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Mathlib/CategoryTheory/Limits/Shapes/CommSq.lean | CategoryTheory.IsPushout.map_iff | [] | [
1067,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1064,
1
] |
Mathlib/MeasureTheory/Function/AEMeasurableSequence.lean | aeSeq.aeSeqSet_measurableSet | [] | [
95,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
94,
1
] |
Mathlib/Analysis/NormedSpace/Pointwise.lean | ediam_smul₀ | [
{
"state_after": "𝕜 : Type u_2\nE : Type u_1\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : BoundedSMul 𝕜 E\nc : 𝕜\ns : Set E\n⊢ ‖c‖₊ • EMetric.diam s ≤ EMetric.diam (c • s)",
"state_before": "𝕜 : Type u_2\nE : Type u_1\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : BoundedSMul 𝕜 E\nc : 𝕜\ns : Set E\n⊢ EMetric.diam (c • s) = ‖c‖₊ • EMetric.diam s",
"tactic": "refine' le_antisymm (ediam_smul_le c s) _"
},
{
"state_after": "case inl\n𝕜 : Type u_2\nE : Type u_1\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : BoundedSMul 𝕜 E\ns : Set E\n⊢ ‖0‖₊ • EMetric.diam s ≤ EMetric.diam (0 • s)\n\ncase inr\n𝕜 : Type u_2\nE : Type u_1\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : BoundedSMul 𝕜 E\nc : 𝕜\ns : Set E\nhc : c ≠ 0\n⊢ ‖c‖₊ • EMetric.diam s ≤ EMetric.diam (c • s)",
"state_before": "𝕜 : Type u_2\nE : Type u_1\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : BoundedSMul 𝕜 E\nc : 𝕜\ns : Set E\n⊢ ‖c‖₊ • EMetric.diam s ≤ EMetric.diam (c • s)",
"tactic": "obtain rfl | hc := eq_or_ne c 0"
},
{
"state_after": "case inl.inl\n𝕜 : Type u_2\nE : Type u_1\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : BoundedSMul 𝕜 E\n⊢ ‖0‖₊ • EMetric.diam ∅ ≤ EMetric.diam (0 • ∅)\n\ncase inl.inr\n𝕜 : Type u_2\nE : Type u_1\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : BoundedSMul 𝕜 E\ns : Set E\nhs : Set.Nonempty s\n⊢ ‖0‖₊ • EMetric.diam s ≤ EMetric.diam (0 • s)",
"state_before": "case inl\n𝕜 : Type u_2\nE : Type u_1\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : BoundedSMul 𝕜 E\ns : Set E\n⊢ ‖0‖₊ • EMetric.diam s ≤ EMetric.diam (0 • s)",
"tactic": "obtain rfl | hs := s.eq_empty_or_nonempty"
},
{
"state_after": "no goals",
"state_before": "case inl.inr\n𝕜 : Type u_2\nE : Type u_1\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : BoundedSMul 𝕜 E\ns : Set E\nhs : Set.Nonempty s\n⊢ ‖0‖₊ • EMetric.diam s ≤ EMetric.diam (0 • s)",
"tactic": "simp [zero_smul_set hs, ← Set.singleton_zero]"
},
{
"state_after": "no goals",
"state_before": "case inl.inl\n𝕜 : Type u_2\nE : Type u_1\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : BoundedSMul 𝕜 E\n⊢ ‖0‖₊ • EMetric.diam ∅ ≤ EMetric.diam (0 • ∅)",
"tactic": "simp"
},
{
"state_after": "case inr\n𝕜 : Type u_2\nE : Type u_1\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : BoundedSMul 𝕜 E\nc : 𝕜\ns : Set E\nhc : c ≠ 0\nthis : EMetric.diam ((fun x x_1 => x • x_1) c⁻¹ '' (c • s)) ≤ ↑‖c⁻¹‖₊ * EMetric.diam (c • s)\n⊢ ‖c‖₊ • EMetric.diam s ≤ EMetric.diam (c • s)",
"state_before": "case inr\n𝕜 : Type u_2\nE : Type u_1\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : BoundedSMul 𝕜 E\nc : 𝕜\ns : Set E\nhc : c ≠ 0\n⊢ ‖c‖₊ • EMetric.diam s ≤ EMetric.diam (c • s)",
"tactic": "have := (lipschitzWith_smul c⁻¹).ediam_image_le (c • s)"
},
{
"state_after": "no goals",
