_after": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nc : Walk G u u\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))\nhv : v ∈ Walk.support c\npuw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw\npwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw\npvu : Walk G v u := Walk.dropUntil c v hv\nhc : List.Nodup (Walk.edges (Walk.append pvu puw) ++ Walk.edges pwv)\nthis : c = Walk.append (Walk.append puw pwv) pvu\nhbq : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges (Walk.append pvu puw)\nhpq' : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges pwv\n⊢ False",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nc : Walk G u u\nhc : Walk.IsTrail c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))\nhv : v ∈ Walk.support c\npuw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw\npwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw\npvu : Walk G v u := Walk.dropUntil c v hv\nthis : c = Walk.append (Walk.append puw pwv) pvu\nhbq : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges (Walk.append pvu puw)\nhpq' : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges pwv\n⊢ False",
"tactic": "rw [Walk.isTrail_def, this, Walk.edges_append, Walk.edges_append, List.nodup_append_comm,\n ← List.append_assoc, ← Walk.edges_append] at hc"
},
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nc : Walk G u u\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))\nhv : v ∈ Walk.support c\npuw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw\npwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw\npvu : Walk G v u := Walk.dropUntil c v hv\nhc : List.Nodup (Walk.edges (Walk.append pvu puw) ++ Walk.edges pwv)\nthis : c = Walk.append (Walk.append puw pwv) pvu\nhbq : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges (Walk.append pvu puw)\nhpq' : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges pwv\n⊢ False",
"tactic": "exact List.disjoint_of_nodup_append hc hbq hpq'"
},
{
"state_after": "no goals",
"state_before": "V : Type u\nV' : Type v\nV'' : Type w\nG : SimpleGraph V\nG' : SimpleGraph V'\nG'' : SimpleGraph V''\ninst✝ : DecidableEq V\nu v w : V\nhb : ∀ (p : Walk G v w), Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges p\nc : Walk G u u\nhc : Walk.IsTrail c\nhe : Quotient.mk (Sym2.Rel.setoid V) (v, w) ∈ Walk.edges c\nhw : w ∈ Walk.support (Walk.takeUntil c v (_ : v ∈ Walk.support c))\nhv : v ∈ Walk.support c\npuw : Walk G u w := Walk.takeUntil (Walk.takeUntil c v hv) w hw\npwv : Walk G w v := Walk.dropUntil (Walk.takeUntil c v hv) w hw\npvu : Walk G v u := Walk.dropUntil c v hv\n⊢ c = Walk.append (Walk.append puw pwv) pvu",
"tactic": "simp"
}
] |
[
2495,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2471,
1
] |
Mathlib/Order/Basic.lean
|
exists_ge_of_linear
|
[] |
[
496,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
493,
1
] |
Mathlib/GroupTheory/Index.lean
|
Subgroup.relindex_eq_zero_of_le_right
|
[] |
[
387,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
386,
1
] |
Mathlib/Data/Subtype.lean
|
Subtype.coind_surjective
|
[] |
[
193,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
190,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Complex.cos_add_int_mul_two_pi
|
[] |
[
1244,
27
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1243,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Localization.mk_le_mk
|
[] |
[
1947,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1946,
1
] |
Mathlib/Algebra/Lie/Submodule.lean
|
LieModuleHom.mem_range
|
[] |
[
1222,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1221,
1
] |
Mathlib/MeasureTheory/Function/StronglyMeasurable/Basic.lean
|
MeasureTheory.AEFinStronglyMeasurable.neg
|
[] |
[
1888,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1886,
11
] |
Mathlib/GroupTheory/MonoidLocalization.lean
|
Submonoid.LocalizationMap.map_mul_left
|
[
{
"state_after": "no goals",
"state_before": "M : Type u_2\ninst✝³ : CommMonoid M\nS : Submonoid M\nN : Type u_4\ninst✝² : CommMonoid N\nP : Type u_3\ninst✝¹ : CommMonoid P\nf : LocalizationMap S N\ng : M →* P\nhg : ∀ (y : { x // x ∈ S }), IsUnit (↑g ↑y)\nT : Submonoid P\nhy : ∀ (y : { x // x ∈ S }), ↑g ↑y ∈ T\nQ : Type u_1\ninst✝ : CommMonoid Q\nk : LocalizationMap T Q\nz : N\n⊢ ↑(toMap k) (↑g ↑(sec f z).snd) * ↑(map f hy k) z = ↑(toMap k) (↑g (sec f z).fst)",
"tactic": "rw [mul_comm, f.map_mul_right]"
}
] |
[
1198,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1197,
1
] |
Mathlib/SetTheory/ZFC/Basic.lean
|
ZFSet.not_mem_empty
|
[] |
[
828,
44
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
827,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
ENNReal.rpow_one_div_le_iff
|
[
{
"state_after": "x y : ℝ≥0∞\nz : ℝ\nhz : 0 < z\n⊢ x ^ (1 / z) ≤ y ^ 1 ↔ x ≤ y ^ z",
"state_before": "x y : ℝ≥0∞\nz : ℝ\nhz : 0 < z\n⊢ x ^ (1 / z) ≤ y ↔ x ≤ y ^ z",
"tactic": "nth_rw 1 [← ENNReal.rpow_one y]"
},
{
"state_after": "x y : ℝ≥0∞\nz : ℝ\nhz : 0 < z\n⊢ x ^ (1 / z) ≤ y ^ (z * z⁻¹) ↔ x ≤ y ^ z",
"state_before": "x y : ℝ≥0∞\nz : ℝ\nhz : 0 < z\n⊢ x ^ (1 / z) ≤ y ^ 1 ↔ x ≤ y ^ z",
"tactic": "nth_rw 2 [← @_root_.mul_inv_cancel _ _ z hz.ne.symm]"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ≥0∞\nz : ℝ\nhz : 0 < z\n⊢ x ^ (1 / z) ≤ y ^ (z * z⁻¹) ↔ x ≤ y ^ z",
"tactic": "rw [ENNReal.rpow_mul, ← one_div, ENNReal.rpow_le_rpow_iff (one_div_pos.2 hz)]"
}
] |
[
596,
80
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
593,
1
] |
Mathlib/NumberTheory/Padics/RingHoms.lean
|
PadicInt.nthHom_zero
|
[
{
"state_after": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\n⊢ (fun n => 0) = 0",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\n⊢ nthHom f 0 = 0",
"tactic": "simp [nthHom]"
},
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp_prime : Fact (Nat.Prime p)\nR : Type u_1\ninst✝ : NonAssocSemiring R\nf : (k : ℕ) → R →+* ZMod (p ^ k)\nf_compat : ∀ (k1 k2 : ℕ) (hk : k1 ≤ k2), RingHom.comp (ZMod.castHom (_ : p ^ k1 ∣ p ^ k2) (ZMod (p ^ k1))) (f k2) = f k1\n⊢ (fun n => 0) = 0",
"tactic": "rfl"
}
] |
[
495,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
495,
1
] |
Mathlib/Algebra/Algebra/Equiv.lean
|
AlgEquiv.invFun_eq_symm
|
[] |
[
344,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
343,
1
] |
Mathlib/CategoryTheory/Abelian/Pseudoelements.lean
|
CategoryTheory.Abelian.Pseudoelement.pseudo_exact_of_exact
|
[
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\nh : Exact f g\na : Pseudoelement P\n⊢ pseudoApply 0 a = 0",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\nh : Exact f g\na : Pseudoelement P\n⊢ pseudoApply g (pseudoApply f a) = 0",
"tactic": "rw [← comp_apply, h.w]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\nh : Exact f g\na : Pseudoelement P\n⊢ pseudoApply 0 a = 0",
"tactic": "exact zero_apply _ _"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\nh : Exact f g\nb' : Pseudoelement Q\nb : Over Q\nhb : pseudoApply g (Quotient.mk (setoid Q) b) = 0\nhb' : b.hom ≫ g = 0\n⊢ ∃ a, pseudoApply f a = Quotient.mk (setoid Q) b",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\nh : Exact f g\nb' : Pseudoelement Q\nb : Over Q\nhb : pseudoApply g (Quotient.mk (setoid Q) b) = 0\n⊢ ∃ a, pseudoApply f a = Quotient.mk (setoid Q) b",
"tactic": "have hb' : b.hom ≫ g = 0 := (pseudoZero_iff _).1 hb"
},
{
