file_path
stringlengths 11
79
| full_name
stringlengths 2
100
| traced_tactics
list | end
list | commit
stringclasses 4
values | url
stringclasses 4
values | start
list |
|---|---|---|---|---|---|---|
Mathlib/Analysis/NormedSpace/MStructure.lean
|
IsLprojection.mul
|
[
{
"state_after": "X : Type u_2\ninst✝³ : NormedAddCommGroup X\nM : Type u_1\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\n⊢ ∀ (x : X), ‖x‖ = ‖(P * Q) • x‖ + ‖(1 - P * Q) • x‖",
"state_before": "X : Type u_2\ninst✝³ : NormedAddCommGroup X\nM : Type u_1\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\n⊢ IsLprojection X (P * Q)",
"tactic": "refine' ⟨IsIdempotentElem.mul_of_commute (h₁.commute h₂) h₁.proj h₂.proj, _⟩"
},
{
"state_after": "X : Type u_2\ninst✝³ : NormedAddCommGroup X\nM : Type u_1\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\nx : X\n⊢ ‖x‖ = ‖(P * Q) • x‖ + ‖(1 - P * Q) • x‖",
"state_before": "X : Type u_2\ninst✝³ : NormedAddCommGroup X\nM : Type u_1\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\n⊢ ∀ (x : X), ‖x‖ = ‖(P * Q) • x‖ + ‖(1 - P * Q) • x‖",
"tactic": "intro x"
},
{
"state_after": "case refine'_1\nX : Type u_2\ninst✝³ : NormedAddCommGroup X\nM : Type u_1\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\nx : X\n⊢ ‖x‖ ≤ ‖(P * Q) • x‖ + ‖(1 - P * Q) • x‖\n\ncase refine'_2\nX : Type u_2\ninst✝³ : NormedAddCommGroup X\nM : Type u_1\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\nx : X\n⊢ ‖(P * Q) • x‖ + ‖(1 - P * Q) • x‖ ≤ ‖x‖",
"state_before": "X : Type u_2\ninst✝³ : NormedAddCommGroup X\nM : Type u_1\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\nx : X\n⊢ ‖x‖ = ‖(P * Q) • x‖ + ‖(1 - P * Q) • x‖",
"tactic": "refine' le_antisymm _ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nX : Type u_2\ninst✝³ : NormedAddCommGroup X\nM : Type u_1\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\nx : X\n⊢ ‖x‖ ≤ ‖(P * Q) • x‖ + ‖(1 - P * Q) • x‖",
"tactic": "calc\n ‖x‖ = ‖(P * Q) • x + (x - (P * Q) • x)‖ := by rw [add_sub_cancel'_right ((P * Q) • x) x]\n _ ≤ ‖(P * Q) • x‖ + ‖x - (P * Q) • x‖ := by apply norm_add_le\n _ = ‖(P * Q) • x‖ + ‖(1 - P * Q) • x‖ := by rw [sub_smul, one_smul]"
},
{
"state_after": "no goals",
"state_before": "X : Type u_2\ninst✝³ : NormedAddCommGroup X\nM : Type u_1\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\nx : X\n⊢ ‖x‖ = ‖(P * Q) • x + (x - (P * Q) • x)‖",
"tactic": "rw [add_sub_cancel'_right ((P * Q) • x) x]"
},
{
"state_after": "no goals",
"state_before": "X : Type u_2\ninst✝³ : NormedAddCommGroup X\nM : Type u_1\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\nx : X\n⊢ ‖(P * Q) • x + (x - (P * Q) • x)‖ ≤ ‖(P * Q) • x‖ + ‖x - (P * Q) • x‖",
"tactic": "apply norm_add_le"
},
{
"state_after": "no goals",
"state_before": "X : Type u_2\ninst✝³ : NormedAddCommGroup X\nM : Type u_1\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\nx : X\n⊢ ‖(P * Q) • x‖ + ‖x - (P * Q) • x‖ = ‖(P * Q) • x‖ + ‖(1 - P * Q) • x‖",
"tactic": "rw [sub_smul, one_smul]"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nX : Type u_2\ninst✝³ : NormedAddCommGroup X\nM : Type u_1\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\nx : X\n⊢ ‖(P * Q) • x‖ + ‖(1 - P * Q) • x‖ ≤ ‖x‖",
"tactic": "calc\n ‖x‖ = ‖P • Q • x‖ + (‖Q • x - P • Q • x‖ + ‖x - Q • x‖) := by\n rw [h₂.Lnorm x, h₁.Lnorm (Q • x), sub_smul, one_smul, sub_smul, one_smul, add_assoc]\n _ ≥ ‖P • Q • x‖ + ‖Q • x - P • Q • x + (x - Q • x)‖ :=\n ((add_le_add_iff_left ‖P • Q • x‖).mpr (norm_add_le (Q • x - P • Q • x) (x - Q • x)))\n _ = ‖(P * Q) • x‖ + ‖(1 - P * Q) • x‖ := by\n rw [sub_add_sub_cancel', sub_smul, one_smul, mul_smul]"
},
{
"state_after": "no goals",
"state_before": "X : Type u_2\ninst✝³ : NormedAddCommGroup X\nM : Type u_1\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\nx : X\n⊢ ‖x‖ = ‖P • Q • x‖ + (‖Q • x - P • Q • x‖ + ‖x - Q • x‖)",
"tactic": "rw [h₂.Lnorm x, h₁.Lnorm (Q • x), sub_smul, one_smul, sub_smul, one_smul, add_assoc]"
},
{
"state_after": "no goals",
"state_before": "X : Type u_2\ninst✝³ : NormedAddCommGroup X\nM : Type u_1\ninst✝² : Ring M\ninst✝¹ : Module M X\ninst✝ : FaithfulSMul M X\nP Q : M\nh₁ : IsLprojection X P\nh₂ : IsLprojection X Q\nx : X\n⊢ ‖P • Q • x‖ + ‖Q • x - P • Q • x + (x - Q • x)‖ = ‖(P * Q) • x‖ + ‖(1 - P * Q) • x‖",
"tactic": "rw [sub_add_sub_cancel', sub_smul, one_smul, mul_smul]"
}
] |
[
165,
63
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
150,
1
] |
Mathlib/Data/Nat/Totient.lean
|
Nat.totient_prime
|
[
{
"state_after": "no goals",
"state_before": "p : ℕ\nhp : Prime p\n⊢ φ p = p - 1",
"tactic": "rw [← pow_one p, totient_prime_pow hp] <;> simp"
}
] |
[
229,
50
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
228,
1
] |
Mathlib/CategoryTheory/Generator.lean
|
CategoryTheory.IsCospearator.isCodetector
|
[] |
[
453,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
452,
1
] |
Mathlib/MeasureTheory/Covering/Differentiation.lean
|
VitaliFamily.withDensity_le_mul
|
[
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\n⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\n⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s",
"tactic": "have t_ne_zero' : t ≠ 0 := (zero_lt_one.trans ht).ne'"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\n⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\n⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s",
"tactic": "have t_ne_zero : (t : ℝ≥0∞) ≠ 0 := by simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\n⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\n⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s",
"tactic": "let ν := μ.withDensity (v.limRatioMeas hρ)"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\n⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\n⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s",
"tactic": "let f := v.limRatioMeas hρ"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\n⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\n⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s",
"tactic": "have f_meas : Measurable f := v.limRatioMeas_measurable hρ"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\n⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\n⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s",
"tactic": "have A : ν (s ∩ f ⁻¹' {0}) ≤ ((t : ℝ≥0∞) ^ 2 • ρ) (s ∩ f ⁻¹' {0}) := by\n apply le_trans _ (zero_le _)\n have M : MeasurableSet (s ∩ f ⁻¹' {0}) := hs.inter (f_meas (measurableSet_singleton _))\n simp only [nonpos_iff_eq_zero, M, withDensity_apply, lintegral_eq_zero_iff f_meas]\n apply (ae_restrict_iff' M).2\n exact eventually_of_forall fun x hx => hx.2"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\n⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\n⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s",
"tactic": "have B : ν (s ∩ f ⁻¹' {∞}) ≤ ((t : ℝ≥0∞) ^ 2 • ρ) (s ∩ f ⁻¹' {∞}) := by\n apply le_trans (le_of_eq _) (zero_le _)\n apply withDensity_absolutelyContinuous μ _\n rw [← nonpos_iff_eq_zero]\n exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top hρ).le"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nC : ∀ (n : ℤ), ↑↑ν (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))\n⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\n⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s",
"tactic": "have C :\n ∀ n : ℤ,\n ν (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) ≤\n ((t : ℝ≥0∞) ^ 2 • ρ) (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) := by\n intro n\n let I := Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))\n have M : MeasurableSet (s ∩ f ⁻¹' I) := hs.inter (f_meas measurableSet_Ico)\n simp only [M, withDensity_apply, coe_nnreal_smul_apply]\n calc\n (∫⁻ x in s ∩ f ⁻¹' I, f x ∂μ) ≤ ∫⁻ x in s ∩ f ⁻¹' I, (t : ℝ≥0∞) ^ (n + 1) ∂μ :=\n lintegral_mono_ae ((ae_restrict_iff' M).2 (eventually_of_forall fun x hx => hx.2.2.le))\n _ = (t : ℝ≥0∞) ^ (n + 1) * μ (s ∩ f ⁻¹' I) := by\n simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]\n _ = (t : ℝ≥0∞) ^ (2 : ℤ) * ((t : ℝ≥0∞) ^ (n - 1) * μ (s ∩ f ⁻¹' I)) := by\n rw [← mul_assoc, ← ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top]\n congr 2\n abel\n _ ≤ (t : ℝ≥0∞) ^ 2 * ρ (s ∩ f ⁻¹' I) := by\n refine' mul_le_mul_left' _ _\n rw [← ENNReal.coe_zpow (zero_lt_one.trans ht).ne']\n apply v.mul_measure_le_of_subset_lt_limRatioMeas hρ\n intro x hx\n apply lt_of_lt_of_le _ hx.2.1\n rw [← ENNReal.coe_zpow (zero_lt_one.trans ht).ne', ENNReal.coe_lt_coe, sub_eq_add_neg,\n zpow_add₀ t_ne_zero']\n conv_rhs => rw [← mul_one (t ^ n)]\n refine' mul_lt_mul' le_rfl _ (zero_le _) (NNReal.zpow_pos t_ne_zero' _)\n rw [zpow_neg_one]\n exact inv_lt_one ht"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nC : ∀ (n : ℤ), ↑↑ν (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))\n⊢ ↑↑(withDensity μ (limRatioMeas v hρ)) s ≤ ↑t ^ 2 * ↑↑ρ s",
"tactic": "calc\n ν s =\n ν (s ∩ f ⁻¹' {0}) + ν (s ∩ f ⁻¹' {∞}) +\n ∑' n : ℤ, ν (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) :=\n measure_eq_measure_preimage_add_measure_tsum_Ico_zpow ν f_meas hs ht\n _ ≤\n ((t : ℝ≥0∞) ^ 2 • ρ) (s ∩ f ⁻¹' {0}) + ((t : ℝ≥0∞) ^ 2 • ρ) (s ∩ f ⁻¹' {∞}) +\n ∑' n : ℤ, ((t : ℝ≥0∞) ^ 2 • ρ) (s ∩ f ⁻¹' Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))) :=\n (add_le_add (add_le_add A B) (ENNReal.tsum_le_tsum C))\n _ = ((t : ℝ≥0∞) ^ 2 • ρ) s :=\n (measure_eq_measure_preimage_add_measure_tsum_Ico_zpow ((t : ℝ≥0∞) ^ 2 • ρ) f_meas hs ht).symm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\n⊢ ↑t ≠ 0",
"tactic": "simpa only [ENNReal.coe_eq_zero, Ne.def] using t_ne_zero'"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\n⊢ ↑↑ν (s ∩ f ⁻¹' {0}) ≤ 0",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\n⊢ ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})",
"tactic": "apply le_trans _ (zero_le _)"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nM : MeasurableSet (s ∩ f ⁻¹' {0})\n⊢ ↑↑ν (s ∩ f ⁻¹' {0}) ≤ 0",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\n⊢ ↑↑ν (s ∩ f ⁻¹' {0}) ≤ 0",
"tactic": "have M : MeasurableSet (s ∩ f ⁻¹' {0}) := hs.inter (f_meas (measurableSet_singleton _))"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nM : MeasurableSet (s ∩ f ⁻¹' {0})\n⊢ limRatioMeas v hρ =ᶠ[ae (Measure.restrict μ (s ∩ limRatioMeas v hρ ⁻¹' {0}))] 0",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nM : MeasurableSet (s ∩ f ⁻¹' {0})\n⊢ ↑↑ν (s ∩ f ⁻¹' {0}) ≤ 0",
"tactic": "simp only [nonpos_iff_eq_zero, M, withDensity_apply, lintegral_eq_zero_iff f_meas]"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nM : MeasurableSet (s ∩ f ⁻¹' {0})\n⊢ ∀ᵐ (x : α) ∂μ, x ∈ s ∩ f ⁻¹' {0} → limRatioMeas v hρ x = OfNat.ofNat 0 x",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nM : MeasurableSet (s ∩ f ⁻¹' {0})\n⊢ limRatioMeas v hρ =ᶠ[ae (Measure.restrict μ (s ∩ limRatioMeas v hρ ⁻¹' {0}))] 0",
"tactic": "apply (ae_restrict_iff' M).2"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nM : MeasurableSet (s ∩ f ⁻¹' {0})\n⊢ ∀ᵐ (x : α) ∂μ, x ∈ s ∩ f ⁻¹' {0} → limRatioMeas v hρ x = OfNat.ofNat 0 x",
"tactic": "exact eventually_of_forall fun x hx => hx.2"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\n⊢ ↑↑ν (s ∩ f ⁻¹' {⊤}) = 0",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\n⊢ ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})",
"tactic": "apply le_trans (le_of_eq _) (zero_le _)"
},
{
"state_after": "case a\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\n⊢ ↑↑μ (s ∩ f ⁻¹' {⊤}) = 0",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\n⊢ ↑↑ν (s ∩ f ⁻¹' {⊤}) = 0",
"tactic": "apply withDensity_absolutelyContinuous μ _"
},
{
"state_after": "case a\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\n⊢ ↑↑μ (s ∩ f ⁻¹' {⊤}) ≤ 0",
"state_before": "case a\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\n⊢ ↑↑μ (s ∩ f ⁻¹' {⊤}) = 0",
"tactic": "rw [← nonpos_iff_eq_zero]"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\n⊢ ↑↑μ (s ∩ f ⁻¹' {⊤}) ≤ 0",
"tactic": "exact (measure_mono (inter_subset_right _ _)).trans (v.measure_limRatioMeas_top hρ).le"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\n⊢ ↑↑ν (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\n⊢ ∀ (n : ℤ), ↑↑ν (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))",
"tactic": "intro n"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\n⊢ ↑↑ν (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\n⊢ ↑↑ν (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))",
"tactic": "let I := Ico ((t : ℝ≥0∞) ^ n) ((t : ℝ≥0∞) ^ (n + 1))"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ ↑↑ν (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\n⊢ ↑↑ν (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))",
"tactic": "have M : MeasurableSet (s ∩ f ⁻¹' I) := hs.inter (f_meas measurableSet_Ico)"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ (∫⁻ (a : α) in s ∩ limRatioMeas v hρ ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)), limRatioMeas v hρ a ∂μ) ≤\n ↑↑(↑t ^ 2 • ρ) (s ∩ limRatioMeas v hρ ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ ↑↑ν (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1))) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))",
"tactic": "simp only [M, withDensity_apply, coe_nnreal_smul_apply]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ (∫⁻ (a : α) in s ∩ limRatioMeas v hρ ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)), limRatioMeas v hρ a ∂μ) ≤\n ↑↑(↑t ^ 2 • ρ) (s ∩ limRatioMeas v hρ ⁻¹' Ico (↑t ^ n) (↑t ^ (n + 1)))",
"tactic": "calc\n (∫⁻ x in s ∩ f ⁻¹' I, f x ∂μ) ≤ ∫⁻ x in s ∩ f ⁻¹' I, (t : ℝ≥0∞) ^ (n + 1) ∂μ :=\n lintegral_mono_ae ((ae_restrict_iff' M).2 (eventually_of_forall fun x hx => hx.2.2.le))\n _ = (t : ℝ≥0∞) ^ (n + 1) * μ (s ∩ f ⁻¹' I) := by\n simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]\n _ = (t : ℝ≥0∞) ^ (2 : ℤ) * ((t : ℝ≥0∞) ^ (n - 1) * μ (s ∩ f ⁻¹' I)) := by\n rw [← mul_assoc, ← ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top]\n congr 2\n abel\n _ ≤ (t : ℝ≥0∞) ^ 2 * ρ (s ∩ f ⁻¹' I) := by\n refine' mul_le_mul_left' _ _\n rw [← ENNReal.coe_zpow (zero_lt_one.trans ht).ne']\n apply v.mul_measure_le_of_subset_lt_limRatioMeas hρ\n intro x hx\n apply lt_of_lt_of_le _ hx.2.1\n rw [← ENNReal.coe_zpow (zero_lt_one.trans ht).ne', ENNReal.coe_lt_coe, sub_eq_add_neg,\n zpow_add₀ t_ne_zero']\n conv_rhs => rw [← mul_one (t ^ n)]\n refine' mul_lt_mul' le_rfl _ (zero_le _) (NNReal.zpow_pos t_ne_zero' _)\n rw [zpow_neg_one]\n exact inv_lt_one ht"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ (∫⁻ (x : α) in s ∩ f ⁻¹' I, ↑t ^ (n + 1) ∂μ) = ↑t ^ (n + 1) * ↑↑μ (s ∩ f ⁻¹' I)",
"tactic": "simp only [lintegral_const, MeasurableSet.univ, Measure.restrict_apply, univ_inter]"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ ↑t ^ (n + 1) * ↑↑μ (s ∩ f ⁻¹' I) = ↑t ^ (2 + (n - 1)) * ↑↑μ (s ∩ f ⁻¹' I)",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ ↑t ^ (n + 1) * ↑↑μ (s ∩ f ⁻¹' I) = ↑t ^ 2 * (↑t ^ (n - 1) * ↑↑μ (s ∩ f ⁻¹' I))",
"tactic": "rw [← mul_assoc, ← ENNReal.zpow_add t_ne_zero ENNReal.coe_ne_top]"
},
{
"state_after": "case e_a.e_a\nα : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ n + 1 = 2 + (n - 1)",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ ↑t ^ (n + 1) * ↑↑μ (s ∩ f ⁻¹' I) = ↑t ^ (2 + (n - 1)) * ↑↑μ (s ∩ f ⁻¹' I)",
"tactic": "congr 2"
},
{
"state_after": "no goals",
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"tactic": "abel"
},
{
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"tactic": "refine' mul_le_mul_left' _ _"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ ↑(t ^ (n - 1)) * ↑↑μ (s ∩ f ⁻¹' I) ≤ ↑↑ρ (s ∩ f ⁻¹' I)",
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"tactic": "rw [← ENNReal.coe_zpow (zero_lt_one.trans ht).ne']"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ s ∩ f ⁻¹' I ⊆ {x | ↑(t ^ (n - 1)) < limRatioMeas v hρ x}",
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"tactic": "apply v.mul_measure_le_of_subset_lt_limRatioMeas hρ"
},
{
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"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\n⊢ s ∩ f ⁻¹' I ⊆ {x | ↑(t ^ (n - 1)) < limRatioMeas v hρ x}",
"tactic": "intro x hx"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\nx : α\nhx : x ∈ s ∩ f ⁻¹' I\n⊢ ↑(t ^ (n - 1)) < ↑t ^ n",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\nx : α\nhx : x ∈ s ∩ f ⁻¹' I\n⊢ x ∈ {x | ↑(t ^ (n - 1)) < limRatioMeas v hρ x}",
"tactic": "apply lt_of_lt_of_le _ hx.2.1"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\nx : α\nhx : x ∈ s ∩ f ⁻¹' I\n⊢ t ^ n * t ^ (-1) < t ^ n",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\nx : α\nhx : x ∈ s ∩ f ⁻¹' I\n⊢ ↑(t ^ (n - 1)) < ↑t ^ n",
"tactic": "rw [← ENNReal.coe_zpow (zero_lt_one.trans ht).ne', ENNReal.coe_lt_coe, sub_eq_add_neg,\n zpow_add₀ t_ne_zero']"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\nx : α\nhx : x ∈ s ∩ f ⁻¹' I\n⊢ t ^ n * t ^ (-1) < t ^ n * 1",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\nx : α\nhx : x ∈ s ∩ f ⁻¹' I\n⊢ t ^ n * t ^ (-1) < t ^ n",
"tactic": "conv_rhs => rw [← mul_one (t ^ n)]"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\nx : α\nhx : x ∈ s ∩ f ⁻¹' I\n⊢ t ^ (-1) < 1",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\nx : α\nhx : x ∈ s ∩ f ⁻¹' I\n⊢ t ^ n * t ^ (-1) < t ^ n * 1",
"tactic": "refine' mul_lt_mul' le_rfl _ (zero_le _) (NNReal.zpow_pos t_ne_zero' _)"
},
{
"state_after": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\nx : α\nhx : x ∈ s ∩ f ⁻¹' I\n⊢ t⁻¹ < 1",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\nx : α\nhx : x ∈ s ∩ f ⁻¹' I\n⊢ t ^ (-1) < 1",
"tactic": "rw [zpow_neg_one]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝⁵ : MetricSpace α\nm0 : MeasurableSpace α\nμ : MeasureTheory.Measure α\nv : VitaliFamily μ\nE : Type ?u.4684926\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : SecondCountableTopology α\ninst✝² : BorelSpace α\ninst✝¹ : IsLocallyFiniteMeasure μ\nρ : MeasureTheory.Measure α\ninst✝ : IsLocallyFiniteMeasure ρ\nhρ : ρ ≪ μ\ns : Set α\nhs : MeasurableSet s\nt : ℝ≥0\nht : 1 < t\nt_ne_zero' : t ≠ 0\nt_ne_zero : ↑t ≠ 0\nν : MeasureTheory.Measure α := withDensity μ (limRatioMeas v hρ)\nf : α → ℝ≥0∞ := limRatioMeas v hρ\nf_meas : Measurable f\nA : ↑↑ν (s ∩ f ⁻¹' {0}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {0})\nB : ↑↑ν (s ∩ f ⁻¹' {⊤}) ≤ ↑↑(↑t ^ 2 • ρ) (s ∩ f ⁻¹' {⊤})\nn : ℤ\nI : Set ℝ≥0∞ := Ico (↑t ^ n) (↑t ^ (n + 1))\nM : MeasurableSet (s ∩ f ⁻¹' I)\nx : α\nhx : x ∈ s ∩ f ⁻¹' I\n⊢ t⁻¹ < 1",
"tactic": "exact inv_lt_one ht"
}
] |
[
608,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
545,
1
] |
Mathlib/Data/Nat/Choose/Basic.lean
|
Nat.factorial_mul_factorial_dvd_factorial_add
|
[
{
"state_after": "i j : ℕ\nthis : i ! * (i + j - i)! ∣ (i + j)!\n⊢ i ! * j ! ∣ (i + j)!\n\ncase this\ni j : ℕ\n⊢ i ! * (i + j - i)! ∣ (i + j)!",
"state_before": "i j : ℕ\n⊢ i ! * j ! ∣ (i + j)!",
"tactic": "suffices : i ! * (i + j - i) ! ∣ (i + j)!"