"state_before": "case inr\n𝕜 : Type u_2\nE : Type u_1\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : BoundedSMul 𝕜 E\nc : 𝕜\ns : Set E\nhc : c ≠ 0\nthis : EMetric.diam ((fun x x_1 => x • x_1) c⁻¹ '' (c • s)) ≤ ↑‖c⁻¹‖₊ * EMetric.diam (c • s)\n⊢ ‖c‖₊ • EMetric.diam s ≤ EMetric.diam (c • s)",
"tactic": "rwa [← smul_eq_mul, ← ENNReal.smul_def, Set.image_smul, inv_smul_smul₀ hc s, nnnorm_inv,\n ENNReal.le_inv_smul_iff (nnnorm_ne_zero_iff.mpr hc)] at this"
}
] | [
56,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
48,
1
] |
Mathlib/LinearAlgebra/Finrank.lean | finrank_eq_zero_of_not_exists_basis_finite | [] | [
204,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
202,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Arctan.lean | Real.image_tan_Ioo | [] | [
106,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
105,
1
] |
Mathlib/MeasureTheory/Constructions/Prod/Basic.lean | MeasureTheory.QuasiMeasurePreserving.prod_of_right | [
{
"state_after": "α : Type u_1\nα' : Type ?u.4534735\nβ : Type u_2\nβ' : Type ?u.4534741\nγ : Type u_3\nE : Type ?u.4534747\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ✝ μ' : Measure α\nν✝ ν' : Measure β\nτ✝ : Measure γ\ninst✝¹ : NormedAddCommGroup E\nf : α × β → γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\nhf : Measurable f\ninst✝ : SigmaFinite ν\nh2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving fun y => f (x, y)\n⊢ map f (Measure.prod μ ν) ≪ τ",
"state_before": "α : Type u_1\nα' : Type ?u.4534735\nβ : Type u_2\nβ' : Type ?u.4534741\nγ : Type u_3\nE : Type ?u.4534747\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ✝ μ' : Measure α\nν✝ ν' : Measure β\nτ✝ : Measure γ\ninst✝¹ : NormedAddCommGroup E\nf : α × β → γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\nhf : Measurable f\ninst✝ : SigmaFinite ν\nh2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving fun y => f (x, y)\n⊢ QuasiMeasurePreserving f",
"tactic": "refine' ⟨hf, _⟩"
},
{
"state_after": "α : Type u_1\nα' : Type ?u.4534735\nβ : Type u_2\nβ' : Type ?u.4534741\nγ : Type u_3\nE : Type ?u.4534747\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ✝ μ' : Measure α\nν✝ ν' : Measure β\nτ✝ : Measure γ\ninst✝¹ : NormedAddCommGroup E\nf : α × β → γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\nhf : Measurable f\ninst✝ : SigmaFinite ν\nh2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving fun y => f (x, y)\ns : Set γ\nhs : MeasurableSet s\nh2s : ↑↑τ s = 0\n⊢ ↑↑(map f (Measure.prod μ ν)) s = 0",
"state_before": "α : Type u_1\nα' : Type ?u.4534735\nβ : Type u_2\nβ' : Type ?u.4534741\nγ : Type u_3\nE : Type ?u.4534747\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ✝ μ' : Measure α\nν✝ ν' : Measure β\nτ✝ : Measure γ\ninst✝¹ : NormedAddCommGroup E\nf : α × β → γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\nhf : Measurable f\ninst✝ : SigmaFinite ν\nh2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving fun y => f (x, y)\n⊢ map f (Measure.prod μ ν) ≪ τ",
"tactic": "refine' AbsolutelyContinuous.mk fun s hs h2s => _"
},
{