"state_after": "case mk\nC : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\nh : Exact f g\nb' : Pseudoelement Q\nb : Over Q\nhb : pseudoApply g (Quotient.mk (setoid Q) b) = 0\nhb' : b.hom ≫ g = 0\nc : (𝟭 C).obj b.left ⟶ (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)).pt\nhc : c ≫ Fork.ι (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)) = b.hom\n⊢ ∃ a, pseudoApply f a = Quotient.mk (setoid Q) b",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\nh : Exact f g\nb' : Pseudoelement Q\nb : Over Q\nhb : pseudoApply g (Quotient.mk (setoid Q) b) = 0\nhb' : b.hom ≫ g = 0\n⊢ ∃ a, pseudoApply f a = Quotient.mk (setoid Q) b",
"tactic": "obtain ⟨c, hc⟩ := KernelFork.IsLimit.lift' (isLimitImage f g h) _ hb'"
},
{
"state_after": "case mk\nC : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\nh : Exact f g\nb' : Pseudoelement Q\nb : Over Q\nhb : pseudoApply g (Quotient.mk (setoid Q) b) = 0\nhb' : b.hom ≫ g = 0\nc : (𝟭 C).obj b.left ⟶ (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)).pt\nhc : c ≫ Fork.ι (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)) = b.hom\n⊢ pseudoApply f (Quot.mk (PseudoEqual P) (Over.mk pullback.fst)) = Quotient.mk (setoid Q) b",
"state_before": "case mk\nC : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\nh : Exact f g\nb' : Pseudoelement Q\nb : Over Q\nhb : pseudoApply g (Quotient.mk (setoid Q) b) = 0\nhb' : b.hom ≫ g = 0\nc : (𝟭 C).obj b.left ⟶ (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)).pt\nhc : c ≫ Fork.ι (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)) = b.hom\n⊢ ∃ a, pseudoApply f a = Quotient.mk (setoid Q) b",
"tactic": "use (pullback.fst : pullback (Abelian.factorThruImage f) c ⟶ P)"
},
{
"state_after": "case mk.a\nC : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\nh : Exact f g\nb' : Pseudoelement Q\nb : Over Q\nhb : pseudoApply g (Quotient.mk (setoid Q) b) = 0\nhb' : b.hom ≫ g = 0\nc : (𝟭 C).obj b.left ⟶ (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)).pt\nhc : c ≫ Fork.ι (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)) = b.hom\n⊢ (fun g => app f g) (Over.mk pullback.fst) ≈ b",
"state_before": "case mk\nC : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\nh : Exact f g\nb' : Pseudoelement Q\nb : Over Q\nhb : pseudoApply g (Quotient.mk (setoid Q) b) = 0\nhb' : b.hom ≫ g = 0\nc : (𝟭 C).obj b.left ⟶ (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)).pt\nhc : c ≫ Fork.ι (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)) = b.hom\n⊢ pseudoApply f (Quot.mk (PseudoEqual P) (Over.mk pullback.fst)) = Quotient.mk (setoid Q) b",
"tactic": "apply Quotient.sound"
},
{
"state_after": "case mk.a\nC : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\nh : Exact f g\nb' : Pseudoelement Q\nb : Over Q\nhb : pseudoApply g (Quotient.mk (setoid Q) b) = 0\nhb' : b.hom ≫ g = 0\nc : (𝟭 C).obj b.left ⟶ (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)).pt\nhc : c ≫ Fork.ι (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)) = b.hom\n⊢ 𝟙 (pullback (Abelian.factorThruImage f) c) ≫ ((fun g => app f g) (Over.mk pullback.fst)).hom = pullback.snd ≫ b.hom",
"state_before": "case mk.a\nC : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\nh : Exact f g\nb' : Pseudoelement Q\nb : Over Q\nhb : pseudoApply g (Quotient.mk (setoid Q) b) = 0\nhb' : b.hom ≫ g = 0\nc : (𝟭 C).obj b.left ⟶ (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)).pt\nhc : c ≫ Fork.ι (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)) = b.hom\n⊢ (fun g => app f g) (Over.mk pullback.fst) ≈ b",
"tactic": "refine'\n ⟨pullback (Abelian.factorThruImage f) c, 𝟙 _, pullback.snd, inferInstance, inferInstance, _⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.a\nC : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\nh : Exact f g\nb' : Pseudoelement Q\nb : Over Q\nhb : pseudoApply g (Quotient.mk (setoid Q) b) = 0\nhb' : b.hom ≫ g = 0\nc : (𝟭 C).obj b.left ⟶ (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)).pt\nhc : c ≫ Fork.ι (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)) = b.hom\n⊢ 𝟙 (pullback (Abelian.factorThruImage f) c) ≫ ((fun g => app f g) (Over.mk pullback.fst)).hom = pullback.snd ≫ b.hom",
"tactic": "calc\n 𝟙 (pullback (Abelian.factorThruImage f) c) ≫ pullback.fst ≫ f = pullback.fst ≫ f :=\n Category.id_comp _\n _ = pullback.fst ≫ Abelian.factorThruImage f ≫ kernel.ι (cokernel.π f) := by\n rw [Abelian.image.fac]\n _ = (pullback.snd ≫ c) ≫ kernel.ι (cokernel.π f) := by\n rw [← Category.assoc, pullback.condition]\n _ = pullback.snd ≫ b.hom := by\n rw [Category.assoc]\n congr"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\nh : Exact f g\nb' : Pseudoelement Q\nb : Over Q\nhb : pseudoApply g (Quotient.mk (setoid Q) b) = 0\nhb' : b.hom ≫ g = 0\nc : (𝟭 C).obj b.left ⟶ (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)).pt\nhc : c ≫ Fork.ι (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)) = b.hom\n⊢ pullback.fst ≫ f = pullback.fst ≫ Abelian.factorThruImage f ≫ kernel.ι (cokernel.π f)",
"tactic": "rw [Abelian.image.fac]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\nh : Exact f g\nb' : Pseudoelement Q\nb : Over Q\nhb : pseudoApply g (Quotient.mk (setoid Q) b) = 0\nhb' : b.hom ≫ g = 0\nc : (𝟭 C).obj b.left ⟶ (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)).pt\nhc : c ≫ Fork.ι (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)) = b.hom\n⊢ pullback.fst ≫ Abelian.factorThruImage f ≫ kernel.ι (cokernel.π f) = (pullback.snd ≫ c) ≫ kernel.ι (cokernel.π f)",
"tactic": "rw [← Category.assoc, pullback.condition]"
},
{
"state_after": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\nh : Exact f g\nb' : Pseudoelement Q\nb : Over Q\nhb : pseudoApply g (Quotient.mk (setoid Q) b) = 0\nhb' : b.hom ≫ g = 0\nc : (𝟭 C).obj b.left ⟶ (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)).pt\nhc : c ≫ Fork.ι (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)) = b.hom\n⊢ pullback.snd ≫ c ≫ kernel.ι (cokernel.π f) = pullback.snd ≫ b.hom",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\nh : Exact f g\nb' : Pseudoelement Q\nb : Over Q\nhb : pseudoApply g (Quotient.mk (setoid Q) b) = 0\nhb' : b.hom ≫ g = 0\nc : (𝟭 C).obj b.left ⟶ (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)).pt\nhc : c ≫ Fork.ι (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)) = b.hom\n⊢ (pullback.snd ≫ c) ≫ kernel.ι (cokernel.π f) = pullback.snd ≫ b.hom",
"tactic": "rw [Category.assoc]"
},
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : Abelian C\nP Q R : C\nf : P ⟶ Q\ng : Q ⟶ R\nh : Exact f g\nb' : Pseudoelement Q\nb : Over Q\nhb : pseudoApply g (Quotient.mk (setoid Q) b) = 0\nhb' : b.hom ≫ g = 0\nc : (𝟭 C).obj b.left ⟶ (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)).pt\nhc : c ≫ Fork.ι (KernelFork.ofι (image.ι f) (_ : image.ι f ≫ g = 0)) = b.hom\n⊢ pullback.snd ≫ c ≫ kernel.ι (cokernel.π f) = pullback.snd ≫ b.hom",
"tactic": "congr"
}
] |
[
389,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
361,
1
] |
Mathlib/Data/Finmap.lean
|
Finmap.lookup_erase
|
[] |
[
450,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
449,
1
] |
Mathlib/Topology/Algebra/Polynomial.lean
|
Polynomial.continuousWithinAt_aeval
|
[] |
[
93,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
91,
11
] |
Mathlib/Data/Polynomial/FieldDivision.lean
|
Polynomial.divByMonic_eq_div