},
{
"state_after": "case this\ni j : ℕ\n⊢ i ! * (i + j - i)! ∣ (i + j)!",
"state_before": "i j : ℕ\nthis : i ! * (i + j - i)! ∣ (i + j)!\n⊢ i ! * j ! ∣ (i + j)!\n\ncase this\ni j : ℕ\n⊢ i ! * (i + j - i)! ∣ (i + j)!",
"tactic": ". rwa [add_tsub_cancel_left i j] at this"
},
{
"state_after": "no goals",
"state_before": "case this\ni j : ℕ\n⊢ i ! * (i + j - i)! ∣ (i + j)!",
"tactic": "exact factorial_mul_factorial_dvd_factorial (Nat.le_add_right _ _)"
},
{
"state_after": "no goals",
"state_before": "i j : ℕ\nthis : i ! * (i + j - i)! ∣ (i + j)!\n⊢ i ! * j ! ∣ (i + j)!",
"tactic": "rwa [add_tsub_cancel_left i j] at this"
}
] |
[
191,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
188,
1
] |
Mathlib/Data/Real/Irrational.lean
|
Irrational.add_cases
|
[
{
"state_after": "q : ℚ\nx y : ℝ\n⊢ ¬x + y ∈ Set.range Rat.cast → ¬x ∈ Set.range Rat.cast ∨ ¬y ∈ Set.range Rat.cast",
"state_before": "q : ℚ\nx y : ℝ\n⊢ Irrational (x + y) → Irrational x ∨ Irrational y",
"tactic": "delta Irrational"
},
{
"state_after": "q : ℚ\nx y : ℝ\n⊢ x ∈ Set.range Rat.cast ∧ y ∈ Set.range Rat.cast → x + y ∈ Set.range Rat.cast",
"state_before": "q : ℚ\nx y : ℝ\n⊢ ¬x + y ∈ Set.range Rat.cast → ¬x ∈ Set.range Rat.cast ∨ ¬y ∈ Set.range Rat.cast",
"tactic": "contrapose!"
},
{
"state_after": "case intro.intro.intro\nq rx ry : ℚ\n⊢ ↑rx + ↑ry ∈ Set.range Rat.cast",
"state_before": "q : ℚ\nx y : ℝ\n⊢ x ∈ Set.range Rat.cast ∧ y ∈ Set.range Rat.cast → x + y ∈ Set.range Rat.cast",
"tactic": "rintro ⟨⟨rx, rfl⟩, ⟨ry, rfl⟩⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro\nq rx ry : ℚ\n⊢ ↑rx + ↑ry ∈ Set.range Rat.cast",
"tactic": "exact ⟨rx + ry, cast_add rx ry⟩"
}
] |
[
209,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
205,
1
] |
Mathlib/MeasureTheory/Integral/Lebesgue.lean
|
MeasureTheory.lintegral_biUnion₀
|
[
{
"state_after": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.1022236\nδ : Type ?u.1022239\nm : MeasurableSpace α\nμ ν : Measure α\nt : Set β\ns : β → Set α\nht : Set.Countable t\nhm : ∀ (i : β), i ∈ t → NullMeasurableSet (s i)\nhd : Set.Pairwise t (AEDisjoint μ on s)\nf : α → ℝ≥0∞\nthis : Encodable ↑t\n⊢ (∫⁻ (a : α) in ⋃ (i : β) (_ : i ∈ t), s i, f a ∂μ) = ∑' (i : ↑t), ∫⁻ (a : α) in s ↑i, f a ∂μ",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.1022236\nδ : Type ?u.1022239\nm : MeasurableSpace α\nμ ν : Measure α\nt : Set β\ns : β → Set α\nht : Set.Countable t\nhm : ∀ (i : β), i ∈ t → NullMeasurableSet (s i)\nhd : Set.Pairwise t (AEDisjoint μ on s)\nf : α → ℝ≥0∞\n⊢ (∫⁻ (a : α) in ⋃ (i : β) (_ : i ∈ t), s i, f a ∂μ) = ∑' (i : ↑t), ∫⁻ (a : α) in s ↑i, f a ∂μ",
"tactic": "haveI := ht.toEncodable"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.1022236\nδ : Type ?u.1022239\nm : MeasurableSpace α\nμ ν : Measure α\nt : Set β\ns : β → Set α\nht : Set.Countable t\nhm : ∀ (i : β), i ∈ t → NullMeasurableSet (s i)\nhd : Set.Pairwise t (AEDisjoint μ on s)\nf : α → ℝ≥0∞\nthis : Encodable ↑t\n⊢ (∫⁻ (a : α) in ⋃ (i : β) (_ : i ∈ t), s i, f a ∂μ) = ∑' (i : ↑t), ∫⁻ (a : α) in s ↑i, f a ∂μ",
"tactic": "rw [biUnion_eq_iUnion, lintegral_iUnion₀ (SetCoe.forall'.1 hm) (hd.subtype _ _)]"
}
] |
[
1200,
83
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1196,
1
] |
Mathlib/LinearAlgebra/DirectSum/Finsupp.lean
|
finsuppTensorFinsupp'_single_tmul_single
|
[
{
"state_after": "case h.mk\nR : Type u\nM : Type v\nN : Type w\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nS : Type u_1\ninst✝ : CommRing S\nα : Type u_2\nβ : Type u_3\na : α\nb : β\nr₁ r₂ : S\na' : α\nb' : β\n⊢ ↑(↑(finsuppTensorFinsupp' S α β) (Finsupp.single a r₁ ⊗ₜ[S] Finsupp.single b r₂)) (a', b') =\n ↑(Finsupp.single (a, b) (r₁ * r₂)) (a', b')",
"state_before": "R : Type u\nM : Type v\nN : Type w\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nS : Type u_1\ninst✝ : CommRing S\nα : Type u_2\nβ : Type u_3\na : α\nb : β\nr₁ r₂ : S\n⊢ ↑(finsuppTensorFinsupp' S α β) (Finsupp.single a r₁ ⊗ₜ[S] Finsupp.single b r₂) = Finsupp.single (a, b) (r₁ * r₂)",
"tactic": "ext ⟨a', b'⟩"
},
{
"state_after": "no goals",
"state_before": "case h.mk\nR : Type u\nM : Type v\nN : Type w\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup M\ninst✝³ : Module R M\ninst✝² : AddCommGroup N\ninst✝¹ : Module R N\nS : Type u_1\ninst✝ : CommRing S\nα : Type u_2\nβ : Type u_3\na : α\nb : β\nr₁ r₂ : S\na' : α\nb' : β\n⊢ ↑(↑(finsuppTensorFinsupp' S α β) (Finsupp.single a r₁ ⊗ₜ[S] Finsupp.single b r₂)) (a', b') =\n ↑(Finsupp.single (a, b) (r₁ * r₂)) (a', b')",
"tactic": "aesop (add norm [Finsupp.single_apply])"
}
] |
[
107,
42
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
103,
1
] |
Mathlib/MeasureTheory/Integral/IntegrableOn.lean
|
MeasureTheory.integrable_indicator_iff
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.1634475\nE : Type u_2\nF : Type ?u.1634481\ninst✝¹ : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nf g : α → E\ns t : Set α\nμ ν : Measure α\nhs : MeasurableSet s\n⊢ Integrable (indicator s f) ↔ IntegrableOn f s",
"tactic": "simp [IntegrableOn, Integrable, HasFiniteIntegral, nnnorm_indicator_eq_indicator_nnnorm,\n ENNReal.coe_indicator, lintegral_indicator _ hs, aestronglyMeasurable_indicator_iff hs]"
}
] |
[
263,
92
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
260,
1
] |
Mathlib/Algebra/GroupPower/Basic.lean
|
pow_ite
|
[
{
"state_after": "no goals",
"state_before": "α : Type ?u.44\nM : Type u\nN : Type v\nG : Type w\nH : Type x\nA : Type y\nB : Type z\nR : Type u₁\nS : Type u₂\ninst✝¹ : Pow M ℕ\nP : Prop\ninst✝ : Decidable P\na : M\nb c : ℕ\n⊢ (a ^ if P then b else c) = if P then a ^ b else a ^ c",
"tactic": "split_ifs <;> rfl"
}
] |
[
56,
82
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
55,
1
] |
Mathlib/Topology/Sets/Compacts.lean
|
TopologicalSpace.Compacts.coe_map
|
[] |
[
141,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
140,
1
] |
Mathlib/Order/LocallyFinite.lean
|
WithBot.Icc_bot_coe
|
[] |
[
1156,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1155,
1
] |
Mathlib/RingTheory/GradedAlgebra/Radical.lean
|
Ideal.IsPrime.homogeneousCore
|
[
{
"state_after": "case I_ne_top\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsPrime I\n⊢ (Ideal.homogeneousCore 𝒜 I).toSubmodule ≠ ⊤\n\ncase homogeneous_mem_or_mem\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsPrime I\n⊢ ∀ {x y : A},\n Homogeneous 𝒜 x →\n Homogeneous 𝒜 y →\n x * y ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule →\n x ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule ∨ y ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule",
"state_before": "ι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsPrime I\n⊢ IsPrime (HomogeneousIdeal.toIdeal (Ideal.homogeneousCore 𝒜 I))",
"tactic": "apply (Ideal.homogeneousCore 𝒜 I).is_homogeneous'.isPrime_of_homogeneous_mem_or_mem"
},
{
"state_after": "case homogeneous_mem_or_mem\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsPrime I\nx y : A\nhx : Homogeneous 𝒜 x\nhy : Homogeneous 𝒜 y\nhxy : x * y ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule\n⊢ x ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule ∨ y ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule",
"state_before": "case homogeneous_mem_or_mem\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsPrime I\n⊢ ∀ {x y : A},\n Homogeneous 𝒜 x →\n Homogeneous 𝒜 y →\n x * y ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule →\n x ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule ∨ y ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule",
"tactic": "rintro x y hx hy hxy"
},
{
"state_after": "case homogeneous_mem_or_mem\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsPrime I\nx y : A\nhx : Homogeneous 𝒜 x\nhy : Homogeneous 𝒜 y\nhxy : x * y ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule\nH : x ∈ I ∨ y ∈ I\n⊢ x ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule ∨ y ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule",
"state_before": "case homogeneous_mem_or_mem\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsPrime I\nx y : A\nhx : Homogeneous 𝒜 x\nhy : Homogeneous 𝒜 y\nhxy : x * y ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule\n⊢ x ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule ∨ y ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule",
"tactic": "have H := h.mem_or_mem (Ideal.toIdeal_homogeneousCore_le 𝒜 I hxy)"
},
{
"state_after": "case homogeneous_mem_or_mem.refine'_1\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsPrime I\nx y : A\nhx : Homogeneous 𝒜 x\nhy : Homogeneous 𝒜 y\nhxy : x * y ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule\nH : x ∈ I ∨ y ∈ I\n⊢ x ∈ I → x ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule\n\ncase homogeneous_mem_or_mem.refine'_2\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsPrime I\nx y : A\nhx : Homogeneous 𝒜 x\nhy : Homogeneous 𝒜 y\nhxy : x * y ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule\nH : x ∈ I ∨ y ∈ I\n⊢ y ∈ I → y ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule",
"state_before": "case homogeneous_mem_or_mem\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsPrime I\nx y : A\nhx : Homogeneous 𝒜 x\nhy : Homogeneous 𝒜 y\nhxy : x * y ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule\nH : x ∈ I ∨ y ∈ I\n⊢ x ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule ∨ y ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule",
"tactic": "refine' H.imp _ _"
},
{
"state_after": "no goals",
"state_before": "case I_ne_top\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsPrime I\n⊢ (Ideal.homogeneousCore 𝒜 I).toSubmodule ≠ ⊤",
"tactic": "exact ne_top_of_le_ne_top h.ne_top (Ideal.toIdeal_homogeneousCore_le 𝒜 I)"
},
{
"state_after": "no goals",
"state_before": "case homogeneous_mem_or_mem.refine'_1\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsPrime I\nx y : A\nhx : Homogeneous 𝒜 x\nhy : Homogeneous 𝒜 y\nhxy : x * y ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule\nH : x ∈ I ∨ y ∈ I\n⊢ x ∈ I → x ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule",
"tactic": "exact Ideal.mem_homogeneousCore_of_homogeneous_of_mem hx"
},
{
"state_after": "no goals",
"state_before": "case homogeneous_mem_or_mem.refine'_2\nι : Type u_2\nσ : Type u_3\nA : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : LinearOrderedCancelAddCommMonoid ι\ninst✝² : SetLike σ A\ninst✝¹ : AddSubmonoidClass σ A\n𝒜 : ι → σ\ninst✝ : GradedRing 𝒜\nI : Ideal A\nh : IsPrime I\nx y : A\nhx : Homogeneous 𝒜 x\nhy : Homogeneous 𝒜 y\nhxy : x * y ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule\nH : x ∈ I ∨ y ∈ I\n⊢ y ∈ I → y ∈ (Ideal.homogeneousCore 𝒜 I).toSubmodule",
"tactic": "exact Ideal.mem_homogeneousCore_of_homogeneous_of_mem hy"
}
] |
[
166,
61
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
158,
1
] |
Mathlib/Data/Set/Basic.lean
|
Set.inter_univ
|
[] |
[
1002,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1001,
1
] |
Mathlib/Topology/QuasiSeparated.lean
|
isQuasiSeparated_univ
|
[] |
[
66,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
64,
1
] |
Mathlib/Algebra/Order/Monoid/WithTop.lean
|
WithTop.one_ne_top
|
[] |
[
107,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/Algebra/Algebra/Hom.lean
|
AlgHom.map_add
|
[] |
[
246,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
11
] |
Mathlib/Algebra/Hom/Ring.lean
|
RingHom.toFun_eq_coe
|
[] |
[
450,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
449,
1
] |
Mathlib/Data/Set/Image.lean
|
Function.Injective.mem_set_image
|
[] |
[
226,
54
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
224,
1
] |
Mathlib/CategoryTheory/Generator.lean
|
CategoryTheory.hasInitial_of_isCoseparating
|
[
{
"state_after": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis : HasProductsOfShape (↑𝒢) C\n⊢ HasInitial C",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\n⊢ HasInitial C",
"tactic": "haveI : HasProductsOfShape 𝒢 C := hasProductsOfShape_of_small C 𝒢"
},
{
"state_after": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝ : HasProductsOfShape (↑𝒢) C\nthis : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\n⊢ HasInitial C",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis : HasProductsOfShape (↑𝒢) C\n⊢ HasInitial C",
"tactic": "haveI := fun A => hasProductsOfShape_of_small.{v₁} C (ΣG : 𝒢, A ⟶ (G : C))"
},
{
"state_after": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\n⊢ HasInitial C",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝ : HasProductsOfShape (↑𝒢) C\nthis : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\n⊢ HasInitial C",
"tactic": "letI := completeLatticeOfCompleteSemilatticeInf (Subobject (piObj (Subtype.val : 𝒢 → C)))"
},
{
"state_after": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\n⊢ (A : C) → Unique (Subobject.underlying.obj ⊥ ⟶ A)",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\n⊢ HasInitial C",
"tactic": "suffices ∀ A : C, Unique (((⊥ : Subobject (piObj (Subtype.val : 𝒢 → C))) : C) ⟶ A) by\n exact hasInitial_of_unique ((⊥ : Subobject (piObj (Subtype.val : 𝒢 → C))) : C)"
},
{
"state_after": "case refine'_1\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\n⊢ Subobject.underlying.obj ⊥ ⟶ A\n\ncase refine'_2\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\nf : Subobject.underlying.obj ⊥ ⟶ A\n⊢ f = default",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\n⊢ (A : C) → Unique (Subobject.underlying.obj ⊥ ⟶ A)",
"tactic": "refine' fun A => ⟨⟨_⟩, fun f => _⟩"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝² : HasProductsOfShape (↑𝒢) C\nthis✝¹ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis✝ : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nthis : (A : C) → Unique (Subobject.underlying.obj ⊥ ⟶ A)\n⊢ HasInitial C",
"tactic": "exact hasInitial_of_unique ((⊥ : Subobject (piObj (Subtype.val : 𝒢 → C))) : C)"
},
{
"state_after": "case refine'_1\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\ns : ∏ Subtype.val ⟶ ∏ fun f => ↑f.fst := Pi.lift fun f => id (Pi.π Subtype.val) f.fst\n⊢ Subobject.underlying.obj ⊥ ⟶ A",
"state_before": "case refine'_1\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\n⊢ Subobject.underlying.obj ⊥ ⟶ A",
"tactic": "let s := Pi.lift fun f : ΣG : 𝒢, A ⟶ (G : C) => id (Pi.π (Subtype.val : 𝒢 → C)) f.1"
},
{
"state_after": "case refine'_1\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\ns : ∏ Subtype.val ⟶ ∏ fun f => ↑f.fst := Pi.lift fun f => id (Pi.π Subtype.val) f.fst\nt : A ⟶ ∏ fun b => ↑b.fst := Pi.lift Sigma.snd\n⊢ Subobject.underlying.obj ⊥ ⟶ A",
"state_before": "case refine'_1\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\ns : ∏ Subtype.val ⟶ ∏ fun f => ↑f.fst := Pi.lift fun f => id (Pi.π Subtype.val) f.fst\n⊢ Subobject.underlying.obj ⊥ ⟶ A",
"tactic": "let t := Pi.lift (@Sigma.snd 𝒢 fun G => A ⟶ (G : C))"
},
{
"state_after": "case refine'_1\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝² : HasProductsOfShape (↑𝒢) C\nthis✝¹ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis✝ : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\ns : ∏ Subtype.val ⟶ ∏ fun f => ↑f.fst := Pi.lift fun f => id (Pi.π Subtype.val) f.fst\nt : A ⟶ ∏ fun b => ↑b.fst := Pi.lift Sigma.snd\nthis : Mono t\n⊢ Subobject.underlying.obj ⊥ ⟶ A",