"state_after": "α : Type u_1\nα' : Type ?u.4534735\nβ : Type u_2\nβ' : Type ?u.4534741\nγ : Type u_3\nE : Type ?u.4534747\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ✝ μ' : Measure α\nν✝ ν' : Measure β\nτ✝ : Measure γ\ninst✝¹ : NormedAddCommGroup E\nf : α × β → γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\nhf : Measurable f\ninst✝ : SigmaFinite ν\nh2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving fun y => f (x, y)\ns : Set γ\nhs : MeasurableSet s\nh2s : ↑↑τ s = 0\n⊢ (∫⁻ (x : α), ↑↑ν (Prod.mk x ⁻¹' (f ⁻¹' s)) ∂μ) = 0",
"state_before": "α : Type u_1\nα' : Type ?u.4534735\nβ : Type u_2\nβ' : Type ?u.4534741\nγ : Type u_3\nE : Type ?u.4534747\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ✝ μ' : Measure α\nν✝ ν' : Measure β\nτ✝ : Measure γ\ninst✝¹ : NormedAddCommGroup E\nf : α × β → γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\nhf : Measurable f\ninst✝ : SigmaFinite ν\nh2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving fun y => f (x, y)\ns : Set γ\nhs : MeasurableSet s\nh2s : ↑↑τ s = 0\n⊢ ↑↑(map f (Measure.prod μ ν)) s = 0",
"tactic": "rw [map_apply hf hs, prod_apply (hf hs)]"
},
{
"state_after": "α : Type u_1\nα' : Type ?u.4534735\nβ : Type u_2\nβ' : Type ?u.4534741\nγ : Type u_3\nE : Type ?u.4534747\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ✝ μ' : Measure α\nν✝ ν' : Measure β\nτ✝ : Measure γ\ninst✝¹ : NormedAddCommGroup E\nf : α × β → γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\nhf : Measurable f\ninst✝ : SigmaFinite ν\nh2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving fun y => f (x, y)\ns : Set γ\nhs : MeasurableSet s\nh2s : ↑↑τ s = 0\n⊢ (∫⁻ (x : α), ↑↑ν ((fun x_1 => f (x, x_1)) ⁻¹' s) ∂μ) = 0",
"state_before": "α : Type u_1\nα' : Type ?u.4534735\nβ : Type u_2\nβ' : Type ?u.4534741\nγ : Type u_3\nE : Type ?u.4534747\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ✝ μ' : Measure α\nν✝ ν' : Measure β\nτ✝ : Measure γ\ninst✝¹ : NormedAddCommGroup E\nf : α × β → γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\nhf : Measurable f\ninst✝ : SigmaFinite ν\nh2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving fun y => f (x, y)\ns : Set γ\nhs : MeasurableSet s\nh2s : ↑↑τ s = 0\n⊢ (∫⁻ (x : α), ↑↑ν (Prod.mk x ⁻¹' (f ⁻¹' s)) ∂μ) = 0",
"tactic": "simp_rw [preimage_preimage]"
},
{
"state_after": "no goals",
"state_before": "α : Type u_1\nα' : Type ?u.4534735\nβ : Type u_2\nβ' : Type ?u.4534741\nγ : Type u_3\nE : Type ?u.4534747\ninst✝⁶ : MeasurableSpace α\ninst✝⁵ : MeasurableSpace α'\ninst✝⁴ : MeasurableSpace β\ninst✝³ : MeasurableSpace β'\ninst✝² : MeasurableSpace γ\nμ✝ μ' : Measure α\nν✝ ν' : Measure β\nτ✝ : Measure γ\ninst✝¹ : NormedAddCommGroup E\nf : α × β → γ\nμ : Measure α\nν : Measure β\nτ : Measure γ\nhf : Measurable f\ninst✝ : SigmaFinite ν\nh2f : ∀ᵐ (x : α) ∂μ, QuasiMeasurePreserving fun y => f (x, y)\ns : Set γ\nhs : MeasurableSet s\nh2s : ↑↑τ s = 0\n⊢ (∫⁻ (x : α), ↑↑ν ((fun x_1 => f (x, x_1)) ⁻¹' s) ∂μ) = 0",
"tactic": "rw [lintegral_congr_ae (h2f.mono fun x hx => hx.preimage_null h2s), lintegral_zero]"
}
] | [
671,
86
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
664,
1
] |
Mathlib/MeasureTheory/Measure/OuterMeasure.lean | MeasureTheory.OuterMeasure.iSup_apply | [
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type ?u.67250\nR : Type ?u.67253\nR' : Type ?u.67256\nms : Set (OuterMeasure α)\nm : OuterMeasure α\nι : Sort u_1\nf : ι → OuterMeasure α\ns : Set α\n⊢ ↑(⨆ (i : ι), f i) s = ⨆ (i : ι), ↑(f i) s",