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nk : Type y\nA : Type z\na b : R\nn : ℕ\ninst✝ : Field R\np✝ q p : R[X]\nhq : Monic q\n⊢ p /ₘ q = ↑C (leadingCoeff q)⁻¹ * (p /ₘ (q * ↑C (leadingCoeff q)⁻¹))",
"tactic": "simp only [Monic.def.1 hq, inv_one, C_1, one_mul, mul_one]"
}
] |
[
207,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Real.cosh_sub_sinh
|
[
{
"state_after": "x y : ℝ\n⊢ ↑(cosh x - sinh x) = ↑(exp (-x))",
"state_before": "x y : ℝ\n⊢ cosh x - sinh x = exp (-x)",
"tactic": "rw [← ofReal_inj]"
},
{
"state_after": "no goals",
"state_before": "x y : ℝ\n⊢ ↑(cosh x - sinh x) = ↑(exp (-x))",
"tactic": "simp"
}
] |
[
1423,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1421,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.nthRoots_zero
|
[
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\nT : Type w\na b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np q : R[X]\nr : R\n⊢ nthRoots 0 r = 0",
"tactic": "simp only [empty_eq_zero, pow_zero, nthRoots, ← C_1, ← C_sub, roots_C]"
}
] |
[
786,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
785,
1
] |
Mathlib/Order/Lattice.lean
|
Prod.sup_def
|
[] |
[
1298,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1297,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_indicator_const
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.952854\nγ : Type ?u.952857\nδ : Type ?u.952860\nm : MeasurableSpace α\nμ ν : Measure α\ns : Set α\nhs : MeasurableSet s\nc : ℝ≥0∞\n⊢ (∫⁻ (a : α), indicator s (fun x => c) a ∂μ) = c * ↑↑μ s",
"tactic": "rw [lintegral_indicator _ hs, set_lintegral_const]"
}
] |
[
793,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
791,
1
] |
Mathlib/RingTheory/Multiplicity.lean
|
multiplicity.multiplicity_lt_iff_neg_dvd
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : Monoid α\ninst✝ : DecidableRel fun x x_1 => x ∣ x_1\na b : α\nk : ℕ\n⊢ multiplicity a b < ↑k ↔ ¬a ^ k ∣ b",
"tactic": "rw [pow_dvd_iff_le_multiplicity, not_le]"
}
] |
[
155,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
154,
1
] |
Mathlib/MeasureTheory/Measure/Stieltjes.lean
|
StieltjesFunction.id_leftLim
|
[] |
[
266,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
264,
1
] |
Mathlib/RingTheory/Artinian.lean
|
IsArtinianRing.localization_surjective
|
[
{
"state_after": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr' : L\n⊢ ∃ a, ↑(algebraMap R L) a = r'",
"state_before": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\n⊢ Surjective ↑(algebraMap R L)",
"tactic": "intro r'"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\n⊢ ∃ a, ↑(algebraMap R L) a = IsLocalization.mk' L r₁ s",
"state_before": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr' : L\n⊢ ∃ a, ↑(algebraMap R L) a = r'",
"tactic": "obtain ⟨r₁, s, rfl⟩ := IsLocalization.mk'_surjective S r'"
},
{
"state_after": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\n⊢ ∃ r, IsLocalization.mk' L 1 s = ↑(algebraMap R L) r\n\ncase intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\nr₂ : R\nh : IsLocalization.mk' L 1 s = ↑(algebraMap R L) r₂\n⊢ ∃ a, ↑(algebraMap R L) a = IsLocalization.mk' L r₁ s",
"state_before": "case intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\n⊢ ∃ a, ↑(algebraMap R L) a = IsLocalization.mk' L r₁ s",
"tactic": "obtain ⟨r₂, h⟩ : ∃ r : R, IsLocalization.mk' L 1 s = algebraMap R L r"
},
{
"state_after": "case intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\nr₂ : R\nh : IsLocalization.mk' L 1 s = ↑(algebraMap R L) r₂\n⊢ ∃ a, ↑(algebraMap R L) a = IsLocalization.mk' L r₁ s\n\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\n⊢ ∃ r, IsLocalization.mk' L 1 s = ↑(algebraMap R L) r",
"state_before": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\n⊢ ∃ r, IsLocalization.mk' L 1 s = ↑(algebraMap R L) r\n\ncase intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\nr₂ : R\nh : IsLocalization.mk' L 1 s = ↑(algebraMap R L) r₂\n⊢ ∃ a, ↑(algebraMap R L) a = IsLocalization.mk' L r₁ s",
"tactic": "swap"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\nn : ℕ\nr : R\nhr : ↑s ^ Nat.succ n • r = ↑s ^ n • 1\n⊢ ∃ r, IsLocalization.mk' L 1 s = ↑(algebraMap R L) r",
"state_before": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\n⊢ ∃ r, IsLocalization.mk' L 1 s = ↑(algebraMap R L) r",
"tactic": "obtain ⟨n, r, hr⟩ := IsArtinian.exists_pow_succ_smul_dvd (s : R) (1 : R)"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\nn : ℕ\nr : R\nhr : ↑s ^ Nat.succ n • r = ↑s ^ n • 1\n⊢ IsLocalization.mk' L 1 s = ↑(algebraMap R L) r",
"state_before": "case intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\nn : ℕ\nr : R\nhr : ↑s ^ Nat.succ n • r = ↑s ^ n • 1\n⊢ ∃ r, IsLocalization.mk' L 1 s = ↑(algebraMap R L) r",
"tactic": "use r"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\nn : ℕ\nr : R\nhr : ↑s ^ n * (↑s * r) = ↑s ^ n * 1\n⊢ IsLocalization.mk' L 1 s = ↑(algebraMap R L) r",
"state_before": "case intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\nn : ℕ\nr : R\nhr : ↑s ^ Nat.succ n • r = ↑s ^ n • 1\n⊢ IsLocalization.mk' L 1 s = ↑(algebraMap R L) r",
"tactic": "rw [smul_eq_mul, smul_eq_mul, pow_succ', mul_assoc] at hr"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\nn : ℕ\nr : R\nhr : ↑(algebraMap R L) (↑s ^ n * (↑s * r)) = ↑(algebraMap R L) (↑s ^ n * 1)\n⊢ IsLocalization.mk' L 1 s = ↑(algebraMap R L) r",
"state_before": "case intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\nn : ℕ\nr : R\nhr : ↑s ^ n * (↑s * r) = ↑s ^ n * 1\n⊢ IsLocalization.mk' L 1 s = ↑(algebraMap R L) r",
"tactic": "apply_fun algebraMap R L at hr"
},
{
"state_after": "case intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\nn : ℕ\nr : R\nhr :\n ↑(algebraMap R L) (↑s ^ n) * (↑(algebraMap R L) ↑s * ↑(algebraMap R L) r) =\n ↑(algebraMap R L) (↑s ^ n) * ↑(algebraMap R L) 1\n⊢ IsLocalization.mk' L 1 s = ↑(algebraMap R L) r",
"state_before": "case intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\nn : ℕ\nr : R\nhr : ↑(algebraMap R L) (↑s ^ n * (↑s * r)) = ↑(algebraMap R L) (↑s ^ n * 1)\n⊢ IsLocalization.mk' L 1 s = ↑(algebraMap R L) r",
"tactic": "simp only [map_mul] at hr"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\nn : ℕ\nr : R\nhr :\n ↑(algebraMap R L) (↑s ^ n) * (↑(algebraMap R L) ↑s * ↑(algebraMap R L) r) =\n ↑(algebraMap R L) (↑s ^ n) * ↑(algebraMap R L) 1\n⊢ IsLocalization.mk' L 1 s = ↑(algebraMap R L) r",
"tactic": "rw [← IsLocalization.mk'_one (M := S) L, IsLocalization.mk'_eq_iff_eq, mul_one, Submonoid.coe_one,\n ← (IsLocalization.map_units L (s ^ n)).mul_left_cancel hr, map_mul]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\nr₂ : R\nh : IsLocalization.mk' L 1 s = ↑(algebraMap R L) r₂\n⊢ ∃ a, ↑(algebraMap R L) a = IsLocalization.mk' L r₁ s",
"tactic": "exact ⟨r₁ * r₂, by rw [IsLocalization.mk'_eq_mul_mk'_one, map_mul, h]⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsArtinianRing R\nS : Submonoid R\nL : Type u_2\ninst✝² : CommRing L\ninst✝¹ : Algebra R L\ninst✝ : IsLocalization S L\nr₁ : R\ns : { x // x ∈ S }\nr₂ : R\nh : IsLocalization.mk' L 1 s = ↑(algebraMap R L) r₂\n⊢ ↑(algebraMap R L) (r₁ * r₂) = IsLocalization.mk' L r₁ s",
"tactic": "rw [IsLocalization.mk'_eq_mul_mk'_one, map_mul, h]"
}
] |
[
441,
72
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
429,
1
] |
Std/Data/RBMap/Lemmas.lean
|
Std.RBNode.Ordered.mem_find?