"state_before": "case refine'_1\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\ns : ∏ Subtype.val ⟶ ∏ fun f => ↑f.fst := Pi.lift fun f => id (Pi.π Subtype.val) f.fst\nt : A ⟶ ∏ fun b => ↑b.fst := Pi.lift Sigma.snd\n⊢ Subobject.underlying.obj ⊥ ⟶ A",
"tactic": "haveI : Mono t := (isCoseparating_iff_mono 𝒢).1 h𝒢 A"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝² : HasProductsOfShape (↑𝒢) C\nthis✝¹ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis✝ : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\ns : ∏ Subtype.val ⟶ ∏ fun f => ↑f.fst := Pi.lift fun f => id (Pi.π Subtype.val) f.fst\nt : A ⟶ ∏ fun b => ↑b.fst := Pi.lift Sigma.snd\nthis : Mono t\n⊢ Subobject.underlying.obj ⊥ ⟶ A",
"tactic": "exact Subobject.ofLEMk _ (pullback.fst : pullback s t ⟶ _) bot_le ≫ pullback.snd"
},
{
"state_after": "case refine'_2\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\nf : Subobject.underlying.obj ⊥ ⟶ A\n⊢ ∀ (g : Subobject.underlying.obj ⊥ ⟶ A), f = g",
"state_before": "case refine'_2\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\nf : Subobject.underlying.obj ⊥ ⟶ A\n⊢ f = default",
"tactic": "suffices ∀ (g : Subobject.underlying.obj ⊥ ⟶ A ), f = g by\n apply this"
},
{
"state_after": "case refine'_2\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\nf : Subobject.underlying.obj ⊥ ⟶ A\ng : Subobject.underlying.obj ⊥ ⟶ A\n⊢ f = g",
"state_before": "case refine'_2\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\nf : Subobject.underlying.obj ⊥ ⟶ A\n⊢ ∀ (g : Subobject.underlying.obj ⊥ ⟶ A), f = g",
"tactic": "intro g"
},
{
"state_after": "case refine'_2\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\nf : Subobject.underlying.obj ⊥ ⟶ A\ng : Subobject.underlying.obj ⊥ ⟶ A\n⊢ IsSplitEpi (equalizer.ι f g)",
"state_before": "case refine'_2\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\nf : Subobject.underlying.obj ⊥ ⟶ A\ng : Subobject.underlying.obj ⊥ ⟶ A\n⊢ f = g",
"tactic": "suffices IsSplitEpi (equalizer.ι f g) by exact eq_of_epi_equalizer"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\nf : Subobject.underlying.obj ⊥ ⟶ A\ng : Subobject.underlying.obj ⊥ ⟶ A\n⊢ IsSplitEpi (equalizer.ι f g)",
"tactic": "exact IsSplitEpi.mk' ⟨Subobject.ofLEMk _ (equalizer.ι f g ≫ Subobject.arrow _) bot_le, by\n ext\n simp⟩"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝² : HasProductsOfShape (↑𝒢) C\nthis✝¹ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis✝ : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\nf : Subobject.underlying.obj ⊥ ⟶ A\nthis : ∀ (g : Subobject.underlying.obj ⊥ ⟶ A), f = g\n⊢ f = default",
"tactic": "apply this"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝² : HasProductsOfShape (↑𝒢) C\nthis✝¹ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis✝ : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\nf : Subobject.underlying.obj ⊥ ⟶ A\ng : Subobject.underlying.obj ⊥ ⟶ A\nthis : IsSplitEpi (equalizer.ι f g)\n⊢ f = g",
"tactic": "exact eq_of_epi_equalizer"
},
{
"state_after": "case h.h\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\nf : Subobject.underlying.obj ⊥ ⟶ A\ng : Subobject.underlying.obj ⊥ ⟶ A\nb✝ : { x // x ∈ 𝒢 }\n⊢ ((Subobject.ofLEMk ⊥ (equalizer.ι f g ≫ Subobject.arrow ⊥)\n (_ : ⊥ ≤ Subobject.mk (equalizer.ι f g ≫ Subobject.arrow ⊥)) ≫\n equalizer.ι f g) ≫\n Subobject.arrow ⊥) ≫\n Pi.π Subtype.val b✝ =\n (𝟙 (Subobject.underlying.obj ⊥) ≫ Subobject.arrow ⊥) ≫ Pi.π Subtype.val b✝",
"state_before": "C : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\nf : Subobject.underlying.obj ⊥ ⟶ A\ng : Subobject.underlying.obj ⊥ ⟶ A\n⊢ Subobject.ofLEMk ⊥ (equalizer.ι f g ≫ Subobject.arrow ⊥)\n (_ : ⊥ ≤ Subobject.mk (equalizer.ι f g ≫ Subobject.arrow ⊥)) ≫\n equalizer.ι f g =\n 𝟙 (Subobject.underlying.obj ⊥)",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h.h\nC : Type u₁\ninst✝⁴ : Category C\nD : Type u₂\ninst✝³ : Category D\ninst✝² : WellPowered C\ninst✝¹ : HasLimits C\n𝒢 : Set C\ninst✝ : Small ↑𝒢\nh𝒢 : IsCoseparating 𝒢\nthis✝¹ : HasProductsOfShape (↑𝒢) C\nthis✝ : ∀ (A : C), HasProductsOfShape ((G : ↑𝒢) × (A ⟶ ↑G)) C\nthis : CompleteLattice (Subobject (∏ Subtype.val)) :=\n completeLatticeOfCompleteSemilatticeInf (Subobject (∏ Subtype.val))\nA : C\nf : Subobject.underlying.obj ⊥ ⟶ A\ng : Subobject.underlying.obj ⊥ ⟶ A\nb✝ : { x // x ∈ 𝒢 }\n⊢ ((Subobject.ofLEMk ⊥ (equalizer.ι f g ≫ Subobject.arrow ⊥)\n (_ : ⊥ ≤ Subobject.mk (equalizer.ι f g ≫ Subobject.arrow ⊥)) ≫\n equalizer.ι f g) ≫\n Subobject.arrow ⊥) ≫\n Pi.π Subtype.val b✝ =\n (𝟙 (Subobject.underlying.obj ⊥) ≫ Subobject.arrow ⊥) ≫ Pi.π Subtype.val b✝",
"tactic": "simp"
}
] |
[
303,
12
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
285,
1
] |
Mathlib/Order/Filter/Archimedean.lean
|
tendsto_nat_cast_atTop_iff
|
[] |
[
37,
65
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
35,
1
] |
Mathlib/NumberTheory/LegendreSymbol/MulCharacter.lean
|
MulChar.map_one
|
[] |
[
270,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
11
] |
Mathlib/Combinatorics/SimpleGraph/Subgraph.lean
|
SimpleGraph.Subgraph.spanningCoe_le_of_le
|
[] |
[
611,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
610,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.one_apply_eq
|
[] |
[
541,
24
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
540,
1
] |
Mathlib/RingTheory/Subring/Basic.lean
|
Subring.mem_sInf
|
[] |
[
732,
18
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
731,
1
] |
Mathlib/Topology/FiberBundle/Trivialization.lean
|
Trivialization.preimage_subset_source
|
[] |
[
445,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
444,
1
] |
Mathlib/Topology/SubsetProperties.lean
|
IsCompact.disjoint_nhdsSet_left
|
[
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.16218\nπ : ι → Type ?u.16223\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nl : Filter α\nhs : IsCompact s\nH : ∀ (x : α), x ∈ s → Disjoint (𝓝 x) l\n⊢ Disjoint (𝓝ˢ s) l",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.16218\nπ : ι → Type ?u.16223\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nl : Filter α\nhs : IsCompact s\n⊢ Disjoint (𝓝ˢ s) l ↔ ∀ (x : α), x ∈ s → Disjoint (𝓝 x) l",
"tactic": "refine' ⟨fun h x hx => h.mono_left <| nhds_le_nhdsSet hx, fun H => _⟩"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.16218\nπ : ι → Type ?u.16223\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nl : Filter α\nhs : IsCompact s\nH : ∀ (x : α), x ∈ s → Disjoint (𝓝 x) l\nU : α → Set α\nhxU : ∀ (x : α), x ∈ s → x ∈ U x ∧ IsOpen (U x)\nhUl : ∀ (x : α), x ∈ s → U xᶜ ∈ l\n⊢ Disjoint (𝓝ˢ s) l",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.16218\nπ : ι → Type ?u.16223\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nl : Filter α\nhs : IsCompact s\nH : ∀ (x : α), x ∈ s → Disjoint (𝓝 x) l\n⊢ Disjoint (𝓝ˢ s) l",
"tactic": "choose! U hxU hUl using fun x hx => (nhds_basis_opens x).disjoint_iff_left.1 (H x hx)"
},
{
"state_after": "α : Type u\nβ : Type v\nι : Type ?u.16218\nπ : ι → Type ?u.16223\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nl : Filter α\nhs : IsCompact s\nH : ∀ (x : α), x ∈ s → Disjoint (𝓝 x) l\nU : α → Set α\nhUl : ∀ (x : α), x ∈ s → U xᶜ ∈ l\nhxU : ∀ (x : α), x ∈ s → x ∈ U x\nhUo : ∀ (x : α), x ∈ s → IsOpen (U x)\n⊢ Disjoint (𝓝ˢ s) l",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.16218\nπ : ι → Type ?u.16223\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nl : Filter α\nhs : IsCompact s\nH : ∀ (x : α), x ∈ s → Disjoint (𝓝 x) l\nU : α → Set α\nhxU : ∀ (x : α), x ∈ s → x ∈ U x ∧ IsOpen (U x)\nhUl : ∀ (x : α), x ∈ s → U xᶜ ∈ l\n⊢ Disjoint (𝓝ˢ s) l",
"tactic": "choose hxU hUo using hxU"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.16218\nπ : ι → Type ?u.16223\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t✝ : Set α\nl : Filter α\nhs : IsCompact s\nH : ∀ (x : α), x ∈ s → Disjoint (𝓝 x) l\nU : α → Set α\nhUl : ∀ (x : α), x ∈ s → U xᶜ ∈ l\nhxU : ∀ (x : α), x ∈ s → x ∈ U x\nhUo : ∀ (x : α), x ∈ s → IsOpen (U x)\nt : Finset α\nhts : ∀ (x : α), x ∈ t → x ∈ s\nhst : s ⊆ ⋃ (x : α) (_ : x ∈ t), U x\n⊢ Disjoint (𝓝ˢ s) l",
"state_before": "α : Type u\nβ : Type v\nι : Type ?u.16218\nπ : ι → Type ?u.16223\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t : Set α\nl : Filter α\nhs : IsCompact s\nH : ∀ (x : α), x ∈ s → Disjoint (𝓝 x) l\nU : α → Set α\nhUl : ∀ (x : α), x ∈ s → U xᶜ ∈ l\nhxU : ∀ (x : α), x ∈ s → x ∈ U x\nhUo : ∀ (x : α), x ∈ s → IsOpen (U x)\n⊢ Disjoint (𝓝ˢ s) l",
"tactic": "rcases hs.elim_nhds_subcover U fun x hx => (hUo x hx).mem_nhds (hxU x hx) with ⟨t, hts, hst⟩"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.16218\nπ : ι → Type ?u.16223\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t✝ : Set α\nl : Filter α\nhs : IsCompact s\nH : ∀ (x : α), x ∈ s → Disjoint (𝓝 x) l\nU : α → Set α\nhUl : ∀ (x : α), x ∈ s → U xᶜ ∈ l\nhxU : ∀ (x : α), x ∈ s → x ∈ U x\nhUo : ∀ (x : α), x ∈ s → IsOpen (U x)\nt : Finset α\nhts : ∀ (x : α), x ∈ t → x ∈ s\nhst : s ⊆ ⋃ (x : α) (_ : x ∈ t), U x\n⊢ (⋃ (x : α) (_ : x ∈ t), U x)ᶜ ∈ l",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.16218\nπ : ι → Type ?u.16223\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t✝ : Set α\nl : Filter α\nhs : IsCompact s\nH : ∀ (x : α), x ∈ s → Disjoint (𝓝 x) l\nU : α → Set α\nhUl : ∀ (x : α), x ∈ s → U xᶜ ∈ l\nhxU : ∀ (x : α), x ∈ s → x ∈ U x\nhUo : ∀ (x : α), x ∈ s → IsOpen (U x)\nt : Finset α\nhts : ∀ (x : α), x ∈ t → x ∈ s\nhst : s ⊆ ⋃ (x : α) (_ : x ∈ t), U x\n⊢ Disjoint (𝓝ˢ s) l",
"tactic": "refine'\n (hasBasis_nhdsSet _).disjoint_iff_left.2\n ⟨⋃ x ∈ t, U x, ⟨isOpen_biUnion fun x hx => hUo x (hts x hx), hst⟩, _⟩"
},
{
"state_after": "case intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.16218\nπ : ι → Type ?u.16223\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t✝ : Set α\nl : Filter α\nhs : IsCompact s\nH : ∀ (x : α), x ∈ s → Disjoint (𝓝 x) l\nU : α → Set α\nhUl : ∀ (x : α), x ∈ s → U xᶜ ∈ l\nhxU : ∀ (x : α), x ∈ s → x ∈ U x\nhUo : ∀ (x : α), x ∈ s → IsOpen (U x)\nt : Finset α\nhts : ∀ (x : α), x ∈ t → x ∈ s\nhst : s ⊆ ⋃ (x : α) (_ : x ∈ t), U x\n⊢ ∀ (i : α), i ∈ t → U iᶜ ∈ l",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.16218\nπ : ι → Type ?u.16223\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t✝ : Set α\nl : Filter α\nhs : IsCompact s\nH : ∀ (x : α), x ∈ s → Disjoint (𝓝 x) l\nU : α → Set α\nhUl : ∀ (x : α), x ∈ s → U xᶜ ∈ l\nhxU : ∀ (x : α), x ∈ s → x ∈ U x\nhUo : ∀ (x : α), x ∈ s → IsOpen (U x)\nt : Finset α\nhts : ∀ (x : α), x ∈ t → x ∈ s\nhst : s ⊆ ⋃ (x : α) (_ : x ∈ t), U x\n⊢ (⋃ (x : α) (_ : x ∈ t), U x)ᶜ ∈ l",
"tactic": "rw [compl_iUnion₂, biInter_finset_mem]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u\nβ : Type v\nι : Type ?u.16218\nπ : ι → Type ?u.16223\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\ns t✝ : Set α\nl : Filter α\nhs : IsCompact s\nH : ∀ (x : α), x ∈ s → Disjoint (𝓝 x) l\nU : α → Set α\nhUl : ∀ (x : α), x ∈ s → U xᶜ ∈ l\nhxU : ∀ (x : α), x ∈ s → x ∈ U x\nhUo : ∀ (x : α), x ∈ s → IsOpen (U x)\nt : Finset α\nhts : ∀ (x : α), x ∈ t → x ∈ s\nhst : s ⊆ ⋃ (x : α) (_ : x ∈ t), U x\n⊢ ∀ (i : α), i ∈ t → U iᶜ ∈ l",
"tactic": "exact fun x hx => hUl x (hts x hx)"
}
] |
[
228,
37
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
218,
1
] |
Mathlib/MeasureTheory/Group/Measure.lean
|
MeasureTheory.Measure.measurePreserving_mul_right_inv
|
[] |
[
475,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
473,
1
] |
Mathlib/GroupTheory/Subgroup/Pointwise.lean
|
AddSubgroup.pointwise_smul_le_iff₀
|
[] |
[
552,
26
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
550,
1
] |
Mathlib/Data/Vector/Basic.lean
|
Vector.not_empty_toList
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\nα : Type u_1\nv : Vector α (n + 1)\n⊢ ¬List.isEmpty (toList v) = true",
"tactic": "simp only [empty_toList_eq_ff, Bool.coe_sort_false, not_false_iff]"
}
] |
[
220,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
219,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Real.lean
|
Real.rpow_lt_rpow
|
[
{
"state_after": "x y z : ℝ\nhx : 0 = x ∨ 0 < x\nhxy : x < y\nhz : 0 < z\n⊢ x ^ z < y ^ z",
"state_before": "x y z : ℝ\nhx : 0 ≤ x\nhxy : x < y\nhz : 0 < z\n⊢ x ^ z < y ^ z",
"tactic": "rw [le_iff_eq_or_lt] at hx"
},
{
"state_after": "case inl\nx y z : ℝ\nhxy : x < y\nhz : 0 < z\nhx : 0 = x\n⊢ x ^ z < y ^ z\n\ncase inr\nx y z : ℝ\nhxy : x < y\nhz : 0 < z\nhx : 0 < x\n⊢ x ^ z < y ^ z",
"state_before": "x y z : ℝ\nhx : 0 = x ∨ 0 < x\nhxy : x < y\nhz : 0 < z\n⊢ x ^ z < y ^ z",
"tactic": "cases' hx with hx hx"
},
{
"state_after": "case inl\nx y z : ℝ\nhxy : x < y\nhz : 0 < z\nhx : 0 = x\n⊢ 0 < y ^ z",
"state_before": "case inl\nx y z : ℝ\nhxy : x < y\nhz : 0 < z\nhx : 0 = x\n⊢ x ^ z < y ^ z",
"tactic": "rw [← hx, zero_rpow (ne_of_gt hz)]"
},
{
"state_after": "no goals",
"state_before": "case inl\nx y z : ℝ\nhxy : x < y\nhz : 0 < z\nhx : 0 = x\n⊢ 0 < y ^ z",
"tactic": "exact rpow_pos_of_pos (by rwa [← hx] at hxy) _"
},
{
"state_after": "no goals",
"state_before": "x y z : ℝ\nhxy : x < y\nhz : 0 < z\nhx : 0 = x\n⊢ 0 < y",
"tactic": "rwa [← hx] at hxy"
},
{
"state_after": "case inr\nx y z : ℝ\nhxy : x < y\nhz : 0 < z\nhx : 0 < x\n⊢ log x * z < log y * z",
"state_before": "case inr\nx y z : ℝ\nhxy : x < y\nhz : 0 < z\nhx : 0 < x\n⊢ x ^ z < y ^ z",
"tactic": "rw [rpow_def_of_pos hx, rpow_def_of_pos (lt_trans hx hxy), exp_lt_exp]"
},
{
"state_after": "no goals",
"state_before": "case inr\nx y z : ℝ\nhxy : x < y\nhz : 0 < z\nhx : 0 < x\n⊢ log x * z < log y * z",
"tactic": "exact mul_lt_mul_of_pos_right (log_lt_log hx hxy) hz"
}
] |
[
419,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
414,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean
|
ContDiffWithinAt.ccosh
|
[] |
[
502,
62
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
500,
1
] |
Mathlib/Data/Finset/Lattice.lean
|
Finset.max_mem_image_coe
|
[] |
[
1525,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1523,
1
] |
Mathlib/Data/Option/NAry.lean
|
Option.map₂_swap
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf✝ : α → β → γ\na✝ : Option α\nb✝ : Option β\nc : Option γ\nf : α → β → γ\na : Option α\nb : Option β\n⊢ map₂ f a b = map₂ (fun a b => f b a) b a",
"tactic": "cases a <;> cases b <;> rfl"
}
] |
[
90,
79
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
89,
1
] |
Mathlib/CategoryTheory/Abelian/NonPreadditive.lean
|
CategoryTheory.NonPreadditiveAbelian.add_comp
|
[
{
"state_after": "no goals",
"state_before": "C : Type u\ninst✝¹ : Category C\ninst✝ : NonPreadditiveAbelian C\nX Y Z : C\nf g : X ⟶ Y\nh : Y ⟶ Z\n⊢ (f + g) ≫ h = f ≫ h + g ≫ h",
"tactic": "rw [add_def, sub_comp, neg_def, sub_comp, zero_comp, add_def, neg_def]"
}
] |
[
444,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
443,
1
] |
Mathlib/CategoryTheory/Sites/Sheaf.lean
|
CategoryTheory.Presheaf.IsSheaf.hom_ext
|
[] |
[
242,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
239,
1
] |
Mathlib/FieldTheory/Fixed.lean
|
FixedPoints.minpoly.irreducible
|
[] |
[
253,
91
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
252,