"tactic": "rw [iSup, sSup_apply, iSup_range, iSup]"
}
] | [
419,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
418,
1
] |
Mathlib/Analysis/Calculus/ContDiffDef.lean | fderiv_iteratedFDeriv | [
{
"state_after": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\n⊢ fderiv 𝕜 (iteratedFDeriv 𝕜 n f) =\n ↑(LinearIsometryEquiv.symm (continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F)) ∘\n ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) ∘ fderiv 𝕜 (iteratedFDeriv 𝕜 n f)",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\n⊢ fderiv 𝕜 (iteratedFDeriv 𝕜 n f) =\n ↑(LinearIsometryEquiv.symm (continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F)) ∘ iteratedFDeriv 𝕜 (n + 1) f",
"tactic": "rw [iteratedFDeriv_succ_eq_comp_left]"
},
{
"state_after": "case h\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nx : E\n⊢ fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x =\n (↑(LinearIsometryEquiv.symm (continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F)) ∘\n ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) ∘ fderiv 𝕜 (iteratedFDeriv 𝕜 n f))\n x",
"state_before": "𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\n⊢ fderiv 𝕜 (iteratedFDeriv 𝕜 n f) =\n ↑(LinearIsometryEquiv.symm (continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F)) ∘\n ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) ∘ fderiv 𝕜 (iteratedFDeriv 𝕜 n f)",
"tactic": "ext1 x"
},
{
"state_after": "no goals",
"state_before": "case h\n𝕜 : Type u\ninst✝⁸ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nG : Type uG\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nX : Type uX\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\ns s₁ t u : Set E\nf f₁ : E → F\ng : F → G\nx✝ x₀ : E\nc : F\nm n✝ : ℕ∞\np : E → FormalMultilinearSeries 𝕜 E F\nn : ℕ\nx : E\n⊢ fderiv 𝕜 (iteratedFDeriv 𝕜 n f) x =\n (↑(LinearIsometryEquiv.symm (continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F)) ∘\n ↑(continuousMultilinearCurryLeftEquiv 𝕜 (fun x => E) F) ∘ fderiv 𝕜 (iteratedFDeriv 𝕜 n f))\n x",
"tactic": "simp only [Function.comp_apply, LinearIsometryEquiv.symm_apply_apply]"
}
] | [
1553,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1547,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean | Polynomial.coeff_pow_mul_natDegree | [
{
"state_after": "case zero\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\n⊢ coeff (p ^ Nat.zero) (Nat.zero * natDegree p) = leadingCoeff p ^ Nat.zero\n\ncase succ\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\n⊢ coeff (p ^ Nat.succ i) (Nat.succ i * natDegree p) = leadingCoeff p ^ Nat.succ i",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn✝ m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\nn : ℕ\n⊢ coeff (p ^ n) (n * natDegree p) = leadingCoeff p ^ n",
"tactic": "induction' n with i hi"
},
{
"state_after": "no goals",
"state_before": "case zero\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\n⊢ coeff (p ^ Nat.zero) (Nat.zero * natDegree p) = leadingCoeff p ^ Nat.zero",
"tactic": "simp"
},
{
"state_after": "case succ\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\n⊢ coeff (p ^ i * p) (i * natDegree p + natDegree p) = leadingCoeff p ^ i * leadingCoeff p",