|
[
{
"state_after": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nt : RBNode α\nx : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsStrictCut cmp cut\nht : Ordered cmp t\nx✝ : x ∈ t ∧ cut x = Ordering.eq\nhx : x ∈ t\ne : cut x = Ordering.eq\n⊢ x ∈ find? cut t",
"state_before": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nt : RBNode α\nx : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsStrictCut cmp cut\nht : Ordered cmp t\n⊢ x ∈ find? cut t ↔ x ∈ t ∧ cut x = Ordering.eq",
"tactic": "refine ⟨fun h => ⟨find?_some_mem h, find?_some_eq_eq h⟩, fun ⟨hx, e⟩ => ?_⟩"
},
{
"state_after": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nt : RBNode α\nx : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsStrictCut cmp cut\nht : Ordered cmp t\nx✝ : x ∈ t ∧ cut x = Ordering.eq\nhx : x ∈ t\ne : cut x = Ordering.eq\ny : α\nhy : y ∈ find? cut t\n⊢ x ∈ find? cut t",
"state_before": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nt : RBNode α\nx : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsStrictCut cmp cut\nht : Ordered cmp t\nx✝ : x ∈ t ∧ cut x = Ordering.eq\nhx : x ∈ t\ne : cut x = Ordering.eq\n⊢ x ∈ find? cut t",
"tactic": "have ⟨y, hy⟩ := ht.memP_iff_find?.1 (memP_def.2 ⟨_, hx, e⟩)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ncmp : α → α → Ordering\ncut : α → Ordering\nt : RBNode α\nx : α\ninst✝¹ : TransCmp cmp\ninst✝ : IsStrictCut cmp cut\nht : Ordered cmp t\nx✝ : x ∈ t ∧ cut x = Ordering.eq\nhx : x ∈ t\ne : cut x = Ordering.eq\ny : α\nhy : y ∈ find? cut t\n⊢ x ∈ find? cut t",
"tactic": "exact ht.unique hx (find?_some_mem hy) ((IsStrictCut.exact e).trans (find?_some_eq_eq hy)) ▸ hy"
}
] |
[
212,
98
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
208,
1
] |
Mathlib/Data/Polynomial/RingDivision.lean
|
Polynomial.leadingCoeff_divByMonic_X_sub_C
|
[
{
"state_after": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\np✝ q p : R[X]\nhp : degree p ≠ 0\na : R\ninst✝ : Nontrivial R\n⊢ leadingCoeff (p /ₘ (X - ↑C a)) = leadingCoeff p",
"state_before": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn : ℕ\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np✝ q p : R[X]\nhp : degree p ≠ 0\na : R\n⊢ leadingCoeff (p /ₘ (X - ↑C a)) = leadingCoeff p",
"tactic": "nontriviality"
},
{
"state_after": "case inl\nR : Type u\nS : Type v\nT : Type w\na✝ b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\np✝ q p : R[X]\nhp : degree p ≠ 0\na : R\ninst✝ : Nontrivial R\nhd : degree p < 0\n⊢ leadingCoeff (p /ₘ (X - ↑C a)) = leadingCoeff p\n\ncase inr\nR : Type u\nS : Type v\nT : Type w\na✝ b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\np✝ q p : R[X]\nhp : degree p ≠ 0\na : R\ninst✝ : Nontrivial R\nhd : 0 < degree p\n⊢ leadingCoeff (p /ₘ (X - ↑C a)) = leadingCoeff p",
"state_before": "R : Type u\nS : Type v\nT : Type w\na✝ b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\np✝ q p : R[X]\nhp : degree p ≠ 0\na : R\ninst✝ : Nontrivial R\n⊢ leadingCoeff (p /ₘ (X - ↑C a)) = leadingCoeff p",
"tactic": "cases' hp.lt_or_lt with hd hd"
},
{
"state_after": "case inr\nR : Type u\nS : Type v\nT : Type w\na✝ b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\np✝ q p : R[X]\nhp : degree p ≠ 0\na : R\ninst✝ : Nontrivial R\nhd : 0 < degree p\n⊢ degree (X - ↑C a) ≤ degree p",
"state_before": "case inr\nR : Type u\nS : Type v\nT : Type w\na✝ b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\np✝ q p : R[X]\nhp : degree p ≠ 0\na : R\ninst✝ : Nontrivial R\nhd : 0 < degree p\n⊢ leadingCoeff (p /ₘ (X - ↑C a)) = leadingCoeff p",
"tactic": "refine' leadingCoeff_divByMonic_of_monic (monic_X_sub_C a) _"
},
{
"state_after": "no goals",
"state_before": "case inr\nR : Type u\nS : Type v\nT : Type w\na✝ b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\np✝ q p : R[X]\nhp : degree p ≠ 0\na : R\ninst✝ : Nontrivial R\nhd : 0 < degree p\n⊢ degree (X - ↑C a) ≤ degree p",
"tactic": "rwa [degree_X_sub_C, Nat.WithBot.one_le_iff_zero_lt]"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u\nS : Type v\nT : Type w\na✝ b : R\nn : ℕ\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\np✝ q p : R[X]\nhp : degree p ≠ 0\na : R\ninst✝ : Nontrivial R\nhd : degree p < 0\n⊢ leadingCoeff (p /ₘ (X - ↑C a)) = leadingCoeff p",
"tactic": "rw [degree_eq_bot.mp <| (Nat.WithBot.lt_zero_iff _).mp hd, zero_divByMonic]"
}
] |
[
1023,
55
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1017,
1
] |
Mathlib/Analysis/LocallyConvex/Basic.lean
|
Set.Finite.absorbs_iUnion
|
[
{
"state_after": "case intro\n𝕜 : Type u_3\n𝕝 : Type ?u.5168\nE : Type u_2\nι✝ : Sort ?u.5174\nκ : ι✝ → Sort ?u.5179\ninst✝¹ : SeminormedRing 𝕜\ninst✝ : SMul 𝕜 E\ns✝ t✝ u v A B : Set E\nι : Type u_1\ns : Set E\nf : ι → Set E\nt : Finset ι\n⊢ Absorbs 𝕜 s (⋃ (i : ι) (_ : i ∈ ↑t), f i) ↔ ∀ (i : ι), i ∈ ↑t → Absorbs 𝕜 s (f i)",
"state_before": "𝕜 : Type u_3\n𝕝 : Type ?u.5168\nE : Type u_2\nι✝ : Sort ?u.5174\nκ : ι✝ → Sort ?u.5179\ninst✝¹ : SeminormedRing 𝕜\ninst✝ : SMul 𝕜 E\ns✝ t✝ u v A B : Set E\nι : Type u_1\ns : Set E\nt : Set ι\nf : ι → Set E\nhi : Set.Finite t\n⊢ Absorbs 𝕜 s (⋃ (i : ι) (_ : i ∈ t), f i) ↔ ∀ (i : ι), i ∈ t → Absorbs 𝕜 s (f i)",
"tactic": "lift t to Finset ι using hi"
},
{
"state_after": "case intro\n𝕜 : Type u_3\n𝕝 : Type ?u.5168\nE : Type u_2\nι✝ : Sort ?u.5174\nκ : ι✝ → Sort ?u.5179\ninst✝¹ : SeminormedRing 𝕜\ninst✝ : SMul 𝕜 E\ns✝ t✝ u v A B : Set E\nι : Type u_1\ns : Set E\nf : ι → Set E\nt : Finset ι\n⊢ Absorbs 𝕜 s (⋃ (i : ι) (_ : i ∈ t), f i) ↔ ∀ (i : ι), i ∈ t → Absorbs 𝕜 s (f i)",
"state_before": "case intro\n𝕜 : Type u_3\n𝕝 : Type ?u.5168\nE : Type u_2\nι✝ : Sort ?u.5174\nκ : ι✝ → Sort ?u.5179\ninst✝¹ : SeminormedRing 𝕜\ninst✝ : SMul 𝕜 E\ns✝ t✝ u v A B : Set E\nι : Type u_1\ns : Set E\nf : ι → Set E\nt : Finset ι\n⊢ Absorbs 𝕜 s (⋃ (i : ι) (_ : i ∈ ↑t), f i) ↔ ∀ (i : ι), i ∈ ↑t → Absorbs 𝕜 s (f i)",
"tactic": "simp only [Finset.mem_coe]"
},
{
"state_after": "no goals",
"state_before": "case intro\n𝕜 : Type u_3\n𝕝 : Type ?u.5168\nE : Type u_2\nι✝ : Sort ?u.5174\nκ : ι✝ → Sort ?u.5179\ninst✝¹ : SeminormedRing 𝕜\ninst✝ : SMul 𝕜 E\ns✝ t✝ u v A B : Set E\nι : Type u_1\ns : Set E\nf : ι → Set E\nt : Finset ι\n⊢ Absorbs 𝕜 s (⋃ (i : ι) (_ : i ∈ t), f i) ↔ ∀ (i : ι), i ∈ t → Absorbs 𝕜 s (f i)",
"tactic": "exact absorbs_iUnion_finset"
}
] |
[
119,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
115,
1
] |
Mathlib/GroupTheory/Submonoid/Operations.lean
|
Submonoid.toAddSubmonoid_closure
|
[] |
[
109,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/RingTheory/ClassGroup.lean
|
ClassGroup.mk_eq_one_iff
|
[
{
"state_after": "R : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\n⊢ ↑(equiv K) (↑mk I) = ↑(equiv K) 1 ↔ Submodule.IsPrincipal ↑↑I",
"state_before": "R : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\n⊢ ↑mk I = 1 ↔ Submodule.IsPrincipal ↑↑I",
"tactic": "rw [← (ClassGroup.equiv K).injective.eq_iff]"
},
{
"state_after": "R : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\n⊢ (∃ x, spanSingleton R⁰ ↑x = ↑I) ↔ Submodule.IsPrincipal ↑↑I",
"state_before": "R : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\n⊢ ↑(equiv K) (↑mk I) = ↑(equiv K) 1 ↔ Submodule.IsPrincipal ↑↑I",
"tactic": "simp only [equiv_mk, canonicalEquiv_self, RingEquiv.coe_mulEquiv_refl, QuotientGroup.mk'_apply,\n _root_.map_one, QuotientGroup.eq_one_iff, MonoidHom.mem_range, ext_iff, coe_toPrincipalIdeal,\n coe_mapEquiv, MulEquiv.refl_apply]"
},
{
"state_after": "R : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\n⊢ Submodule.IsPrincipal ↑↑I → ∃ x, spanSingleton R⁰ ↑x = ↑I",
"state_before": "R : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\n⊢ (∃ x, spanSingleton R⁰ ↑x = ↑I) ↔ Submodule.IsPrincipal ↑↑I",
"tactic": "refine ⟨fun ⟨x, hx⟩ => ⟨⟨x, by rw [← hx, coe_spanSingleton]⟩⟩, ?_⟩"
},
{
"state_after": "R : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : Submodule.IsPrincipal ↑↑I\n⊢ ∃ x, spanSingleton R⁰ ↑x = ↑I",
"state_before": "R : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\n⊢ Submodule.IsPrincipal ↑↑I → ∃ x, spanSingleton R⁰ ↑x = ↑I",
"tactic": "intro hI"
},
{
"state_after": "case intro\nR : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : Submodule.IsPrincipal ↑↑I\nx : K\nhx : ↑↑I = Submodule.span R {x}\n⊢ ∃ x, spanSingleton R⁰ ↑x = ↑I",