1
] |
Mathlib/Data/Real/Hyperreal.lean
|
Hyperreal.InfiniteNeg.neg
|
[] |
[
463,
22
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
462,
1
] |
Mathlib/RingTheory/UniqueFactorizationDomain.lean
|
Associates.dvd_count_of_dvd_count_mul
|
[
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na b : Associates α\nhb : b ≠ 0\np : Associates α\nhp : Irreducible p\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nhabk : k ∣ count p (factors (a * b))\nha : a = 0\n⊢ k ∣ count p (factors a)\n\ncase neg\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na b : Associates α\nhb : b ≠ 0\np : Associates α\nhp : Irreducible p\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nhabk : k ∣ count p (factors (a * b))\nha : ¬a = 0\n⊢ k ∣ count p (factors a)",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na b : Associates α\nhb : b ≠ 0\np : Associates α\nhp : Irreducible p\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nhabk : k ∣ count p (factors (a * b))\n⊢ k ∣ count p (factors a)",
"tactic": "by_cases ha : a = 0"
},
{
"state_after": "case neg.inl\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na b : Associates α\nhb : b ≠ 0\np : Associates α\nhp : Irreducible p\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nhabk : k ∣ count p (factors (a * b))\nha : ¬a = 0\nhz : count p (factors a) = 0\n⊢ k ∣ count p (factors a)\n\ncase neg.inr\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na b : Associates α\nhb : b ≠ 0\np : Associates α\nhp : Irreducible p\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nhabk : k ∣ count p (factors (a * b))\nha : ¬a = 0\nh : count p (factors b) = 0\n⊢ k ∣ count p (factors a)",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na b : Associates α\nhb : b ≠ 0\np : Associates α\nhp : Irreducible p\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nhabk : k ∣ count p (factors (a * b))\nha : ¬a = 0\n⊢ k ∣ count p (factors a)",
"tactic": "cases' count_of_coprime ha hb hab hp with hz h"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na b : Associates α\nhb : b ≠ 0\np : Associates α\nhp : Irreducible p\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nhabk : k ∣ count p (factors (a * b))\nha : a = 0\n⊢ k ∣ count p (factors a)",
"tactic": "simpa [*] using habk"
},
{
"state_after": "case neg.inl\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na b : Associates α\nhb : b ≠ 0\np : Associates α\nhp : Irreducible p\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nhabk : k ∣ count p (factors (a * b))\nha : ¬a = 0\nhz : count p (factors a) = 0\n⊢ k ∣ 0",
"state_before": "case neg.inl\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na b : Associates α\nhb : b ≠ 0\np : Associates α\nhp : Irreducible p\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nhabk : k ∣ count p (factors (a * b))\nha : ¬a = 0\nhz : count p (factors a) = 0\n⊢ k ∣ count p (factors a)",
"tactic": "rw [hz]"
},
{
"state_after": "no goals",
"state_before": "case neg.inl\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na b : Associates α\nhb : b ≠ 0\np : Associates α\nhp : Irreducible p\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nhabk : k ∣ count p (factors (a * b))\nha : ¬a = 0\nhz : count p (factors a) = 0\n⊢ k ∣ 0",
"tactic": "exact dvd_zero k"
},
{
"state_after": "case neg.inr\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na b : Associates α\nhb : b ≠ 0\np : Associates α\nhp : Irreducible p\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nhabk : k ∣ count p (factors a) + 0\nha : ¬a = 0\nh : count p (factors b) = 0\n⊢ k ∣ count p (factors a)",
"state_before": "case neg.inr\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na b : Associates α\nhb : b ≠ 0\np : Associates α\nhp : Irreducible p\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nhabk : k ∣ count p (factors (a * b))\nha : ¬a = 0\nh : count p (factors b) = 0\n⊢ k ∣ count p (factors a)",
"tactic": "rw [count_mul ha hb hp, h] at habk"
},
{
"state_after": "no goals",
"state_before": "case neg.inr\nα : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ndec_irr : (p : Associates α) → Decidable (Irreducible p)\ninst✝ : UniqueFactorizationMonoid α\ndec : DecidableEq α\ndec' : DecidableEq (Associates α)\na b : Associates α\nhb : b ≠ 0\np : Associates α\nhp : Irreducible p\nhab : ∀ (d : Associates α), d ∣ a → d ∣ b → ¬Prime d\nk : ℕ\nhabk : k ∣ count p (factors a) + 0\nha : ¬a = 0\nh : count p (factors b) = 0\n⊢ k ∣ count p (factors a)",
"tactic": "exact habk"
}
] |
[
1785,
15
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1776,
1
] |
Mathlib/LinearAlgebra/Basic.lean
|
Submodule.mem_left_iff_eq_zero_of_disjoint
|
[] |
[
679,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
677,
1
] |
Mathlib/RingTheory/MvPolynomial/Homogeneous.lean
|
MvPolynomial.IsHomogeneous.prod
|
[
{
"state_after": "σ : Type u_2\nτ : Type ?u.161634\nR : Type u_3\nS : Type ?u.161640\ninst✝ : CommSemiring R\nφ✝ ψ : MvPolynomial σ R\nm n✝ : ℕ\nι : Type u_1\ns : Finset ι\nφ : ι → MvPolynomial σ R\nn : ι → ℕ\n⊢ (∀ (i : ι), i ∈ s → IsHomogeneous (φ i) (n i)) → IsHomogeneous (∏ i in s, φ i) (∑ i in s, n i)",
"state_before": "σ : Type u_2\nτ : Type ?u.161634\nR : Type u_3\nS : Type ?u.161640\ninst✝ : CommSemiring R\nφ✝ ψ : MvPolynomial σ R\nm n✝ : ℕ\nι : Type u_1\ns : Finset ι\nφ : ι → MvPolynomial σ R\nn : ι → ℕ\nh : ∀ (i : ι), i ∈ s → IsHomogeneous (φ i) (n i)\n⊢ IsHomogeneous (∏ i in s, φ i) (∑ i in s, n i)",
"tactic": "revert h"
},
{
"state_after": "case refine'_1\nσ : Type u_2\nτ : Type ?u.161634\nR : Type u_3\nS : Type ?u.161640\ninst✝ : CommSemiring R\nφ✝ ψ : MvPolynomial σ R\nm n✝ : ℕ\nι : Type u_1\ns : Finset ι\nφ : ι → MvPolynomial σ R\nn : ι → ℕ\n⊢ (∀ (i : ι), i ∈ ∅ → IsHomogeneous (φ i) (n i)) → IsHomogeneous (∏ i in ∅, φ i) (∑ i in ∅, n i)\n\ncase refine'_2\nσ : Type u_2\nτ : Type ?u.161634\nR : Type u_3\nS : Type ?u.161640\ninst✝ : CommSemiring R\nφ✝ ψ : MvPolynomial σ R\nm n✝ : ℕ\nι : Type u_1\ns : Finset ι\nφ : ι → MvPolynomial σ R\nn : ι → ℕ\n⊢ ∀ ⦃a : ι⦄ {s : Finset ι},\n ¬a ∈ s →\n ((∀ (i : ι), i ∈ s → IsHomogeneous (φ i) (n i)) → IsHomogeneous (∏ i in s, φ i) (∑ i in s, n i)) →\n (∀ (i : ι), i ∈ insert a s → IsHomogeneous (φ i) (n i)) →\n IsHomogeneous (∏ i in insert a s, φ i) (∑ i in insert a s, n i)",
"state_before": "σ : Type u_2\nτ : Type ?u.161634\nR : Type u_3\nS : Type ?u.161640\ninst✝ : CommSemiring R\nφ✝ ψ : MvPolynomial σ R\nm n✝ : ℕ\nι : Type u_1\ns : Finset ι\nφ : ι → MvPolynomial σ R\nn : ι → ℕ\n⊢ (∀ (i : ι), i ∈ s → IsHomogeneous (φ i) (n i)) → IsHomogeneous (∏ i in s, φ i) (∑ i in s, n i)",
"tactic": "refine' Finset.induction_on s _ _"
},
{
"state_after": "case refine'_1\nσ : Type u_2\nτ : Type ?u.161634\nR : Type u_3\nS : Type ?u.161640\ninst✝ : CommSemiring R\nφ✝ ψ : MvPolynomial σ R\nm n✝ : ℕ\nι : Type u_1\ns : Finset ι\nφ : ι → MvPolynomial σ R\nn : ι → ℕ\nh✝ : ∀ (i : ι), i ∈ ∅ → IsHomogeneous (φ i) (n i)\n⊢ IsHomogeneous (∏ i in ∅, φ i) (∑ i in ∅, n i)",
"state_before": "case refine'_1\nσ : Type u_2\nτ : Type ?u.161634\nR : Type u_3\nS : Type ?u.161640\ninst✝ : CommSemiring R\nφ✝ ψ : MvPolynomial σ R\nm n✝ : ℕ\nι : Type u_1\ns : Finset ι\nφ : ι → MvPolynomial σ R\nn : ι → ℕ\n⊢ (∀ (i : ι), i ∈ ∅ → IsHomogeneous (φ i) (n i)) → IsHomogeneous (∏ i in ∅, φ i) (∑ i in ∅, n i)",
"tactic": "intro"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nσ : Type u_2\nτ : Type ?u.161634\nR : Type u_3\nS : Type ?u.161640\ninst✝ : CommSemiring R\nφ✝ ψ : MvPolynomial σ R\nm n✝ : ℕ\nι : Type u_1\ns : Finset ι\nφ : ι → MvPolynomial σ R\nn : ι → ℕ\nh✝ : ∀ (i : ι), i ∈ ∅ → IsHomogeneous (φ i) (n i)\n⊢ IsHomogeneous (∏ i in ∅, φ i) (∑ i in ∅, n i)",
"tactic": "simp only [isHomogeneous_one, Finset.sum_empty, Finset.prod_empty]"
},
{
"state_after": "case refine'_2\nσ : Type u_2\nτ : Type ?u.161634\nR : Type u_3\nS : Type ?u.161640\ninst✝ : CommSemiring R\nφ✝ ψ : MvPolynomial σ R\nm n✝ : ℕ\nι : Type u_1\ns✝ : Finset ι\nφ : ι → MvPolynomial σ R\nn : ι → ℕ\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nIH : (∀ (i : ι), i ∈ s → IsHomogeneous (φ i) (n i)) → IsHomogeneous (∏ i in s, φ i) (∑ i in s, n i)\nh : ∀ (i_1 : ι), i_1 ∈ insert i s → IsHomogeneous (φ i_1) (n i_1)\n⊢ IsHomogeneous (∏ i in insert i s, φ i) (∑ i in insert i s, n i)",
"state_before": "case refine'_2\nσ : Type u_2\nτ : Type ?u.161634\nR : Type u_3\nS : Type ?u.161640\ninst✝ : CommSemiring R\nφ✝ ψ : MvPolynomial σ R\nm n✝ : ℕ\nι : Type u_1\ns : Finset ι\nφ : ι → MvPolynomial σ R\nn : ι → ℕ\n⊢ ∀ ⦃a : ι⦄ {s : Finset ι},\n ¬a ∈ s →\n ((∀ (i : ι), i ∈ s → IsHomogeneous (φ i) (n i)) → IsHomogeneous (∏ i in s, φ i) (∑ i in s, n i)) →\n (∀ (i : ι), i ∈ insert a s → IsHomogeneous (φ i) (n i)) →\n IsHomogeneous (∏ i in insert a s, φ i) (∑ i in insert a s, n i)",
"tactic": "intro i s his IH h"
},
{
"state_after": "case refine'_2\nσ : Type u_2\nτ : Type ?u.161634\nR : Type u_3\nS : Type ?u.161640\ninst✝ : CommSemiring R\nφ✝ ψ : MvPolynomial σ R\nm n✝ : ℕ\nι : Type u_1\ns✝ : Finset ι\nφ : ι → MvPolynomial σ R\nn : ι → ℕ\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nIH : (∀ (i : ι), i ∈ s → IsHomogeneous (φ i) (n i)) → IsHomogeneous (∏ i in s, φ i) (∑ i in s, n i)\nh : ∀ (i_1 : ι), i_1 ∈ insert i s → IsHomogeneous (φ i_1) (n i_1)\n⊢ IsHomogeneous (φ i * ∏ i in s, φ i) (n i + ∑ i in s, n i)",
"state_before": "case refine'_2\nσ : Type u_2\nτ : Type ?u.161634\nR : Type u_3\nS : Type ?u.161640\ninst✝ : CommSemiring R\nφ✝ ψ : MvPolynomial σ R\nm n✝ : ℕ\nι : Type u_1\ns✝ : Finset ι\nφ : ι → MvPolynomial σ R\nn : ι → ℕ\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nIH : (∀ (i : ι), i ∈ s → IsHomogeneous (φ i) (n i)) → IsHomogeneous (∏ i in s, φ i) (∑ i in s, n i)\nh : ∀ (i_1 : ι), i_1 ∈ insert i s → IsHomogeneous (φ i_1) (n i_1)\n⊢ IsHomogeneous (∏ i in insert i s, φ i) (∑ i in insert i s, n i)",
"tactic": "simp only [his, Finset.prod_insert, Finset.sum_insert, not_false_iff]"
},
{
"state_after": "σ : Type u_2\nτ : Type ?u.161634\nR : Type u_3\nS : Type ?u.161640\ninst✝ : CommSemiring R\nφ✝ ψ : MvPolynomial σ R\nm n✝ : ℕ\nι : Type u_1\ns✝ : Finset ι\nφ : ι → MvPolynomial σ R\nn : ι → ℕ\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nIH : (∀ (i : ι), i ∈ s → IsHomogeneous (φ i) (n i)) → IsHomogeneous (∏ i in s, φ i) (∑ i in s, n i)\nh : ∀ (i_1 : ι), i_1 ∈ insert i s → IsHomogeneous (φ i_1) (n i_1)\n⊢ ∀ (i : ι), i ∈ s → IsHomogeneous (φ i) (n i)",
"state_before": "case refine'_2\nσ : Type u_2\nτ : Type ?u.161634\nR : Type u_3\nS : Type ?u.161640\ninst✝ : CommSemiring R\nφ✝ ψ : MvPolynomial σ R\nm n✝ : ℕ\nι : Type u_1\ns✝ : Finset ι\nφ : ι → MvPolynomial σ R\nn : ι → ℕ\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nIH : (∀ (i : ι), i ∈ s → IsHomogeneous (φ i) (n i)) → IsHomogeneous (∏ i in s, φ i) (∑ i in s, n i)\nh : ∀ (i_1 : ι), i_1 ∈ insert i s → IsHomogeneous (φ i_1) (n i_1)\n⊢ IsHomogeneous (φ i * ∏ i in s, φ i) (n i + ∑ i in s, n i)",
"tactic": "apply (h i (Finset.mem_insert_self _ _)).mul (IH _)"
},
{
"state_after": "σ : Type u_2\nτ : Type ?u.161634\nR : Type u_3\nS : Type ?u.161640\ninst✝ : CommSemiring R\nφ✝ ψ : MvPolynomial σ R\nm n✝ : ℕ\nι : Type u_1\ns✝ : Finset ι\nφ : ι → MvPolynomial σ R\nn : ι → ℕ\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nIH : (∀ (i : ι), i ∈ s → IsHomogeneous (φ i) (n i)) → IsHomogeneous (∏ i in s, φ i) (∑ i in s, n i)\nh : ∀ (i_1 : ι), i_1 ∈ insert i s → IsHomogeneous (φ i_1) (n i_1)\nj : ι\nhjs : j ∈ s\n⊢ IsHomogeneous (φ j) (n j)",
"state_before": "σ : Type u_2\nτ : Type ?u.161634\nR : Type u_3\nS : Type ?u.161640\ninst✝ : CommSemiring R\nφ✝ ψ : MvPolynomial σ R\nm n✝ : ℕ\nι : Type u_1\ns✝ : Finset ι\nφ : ι → MvPolynomial σ R\nn : ι → ℕ\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nIH : (∀ (i : ι), i ∈ s → IsHomogeneous (φ i) (n i)) → IsHomogeneous (∏ i in s, φ i) (∑ i in s, n i)\nh : ∀ (i_1 : ι), i_1 ∈ insert i s → IsHomogeneous (φ i_1) (n i_1)\n⊢ ∀ (i : ι), i ∈ s → IsHomogeneous (φ i) (n i)",
"tactic": "intro j hjs"
},
{
"state_after": "no goals",
"state_before": "σ : Type u_2\nτ : Type ?u.161634\nR : Type u_3\nS : Type ?u.161640\ninst✝ : CommSemiring R\nφ✝ ψ : MvPolynomial σ R\nm n✝ : ℕ\nι : Type u_1\ns✝ : Finset ι\nφ : ι → MvPolynomial σ R\nn : ι → ℕ\ni : ι\ns : Finset ι\nhis : ¬i ∈ s\nIH : (∀ (i : ι), i ∈ s → IsHomogeneous (φ i) (n i)) → IsHomogeneous (∏ i in s, φ i) (∑ i in s, n i)\nh : ∀ (i_1 : ι), i_1 ∈ insert i s → IsHomogeneous (φ i_1) (n i_1)\nj : ι\nhjs : j ∈ s\n⊢ IsHomogeneous (φ j) (n j)",
"tactic": "exact h j (Finset.mem_insert_of_mem hjs)"
}
] |
[
205,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
194,
1
] |
Mathlib/Analysis/BoxIntegral/Partition/Basic.lean
|
BoxIntegral.Prepartition.ext
|
[] |
[
113,
34
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
112,
1
] |
Mathlib/Algebra/Order/Monoid/Lemmas.lean
|
eq_one_of_mul_le_one_right
|
[] |
[
1180,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1179,
1
] |
Mathlib/Data/QPF/Univariate/Basic.lean
|
Qpf.has_good_supp_iff
|
[
{
"state_after": "case mp\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\n⊢ (∀ (p : α → Prop), Liftp p x ↔ ∀ (u : α), u ∈ supp x → p u) →\n ∃ a f,\n abs { fst := a, snd := f } = x ∧\n ∀ (a' : (P F).A) (f' : PFunctor.B (P F) a' → α), abs { fst := a', snd := f' } = x → f '' univ ⊆ f' '' univ\n\ncase mpr\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\n⊢ (∃ a f,\n abs { fst := a, snd := f } = x ∧\n ∀ (a' : (P F).A) (f' : PFunctor.B (P F) a' → α), abs { fst := a', snd := f' } = x → f '' univ ⊆ f' '' univ) →\n ∀ (p : α → Prop), Liftp p x ↔ ∀ (u : α), u ∈ supp x → p u",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\n⊢ (∀ (p : α → Prop), Liftp p x ↔ ∀ (u : α), u ∈ supp x → p u) ↔\n ∃ a f,\n abs { fst := a, snd := f } = x ∧\n ∀ (a' : (P F).A) (f' : PFunctor.B (P F) a' → α), abs { fst := a', snd := f' } = x → f '' univ ⊆ f' '' univ",
"tactic": "constructor"
},
{
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{
"state_after": "case mp.intro.intro.intro.intro.intro\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\nh : ∀ (p : α → Prop), Liftp p x ↔ ∀ (u : α), u ∈ supp x → p u\nthis : Liftp (supp x) x\na : (P F).A\nf : PFunctor.B (P F) a → α\nxeq : x = abs { fst := a, snd := f }\nh' : ∀ (i : PFunctor.B (P F) a), supp x (f i)\na' : (P F).A\nf' : PFunctor.B (P F) a' → α\nh'' : abs { fst := a', snd := f' } = x\nu : α\ni : PFunctor.B (P F) a\nleft✝ : i ∈ univ\nhfi : f i = u\n⊢ u ∈ f' '' univ",
"state_before": "case mp.intro.intro.intro\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\nh : ∀ (p : α → Prop), Liftp p x ↔ ∀ (u : α), u ∈ supp x → p u\nthis : Liftp (supp x) x\na : (P F).A\nf : PFunctor.B (P F) a → α\nxeq : x = abs { fst := a, snd := f }\nh' : ∀ (i : PFunctor.B (P F) a), supp x (f i)\na' : (P F).A\nf' : PFunctor.B (P F) a' → α\nh'' : abs { fst := a', snd := f' } = x\n⊢ f '' univ ⊆ f' '' univ",
"tactic": "rintro u ⟨i, _, hfi⟩"
},
{
"state_after": "case mp.intro.intro.intro.intro.intro\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\nh : ∀ (p : α → Prop), Liftp p x ↔ ∀ (u : α), u ∈ supp x → p u\nthis✝ : Liftp (supp x) x\na : (P F).A\nf : PFunctor.B (P F) a → α\nxeq : x = abs { fst := a, snd := f }\nh' : ∀ (i : PFunctor.B (P F) a), supp x (f i)\na' : (P F).A\nf' : PFunctor.B (P F) a' → α\nh'' : abs { fst := a', snd := f' } = x\nu : α\ni : PFunctor.B (P F) a\nleft✝ : i ∈ univ\nhfi : f i = u\nthis : u ∈ supp x\n⊢ u ∈ f' '' univ",