"state_before": "R : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : Submodule.IsPrincipal ↑↑I\n⊢ ∃ x, spanSingleton R⁰ ↑x = ↑I",
"tactic": "obtain ⟨x, hx⟩ := @Submodule.IsPrincipal.principal _ _ _ _ _ _ hI"
},
{
"state_after": "case intro\nR : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : Submodule.IsPrincipal ↑↑I\nx : K\nhx : ↑↑I = Submodule.span R {x}\nhx' : ↑I = spanSingleton R⁰ x\n⊢ ∃ x, spanSingleton R⁰ ↑x = ↑I",
"state_before": "case intro\nR : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : Submodule.IsPrincipal ↑↑I\nx : K\nhx : ↑↑I = Submodule.span R {x}\n⊢ ∃ x, spanSingleton R⁰ ↑x = ↑I",
"tactic": "have hx' : (I : FractionalIdeal R⁰ K) = spanSingleton R⁰ x := by\n apply Subtype.coe_injective\n simp only [val_eq_coe, hx, coe_spanSingleton]"
},
{
"state_after": "case intro.refine_1\nR : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : Submodule.IsPrincipal ↑↑I\nx : K\nhx : ↑↑I = Submodule.span R {x}\nhx' : ↑I = spanSingleton R⁰ x\n⊢ x ≠ 0\n\ncase intro.refine_2\nR : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : Submodule.IsPrincipal ↑↑I\nx : K\nhx : ↑↑I = Submodule.span R {x}\nhx' : ↑I = spanSingleton R⁰ x\n⊢ spanSingleton R⁰ ↑(Units.mk0 x ?intro.refine_1) = ↑I",
"state_before": "case intro\nR : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : Submodule.IsPrincipal ↑↑I\nx : K\nhx : ↑↑I = Submodule.span R {x}\nhx' : ↑I = spanSingleton R⁰ x\n⊢ ∃ x, spanSingleton R⁰ ↑x = ↑I",
"tactic": "refine ⟨Units.mk0 x ?_, ?_⟩"
},
{
"state_after": "no goals",
"state_before": "R : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nx✝ : ∃ x, spanSingleton R⁰ ↑x = ↑I\nx : Kˣ\nhx : spanSingleton R⁰ ↑x = ↑I\n⊢ ↑↑I = Submodule.span R {↑x}",
"tactic": "rw [← hx, coe_spanSingleton]"
},
{
"state_after": "case a\nR : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : Submodule.IsPrincipal ↑↑I\nx : K\nhx : ↑↑I = Submodule.span R {x}\n⊢ (fun a => ↑a) ↑I = (fun a => ↑a) (spanSingleton R⁰ x)",
"state_before": "R : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : Submodule.IsPrincipal ↑↑I\nx : K\nhx : ↑↑I = Submodule.span R {x}\n⊢ ↑I = spanSingleton R⁰ x",
"tactic": "apply Subtype.coe_injective"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : Submodule.IsPrincipal ↑↑I\nx : K\nhx : ↑↑I = Submodule.span R {x}\n⊢ (fun a => ↑a) ↑I = (fun a => ↑a) (spanSingleton R⁰ x)",
"tactic": "simp only [val_eq_coe, hx, coe_spanSingleton]"
},
{
"state_after": "case intro.refine_1\nR : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : Submodule.IsPrincipal ↑↑I\nx : K\nhx : ↑↑I = Submodule.span R {x}\nhx' : ↑I = spanSingleton R⁰ x\nx_eq : x = 0\n⊢ False",
"state_before": "case intro.refine_1\nR : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : Submodule.IsPrincipal ↑↑I\nx : K\nhx : ↑↑I = Submodule.span R {x}\nhx' : ↑I = spanSingleton R⁰ x\n⊢ x ≠ 0",
"tactic": "intro x_eq"
},
{
"state_after": "case intro.refine_1\nR : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : Submodule.IsPrincipal ↑↑I\nx : K\nhx : ↑↑I = Submodule.span R {x}\nhx' : ↑I = spanSingleton R⁰ x\nx_eq : x = 0\n⊢ ↑I = 0",
"state_before": "case intro.refine_1\nR : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : Submodule.IsPrincipal ↑↑I\nx : K\nhx : ↑↑I = Submodule.span R {x}\nhx' : ↑I = spanSingleton R⁰ x\nx_eq : x = 0\n⊢ False",
"tactic": "apply Units.ne_zero I"
},
{
"state_after": "no goals",
"state_before": "case intro.refine_1\nR : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : Submodule.IsPrincipal ↑↑I\nx : K\nhx : ↑↑I = Submodule.span R {x}\nhx' : ↑I = spanSingleton R⁰ x\nx_eq : x = 0\n⊢ ↑I = 0",
"tactic": "simp [hx', x_eq]"
},
{
"state_after": "no goals",
"state_before": "case intro.refine_2\nR : Type u_2\nK : Type u_1\nL : Type ?u.1775691\ninst✝¹⁰ : CommRing R\ninst✝⁹ : Field K\ninst✝⁸ : Field L\ninst✝⁷ : DecidableEq L\ninst✝⁶ : Algebra R K\ninst✝⁵ : IsFractionRing R K\ninst✝⁴ : Algebra K L\ninst✝³ : FiniteDimensional K L\ninst✝² : Algebra R L\ninst✝¹ : IsScalarTower R K L\ninst✝ : IsDomain R\nI : (FractionalIdeal R⁰ K)ˣ\nhI : Submodule.IsPrincipal ↑↑I\nx : K\nhx : ↑↑I = Submodule.span R {x}\nhx' : ↑I = spanSingleton R⁰ x\n⊢ spanSingleton R⁰ ↑(Units.mk0 x (_ : x = 0 → False)) = ↑I",
"tactic": "simp [hx']"
}
] |
[
381,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
367,
1
] |
Mathlib/Algebra/Regular/SMul.lean
|
isLeftRegular_iff
|
[] |
[
48,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
47,
1
] |
Mathlib/Algebra/Order/Ring/Defs.lean
|
mul_nonneg_iff_left_nonneg_of_pos
|
[] |
[
935,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
934,
1
] |
Mathlib/Analysis/Calculus/ParametricIntegral.lean
|
hasFDerivAt_integral_of_dominated_loc_of_lip'
|
[
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\n⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\n⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"tactic": "have x₀_in : x₀ ∈ ball x₀ ε := mem_ball_self ε_pos"
},
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\n⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\n⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"tactic": "have nneg : ∀ x, 0 ≤ ‖x - x₀‖⁻¹ := fun x => inv_nonneg.mpr (norm_nonneg _)"
},
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\n⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\n⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"tactic": "set b : α → ℝ := fun a => |bound a|"
},
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\n⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\n⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"tactic": "have b_int : Integrable b μ := bound_integrable.norm"
},
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\n⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\n⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"tactic": "have b_nonneg : ∀ a, 0 ≤ b a := fun a => abs_nonneg _"
},
{
"state_after": "case h_lipsch\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\n⊢ ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\n⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"tactic": "replace h_lipsch : ∀ᵐ a ∂μ, ∀ x ∈ ball x₀ ε, ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖"
},
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"state_before": "case h_lipsch\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ bound a * ‖x - x₀‖\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\n⊢ ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"tactic": "exact h_lipsch.mono fun a ha x hx =>\n (ha x hx).trans <| mul_le_mul_of_nonneg_right (le_abs_self _) (norm_nonneg _)"
},
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\n⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"tactic": "have hF_int' : ∀ x ∈ ball x₀ ε, Integrable (F x) μ := fun x x_in ↦ by\n have : ∀ᵐ a ∂μ, ‖F x₀ a - F x a‖ ≤ ε * b a := by\n simp only [norm_sub_rev (F x₀ _)]\n refine' h_lipsch.mono fun a ha => (ha x x_in).trans _\n rw [mul_comm ε]\n rw [mem_ball, dist_eq_norm] at x_in \n exact mul_le_mul_of_nonneg_left x_in.le (b_nonneg _)\n exact integrable_of_norm_sub_le (hF_meas x x_in) hF_int\n (bound_integrable.norm.const_mul ε) this"
},
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\n⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\n⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"tactic": "have hF'_int : Integrable F' μ :=\n have : ∀ᵐ a ∂μ, ‖F' a‖ ≤ b a := by\n apply (h_diff.and h_lipsch).mono\n rintro a ⟨ha_diff, ha_lip⟩\n refine' ha_diff.le_of_lip' (b_nonneg a) (mem_of_superset (ball_mem_nhds _ ε_pos) <| ha_lip)\n b_int.mono' hF'_meas this"
},
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\n⊢ Integrable F' ∧ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"tactic": "refine' ⟨hF'_int, _⟩"
},
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"tactic": "have h_ball : ball x₀ ε ∈ 𝓝 x₀ := ball_mem_nhds x₀ ε_pos"
},
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"tactic": "have : ∀ᶠ x in 𝓝 x₀, ‖x - x₀‖⁻¹ * ‖((∫ a, F x a ∂μ) - ∫ a, F x₀ a ∂μ) - (∫ a, F' a ∂μ) (x - x₀)‖ =\n ‖∫ a, ‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀)) ∂μ‖ := by\n apply mem_of_superset (ball_mem_nhds _ ε_pos)\n intro x x_in; simp only\n rw [Set.mem_setOf_eq, ← norm_smul_of_nonneg (nneg _), integral_smul, integral_sub, integral_sub,\n ← ContinuousLinearMap.integral_apply hF'_int]\n exacts [hF_int' x x_in, hF_int, (hF_int' x x_in).sub hF_int,\n hF'_int.apply_continuousLinearMap _]"
},
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\n⊢ Tendsto (fun x => ∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ) (𝓝 x₀)\n (𝓝 (∫ (a : α), ‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - ↑(F' a) (x₀ - x₀)) ∂μ))",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\n⊢ HasFDerivAt (fun x => ∫ (a : α), F x a ∂μ) (∫ (a : α), F' a ∂μ) x₀",
"tactic": "rw [hasFDerivAt_iff_tendsto, tendsto_congr' this, ← tendsto_zero_iff_norm_tendsto_zero, ←\n show (∫ a : α, ‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - (F' a) (x₀ - x₀)) ∂μ) = 0 by simp]"
},
{
"state_after": "case hF_meas\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\n⊢ ∀ᶠ (n : H) in 𝓝 x₀, AEStronglyMeasurable (fun a => ‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))) μ\n\ncase h_bound\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\n⊢ ∀ᶠ (n : H) in 𝓝 x₀, ∀ᵐ (a : α) ∂μ, ‖‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))‖ ≤ ?bound a\n\ncase bound_integrable\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\n⊢ Integrable ?bound\n\ncase h_lim\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\n⊢ ∀ᵐ (a : α) ∂μ,\n Tendsto (fun n => ‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))) (𝓝 x₀)\n (𝓝 (‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - ↑(F' a) (x₀ - x₀))))\n\ncase bound\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\n⊢ α → ℝ",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\n⊢ Tendsto (fun x => ∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ) (𝓝 x₀)\n (𝓝 (∫ (a : α), ‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - ↑(F' a) (x₀ - x₀)) ∂μ))",
"tactic": "apply tendsto_integral_filter_of_dominated_convergence"
},
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nx : H\nx_in : x ∈ ball x₀ ε\nthis : ∀ᵐ (a : α) ∂μ, ‖F x₀ a - F x a‖ ≤ ε * b a\n⊢ Integrable (F x)",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nx : H\nx_in : x ∈ ball x₀ ε\n⊢ Integrable (F x)",
"tactic": "have : ∀ᵐ a ∂μ, ‖F x₀ a - F x a‖ ≤ ε * b a := by\n simp only [norm_sub_rev (F x₀ _)]\n refine' h_lipsch.mono fun a ha => (ha x x_in).trans _\n rw [mul_comm ε]\n rw [mem_ball, dist_eq_norm] at x_in \n exact mul_le_mul_of_nonneg_left x_in.le (b_nonneg _)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nx : H\nx_in : x ∈ ball x₀ ε\nthis : ∀ᵐ (a : α) ∂μ, ‖F x₀ a - F x a‖ ≤ ε * b a\n⊢ Integrable (F x)",
"tactic": "exact integrable_of_norm_sub_le (hF_meas x x_in) hF_int\n (bound_integrable.norm.const_mul ε) this"
},
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nx : H\nx_in : x ∈ ball x₀ ε\n⊢ ∀ᵐ (a : α) ∂μ, ‖F x a - F x₀ a‖ ≤ ε * abs (bound a)",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nx : H\nx_in : x ∈ ball x₀ ε\n⊢ ∀ᵐ (a : α) ∂μ, ‖F x₀ a - F x a‖ ≤ ε * b a",
"tactic": "simp only [norm_sub_rev (F x₀ _)]"
},
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nx : H\nx_in : x ∈ ball x₀ ε\na : α\nha : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ b a * ‖x - x₀‖ ≤ ε * abs (bound a)",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nx : H\nx_in : x ∈ ball x₀ ε\n⊢ ∀ᵐ (a : α) ∂μ, ‖F x a - F x₀ a‖ ≤ ε * abs (bound a)",
"tactic": "refine' h_lipsch.mono fun a ha => (ha x x_in).trans _"
},
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nx : H\nx_in : x ∈ ball x₀ ε\na : α\nha : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ b a * ‖x - x₀‖ ≤ abs (bound a) * ε",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nx : H\nx_in : x ∈ ball x₀ ε\na : α\nha : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ b a * ‖x - x₀‖ ≤ ε * abs (bound a)",
"tactic": "rw [mul_comm ε]"
},
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nx : H\nx_in : ‖x - x₀‖ < ε\na : α\nha : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ b a * ‖x - x₀‖ ≤ abs (bound a) * ε",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nx : H\nx_in : x ∈ ball x₀ ε\na : α\nha : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ b a * ‖x - x₀‖ ≤ abs (bound a) * ε",
"tactic": "rw [mem_ball, dist_eq_norm] at x_in"
},
{
"state_after": "no goals",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nx : H\nx_in : ‖x - x₀‖ < ε\na : α\nha : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ b a * ‖x - x₀‖ ≤ abs (bound a) * ε",
"tactic": "exact mul_le_mul_of_nonneg_left x_in.le (b_nonneg _)"
},
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\n⊢ ∀ (x : α),\n (HasFDerivAt (fun x_1 => F x_1 x) (F' x) x₀ ∧\n ∀ (x_1 : H), x_1 ∈ ball x₀ ε → ‖F x_1 x - F x₀ x‖ ≤ b x * ‖x_1 - x₀‖) →\n ‖F' x‖ ≤ b x",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\n⊢ ∀ᵐ (a : α) ∂μ, ‖F' a‖ ≤ b a",
"tactic": "apply (h_diff.and h_lipsch).mono"
},
{
"state_after": "case intro\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\na : α\nha_diff : HasFDerivAt (fun x => F x a) (F' a) x₀\nha_lip : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ ‖F' a‖ ≤ b a",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\n⊢ ∀ (x : α),\n (HasFDerivAt (fun x_1 => F x_1 x) (F' x) x₀ ∧\n ∀ (x_1 : H), x_1 ∈ ball x₀ ε → ‖F x_1 x - F x₀ x‖ ≤ b x * ‖x_1 - x₀‖) →\n ‖F' x‖ ≤ b x",
"tactic": "rintro a ⟨ha_diff, ha_lip⟩"
},
{
"state_after": "no goals",
"state_before": "case intro\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\na : α\nha_diff : HasFDerivAt (fun x => F x a) (F' a) x₀\nha_lip : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ ‖F' a‖ ≤ b a",
"tactic": "refine' ha_diff.le_of_lip' (b_nonneg a) (mem_of_superset (ball_mem_nhds _ ε_pos) <| ha_lip)"
},
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\n⊢ ball x₀ ε ⊆\n {x |\n (fun x =>\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖)\n x}",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\n⊢ ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖",
"tactic": "apply mem_of_superset (ball_mem_nhds _ ε_pos)"
},
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nx : H\nx_in : x ∈ ball x₀ ε\n⊢ x ∈\n {x |\n (fun x =>\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖)\n x}",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\n⊢ ball x₀ ε ⊆\n {x |\n (fun x =>\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖)\n x}",
"tactic": "intro x x_in"
},
{
"state_after": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nx : H\nx_in : x ∈ ball x₀ ε\n⊢ x ∈\n {x |\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖}",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nx : H\nx_in : x ∈ ball x₀ ε\n⊢ x ∈\n {x |\n (fun x =>\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖)\n x}",
"tactic": "simp only"
},
{
"state_after": "case hf\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nx : H\nx_in : x ∈ ball x₀ ε\n⊢ Integrable fun a => F x a\n\ncase hg\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nx : H\nx_in : x ∈ ball x₀ ε\n⊢ Integrable fun a => F x₀ a\n\ncase hf\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nx : H\nx_in : x ∈ ball x₀ ε\n⊢ Integrable fun a => F x a - F x₀ a\n\ncase hg\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nx : H\nx_in : x ∈ ball x₀ ε\n⊢ Integrable fun a => ↑(F' a) (x - x₀)",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nx : H\nx_in : x ∈ ball x₀ ε\n⊢ x ∈\n {x |\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖}",
"tactic": "rw [Set.mem_setOf_eq, ← norm_smul_of_nonneg (nneg _), integral_smul, integral_sub, integral_sub,\n ← ContinuousLinearMap.integral_apply hF'_int]"
},
{
"state_after": "no goals",
"state_before": "case hf\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nx : H\nx_in : x ∈ ball x₀ ε\n⊢ Integrable fun a => F x a\n\ncase hg\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nx : H\nx_in : x ∈ ball x₀ ε\n⊢ Integrable fun a => F x₀ a\n\ncase hf\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nx : H\nx_in : x ∈ ball x₀ ε\n⊢ Integrable fun a => F x a - F x₀ a\n\ncase hg\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nx : H\nx_in : x ∈ ball x₀ ε\n⊢ Integrable fun a => ↑(F' a) (x - x₀)",
"tactic": "exacts [hF_int' x x_in, hF_int, (hF_int' x x_in).sub hF_int,\n hF'_int.apply_continuousLinearMap _]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\n⊢ (∫ (a : α), ‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - ↑(F' a) (x₀ - x₀)) ∂μ) = 0",