"state_before": "case mp.intro.intro.intro.intro.intro\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\nh : ∀ (p : α → Prop), Liftp p x ↔ ∀ (u : α), u ∈ supp x → p u\nthis : Liftp (supp x) x\na : (P F).A\nf : PFunctor.B (P F) a → α\nxeq : x = abs { fst := a, snd := f }\nh' : ∀ (i : PFunctor.B (P F) a), supp x (f i)\na' : (P F).A\nf' : PFunctor.B (P F) a' → α\nh'' : abs { fst := a', snd := f' } = x\nu : α\ni : PFunctor.B (P F) a\nleft✝ : i ∈ univ\nhfi : f i = u\n⊢ u ∈ f' '' univ",
"tactic": "have : u ∈ supp x := by rw [← hfi]; apply h'"
},
{
"state_after": "no goals",
"state_before": "case mp.intro.intro.intro.intro.intro\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\nh : ∀ (p : α → Prop), Liftp p x ↔ ∀ (u : α), u ∈ supp x → p u\nthis✝ : Liftp (supp x) x\na : (P F).A\nf : PFunctor.B (P F) a → α\nxeq : x = abs { fst := a, snd := f }\nh' : ∀ (i : PFunctor.B (P F) a), supp x (f i)\na' : (P F).A\nf' : PFunctor.B (P F) a' → α\nh'' : abs { fst := a', snd := f' } = x\nu : α\ni : PFunctor.B (P F) a\nleft✝ : i ∈ univ\nhfi : f i = u\nthis : u ∈ supp x\n⊢ u ∈ f' '' univ",
"tactic": "exact (mem_supp x u).mp this _ _ h''"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\nh : ∀ (p : α → Prop), Liftp p x ↔ ∀ (u : α), u ∈ supp x → p u\n⊢ ∀ (u : α), u ∈ supp x → supp x u",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\nh : ∀ (p : α → Prop), Liftp p x ↔ ∀ (u : α), u ∈ supp x → p u\n⊢ Liftp (supp x) x",
"tactic": "rw [h]"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\nh : ∀ (p : α → Prop), Liftp p x ↔ ∀ (u : α), u ∈ supp x → p u\nu : α\n⊢ u ∈ supp x → supp x u",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\nh : ∀ (p : α → Prop), Liftp p x ↔ ∀ (u : α), u ∈ supp x → p u\n⊢ ∀ (u : α), u ∈ supp x → supp x u",
"tactic": "intro u"
},
{
"state_after": "no goals",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\nh : ∀ (p : α → Prop), Liftp p x ↔ ∀ (u : α), u ∈ supp x → p u\nu : α\n⊢ u ∈ supp x → supp x u",
"tactic": "exact id"
},
{
"state_after": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\nh : ∀ (p : α → Prop), Liftp p x ↔ ∀ (u : α), u ∈ supp x → p u\nthis : Liftp (supp x) x\na : (P F).A\nf : PFunctor.B (P F) a → α\nxeq : x = abs { fst := a, snd := f }\nh' : ∀ (i : PFunctor.B (P F) a), supp x (f i)\na' : (P F).A\nf' : PFunctor.B (P F) a' → α\nh'' : abs { fst := a', snd := f' } = x\nu : α\ni : PFunctor.B (P F) a\nleft✝ : i ∈ univ\nhfi : f i = u\n⊢ f i ∈ supp x",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\nh : ∀ (p : α → Prop), Liftp p x ↔ ∀ (u : α), u ∈ supp x → p u\nthis : Liftp (supp x) x\na : (P F).A\nf : PFunctor.B (P F) a → α\nxeq : x = abs { fst := a, snd := f }\nh' : ∀ (i : PFunctor.B (P F) a), supp x (f i)\na' : (P F).A\nf' : PFunctor.B (P F) a' → α\nh'' : abs { fst := a', snd := f' } = x\nu : α\ni : PFunctor.B (P F) a\nleft✝ : i ∈ univ\nhfi : f i = u\n⊢ u ∈ supp x",
"tactic": "rw [← hfi]"
},
{
"state_after": "no goals",
"state_before": "F : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\nh : ∀ (p : α → Prop), Liftp p x ↔ ∀ (u : α), u ∈ supp x → p u\nthis : Liftp (supp x) x\na : (P F).A\nf : PFunctor.B (P F) a → α\nxeq : x = abs { fst := a, snd := f }\nh' : ∀ (i : PFunctor.B (P F) a), supp x (f i)\na' : (P F).A\nf' : PFunctor.B (P F) a' → α\nh'' : abs { fst := a', snd := f' } = x\nu : α\ni : PFunctor.B (P F) a\nleft✝ : i ∈ univ\nhfi : f i = u\n⊢ f i ∈ supp x",
"tactic": "apply h'"
},
{
"state_after": "case mpr.intro.intro.intro.mp.intro.intro.intro\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\na : (P F).A\nf : PFunctor.B (P F) a → α\nxeq : abs { fst := a, snd := f } = x\nh : ∀ (a' : (P F).A) (f' : PFunctor.B (P F) a' → α), abs { fst := a', snd := f' } = x → f '' univ ⊆ f' '' univ\np : α → Prop\na' : (P F).A\nf' : PFunctor.B (P F) a' → α\nxeq' : x = abs { fst := a', snd := f' }\nh' : ∀ (i : PFunctor.B (P F) a'), p (f' i)\nu : α\nusuppx : u ∈ supp x\n⊢ p u",
"state_before": "case mpr.intro.intro.intro.mp\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\na : (P F).A\nf : PFunctor.B (P F) a → α\nxeq : abs { fst := a, snd := f } = x\nh : ∀ (a' : (P F).A) (f' : PFunctor.B (P F) a' → α), abs { fst := a', snd := f' } = x → f '' univ ⊆ f' '' univ\np : α → Prop\n⊢ (∃ a f, x = abs { fst := a, snd := f } ∧ ∀ (i : PFunctor.B (P F) a), p (f i)) → ∀ (u : α), u ∈ supp x → p u",
"tactic": "rintro ⟨a', f', xeq', h'⟩ u usuppx"
},
{
"state_after": "case mpr.intro.intro.intro.mp.intro.intro.intro.intro.intro\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\na : (P F).A\nf : PFunctor.B (P F) a → α\nxeq : abs { fst := a, snd := f } = x\nh : ∀ (a' : (P F).A) (f' : PFunctor.B (P F) a' → α), abs { fst := a', snd := f' } = x → f '' univ ⊆ f' '' univ\np : α → Prop\na' : (P F).A\nf' : PFunctor.B (P F) a' → α\nxeq' : x = abs { fst := a', snd := f' }\nh' : ∀ (i : PFunctor.B (P F) a'), p (f' i)\nu : α\nusuppx : u ∈ supp x\ni : PFunctor.B (P F) a'\nleft✝ : i ∈ univ\nf'ieq : f' i = u\n⊢ p u",
"state_before": "case mpr.intro.intro.intro.mp.intro.intro.intro\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\na : (P F).A\nf : PFunctor.B (P F) a → α\nxeq : abs { fst := a, snd := f } = x\nh : ∀ (a' : (P F).A) (f' : PFunctor.B (P F) a' → α), abs { fst := a', snd := f' } = x → f '' univ ⊆ f' '' univ\np : α → Prop\na' : (P F).A\nf' : PFunctor.B (P F) a' → α\nxeq' : x = abs { fst := a', snd := f' }\nh' : ∀ (i : PFunctor.B (P F) a'), p (f' i)\nu : α\nusuppx : u ∈ supp x\n⊢ p u",
"tactic": "rcases (mem_supp x u).mp usuppx a' f' xeq'.symm with ⟨i, _, f'ieq⟩"
},
{
"state_after": "case mpr.intro.intro.intro.mp.intro.intro.intro.intro.intro\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\na : (P F).A\nf : PFunctor.B (P F) a → α\nxeq : abs { fst := a, snd := f } = x\nh : ∀ (a' : (P F).A) (f' : PFunctor.B (P F) a' → α), abs { fst := a', snd := f' } = x → f '' univ ⊆ f' '' univ\np : α → Prop\na' : (P F).A\nf' : PFunctor.B (P F) a' → α\nxeq' : x = abs { fst := a', snd := f' }\nh' : ∀ (i : PFunctor.B (P F) a'), p (f' i)\nu : α\nusuppx : u ∈ supp x\ni : PFunctor.B (P F) a'\nleft✝ : i ∈ univ\nf'ieq : f' i = u\n⊢ p (f' i)",
"state_before": "case mpr.intro.intro.intro.mp.intro.intro.intro.intro.intro\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\na : (P F).A\nf : PFunctor.B (P F) a → α\nxeq : abs { fst := a, snd := f } = x\nh : ∀ (a' : (P F).A) (f' : PFunctor.B (P F) a' → α), abs { fst := a', snd := f' } = x → f '' univ ⊆ f' '' univ\np : α → Prop\na' : (P F).A\nf' : PFunctor.B (P F) a' → α\nxeq' : x = abs { fst := a', snd := f' }\nh' : ∀ (i : PFunctor.B (P F) a'), p (f' i)\nu : α\nusuppx : u ∈ supp x\ni : PFunctor.B (P F) a'\nleft✝ : i ∈ univ\nf'ieq : f' i = u\n⊢ p u",
"tactic": "rw [← f'ieq]"
},
{
"state_after": "no goals",
"state_before": "case mpr.intro.intro.intro.mp.intro.intro.intro.intro.intro\nF : Type u → Type u\ninst✝ : Functor F\nq : Qpf F\nα : Type u\nx : F α\na : (P F).A\nf : PFunctor.B (P F) a → α\nxeq : abs { fst := a, snd := f } = x\nh : ∀ (a' : (P F).A) (f' : PFunctor.B (P F) a' → α), abs { fst := a', snd := f' } = x → f '' univ ⊆ f' '' univ\np : α → Prop\na' : (P F).A\nf' : PFunctor.B (P F) a' → α\nxeq' : x = abs { fst := a', snd := f' }\nh' : ∀ (i : PFunctor.B (P F) a'), p (f' i)\nu : α\nusuppx : u ∈ supp x\ni : PFunctor.B (P F) a'\nleft✝ : i ∈ univ\nf'ieq : f' i = u\n⊢ p (f' i)",
"tactic": "apply h'"
}
] |
[
655,
40
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
632,
1
] |
Mathlib/Init/Algebra/Order.lean
|
lt_iff_not_ge
|
[] |
[
368,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
367,
1
] |
Mathlib/Tactic/Ring/Basic.lean
|
Mathlib.Tactic.Ring.add_pos_right
|
[] |
[
613,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
613,
1
] |
Mathlib/Topology/Algebra/Module/FiniteDimension.lean
|
FiniteDimensional.nonempty_continuousLinearEquiv_of_finrank_eq
|
[] |
[
417,
84
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
415,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean
|
Subgroup.map_eq_bot_iff_of_injective
|
[
{
"state_after": "no goals",
"state_before": "G : Type u_1\nG' : Type ?u.556305\ninst✝³ : Group G\ninst✝² : Group G'\nA : Type ?u.556314\ninst✝¹ : AddGroup A\nN : Type u_2\ninst✝ : Group N\nH : Subgroup G\nf : G →* N\nhf : Injective ↑f\n⊢ map f H = ⊥ ↔ H = ⊥",
"tactic": "rw [map_eq_bot_iff, f.ker_eq_bot_iff.mpr hf, le_bot_iff]"
}
] |
[
2964,
87
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2963,
1
] |
Mathlib/CategoryTheory/Subobject/Basic.lean
|
CategoryTheory.Subobject.ofLEMk_comp_ofMkLE
|
[
{
"state_after": "C : Type u₁\ninst✝² : Category C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝¹ : Category D\nB A : C\nX : Subobject B\nf : A ⟶ B\ninst✝ : Mono f\nY : Subobject B\nh₁ : X ≤ mk f\nh₂ : mk f ≤ Y\n⊢ underlying.map (LE.le.hom h₁ ≫ LE.le.hom h₂) = underlying.map (LE.le.hom (_ : X ≤ Y))",
"state_before": "C : Type u₁\ninst✝² : Category C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝¹ : Category D\nB A : C\nX : Subobject B\nf : A ⟶ B\ninst✝ : Mono f\nY : Subobject B\nh₁ : X ≤ mk f\nh₂ : mk f ≤ Y\n⊢ ofLEMk X f h₁ ≫ ofMkLE f Y h₂ = ofLE X Y (_ : X ≤ Y)",
"tactic": "simp only [ofMkLE, ofLEMk, ofLE, ← Functor.map_comp underlying, assoc, Iso.hom_inv_id_assoc]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝² : Category C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝¹ : Category D\nB A : C\nX : Subobject B\nf : A ⟶ B\ninst✝ : Mono f\nY : Subobject B\nh₁ : X ≤ mk f\nh₂ : mk f ≤ Y\n⊢ underlying.map (LE.le.hom h₁ ≫ LE.le.hom h₂) = underlying.map (LE.le.hom (_ : X ≤ Y))",
"tactic": "congr 1"
}
] |
[
406,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
403,
1
] |
Mathlib/Algebra/MonoidAlgebra/Support.lean
|
AddMonoidAlgebra.support_single_mul
|
[] |
[
137,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
134,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/NNReal.lean
|
NNReal.rpow_le_one
|
[] |
[
205,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
204,
1
] |
Mathlib/Data/Real/ENNReal.lean
|
ENNReal.toNNReal_sum
|
[
{
"state_after": "α : Type u_1\nβ : Type ?u.202901\na b c d : ℝ≥0∞\nr p q : ℝ≥0\ns : Finset α\nf : α → ℝ≥0∞\nhf : ∀ (a : α), a ∈ s → f a ≠ ⊤\n⊢ ∀ (x : α), x ∈ s → f x = ↑(ENNReal.toNNReal (f x))\n\nα : Type u_1\nβ : Type ?u.202901\na b c d : ℝ≥0∞\nr p q : ℝ≥0\ns : Finset α\nf : α → ℝ≥0∞\nhf : ∀ (a : α), a ∈ s → f a ≠ ⊤\n⊢ ∑ a in s, f a ≠ ⊤",
"state_before": "α : Type u_1\nβ : Type ?u.202901\na b c d : ℝ≥0∞\nr p q : ℝ≥0\ns : Finset α\nf : α → ℝ≥0∞\nhf : ∀ (a : α), a ∈ s → f a ≠ ⊤\n⊢ ENNReal.toNNReal (∑ a in s, f a) = ∑ a in s, ENNReal.toNNReal (f a)",
"tactic": "rw [← coe_eq_coe, coe_toNNReal, coe_finset_sum, sum_congr rfl]"
},
{
"state_after": "α : Type u_1\nβ : Type ?u.202901\na b c d : ℝ≥0∞\nr p q : ℝ≥0\ns : Finset α\nf : α → ℝ≥0∞\nhf : ∀ (a : α), a ∈ s → f a ≠ ⊤\nx : α\nhx : x ∈ s\n⊢ f x = ↑(ENNReal.toNNReal (f x))",
"state_before": "α : Type u_1\nβ : Type ?u.202901\na b c d : ℝ≥0∞\nr p q : ℝ≥0\ns : Finset α\nf : α → ℝ≥0∞\nhf : ∀ (a : α), a ∈ s → f a ≠ ⊤\n⊢ ∀ (x : α), x ∈ s → f x = ↑(ENNReal.toNNReal (f x))",
"tactic": "intro x hx"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.202901\na b c d : ℝ≥0∞\nr p q : ℝ≥0\ns : Finset α\nf : α → ℝ≥0∞\nhf : ∀ (a : α), a ∈ s → f a ≠ ⊤\nx : α\nhx : x ∈ s\n⊢ f x = ↑(ENNReal.toNNReal (f x))",
"tactic": "exact (coe_toNNReal (hf x hx)).symm"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nβ : Type ?u.202901\na b c d : ℝ≥0∞\nr p q : ℝ≥0\ns : Finset α\nf : α → ℝ≥0∞\nhf : ∀ (a : α), a ∈ s → f a ≠ ⊤\n⊢ ∑ a in s, f a ≠ ⊤",
"tactic": "exact (sum_lt_top hf).ne"
}
] |
[
1247,
29
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1242,
1
] |
Mathlib/Analysis/Calculus/FDerivMeasurable.lean
|
FDerivMeasurableAux.differentiable_set_subset_d
|
[
{
"state_after": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\n⊢ x ∈ D f K",
"state_before": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\n⊢ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K} ⊆ D f K",
"tactic": "intro x hx"
},
{
"state_after": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\n⊢ ∀ (i : ℕ), x ∈ ⋃ (n : ℕ), ⋂ (p : ℕ) (_ : p ≥ n) (q : ℕ) (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ i)",
"state_before": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\n⊢ x ∈ D f K",
"tactic": "rw [D, mem_iInter]"
},
{
"state_after": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\ne : ℕ\n⊢ x ∈ ⋃ (n : ℕ), ⋂ (p : ℕ) (_ : p ≥ n) (q : ℕ) (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)",
"state_before": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\n⊢ ∀ (i : ℕ), x ∈ ⋃ (n : ℕ), ⋂ (p : ℕ) (_ : p ≥ n) (q : ℕ) (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ i)",
"tactic": "intro e"
},
{
"state_after": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\n⊢ x ∈ ⋃ (n : ℕ), ⋂ (p : ℕ) (_ : p ≥ n) (q : ℕ) (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)",
"state_before": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\ne : ℕ\n⊢ x ∈ ⋃ (n : ℕ), ⋂ (p : ℕ) (_ : p ≥ n) (q : ℕ) (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)",
"tactic": "have : (0 : ℝ) < (1 / 2) ^ e := pow_pos (by norm_num) _"
},
{
"state_after": "case intro.intro\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)\n⊢ x ∈ ⋃ (n : ℕ), ⋂ (p : ℕ) (_ : p ≥ n) (q : ℕ) (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)",
"state_before": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\n⊢ x ∈ ⋃ (n : ℕ), ⋂ (p : ℕ) (_ : p ≥ n) (q : ℕ) (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)",
"tactic": "rcases mem_a_of_differentiable this hx.1 with ⟨R, R_pos, hR⟩"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)\nn : ℕ\nhn : (1 / 2) ^ n < R\n⊢ x ∈ ⋃ (n : ℕ), ⋂ (p : ℕ) (_ : p ≥ n) (q : ℕ) (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)",
"state_before": "case intro.intro\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)\n⊢ x ∈ ⋃ (n : ℕ), ⋂ (p : ℕ) (_ : p ≥ n) (q : ℕ) (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)",
"tactic": "obtain ⟨n, hn⟩ : ∃ n : ℕ, (1 / 2) ^ n < R :=\n exists_pow_lt_of_lt_one R_pos (by norm_num : (1 : ℝ) / 2 < 1)"
},
{
"state_after": "case intro.intro.intro\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)\nn : ℕ\nhn : (1 / 2) ^ n < R\n⊢ ∃ i,\n ∀ (i_1 : ℕ),\n i_1 ≥ i →\n ∀ (i_3 : ℕ), i_3 ≥ i → ∃ i h, x ∈ A f i ((1 / 2) ^ i_1) ((1 / 2) ^ e) ∧ x ∈ A f i ((1 / 2) ^ i_3) ((1 / 2) ^ e)",
"state_before": "case intro.intro.intro\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)\nn : ℕ\nhn : (1 / 2) ^ n < R\n⊢ x ∈ ⋃ (n : ℕ), ⋂ (p : ℕ) (_ : p ≥ n) (q : ℕ) (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)",
"tactic": "simp only [mem_iUnion, mem_iInter, B, mem_inter_iff]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\ne : ℕ\n⊢ 0 < 1 / 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)\n⊢ 1 / 2 < 1",
"tactic": "norm_num"
},
{
"state_after": "case intro.intro.intro.refine'_2\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)\nn : ℕ\nhn : (1 / 2) ^ n < R\np : ℕ\nhp : p ≥ n\nq : ℕ\nhq : q ≥ n\n⊢ (1 / 2) ^ q ≤ (1 / 2) ^ n",
"state_before": "case intro.intro.intro.refine'_2\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)\nn : ℕ\nhn : (1 / 2) ^ n < R\np : ℕ\nhp : p ≥ n\nq : ℕ\nhq : q ≥ n\n⊢ x ∈ A f (fderiv 𝕜 f x) ((1 / 2) ^ q) ((1 / 2) ^ e)",
"tactic": "refine' hR _ ⟨pow_pos (by norm_num) _, lt_of_le_of_lt _ hn⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.refine'_2\n𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)\nn : ℕ\nhn : (1 / 2) ^ n < R\np : ℕ\nhp : p ≥ n\nq : ℕ\nhq : q ≥ n\n⊢ (1 / 2) ^ q ≤ (1 / 2) ^ n",
"tactic": "exact pow_le_pow_of_le_one (by norm_num) (by norm_num) (by assumption)"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)\nn : ℕ\nhn : (1 / 2) ^ n < R\np : ℕ\nhp : p ≥ n\nq : ℕ\nhq : q ≥ n\n⊢ 0 < 1 / 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)\nn : ℕ\nhn : (1 / 2) ^ n < R\np : ℕ\nhp : p ≥ n\nq : ℕ\nhq : q ≥ n\n⊢ 0 ≤ 1 / 2",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)\nn : ℕ\nhn : (1 / 2) ^ n < R\np : ℕ\nhp : p ≥ n\nq : ℕ\nhq : q ≥ n\n⊢ 1 / 2 ≤ 1",
"tactic": "norm_num"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nx : E\nhx : x ∈ {x | DifferentiableAt 𝕜 f x ∧ fderiv 𝕜 f x ∈ K}\ne : ℕ\nthis : 0 < (1 / 2) ^ e\nR : ℝ\nR_pos : R > 0\nhR : ∀ (r : ℝ), r ∈ Ioo 0 R → x ∈ A f (fderiv 𝕜 f x) r ((1 / 2) ^ e)\nn : ℕ\nhn : (1 / 2) ^ n < R\np : ℕ\nhp : p ≥ n\nq : ℕ\nhq : q ≥ n\n⊢ n ≤ q",
"tactic": "assumption"
}
] |
[
223,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
211,
1
] |
Mathlib/Order/WithBot.lean
|
WithTop.le_toDual_iff
|
[] |
[
803,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
801,
1
] |
Mathlib/Combinatorics/Additive/RuzsaCovering.lean
|
Finset.exists_subset_mul_div
|
[
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ∈ C\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\n⊢ ∃ u, card u * card t ≤ card (s * t) ∧ s ⊆ u * t / t",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\n⊢ ∃ u, card u * card t ≤ card (s * t) ∧ s ⊆ u * t / t",
"tactic": "obtain ⟨u, hu, hCmax⟩ := C.exists_maximal (filter_nonempty_iff.2\n ⟨∅, empty_mem_powerset _, by rw [coe_empty]; exact Set.pairwiseDisjoint_empty⟩)"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\n⊢ ∃ u, card u * card t ≤ card (s * t) ∧ s ⊆ u * t / t",
"state_before": "case intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ∈ C\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\n⊢ ∃ u, card u * card t ≤ card (s * t) ∧ s ⊆ u * t / t",
"tactic": "rw [mem_filter, mem_powerset] at hu"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\n⊢ a ∈ u * t / t",
"state_before": "case intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\n⊢ ∃ u, card u * card t ≤ card (s * t) ∧ s ⊆ u * t / t",
"tactic": "refine' ⟨u,\n (card_mul_iff.2 <| pairwiseDisjoint_smul_iff.1 hu.2).ge.trans\n (card_le_of_subset <| mul_subset_mul_right hu.1),\n fun a ha ↦ _⟩"
},
{
"state_after": "case intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\n⊢ a ∈ u * (t / t)",
"state_before": "case intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\n⊢ a ∈ u * t / t",
"tactic": "rw [mul_div_assoc]"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : a ∈ u\n⊢ a ∈ u * (t / t)\n\ncase neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : ¬a ∈ u\n⊢ a ∈ u * (t / t)",
"state_before": "case intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\n⊢ a ∈ u * (t / t)",
"tactic": "by_cases hau : a ∈ u"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : ¬a ∈ u\nH : ∀ (b : α), b ∈ u → Disjoint (a • t) (b • t)\n⊢ a ∈ u * (t / t)\n\ncase neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : ¬a ∈ u\nH : ¬∀ (b : α), b ∈ u → Disjoint (a • t) (b • t)\n⊢ a ∈ u * (t / t)",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : ¬a ∈ u\n⊢ a ∈ u * (t / t)",
"tactic": "by_cases H : ∀ b ∈ u, Disjoint (a • t) (b • t)"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : ¬a ∈ u\nH : ∃ b, b ∈ u ∧ ¬Disjoint (a • t) (b • t)\n⊢ a ∈ u * (t / t)",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : ¬a ∈ u\nH : ¬∀ (b : α), b ∈ u → Disjoint (a • t) (b • t)\n⊢ a ∈ u * (t / t)",
"tactic": "push_neg at H"
},
{
"state_after": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : ¬a ∈ u\nH : ∃ b, b ∈ u ∧ ∃ a_1, a⁻¹ • a_1 ∈ t ∧ b⁻¹ • a_1 ∈ t\n⊢ a ∈ u * (t / t)",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : ¬a ∈ u\nH : ∃ b, b ∈ u ∧ ¬Disjoint (a • t) (b • t)\n⊢ a ∈ u * (t / t)",
"tactic": "simp_rw [not_disjoint_iff, ← inv_smul_mem_iff] at H"
},
{
"state_after": "case neg.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : ¬a ∈ u\nb : α\nhb : b ∈ u\nc : α\nhc₁ : a⁻¹ • c ∈ t\nhc₂ : b⁻¹ • c ∈ t\n⊢ a ∈ u * (t / t)",
"state_before": "case neg\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : ¬a ∈ u\nH : ∃ b, b ∈ u ∧ ∃ a_1, a⁻¹ • a_1 ∈ t ∧ b⁻¹ • a_1 ∈ t\n⊢ a ∈ u * (t / t)",
"tactic": "obtain ⟨b, hb, c, hc₁, hc₂⟩ := H"
},
{
"state_after": "case neg.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : ¬a ∈ u\nb : α\nhb : b ∈ u\nc : α\nhc₁ : a⁻¹ • c ∈ t\nhc₂ : b⁻¹ • c ∈ t\n⊢ a / b ∈ t / t",
"state_before": "case neg.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : ¬a ∈ u\nb : α\nhb : b ∈ u\nc : α\nhc₁ : a⁻¹ • c ∈ t\nhc₂ : b⁻¹ • c ∈ t\n⊢ a ∈ u * (t / t)",
"tactic": "refine' mem_mul.2 ⟨b, a / b, hb, _, by simp⟩"
},
{
"state_after": "no goals",
"state_before": "case neg.intro.intro.intro.intro\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : ¬a ∈ u\nb : α\nhb : b ∈ u\nc : α\nhc₁ : a⁻¹ • c ∈ t\nhc₂ : b⁻¹ • c ∈ t\n⊢ a / b ∈ t / t",
"tactic": "exact mem_div.2 ⟨_, _, hc₂, hc₁, by simp [div_eq_mul_inv a b, mul_comm]⟩"
},
{
"state_after": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\n⊢ Set.PairwiseDisjoint ∅ fun x => x • t",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\n⊢ Set.PairwiseDisjoint ↑∅ fun x => x • t",
"tactic": "rw [coe_empty]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\n⊢ Set.PairwiseDisjoint ∅ fun x => x • t",
"tactic": "exact Set.pairwiseDisjoint_empty"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : a ∈ u\n⊢ a ∈ u * (t / t)",
"tactic": "exact subset_mul_left _ ht.one_mem_div hau"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : ¬a ∈ u\nH : ∀ (b : α), b ∈ u → Disjoint (a • t) (b • t)\n⊢ insert a u ∈ C",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : ¬a ∈ u\nH : ∀ (b : α), b ∈ u → Disjoint (a • t) (b • t)\n⊢ a ∈ u * (t / t)",
"tactic": "refine' (hCmax _ _ <| ssubset_insert hau).elim"
},
{
"state_after": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : ¬a ∈ u\nH : ∀ (b : α), b ∈ u → Disjoint (a • t) (b • t)\n⊢ (a ∈ s ∧ u ⊆ s) ∧ Set.PairwiseDisjoint (insert a ↑u) fun x => x • t",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : ¬a ∈ u\nH : ∀ (b : α), b ∈ u → Disjoint (a • t) (b • t)\n⊢ insert a u ∈ C",
"tactic": "rw [mem_filter, mem_powerset, insert_subset, coe_insert]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : ¬a ∈ u\nH : ∀ (b : α), b ∈ u → Disjoint (a • t) (b • t)\n⊢ (a ∈ s ∧ u ⊆ s) ∧ Set.PairwiseDisjoint (insert a ↑u) fun x => x • t",
"tactic": "exact ⟨⟨ha, hu.1⟩, hu.2.insert fun _ hb _ ↦ H _ hb⟩"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : ¬a ∈ u\nb : α\nhb : b ∈ u\nc : α\nhc₁ : a⁻¹ • c ∈ t\nhc₂ : b⁻¹ • c ∈ t\n⊢ b * (a / b) = a",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : CommGroup α\ns t : Finset α\nht : Finset.Nonempty t\nthis : (u : Set α) → Decidable (Set.PairwiseDisjoint u fun x => x • t)\nC : Finset (Finset α) := filter (fun u => Set.PairwiseDisjoint ↑u fun x => x • t) (powerset s)\nu : Finset α\nhu : u ⊆ s ∧ Set.PairwiseDisjoint ↑u fun x => x • t\nhCmax : ∀ (x : Finset α), x ∈ C → ¬u < x\na : α\nha : a ∈ s\nhau : ¬a ∈ u\nb : α\nhb : b ∈ u\nc : α\nhc₁ : a⁻¹ • c ∈ t\nhc₂ : b⁻¹ • c ∈ t\n⊢ b⁻¹ • c / a⁻¹ • c = a / b",
"tactic": "simp [div_eq_mul_inv a b, mul_comm]"
}
] |
[
55,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
33,
1
] |
Mathlib/ModelTheory/Satisfiability.lean
|
FirstOrder.Language.Theory.SemanticallyEquivalent.not
|
[
{
"state_after": "L : Language\nT : Theory L\nα : Type w\nn : ℕ\nφ ψ : BoundedFormula L α n\nh : SemanticallyEquivalent T φ ψ\n⊢ ∀ (M : ModelType T) (v : α → ↑M) (xs : Fin n → ↑M), ¬BoundedFormula.Realize φ v xs ↔ ¬BoundedFormula.Realize ψ v xs",
"state_before": "L : Language\nT : Theory L\nα : Type w\nn : ℕ\nφ ψ : BoundedFormula L α n\nh : SemanticallyEquivalent T φ ψ\n⊢ SemanticallyEquivalent T (∼φ) ∼ψ",
"tactic": "simp_rw [SemanticallyEquivalent, ModelsBoundedFormula, BoundedFormula.realize_iff,\n BoundedFormula.realize_not]"
},
{
"state_after": "no goals",
"state_before": "L : Language\nT : Theory L\nα : Type w\nn : ℕ\nφ ψ : BoundedFormula L α n\nh : SemanticallyEquivalent T φ ψ\n⊢ ∀ (M : ModelType T) (v : α → ↑M) (xs : Fin n → ↑M), ¬BoundedFormula.Realize φ v xs ↔ ¬BoundedFormula.Realize ψ v xs",
"tactic": "exact fun M v xs => not_congr h.realize_bd_iff"
}
] |
[
482,
49
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
478,
11
] |
Mathlib/Topology/Order.lean
|
eq_of_nhds_eq_nhds
|
[] |
[
311,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
309,
1
] |
Mathlib/MeasureTheory/Measure/MeasureSpaceDef.lean
|
MeasureTheory.ae_eq_set_union
|
[] |
[
508,
13
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
506,
1
] |
Mathlib/Topology/Basic.lean
|
ClusterPt.map
|
[] |
[
1618,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1616,
1
] |
Mathlib/NumberTheory/BernoulliPolynomials.lean
|
Polynomial.bernoulli_eq_sub_sum
|
[
{
"state_after": "no goals",
"state_before": "n : ℕ\n⊢ ↑(succ n) • bernoulli n = ↑(monomial n) ↑(succ n) - ∑ k in range n, ↑(Nat.choose (n + 1) k) • bernoulli k",
"tactic": "rw [Nat.cast_succ, ← sum_bernoulli n, sum_range_succ, add_sub_cancel', choose_succ_self_right,\n Nat.cast_succ]"
}
] |
[
160,
19
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
156,
1
] |
Mathlib/Topology/Sequences.lean
|
IsSeqCompact.exists_tendsto_of_frequently_mem
|
[] |
[
331,
76
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean
|
CategoryTheory.Limits.coprod.desc_comp_assoc
|
[
{
"state_after": "no goals",
"state_before": "C✝ : Type u\ninst✝² : Category C✝\nX✝ Y✝ : C✝\nC : Type u\ninst✝¹ : Category C\nV W X Y : C\ninst✝ : HasBinaryCoproduct X Y\nf : V ⟶ W\ng : X ⟶ V\nh : Y ⟶ V\nZ : C\nl : W ⟶ Z\n⊢ desc g h ≫ f ≫ l = desc (g ≫ f) (h ≫ f) ≫ l",
"tactic": "simp"
}
] |
[
835,
73
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
833,
1
] |
Mathlib/LinearAlgebra/Isomorphisms.lean
|
Submodule.card_quotient_mul_card_quotient
|
[
{
"state_after": "no goals",
"state_before": "R : Type u_1\nM : Type u_2\nM₂ : Type ?u.157808\nM₃ : Type ?u.157811\ninst✝⁹ : Ring R\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : AddCommGroup M₂\ninst✝⁶ : AddCommGroup M₃\ninst✝⁵ : Module R M\ninst✝⁴ : Module R M₂\ninst✝³ : Module R M₃\nf : M →ₗ[R] M₂\nS✝ T✝ : Submodule R M\nh : S✝ ≤ T✝\nS T : Submodule R M\nhST : T ≤ S\ninst✝² : DecidablePred fun x => x ∈ map (mkQ T) S\ninst✝¹ : Fintype (M ⧸ S)\ninst✝ : Fintype (M ⧸ T)\n⊢ Fintype.card { x // x ∈ map (mkQ T) S } * Fintype.card (M ⧸ S) = Fintype.card (M ⧸ T)",
"tactic": "rw [Submodule.card_eq_card_quotient_mul_card (map T.mkQ S),\n Fintype.card_eq.mpr ⟨(quotientQuotientEquivQuotient T S hST).toEquiv⟩]"
}
] |
[
195,
75
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
191,
1
] |
Mathlib/MeasureTheory/Measure/AEMeasurable.lean
|
AEMeasurable.comp_measurable
|
[] |
[
166,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
164,
1
] |
Mathlib/Logic/Basic.lean
|
exists_or_eq_right
|
[] |
[
793,
98
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
793,
9
] |
Mathlib/CategoryTheory/Adjunction/Mates.lean
|
CategoryTheory.transferNatTransSelf_adjunction_id
|
[
{
"state_after": "C : Type u₁\nD : Type u₂\ninst✝¹ : Category C\ninst✝ : Category D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nL R : C ⥤ C\nadj : L ⊣ R\nf : 𝟭 C ⟶ L\nX : C\n⊢ 𝟙 (R.obj X) ≫ (𝟙 (R.obj X) ≫ (𝟙 (R.obj X) ≫ f.app (R.obj X) ≫ 𝟙 (L.obj (R.obj X))) ≫ adj.counit.app X) ≫ 𝟙 X =\n f.app (R.obj X) ≫ adj.counit.app X",
"state_before": "C : Type u₁\nD : Type u₂\ninst✝¹ : Category C\ninst✝ : Category D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nL R : C ⥤ C\nadj : L ⊣ R\nf : 𝟭 C ⟶ L\nX : C\n⊢ (↑(transferNatTransSelf adj Adjunction.id) f).app X = f.app (R.obj X) ≫ adj.counit.app X",
"tactic": "dsimp [transferNatTransSelf, transferNatTrans, Adjunction.id]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\nD : Type u₂\ninst✝¹ : Category C\ninst✝ : Category D\nL₁ L₂ L₃ : C ⥤ D\nR₁ R₂ R₃ : D ⥤ C\nadj₁ : L₁ ⊣ R₁\nadj₂ : L₂ ⊣ R₂\nadj₃ : L₃ ⊣ R₃\nL R : C ⥤ C\nadj : L ⊣ R\nf : 𝟭 C ⟶ L\nX : C\n⊢ 𝟙 (R.obj X) ≫ (𝟙 (R.obj X) ≫ (𝟙 (R.obj X) ≫ f.app (R.obj X) ≫ 𝟙 (L.obj (R.obj X))) ≫ adj.counit.app X) ≫ 𝟙 X =\n f.app (R.obj X) ≫ adj.counit.app X",
"tactic": "simp only [comp_id, id_comp]"
}
] |
[
211,
31
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Mathlib/SetTheory/Ordinal/Exponential.lean
|
Ordinal.log_nonempty
|
[] |
[
263,
41
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
262,
1
] |
Mathlib/Data/SetLike/Basic.lean
|
SetLike.forall
|
[] |
[
133,
16
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
132,
11
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.sumCoords_reindex
|
[
{
"state_after": "case h\nι : Type u_4\nι' : Type u_3\nR : Type u_1\nR₂ : Type ?u.370286\nK : Type ?u.370289\nM : Type u_2\nM' : Type ?u.370295\nM'' : Type ?u.370298\nV : Type u\nV' : Type ?u.370303\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ : M\nb' : Basis ι' R M'\ne : ι ≃ ι'\nx : M\n⊢ ↑(sumCoords (reindex b e)) x = ↑(sumCoords b) x",
"state_before": "ι : Type u_4\nι' : Type u_3\nR : Type u_1\nR₂ : Type ?u.370286\nK : Type ?u.370289\nM : Type u_2\nM' : Type ?u.370295\nM'' : Type ?u.370298\nV : Type u\nV' : Type ?u.370303\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx : M\nb' : Basis ι' R M'\ne : ι ≃ ι'\n⊢ sumCoords (reindex b e) = sumCoords b",
"tactic": "ext x"
},
{
"state_after": "case h\nι : Type u_4\nι' : Type u_3\nR : Type u_1\nR₂ : Type ?u.370286\nK : Type ?u.370289\nM : Type u_2\nM' : Type ?u.370295\nM'' : Type ?u.370298\nV : Type u\nV' : Type ?u.370303\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ : M\nb' : Basis ι' R M'\ne : ι ≃ ι'\nx : M\n⊢ (Finsupp.sum (Finsupp.mapDomain (↑e) (↑b.repr x)) fun x => id) = Finsupp.sum (↑b.repr x) fun x => id",
"state_before": "case h\nι : Type u_4\nι' : Type u_3\nR : Type u_1\nR₂ : Type ?u.370286\nK : Type ?u.370289\nM : Type u_2\nM' : Type ?u.370295\nM'' : Type ?u.370298\nV : Type u\nV' : Type ?u.370303\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ : M\nb' : Basis ι' R M'\ne : ι ≃ ι'\nx : M\n⊢ ↑(sumCoords (reindex b e)) x = ↑(sumCoords b) x",
"tactic": "simp only [coe_sumCoords, repr_reindex]"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Type u_4\nι' : Type u_3\nR : Type u_1\nR₂ : Type ?u.370286\nK : Type ?u.370289\nM : Type u_2\nM' : Type ?u.370295\nM'' : Type ?u.370298\nV : Type u\nV' : Type ?u.370303\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\ninst✝¹ : AddCommMonoid M'\ninst✝ : Module R M'\nb b₁ : Basis ι R M\ni : ι\nc : R\nx✝ : M\nb' : Basis ι' R M'\ne : ι ≃ ι'\nx : M\n⊢ (Finsupp.sum (Finsupp.mapDomain (↑e) (↑b.repr x)) fun x => id) = Finsupp.sum (↑b.repr x) fun x => id",