"tactic": "simp"
},
{
"state_after": "case h\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\na✝ : H\nx_in : a✝ ∈ ball x₀ ε\n⊢ AEStronglyMeasurable (fun a => ‖a✝ - x₀‖⁻¹ • (F a✝ a - F x₀ a - ↑(F' a) (a✝ - x₀))) μ",
"state_before": "case hF_meas\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\n⊢ ∀ᶠ (n : H) in 𝓝 x₀, AEStronglyMeasurable (fun a => ‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))) μ",
"tactic": "filter_upwards [h_ball] with _ x_in"
},
{
"state_after": "case h.hf\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\na✝ : H\nx_in : a✝ ∈ ball x₀ ε\n⊢ AEStronglyMeasurable (fun a => F a✝ a - F x₀ a - ↑(F' a) (a✝ - x₀)) μ",
"state_before": "case h\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\na✝ : H\nx_in : a✝ ∈ ball x₀ ε\n⊢ AEStronglyMeasurable (fun a => ‖a✝ - x₀‖⁻¹ • (F a✝ a - F x₀ a - ↑(F' a) (a✝ - x₀))) μ",
"tactic": "apply AEStronglyMeasurable.const_smul"
},
{
"state_after": "no goals",
"state_before": "case h.hf\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\na✝ : H\nx_in : a✝ ∈ ball x₀ ε\n⊢ AEStronglyMeasurable (fun a => F a✝ a - F x₀ a - ↑(F' a) (a✝ - x₀)) μ",
"tactic": "exact ((hF_meas _ x_in).sub (hF_meas _ x₀_in)).sub (hF'_meas.apply_continuousLinearMap _)"
},
{
"state_after": "case h_bound\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\n⊢ x ∈ {x | (fun n => ∀ᵐ (a : α) ∂μ, ‖‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))‖ ≤ ?bound a) x}",
"state_before": "case h_bound\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\n⊢ ∀ᶠ (n : H) in 𝓝 x₀, ∀ᵐ (a : α) ∂μ, ‖‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))‖ ≤ ?bound a",
"tactic": "refine mem_of_superset h_ball fun x hx ↦ ?_"
},
{
"state_after": "case h_bound\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\n⊢ ∀ (x_1 : α),\n (HasFDerivAt (fun x => F x x_1) (F' x_1) x₀ ∧ ∀ (x : H), x ∈ ball x₀ ε → ‖F x x_1 - F x₀ x_1‖ ≤ b x_1 * ‖x - x₀‖) →\n ‖‖x - x₀‖⁻¹ • (F x x_1 - F x₀ x_1 - ↑(F' x_1) (x - x₀))‖ ≤ ?bound x_1\n\ncase bound\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\n⊢ α → ℝ",
"state_before": "case h_bound\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\n⊢ x ∈ {x | (fun n => ∀ᵐ (a : α) ∂μ, ‖‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))‖ ≤ ?bound a) x}",
"tactic": "apply (h_diff.and h_lipsch).mono"
},
{
"state_after": "case h_bound.intro\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\na : α\nha_bound : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖ ≤ ?bound a\n\ncase bound\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\n⊢ α → ℝ",
"state_before": "case h_bound\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\n⊢ ∀ (x_1 : α),\n (HasFDerivAt (fun x => F x x_1) (F' x_1) x₀ ∧ ∀ (x : H), x ∈ ball x₀ ε → ‖F x x_1 - F x₀ x_1‖ ≤ b x_1 * ‖x - x₀‖) →\n ‖‖x - x₀‖⁻¹ • (F x x_1 - F x₀ x_1 - ↑(F' x_1) (x - x₀))‖ ≤ ?bound x_1\n\ncase bound\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\n⊢ α → ℝ",
"tactic": "rintro a ⟨-, ha_bound⟩"
},
{
"state_after": "case h_bound.intro\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\na : α\nha_bound : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖ ≤ b a + ‖F' a‖",
"state_before": "case h_bound.intro\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\na : α\nha_bound : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖ ≤ ?bound a\n\ncase bound\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\n⊢ α → ℝ",
"tactic": "show ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))‖ ≤ b a + ‖F' a‖"
},
{
"state_after": "case h_bound.intro\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\na : α\nha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖ ≤ b a + ‖F' a‖",
"state_before": "case h_bound.intro\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\na : α\nha_bound : ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖ ≤ b a + ‖F' a‖",
"tactic": "replace ha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖ := ha_bound x hx"
},
{
"state_after": "case h_bound.intro\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\na : α\nha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ ‖x - x₀‖⁻¹ * (b a * ‖x - x₀‖) + ‖x - x₀‖⁻¹ * (‖F' a‖ * ‖x - x₀‖) ≤ b a + ‖F' a‖",
"state_before": "case h_bound.intro\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\na : α\nha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖ ≤ b a + ‖F' a‖",
"tactic": "calc\n ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))‖ =\n ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a) - ‖x - x₀‖⁻¹ • F' a (x - x₀)‖ :=\n by rw [smul_sub]\n _ ≤ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a)‖ + ‖‖x - x₀‖⁻¹ • F' a (x - x₀)‖ := (norm_sub_le _ _)\n _ = ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a‖ + ‖x - x₀‖⁻¹ * ‖F' a (x - x₀)‖ := by\n rw [norm_smul_of_nonneg, norm_smul_of_nonneg] <;> exact nneg _\n _ ≤ ‖x - x₀‖⁻¹ * (b a * ‖x - x₀‖) + ‖x - x₀‖⁻¹ * (‖F' a‖ * ‖x - x₀‖) := by\n gcongr; exact (F' a).le_op_norm _\n _ ≤ b a + ‖F' a‖ := ?_"
},
{
"state_after": "case h_bound.intro\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\na : α\nha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ abs (bound a) * ‖x - x₀‖ / ‖x - x₀‖ + ‖F' a‖ * ‖x - x₀‖ / ‖x - x₀‖ ≤ abs (bound a) + ‖F' a‖",
"state_before": "case h_bound.intro\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\na : α\nha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ ‖x - x₀‖⁻¹ * (b a * ‖x - x₀‖) + ‖x - x₀‖⁻¹ * (‖F' a‖ * ‖x - x₀‖) ≤ b a + ‖F' a‖",
"tactic": "simp only [← div_eq_inv_mul]"
},
{
"state_after": "no goals",
"state_before": "case h_bound.intro\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\na : α\nha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ abs (bound a) * ‖x - x₀‖ / ‖x - x₀‖ + ‖F' a‖ * ‖x - x₀‖ / ‖x - x₀‖ ≤ abs (bound a) + ‖F' a‖",
"tactic": "apply_rules [add_le_add, div_le_of_nonneg_of_le_mul] <;> first | rfl | positivity"
},
{
"state_after": "no goals",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\na : α\nha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖ = ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a) - ‖x - x₀‖⁻¹ • ↑(F' a) (x - x₀)‖",
"tactic": "rw [smul_sub]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\na : α\nha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a)‖ + ‖‖x - x₀‖⁻¹ • ↑(F' a) (x - x₀)‖ =\n ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a‖ + ‖x - x₀‖⁻¹ * ‖↑(F' a) (x - x₀)‖",
"tactic": "rw [norm_smul_of_nonneg, norm_smul_of_nonneg] <;> exact nneg _"
},
{
"state_after": "case h₂.h\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\na : α\nha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ ‖↑(F' a) (x - x₀)‖ ≤ ‖F' a‖ * ‖x - x₀‖",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\na : α\nha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a‖ + ‖x - x₀‖⁻¹ * ‖↑(F' a) (x - x₀)‖ ≤\n ‖x - x₀‖⁻¹ * (b a * ‖x - x₀‖) + ‖x - x₀‖⁻¹ * (‖F' a‖ * ‖x - x₀‖)",
"tactic": "gcongr"
},
{
"state_after": "no goals",
"state_before": "case h₂.h\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\na : α\nha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ ‖↑(F' a) (x - x₀)‖ ≤ ‖F' a‖ * ‖x - x₀‖",
"tactic": "exact (F' a).le_op_norm _"
},
{
"state_after": "no goals",
"state_before": "case h_bound.intro.h₂.h\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\na : α\nha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ ‖F' a‖ * ‖x - x₀‖ ≤ ‖F' a‖ * ‖x - x₀‖",
"tactic": "rfl"
},
{
"state_after": "no goals",
"state_before": "case h_bound.intro.h₂.hc\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\nx : H\nhx : x ∈ ball x₀ ε\na : α\nha_bound : ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\n⊢ 0 ≤ ‖F' a‖",
"tactic": "positivity"
},
{
"state_after": "no goals",
"state_before": "case bound_integrable\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\n⊢ Integrable fun a => b a + ‖F' a‖",
"tactic": "exact b_int.add hF'_int.norm"
},
{
"state_after": "case h_lim\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\n⊢ ∀ (x : α),\n HasFDerivAt (fun x_1 => F x_1 x) (F' x) x₀ →\n Tendsto (fun n => ‖n - x₀‖⁻¹ • (F n x - F x₀ x - ↑(F' x) (n - x₀))) (𝓝 x₀)\n (𝓝 (‖x₀ - x₀‖⁻¹ • (F x₀ x - F x₀ x - ↑(F' x) (x₀ - x₀))))",