"tactic": "exact Finsupp.sum_mapDomain_index (fun _ => rfl) fun _ _ _ => rfl"
}
] |
[
454,
68
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
451,
1
] |
Mathlib/Geometry/Euclidean/Angle/Unoriented/Basic.lean
|
InnerProductGeometry.norm_sub_eq_abs_sub_norm_of_angle_eq_zero
|
[
{
"state_after": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\nh : angle x y = 0\n⊢ ‖x‖ ^ 2 - 2 * (‖x‖ * ‖y‖) + ‖y‖ ^ 2 = (‖x‖ - ‖y‖) ^ 2",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\nh : angle x y = 0\n⊢ ‖x - y‖ = abs (‖x‖ - ‖y‖)",
"tactic": "rw [← sq_eq_sq (norm_nonneg (x - y)) (abs_nonneg (‖x‖ - ‖y‖)), norm_sub_pow_two_real,\n inner_eq_mul_norm_of_angle_eq_zero h, sq_abs (‖x‖ - ‖y‖)]"
},
{
"state_after": "no goals",
"state_before": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx✝ y✝ x y : V\nh : angle x y = 0\n⊢ ‖x‖ ^ 2 - 2 * (‖x‖ * ‖y‖) + ‖y‖ ^ 2 = (‖x‖ - ‖y‖) ^ 2",
"tactic": "ring"
}
] |
[
292,
7
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
288,
1
] |
Mathlib/Data/Fin/Basic.lean
|
Fin.subNat_mk
|
[] |
[
1586,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1584,
1
] |
Mathlib/Data/List/Basic.lean
|
List.indexOf_le_length
|
[
{
"state_after": "case nil\nι : Type ?u.80422\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na : α\n⊢ indexOf a [] ≤ length []\n\ncase cons\nι : Type ?u.80422\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l ≤ length l\n⊢ indexOf a (b :: l) ≤ length (b :: l)",
"state_before": "ι : Type ?u.80422\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na : α\nl : List α\n⊢ indexOf a l ≤ length l",
"tactic": "induction' l with b l ih"
},
{
"state_after": "case cons\nι : Type ?u.80422\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l ≤ length l\n⊢ (if a = b then 0 else succ (indexOf a l)) ≤ length l + 1",
"state_before": "case cons\nι : Type ?u.80422\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l ≤ length l\n⊢ indexOf a (b :: l) ≤ length (b :: l)",
"tactic": "simp only [length, indexOf_cons]"
},
{
"state_after": "case pos\nι : Type ?u.80422\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l ≤ length l\nh : a = b\n⊢ (if a = b then 0 else succ (indexOf a l)) ≤ length l + 1\n\ncase neg\nι : Type ?u.80422\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l ≤ length l\nh : ¬a = b\n⊢ (if a = b then 0 else succ (indexOf a l)) ≤ length l + 1",
"state_before": "case cons\nι : Type ?u.80422\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l ≤ length l\n⊢ (if a = b then 0 else succ (indexOf a l)) ≤ length l + 1",
"tactic": "by_cases h : a = b"
},
{
"state_after": "case neg\nι : Type ?u.80422\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l ≤ length l\nh : ¬a = b\n⊢ succ (indexOf a l) ≤ length l + 1",
"state_before": "case neg\nι : Type ?u.80422\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l ≤ length l\nh : ¬a = b\n⊢ (if a = b then 0 else succ (indexOf a l)) ≤ length l + 1",
"tactic": "rw [if_neg h]"
},
{
"state_after": "no goals",
"state_before": "case neg\nι : Type ?u.80422\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l ≤ length l\nh : ¬a = b\n⊢ succ (indexOf a l) ≤ length l + 1",
"tactic": "exact succ_le_succ ih"
},
{
"state_after": "no goals",
"state_before": "case nil\nι : Type ?u.80422\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na : α\n⊢ indexOf a [] ≤ length []",
"tactic": "rfl"
},
{
"state_after": "case pos\nι : Type ?u.80422\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l ≤ length l\nh : a = b\n⊢ 0 ≤ length l + 1",
"state_before": "case pos\nι : Type ?u.80422\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l ≤ length l\nh : a = b\n⊢ (if a = b then 0 else succ (indexOf a l)) ≤ length l + 1",
"tactic": "rw [if_pos h]"
},
{
"state_after": "no goals",
"state_before": "case pos\nι : Type ?u.80422\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nl₁ l₂ : List α\ninst✝ : DecidableEq α\na b : α\nl : List α\nih : indexOf a l ≤ length l\nh : a = b\n⊢ 0 ≤ length l + 1",
"tactic": "exact Nat.zero_le _"
}
] |
[
1204,
39
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1198,
1
] |
Mathlib/Algebra/DirectSum/Module.lean
|
DirectSum.IsInternal.addSubgroup_independent
|
[] |
[
434,
77
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
432,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Basic.lean
|
Real.sin_pi_sub
|
[] |
[
259,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
258,
1
] |
Mathlib/LinearAlgebra/Eigenspace/Basic.lean
|
Module.End.generalizedEigenspace_le_generalizedEigenspace_finrank
|
[] |
[
375,
33
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
373,
1
] |
Mathlib/RingTheory/FreeCommRing.lean
|
FreeRing.coe_add
|
[] |
[
350,
32
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
349,
11
] |
Mathlib/Algebra/Order/UpperLower.lean
|
upperClosure_mul_distrib
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝ : OrderedCommGroup α\ns t : Set α\na : α\n⊢ ↑(upperClosure (s * t)) = ↑(upperClosure s * upperClosure t)",
"tactic": "rw [UpperSet.coe_mul, mul_upperClosure, upperClosure_mul, UpperSet.upperClosure]"
}
] |
[
320,
85
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
318,
1
] |
Mathlib/LinearAlgebra/Basis.lean
|
Basis.groupSmul_apply
|
[] |
[
1232,
43
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1228,
1
] |
Mathlib/Algebra/Support.lean
|
Function.mulSupport_one_sub'
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.118394\nA : Type ?u.118397\nB : Type ?u.118400\nM : Type ?u.118403\nN : Type ?u.118406\nP : Type ?u.118409\nR : Type u_1\nS : Type ?u.118415\nG : Type ?u.118418\nM₀ : Type ?u.118421\nG₀ : Type ?u.118424\nι : Sort ?u.118427\ninst✝¹ : One R\ninst✝ : AddGroup R\nf : α → R\n⊢ mulSupport (1 - f) = support f",
"tactic": "rw [sub_eq_add_neg, mulSupport_one_add', support_neg']"
}
] |
[
434,
57
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
433,
1
] |
Mathlib/Combinatorics/SetFamily/Intersecting.lean
|
Set.Intersecting.isUpperSet'
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderBot α\ns✝ t : Set α\na✝ b✝ c : α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\na b : α\nhab : a ≤ b\nha : a ∈ ↑s\n⊢ b ∈ ↑s",
"state_before": "α : Type u_1\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderBot α\ns✝ t : Set α\na b c : α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\n⊢ IsUpperSet ↑s",
"tactic": "rintro a b hab ha"
},
{
"state_after": "α : Type u_1\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderBot α\ns✝ t : Set α\na✝ b✝ c : α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\na b : α\nhab : a ≤ b\nha : a ∈ ↑s\n⊢ b ∈ ↑(Insert.insert b s)\n\nα : Type u_1\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderBot α\ns✝ t : Set α\na✝ b✝ c : α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\na b : α\nhab : a ≤ b\nha : a ∈ ↑s\n⊢ Intersecting ↑(Insert.insert b s)",
"state_before": "α : Type u_1\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderBot α\ns✝ t : Set α\na✝ b✝ c : α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\na b : α\nhab : a ≤ b\nha : a ∈ ↑s\n⊢ b ∈ ↑s",
"tactic": "rw [h (Insert.insert b s) _ (Finset.subset_insert _ _)]"
},
{
"state_after": "α : Type u_1\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderBot α\ns✝ t : Set α\na✝ b✝ c : α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\na b : α\nhab : a ≤ b\nha : a ∈ ↑s\n⊢ Intersecting (Insert.insert b ↑s)",
"state_before": "α : Type u_1\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderBot α\ns✝ t : Set α\na✝ b✝ c : α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\na b : α\nhab : a ≤ b\nha : a ∈ ↑s\n⊢ Intersecting ↑(Insert.insert b s)",
"tactic": "rw [coe_insert]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderBot α\ns✝ t : Set α\na✝ b✝ c : α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\na b : α\nhab : a ≤ b\nha : a ∈ ↑s\n⊢ Intersecting (Insert.insert b ↑s)",
"tactic": "exact\n hs.insert (mt (eq_bot_mono hab) <| hs.ne_bot ha) fun c hc hbc => hs ha hc <| hbc.mono_left hab"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : SemilatticeInf α\ninst✝ : OrderBot α\ns✝ t : Set α\na✝ b✝ c : α\ns : Finset α\nhs : Intersecting ↑s\nh : ∀ (t : Finset α), Intersecting ↑t → s ⊆ t → s = t\na b : α\nhab : a ≤ b\nha : a ∈ ↑s\n⊢ b ∈ ↑(Insert.insert b s)",
"tactic": "exact mem_insert_self _ _"
}
] |
[
133,
101
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
125,
1
] |
Mathlib/CategoryTheory/Equivalence.lean
|
CategoryTheory.Equivalence.inverse_counitInv_comp
|
[
{
"state_after": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (e.unitIso.app (e.inverse.obj Y) ≪≫ e.inverse.mapIso (e.counitIso.app Y)).hom =\n (Iso.refl ((𝟭 C).obj (e.inverse.obj Y))).hom",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ e.inverse.map ((counitInv e).app Y) ≫ (unitInv e).app (e.inverse.obj Y) = 𝟙 (e.inverse.obj Y)",
"tactic": "erw [Iso.inv_eq_inv (e.unitIso.app (e.inverse.obj Y) ≪≫ e.inverse.mapIso (e.counitIso.app Y))\n (Iso.refl _)]"
},
{
"state_after": "no goals",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nD : Type u₂\ninst✝ : Category D\ne : C ≌ D\nY : D\n⊢ (e.unitIso.app (e.inverse.obj Y) ≪≫ e.inverse.mapIso (e.counitIso.app Y)).hom =\n (Iso.refl ((𝟭 C).obj (e.inverse.obj Y))).hom",
"tactic": "exact e.unit_inverse_comp Y"
}
] |
[
212,
30
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
208,
1
] |
Std/Data/List/Lemmas.lean
|
List.chain_cons
|
[
{
"state_after": "no goals",
"state_before": "α : Type u_1\nR : α → α → Prop\na b : α\nl : List α\np : Chain R a (b :: l)\n⊢ R a b ∧ Chain R b l",
"tactic": "cases p with | cons n p => exact ⟨n, p⟩"
},
{
"state_after": "no goals",
"state_before": "case cons\nα : Type u_1\nR : α → α → Prop\na b : α\nl : List α\nn : R a b\np : Chain R b l\n⊢ R a b ∧ Chain R b l",
"tactic": "exact ⟨n, p⟩"
}
] |
[
1778,
26
] |
e68aa8f5fe47aad78987df45f99094afbcb5e936
|
https://github.com/leanprover/std4
|
[
1776,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean
|
Equiv.Perm.isCycleOn_swap
|
[
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1649642\nα : Type u_1\nβ : Type ?u.1649648\nf g : Perm α\ns t : Set α\na b x y : α\ninst✝ : DecidableEq α\nhab : a ≠ b\n⊢ a ∈ {a, b}",
"tactic": "simp"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1649642\nα : Type u_1\nβ : Type ?u.1649648\nf g : Perm α\ns t : Set α\na b x y : α\ninst✝ : DecidableEq α\nhab : a ≠ b\n⊢ b ∈ {a, b}",
"tactic": "simp"
},
{
"state_after": "ι : Type ?u.1649642\nα : Type u_1\nβ : Type ?u.1649648\nf g : Perm α\ns t : Set α\na b x✝ y✝ : α\ninst✝ : DecidableEq α\nhab : a ≠ b\nx : α\nhx : x = a ∨ x = b\ny : α\nhy : y = a ∨ y = b\n⊢ SameCycle (swap a b) x y",
"state_before": "ι : Type ?u.1649642\nα : Type u_1\nβ : Type ?u.1649648\nf g : Perm α\ns t : Set α\na b x✝ y✝ : α\ninst✝ : DecidableEq α\nhab : a ≠ b\nx : α\nhx : x ∈ {a, b}\ny : α\nhy : y ∈ {a, b}\n⊢ SameCycle (swap a b) x y",
"tactic": "rw [Set.mem_insert_iff, Set.mem_singleton_iff] at hx hy"
},
{
"state_after": "case inl.inl\nι : Type ?u.1649642\nα : Type u_1\nβ : Type ?u.1649648\nf g : Perm α\ns t : Set α\nb x y✝ : α\ninst✝ : DecidableEq α\ny : α\nhab : y ≠ b\n⊢ SameCycle (swap y b) y y\n\ncase inl.inr\nι : Type ?u.1649642\nα : Type u_1\nβ : Type ?u.1649648\nf g : Perm α\ns t : Set α\nx✝ y✝ : α\ninst✝ : DecidableEq α\nx y : α\nhab : x ≠ y\n⊢ SameCycle (swap x y) x y\n\ncase inr.inl\nι : Type ?u.1649642\nα : Type u_1\nβ : Type ?u.1649648\nf g : Perm α\ns t : Set α\nx✝ y✝ : α\ninst✝ : DecidableEq α\nx y : α\nhab : y ≠ x\n⊢ SameCycle (swap y x) x y\n\ncase inr.inr\nι : Type ?u.1649642\nα : Type u_1\nβ : Type ?u.1649648\nf g : Perm α\ns t : Set α\na x y✝ : α\ninst✝ : DecidableEq α\ny : α\nhab : a ≠ y\n⊢ SameCycle (swap a y) y y",
"state_before": "ι : Type ?u.1649642\nα : Type u_1\nβ : Type ?u.1649648\nf g : Perm α\ns t : Set α\na b x✝ y✝ : α\ninst✝ : DecidableEq α\nhab : a ≠ b\nx : α\nhx : x = a ∨ x = b\ny : α\nhy : y = a ∨ y = b\n⊢ SameCycle (swap a b) x y",
"tactic": "obtain rfl | rfl := hx <;> obtain rfl | rfl := hy"
},
{
"state_after": "no goals",
"state_before": "case inl.inl\nι : Type ?u.1649642\nα : Type u_1\nβ : Type ?u.1649648\nf g : Perm α\ns t : Set α\nb x y✝ : α\ninst✝ : DecidableEq α\ny : α\nhab : y ≠ b\n⊢ SameCycle (swap y b) y y",
"tactic": "exact ⟨0, by rw [zpow_zero, coe_one, id.def]⟩"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1649642\nα : Type u_1\nβ : Type ?u.1649648\nf g : Perm α\ns t : Set α\nb x y✝ : α\ninst✝ : DecidableEq α\ny : α\nhab : y ≠ b\n⊢ ↑(swap y b ^ 0) y = y",
"tactic": "rw [zpow_zero, coe_one, id.def]"
},
{
"state_after": "no goals",
"state_before": "case inl.inr\nι : Type ?u.1649642\nα : Type u_1\nβ : Type ?u.1649648\nf g : Perm α\ns t : Set α\nx✝ y✝ : α\ninst✝ : DecidableEq α\nx y : α\nhab : x ≠ y\n⊢ SameCycle (swap x y) x y",
"tactic": "exact ⟨1, by rw [zpow_one, swap_apply_left]⟩"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1649642\nα : Type u_1\nβ : Type ?u.1649648\nf g : Perm α\ns t : Set α\nx✝ y✝ : α\ninst✝ : DecidableEq α\nx y : α\nhab : x ≠ y\n⊢ ↑(swap x y ^ 1) x = y",
"tactic": "rw [zpow_one, swap_apply_left]"
},
{
"state_after": "no goals",
"state_before": "case inr.inl\nι : Type ?u.1649642\nα : Type u_1\nβ : Type ?u.1649648\nf g : Perm α\ns t : Set α\nx✝ y✝ : α\ninst✝ : DecidableEq α\nx y : α\nhab : y ≠ x\n⊢ SameCycle (swap y x) x y",
"tactic": "exact ⟨1, by rw [zpow_one, swap_apply_right]⟩"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1649642\nα : Type u_1\nβ : Type ?u.1649648\nf g : Perm α\ns t : Set α\nx✝ y✝ : α\ninst✝ : DecidableEq α\nx y : α\nhab : y ≠ x\n⊢ ↑(swap y x ^ 1) x = y",
"tactic": "rw [zpow_one, swap_apply_right]"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nι : Type ?u.1649642\nα : Type u_1\nβ : Type ?u.1649648\nf g : Perm α\ns t : Set α\na x y✝ : α\ninst✝ : DecidableEq α\ny : α\nhab : a ≠ y\n⊢ SameCycle (swap a y) y y",
"tactic": "exact ⟨0, by rw [zpow_zero, coe_one, id.def]⟩"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.1649642\nα : Type u_1\nβ : Type ?u.1649648\nf g : Perm α\ns t : Set α\na x y✝ : α\ninst✝ : DecidableEq α\ny : α\nhab : a ≠ y\n⊢ ↑(swap a y ^ 0) y = y",
"tactic": "rw [zpow_zero, coe_one, id.def]"
}
] |
[
802,
53
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
795,
1
] |
Mathlib/ModelTheory/LanguageMap.lean
|
FirstOrder.Language.withConstants_relMap_sum_inl
|
[] |
[
518,
45
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
516,
1
] |
Mathlib/Data/Multiset/Basic.lean
|
Multiset.count_le_of_le
|
[] |
[
2367,
20
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
2366,
1
] |
Mathlib/LinearAlgebra/Matrix/Adjugate.lean
|
Matrix.cramer_transpose_apply
|
[
{
"state_after": "no goals",
"state_before": "m : Type u\nn : Type v\nα : Type w\ninst✝⁴ : DecidableEq n\ninst✝³ : Fintype n\ninst✝² : DecidableEq m\ninst✝¹ : Fintype m\ninst✝ : CommRing α\nA : Matrix n n α\nb : n → α\ni : n\n⊢ ↑(cramer Aᵀ) b i = det (updateRow A i b)",
"tactic": "rw [cramer_apply, updateColumn_transpose, det_transpose]"
}
] |
[
107,
59
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
106,
1
] |