"state_before": "case h_lim\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\n⊢ ∀ᵐ (a : α) ∂μ,\n Tendsto (fun n => ‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))) (𝓝 x₀)\n (𝓝 (‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - ↑(F' a) (x₀ - x₀))))",
"tactic": "apply h_diff.mono"
},
{
"state_after": "case h_lim\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\na : α\nha : HasFDerivAt (fun x => F x a) (F' a) x₀\n⊢ Tendsto (fun n => ‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))) (𝓝 x₀)\n (𝓝 (‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - ↑(F' a) (x₀ - x₀))))",
"state_before": "case h_lim\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\n⊢ ∀ (x : α),\n HasFDerivAt (fun x_1 => F x_1 x) (F' x) x₀ →\n Tendsto (fun n => ‖n - x₀‖⁻¹ • (F n x - F x₀ x - ↑(F' x) (n - x₀))) (𝓝 x₀)\n (𝓝 (‖x₀ - x₀‖⁻¹ • (F x₀ x - F x₀ x - ↑(F' x) (x₀ - x₀))))",
"tactic": "intro a ha"
},
{
"state_after": "case h_lim\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\na : α\nha : HasFDerivAt (fun x => F x a) (F' a) x₀\n⊢ Tendsto (fun x => ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))) (𝓝 x₀) (𝓝 0)",
"state_before": "case h_lim\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\na : α\nha : HasFDerivAt (fun x => F x a) (F' a) x₀\n⊢ Tendsto (fun n => ‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))) (𝓝 x₀)\n (𝓝 (‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - ↑(F' a) (x₀ - x₀))))",
"tactic": "suffices Tendsto (fun x => ‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))) (𝓝 x₀) (𝓝 0) by simpa"
},
{
"state_after": "case h_lim\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\na : α\nha : HasFDerivAt (fun x => F x a) (F' a) x₀\n⊢ Tendsto (fun e => ‖‖e - x₀‖⁻¹ • (F e a - F x₀ a - ↑(F' a) (e - x₀))‖) (𝓝 x₀) (𝓝 0)",
"state_before": "case h_lim\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\na : α\nha : HasFDerivAt (fun x => F x a) (F' a) x₀\n⊢ Tendsto (fun x => ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))) (𝓝 x₀) (𝓝 0)",
"tactic": "rw [tendsto_zero_iff_norm_tendsto_zero]"
},
{
"state_after": "case h_lim\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis✝ :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\na : α\nha : HasFDerivAt (fun x => F x a) (F' a) x₀\nthis :\n (fun x => ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a - ↑(F' a) (x - x₀)‖) = fun x =>\n ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖\n⊢ Tendsto (fun e => ‖‖e - x₀‖⁻¹ • (F e a - F x₀ a - ↑(F' a) (e - x₀))‖) (𝓝 x₀) (𝓝 0)",
"state_before": "case h_lim\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\na : α\nha : HasFDerivAt (fun x => F x a) (F' a) x₀\n⊢ Tendsto (fun e => ‖‖e - x₀‖⁻¹ • (F e a - F x₀ a - ↑(F' a) (e - x₀))‖) (𝓝 x₀) (𝓝 0)",
"tactic": "have : (fun x => ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a - F' a (x - x₀)‖) = fun x =>\n ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - F' a (x - x₀))‖ := by\n ext x\n rw [norm_smul_of_nonneg (nneg _)]"
},
{
"state_after": "no goals",
"state_before": "case h_lim\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis✝ :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\na : α\nha : HasFDerivAt (fun x => F x a) (F' a) x₀\nthis :\n (fun x => ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a - ↑(F' a) (x - x₀)‖) = fun x =>\n ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖\n⊢ Tendsto (fun e => ‖‖e - x₀‖⁻¹ • (F e a - F x₀ a - ↑(F' a) (e - x₀))‖) (𝓝 x₀) (𝓝 0)",
"tactic": "rwa [hasFDerivAt_iff_tendsto, this] at ha"
},
{
"state_after": "no goals",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis✝ :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\na : α\nha : HasFDerivAt (fun x => F x a) (F' a) x₀\nthis : Tendsto (fun x => ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))) (𝓝 x₀) (𝓝 0)\n⊢ Tendsto (fun n => ‖n - x₀‖⁻¹ • (F n a - F x₀ a - ↑(F' a) (n - x₀))) (𝓝 x₀)\n (𝓝 (‖x₀ - x₀‖⁻¹ • (F x₀ a - F x₀ a - ↑(F' a) (x₀ - x₀))))",
"tactic": "simpa"
},
{
"state_after": "case h\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\na : α\nha : HasFDerivAt (fun x => F x a) (F' a) x₀\nx : H\n⊢ ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a - ↑(F' a) (x - x₀)‖ = ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖",
"state_before": "α : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\na : α\nha : HasFDerivAt (fun x => F x a) (F' a) x₀\n⊢ (fun x => ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a - ↑(F' a) (x - x₀)‖) = fun x =>\n ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖",
"tactic": "ext x"
},
{
"state_after": "no goals",
"state_before": "case h\nα : Type u_4\ninst✝⁷ : MeasurableSpace α\nμ : MeasureTheory.Measure α\n𝕜 : Type u_1\ninst✝⁶ : IsROrC 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : CompleteSpace E\nH : Type u_2\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nF : H → α → E\nF' : α → H →L[𝕜] E\nx₀ : H\nbound : α → ℝ\nε : ℝ\nε_pos : 0 < ε\nhF_meas : ∀ (x : H), x ∈ ball x₀ ε → AEStronglyMeasurable (F x) μ\nhF_int : Integrable (F x₀)\nhF'_meas : AEStronglyMeasurable F' μ\nbound_integrable : Integrable bound\nh_diff : ∀ᵐ (a : α) ∂μ, HasFDerivAt (fun x => F x a) (F' a) x₀\nx₀_in : x₀ ∈ ball x₀ ε\nnneg : ∀ (x : H), 0 ≤ ‖x - x₀‖⁻¹\nb : α → ℝ := fun a => abs (bound a)\nb_int : Integrable b\nb_nonneg : ∀ (a : α), 0 ≤ b a\nh_lipsch : ∀ᵐ (a : α) ∂μ, ∀ (x : H), x ∈ ball x₀ ε → ‖F x a - F x₀ a‖ ≤ b a * ‖x - x₀‖\nhF_int' : ∀ (x : H), x ∈ ball x₀ ε → Integrable (F x)\nhF'_int : Integrable F'\nh_ball : ball x₀ ε ∈ 𝓝 x₀\nthis :\n ∀ᶠ (x : H) in 𝓝 x₀,\n ‖x - x₀‖⁻¹ * ‖((∫ (a : α), F x a ∂μ) - ∫ (a : α), F x₀ a ∂μ) - ↑(∫ (a : α), F' a ∂μ) (x - x₀)‖ =\n ‖∫ (a : α), ‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀)) ∂μ‖\na : α\nha : HasFDerivAt (fun x => F x a) (F' a) x₀\nx : H\n⊢ ‖x - x₀‖⁻¹ * ‖F x a - F x₀ a - ↑(F' a) (x - x₀)‖ = ‖‖x - x₀‖⁻¹ • (F x a - F x₀ a - ↑(F' a) (x - x₀))‖",
"tactic": "rw [norm_smul_of_nonneg (nneg _)]"
}
] |
[
147,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
74,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/Pullbacks.lean
|
CategoryTheory.Limits.PullbackCone.mk_fst
|
[] |
[
601,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
600,
1
] |
Mathlib/Analysis/Calculus/FormalMultilinearSeries.lean
|
FormalMultilinearSeries.ext
|
[] |
[
83,
11
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
82,
11
] |
Mathlib/Order/UpperLower/Hom.lean
|
UpperSet.coe_iciSupHom
|
[] |
[
39,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
38,
1
] |
Mathlib/Data/List/MinMax.lean
|
List.le_minimum_of_mem'
|
[] |
[
367,
35
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
366,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.subset_closure
|
[] |
[
807,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
807,
1
] |
Mathlib/Topology/Semicontinuous.lean
|
UpperSemicontinuousOn.add'
|
[] |
[
919,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
915,
1
] |
Mathlib/Algebra/Module/Submodule/Pointwise.lean
|
Submodule.coe_set_neg
|
[] |
[
68,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
67,
1
] |
src/lean/Init/Data/Nat/Div.lean
|
Nat.div_rec_lemma
|
[] |
[
13,
65
] |
d5348dfac847a56a4595fb6230fd0708dcb4e7e9
|
https://github.com/leanprover/lean4
|
[
12,
1
] |
Mathlib/Algebra/Order/Floor.lean
|
Nat.lt_ceil
|
[] |
[
283,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
282,
1
] |
Mathlib/Order/Filter/Ultrafilter.lean
|
Ultrafilter.map_id
|
[] |
[
229,
23
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
8
] |
Mathlib/Data/Dfinsupp/Basic.lean
|
Dfinsupp.subtypeDomain_smul
|
[] |
[
517,
28
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
514,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.sup_mono_fun
|
[] |
[
151,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Data/Complex/Exponential.lean
|
Complex.abs_exp_ofReal
|
[
{
"state_after": "x : ℝ\n⊢ ↑abs ↑(Real.exp x) = Real.exp x",
"state_before": "x : ℝ\n⊢ ↑abs (exp ↑x) = Real.exp x",
"tactic": "rw [← ofReal_exp]"
},
{
"state_after": "no goals",
"state_before": "x : ℝ\n⊢ ↑abs ↑(Real.exp x) = Real.exp x",
"tactic": "exact abs_of_nonneg (le_of_lt (Real.exp_pos _))"
}
] |
[
2032,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2030,
1
] |
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