Mathlib/MeasureTheory/Group/Prod.lean
|
MeasureTheory.measurePreserving_mul_prod_inv
|
[
{
"state_after": "case h.e'_5\nG : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\n⊢ (fun z => (z.snd * z.fst, z.fst⁻¹)) = (fun z => (z.snd, z.snd⁻¹ * z.fst)) ∘ fun z => (z.snd, z.snd * z.fst)",
"state_before": "G : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\n⊢ MeasurePreserving fun z => (z.snd * z.fst, z.fst⁻¹)",
"tactic": "convert (measurePreserving_prod_inv_mul_swap ν μ).comp (measurePreserving_prod_mul_swap μ ν)\n using 1"
},
{
"state_after": "case h.e'_5.h.mk\nG : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\nx y : G\n⊢ ((x, y).snd * (x, y).fst, (x, y).fst⁻¹) =\n ((fun z => (z.snd, z.snd⁻¹ * z.fst)) ∘ fun z => (z.snd, z.snd * z.fst)) (x, y)",
"state_before": "case h.e'_5\nG : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\n⊢ (fun z => (z.snd * z.fst, z.fst⁻¹)) = (fun z => (z.snd, z.snd⁻¹ * z.fst)) ∘ fun z => (z.snd, z.snd * z.fst)",
"tactic": "ext1 ⟨x, y⟩"
},
{
"state_after": "no goals",
"state_before": "case h.e'_5.h.mk\nG : Type u_1\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : Group G\ninst✝⁵ : MeasurableMul₂ G\nμ ν : Measure G\ninst✝⁴ : SigmaFinite ν\ninst✝³ : SigmaFinite μ\ns : Set G\ninst✝² : MeasurableInv G\ninst✝¹ : IsMulLeftInvariant μ\ninst✝ : IsMulLeftInvariant ν\nx y : G\n⊢ ((x, y).snd * (x, y).fst, (x, y).fst⁻¹) =\n ((fun z => (z.snd, z.snd⁻¹ * z.fst)) ∘ fun z => (z.snd, z.snd * z.fst)) (x, y)",
"tactic": "simp_rw [Function.comp_apply, mul_inv_rev, inv_mul_cancel_right]"
}
] |
[
156,
67
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
151,
1
] |
Mathlib/Topology/Basic.lean
|
DenseRange.dense_image
|
[] |
[
1827,
66
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1825,
1
] |
Mathlib/Order/Antichain.lean
|
IsStrongAntichain.eq
|
[] |
[
295,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
292,
1
] |
Mathlib/Algebra/GCDMonoid/Basic.lean
|
lcm_mul_left
|
[
{
"state_after": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\nb c : α\n⊢ lcm (0 * b) (0 * c) = ↑normalize 0 * lcm b c",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\na b c : α\n⊢ a = 0 → lcm (a * b) (a * c) = ↑normalize a * lcm b c",
"tactic": "rintro rfl"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\nb c : α\n⊢ lcm (0 * b) (0 * c) = ↑normalize 0 * lcm b c",
"tactic": "simp only [zero_mul, lcm_zero_left, normalize_zero]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\ninst✝¹ : CancelCommMonoidWithZero α\ninst✝ : NormalizedGCDMonoid α\na b c : α\nha : a ≠ 0\nthis : lcm (a * b) (a * c) = ↑normalize (a * lcm b c)\n⊢ lcm (a * b) (a * c) = ↑normalize a * lcm b c",
"tactic": "simpa"
}
] |
[
830,
70
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
818,
1
] |
Mathlib/Topology/MetricSpace/HausdorffDistance.lean
|
Metric.Bounded.thickening
|
[
{
"state_after": "case inl\nι : Sort ?u.97539\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\nδ✝ : ℝ\ns : Set α\nx : α\nX : Type u\ninst✝ : PseudoMetricSpace X\nδ : ℝ\nh : Bounded ∅\n⊢ Bounded (thickening δ ∅)\n\ncase inr.intro\nι : Sort ?u.97539\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\nδ✝ : ℝ\ns : Set α\nx✝ : α\nX : Type u\ninst✝ : PseudoMetricSpace X\nδ : ℝ\nE : Set X\nh : Bounded E\nx : X\nhx : x ∈ E\n⊢ Bounded (thickening δ E)",
"state_before": "ι : Sort ?u.97539\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\nδ✝ : ℝ\ns : Set α\nx : α\nX : Type u\ninst✝ : PseudoMetricSpace X\nδ : ℝ\nE : Set X\nh : Bounded E\n⊢ Bounded (thickening δ E)",
"tactic": "rcases E.eq_empty_or_nonempty with rfl | ⟨x, hx⟩"
},
{
"state_after": "no goals",
"state_before": "case inl\nι : Sort ?u.97539\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\nδ✝ : ℝ\ns : Set α\nx : α\nX : Type u\ninst✝ : PseudoMetricSpace X\nδ : ℝ\nh : Bounded ∅\n⊢ Bounded (thickening δ ∅)",
"tactic": "simp"
},
{
"state_after": "case inr.intro\nι : Sort ?u.97539\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\nδ✝ : ℝ\ns : Set α\nx✝ : α\nX : Type u\ninst✝ : PseudoMetricSpace X\nδ : ℝ\nE : Set X\nh : Bounded E\nx : X\nhx : x ∈ E\ny : X\nhy : y ∈ thickening δ E\n⊢ y ∈ closedBall x (δ + diam E)",
"state_before": "case inr.intro\nι : Sort ?u.97539\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\nδ✝ : ℝ\ns : Set α\nx✝ : α\nX : Type u\ninst✝ : PseudoMetricSpace X\nδ : ℝ\nE : Set X\nh : Bounded E\nx : X\nhx : x ∈ E\n⊢ Bounded (thickening δ E)",
"tactic": "refine (bounded_iff_subset_ball x).2 ⟨δ + diam E, fun y hy ↦ ?_⟩"
},
{
"state_after": "no goals",
"state_before": "case inr.intro\nι : Sort ?u.97539\nα : Type u\nβ : Type v\ninst✝¹ : PseudoEMetricSpace α\nδ✝ : ℝ\ns : Set α\nx✝ : α\nX : Type u\ninst✝ : PseudoMetricSpace X\nδ : ℝ\nE : Set X\nh : Bounded E\nx : X\nhx : x ∈ E\ny : X\nhy : y ∈ thickening δ E\n⊢ y ∈ closedBall x (δ + diam E)",
"tactic": "calc\n dist y x ≤ infDist y E + diam E := dist_le_infDist_add_diam (x := y) h hx\n _ ≤ δ + diam E := add_le_add_right ((mem_thickening_iff_infDist_lt ⟨x, hx⟩).1 hy).le _"
}
] |
[
1009,
93
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1002,
11
] |
Mathlib/Algebra/Regular/Basic.lean
|
not_isRegular_zero
|
[] |
[
269,
99
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
269,
1
] |
Mathlib/SetTheory/Game/PGame.lean
|
PGame.lt_iff_le_and_lf
|
[] |
[
536,
10
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
535,
1
] |
Mathlib/Data/Matrix/Basic.lean
|
Matrix.mul_apply
|
[] |
[
926,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
924,
1
] |
Mathlib/Order/ConditionallyCompleteLattice/Basic.lean
|
WithTop.iSup_coe_eq_top
|
[
{
"state_after": "α✝ : Type ?u.130874\nβ : Type ?u.130877\nγ : Type ?u.130880\nι✝ : Sort ?u.130883\nι : Sort u_1\nα : Type u_2\ninst✝ : ConditionallyCompleteLinearOrderBot α\nf : ι → α\n⊢ (∀ (b : WithTop α), b < ⊤ → ∃ i, b < ↑(f i)) ↔ ∀ (x : α), ∃ y, y ∈ range f ∧ x < y",
"state_before": "α✝ : Type ?u.130874\nβ : Type ?u.130877\nγ : Type ?u.130880\nι✝ : Sort ?u.130883\nι : Sort u_1\nα : Type u_2\ninst✝ : ConditionallyCompleteLinearOrderBot α\nf : ι → α\n⊢ (⨆ (x : ι), ↑(f x)) = ⊤ ↔ ¬BddAbove (range f)",
"tactic": "rw [iSup_eq_top, not_bddAbove_iff]"
},
{
"state_after": "case refine'_1\nα✝ : Type ?u.130874\nβ : Type ?u.130877\nγ : Type ?u.130880\nι✝ : Sort ?u.130883\nι : Sort u_1\nα : Type u_2\ninst✝ : ConditionallyCompleteLinearOrderBot α\nf : ι → α\nhf : ∀ (b : WithTop α), b < ⊤ → ∃ i, b < ↑(f i)\nr : α\n⊢ ∃ y, y ∈ range f ∧ r < y\n\ncase refine'_2\nα✝ : Type ?u.130874\nβ : Type ?u.130877\nγ : Type ?u.130880\nι✝ : Sort ?u.130883\nι : Sort u_1\nα : Type u_2\ninst✝ : ConditionallyCompleteLinearOrderBot α\nf : ι → α\nhf : ∀ (x : α), ∃ y, y ∈ range f ∧ x < y\na : WithTop α\nha : a < ⊤\n⊢ ∃ i, a < ↑(f i)",
"state_before": "α✝ : Type ?u.130874\nβ : Type ?u.130877\nγ : Type ?u.130880\nι✝ : Sort ?u.130883\nι : Sort u_1\nα : Type u_2\ninst✝ : ConditionallyCompleteLinearOrderBot α\nf : ι → α\n⊢ (∀ (b : WithTop α), b < ⊤ → ∃ i, b < ↑(f i)) ↔ ∀ (x : α), ∃ y, y ∈ range f ∧ x < y",
"tactic": "refine' ⟨fun hf r => _, fun hf a ha => _⟩"
},
{
"state_after": "case refine'_1.intro\nα✝ : Type ?u.130874\nβ : Type ?u.130877\nγ : Type ?u.130880\nι✝ : Sort ?u.130883\nι : Sort u_1\nα : Type u_2\ninst✝ : ConditionallyCompleteLinearOrderBot α\nf : ι → α\nhf : ∀ (b : WithTop α), b < ⊤ → ∃ i, b < ↑(f i)\nr : α\ni : ι\nhi : ↑r < ↑(f i)\n⊢ ∃ y, y ∈ range f ∧ r < y",
"state_before": "case refine'_1\nα✝ : Type ?u.130874\nβ : Type ?u.130877\nγ : Type ?u.130880\nι✝ : Sort ?u.130883\nι : Sort u_1\nα : Type u_2\ninst✝ : ConditionallyCompleteLinearOrderBot α\nf : ι → α\nhf : ∀ (b : WithTop α), b < ⊤ → ∃ i, b < ↑(f i)\nr : α\n⊢ ∃ y, y ∈ range f ∧ r < y",
"tactic": "rcases hf r (WithTop.coe_lt_top r) with ⟨i, hi⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_1.intro\nα✝ : Type ?u.130874\nβ : Type ?u.130877\nγ : Type ?u.130880\nι✝ : Sort ?u.130883\nι : Sort u_1\nα : Type u_2\ninst✝ : ConditionallyCompleteLinearOrderBot α\nf : ι → α\nhf : ∀ (b : WithTop α), b < ⊤ → ∃ i, b < ↑(f i)\nr : α\ni : ι\nhi : ↑r < ↑(f i)\n⊢ ∃ y, y ∈ range f ∧ r < y",
"tactic": "exact ⟨f i, ⟨i, rfl⟩, WithTop.coe_lt_coe.mp hi⟩"
},
{
"state_after": "case refine'_2.intro.intro.intro\nα✝ : Type ?u.130874\nβ : Type ?u.130877\nγ : Type ?u.130880\nι✝ : Sort ?u.130883\nι : Sort u_1\nα : Type u_2\ninst✝ : ConditionallyCompleteLinearOrderBot α\nf : ι → α\nhf : ∀ (x : α), ∃ y, y ∈ range f ∧ x < y\na : WithTop α\nha : a < ⊤\ni : ι\nhi : untop a (_ : a ≠ ⊤) < f i\n⊢ ∃ i, a < ↑(f i)",
"state_before": "case refine'_2\nα✝ : Type ?u.130874\nβ : Type ?u.130877\nγ : Type ?u.130880\nι✝ : Sort ?u.130883\nι : Sort u_1\nα : Type u_2\ninst✝ : ConditionallyCompleteLinearOrderBot α\nf : ι → α\nhf : ∀ (x : α), ∃ y, y ∈ range f ∧ x < y\na : WithTop α\nha : a < ⊤\n⊢ ∃ i, a < ↑(f i)",
"tactic": "rcases hf (a.untop ha.ne) with ⟨-, ⟨i, rfl⟩, hi⟩"
},
{
"state_after": "no goals",
"state_before": "case refine'_2.intro.intro.intro\nα✝ : Type ?u.130874\nβ : Type ?u.130877\nγ : Type ?u.130880\nι✝ : Sort ?u.130883\nι : Sort u_1\nα : Type u_2\ninst✝ : ConditionallyCompleteLinearOrderBot α\nf : ι → α\nhf : ∀ (x : α), ∃ y, y ∈ range f ∧ x < y\na : WithTop α\nha : a < ⊤\ni : ι\nhi : untop a (_ : a ≠ ⊤) < f i\n⊢ ∃ i, a < ↑(f i)",
"tactic": "exact ⟨i, by simpa only [WithTop.coe_untop _ ha.ne] using WithTop.coe_lt_coe.mpr hi⟩"
},
{
"state_after": "no goals",
"state_before": "α✝ : Type ?u.130874\nβ : Type ?u.130877\nγ : Type ?u.130880\nι✝ : Sort ?u.130883\nι : Sort u_1\nα : Type u_2\ninst✝ : ConditionallyCompleteLinearOrderBot α\nf : ι → α\nhf : ∀ (x : α), ∃ y, y ∈ range f ∧ x < y\na : WithTop α\nha : a < ⊤\ni : ι\nhi : untop a (_ : a ≠ ⊤) < f i\n⊢ a < ↑(f i)",
"tactic": "simpa only [WithTop.coe_untop _ ha.ne] using WithTop.coe_lt_coe.mpr hi"
}
] |
[
1541,
89
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
1534,
1
] |
Mathlib/Analysis/MeanInequalitiesPow.lean
|
ENNReal.rpow_add_rpow_le
|
[
{
"state_after": "ι : Type u\ns : Finset ι\np q : ℝ\na b : ℝ≥0∞\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0∞), a ^ q = (a ^ p) ^ (q / p)\n⊢ (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)",
"state_before": "ι : Type u\ns : Finset ι\np q : ℝ\na b : ℝ≥0∞\nhp_pos : 0 < p\nhpq : p ≤ q\n⊢ (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)",
"tactic": "have h_rpow : ∀ a : ℝ≥0∞, a ^ q = (a ^ p) ^ (q / p) := fun a => by\n rw [← ENNReal.rpow_mul, _root_.mul_div_cancel' _ hp_pos.ne']"
},
{
"state_after": "ι : Type u\ns : Finset ι\np q : ℝ\na b : ℝ≥0∞\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0∞), a ^ q = (a ^ p) ^ (q / p)\nh_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p\n⊢ (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)",
"state_before": "ι : Type u\ns : Finset ι\np q : ℝ\na b : ℝ≥0∞\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0∞), a ^ q = (a ^ p) ^ (q / p)\n⊢ (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)",
"tactic": "have h_rpow_add_rpow_le_add :\n ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p := by\n refine' rpow_add_rpow_le_add (a ^ p) (b ^ p) _\n rwa [one_le_div hp_pos]"
},
{
"state_after": "ι : Type u\ns : Finset ι\np q : ℝ\na b : ℝ≥0∞\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0∞), a ^ q = (a ^ p) ^ (q / p)\nh_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p\n⊢ ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (p / q) ≤ a ^ p + b ^ p",
"state_before": "ι : Type u\ns : Finset ι\np q : ℝ\na b : ℝ≥0∞\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0∞), a ^ q = (a ^ p) ^ (q / p)\nh_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p\n⊢ (a ^ q + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)",
"tactic": "rw [h_rpow a, h_rpow b, ENNReal.le_rpow_one_div_iff hp_pos, ← ENNReal.rpow_mul, mul_comm,\n mul_one_div]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\ns : Finset ι\np q : ℝ\na b : ℝ≥0∞\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0∞), a ^ q = (a ^ p) ^ (q / p)\nh_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p\n⊢ ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (p / q) ≤ a ^ p + b ^ p",
"tactic": "rwa [one_div_div] at h_rpow_add_rpow_le_add"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\ns : Finset ι\np q : ℝ\na✝ b : ℝ≥0∞\nhp_pos : 0 < p\nhpq : p ≤ q\na : ℝ≥0∞\n⊢ a ^ q = (a ^ p) ^ (q / p)",
"tactic": "rw [← ENNReal.rpow_mul, _root_.mul_div_cancel' _ hp_pos.ne']"
},
{
"state_after": "ι : Type u\ns : Finset ι\np q : ℝ\na b : ℝ≥0∞\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0∞), a ^ q = (a ^ p) ^ (q / p)\n⊢ 1 ≤ q / p",
"state_before": "ι : Type u\ns : Finset ι\np q : ℝ\na b : ℝ≥0∞\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0∞), a ^ q = (a ^ p) ^ (q / p)\n⊢ ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p",
"tactic": "refine' rpow_add_rpow_le_add (a ^ p) (b ^ p) _"
},
{
"state_after": "no goals",
"state_before": "ι : Type u\ns : Finset ι\np q : ℝ\na b : ℝ≥0∞\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0∞), a ^ q = (a ^ p) ^ (q / p)\n⊢ 1 ≤ q / p",
"tactic": "rwa [one_le_div hp_pos]"
}
] |
[
338,
46
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
328,
1
] |
Mathlib/RingTheory/Subsemiring/Basic.lean
|
Subsemiring.closure_le
|
[] |
[
817,
56
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
816,
1
] |
Mathlib/Order/WithBot.lean
|
WithBot.wellFounded_lt
|
[] |
[
510,
58
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
493,
1
] |
Mathlib/Algebra/Order/Hom/Ring.lean
|
OrderRingHom.coe_orderMonoidWithZeroHom_apply
|
[] |
[
246,
6
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
245,
1
] |
Mathlib/Analysis/InnerProductSpace/Positive.lean
|
ContinuousLinearMap.IsPositive.add
|
[
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.147588\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT S : E →L[𝕜] E\nhT : IsPositive T\nhS : IsPositive S\nx : E\n⊢ 0 ≤ reApplyInnerSelf (T + S) x",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.147588\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT S : E →L[𝕜] E\nhT : IsPositive T\nhS : IsPositive S\n⊢ IsPositive (T + S)",
"tactic": "refine' ⟨hT.isSelfAdjoint.add hS.isSelfAdjoint, fun x => _⟩"
},
{
"state_after": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.147588\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT S : E →L[𝕜] E\nhT : IsPositive T\nhS : IsPositive S\nx : E\n⊢ 0 ≤ ↑re (inner (↑T x) x) + ↑re (inner (↑S x) x)",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.147588\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT S : E →L[𝕜] E\nhT : IsPositive T\nhS : IsPositive S\nx : E\n⊢ 0 ≤ reApplyInnerSelf (T + S) x",
"tactic": "rw [reApplyInnerSelf, add_apply, inner_add_left, map_add]"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\nE : Type u_2\nF : Type ?u.147588\ninst✝⁶ : IsROrC 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nT S : E →L[𝕜] E\nhT : IsPositive T\nhS : IsPositive S\nx : E\n⊢ 0 ≤ ↑re (inner (↑T x) x) + ↑re (inner (↑S x) x)",
"tactic": "exact add_nonneg (hT.inner_nonneg_left x) (hS.inner_nonneg_left x)"
}
] |
[
91,
69
] |
5a919533f110b7d76410134a237ee374f24eaaad
|
https://github.com/leanprover-community/mathlib4
|
[
87,
1
] |
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