"state_before": "case succ\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\n⊢ coeff (p ^ Nat.succ i) (Nat.succ i * natDegree p) = leadingCoeff p ^ Nat.succ i",
"tactic": "rw [pow_succ', pow_succ', Nat.succ_mul]"
},
{
"state_after": "case pos\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\n⊢ coeff (p ^ i * p) (i * natDegree p + natDegree p) = leadingCoeff p ^ i * leadingCoeff p\n\ncase neg\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : ¬leadingCoeff p ^ i = 0\n⊢ coeff (p ^ i * p) (i * natDegree p + natDegree p) = leadingCoeff p ^ i * leadingCoeff p",
"state_before": "case succ\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\n⊢ coeff (p ^ i * p) (i * natDegree p + natDegree p) = leadingCoeff p ^ i * leadingCoeff p",
"tactic": "by_cases hp1 : p.leadingCoeff ^ i = 0"
},
{
"state_after": "case pos\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\n⊢ coeff (p ^ i * p) (i * natDegree p + natDegree p) = 0",
"state_before": "case pos\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\n⊢ coeff (p ^ i * p) (i * natDegree p + natDegree p) = leadingCoeff p ^ i * leadingCoeff p",
"tactic": "rw [hp1, zero_mul]"
},
{
"state_after": "case pos\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : p ^ i = 0\n⊢ coeff (p ^ i * p) (i * natDegree p + natDegree p) = 0\n\ncase neg\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : ¬p ^ i = 0\n⊢ coeff (p ^ i * p) (i * natDegree p + natDegree p) = 0",
"state_before": "case pos\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\n⊢ coeff (p ^ i * p) (i * natDegree p + natDegree p) = 0",
"tactic": "by_cases hp2 : p ^ i = 0"
},
{
"state_after": "no goals",
"state_before": "case pos\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : p ^ i = 0\n⊢ coeff (p ^ i * p) (i * natDegree p + natDegree p) = 0",
"tactic": "rw [hp2, zero_mul, coeff_zero]"
},
{
"state_after": "case neg.h\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : ¬p ^ i = 0\n⊢ natDegree (p ^ i * p) < i * natDegree p + natDegree p",
"state_before": "case neg\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : ¬p ^ i = 0\n⊢ coeff (p ^ i * p) (i * natDegree p + natDegree p) = 0",
"tactic": "apply coeff_eq_zero_of_natDegree_lt"
},
{
"state_after": "case neg.h\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : ¬p ^ i = 0\nh1 : natDegree (p ^ i) < i * natDegree p\n⊢ natDegree (p ^ i * p) < i * natDegree p + natDegree p",
"state_before": "case neg.h\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : ¬p ^ i = 0\n⊢ natDegree (p ^ i * p) < i * natDegree p + natDegree p",
"tactic": "have h1 : (p ^ i).natDegree < i * p.natDegree := by\n refine lt_of_le_of_ne natDegree_pow_le fun h => hp2 ?_\n rw [← h, hp1] at hi\n exact leadingCoeff_eq_zero.mp hi"
},
{
"state_after": "no goals",
"state_before": "case neg.h\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : ¬p ^ i = 0\nh1 : natDegree (p ^ i) < i * natDegree p\n⊢ natDegree (p ^ i * p) < i * natDegree p + natDegree p",
"tactic": "calc\n (p ^ i * p).natDegree ≤ (p ^ i).natDegree + p.natDegree := natDegree_mul_le\n _ < i * p.natDegree + p.natDegree := add_lt_add_right h1 _"
},
{
"state_after": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : ¬p ^ i = 0\nh : natDegree (p ^ i) = i * natDegree p\n⊢ p ^ i = 0",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : ¬p ^ i = 0\n⊢ natDegree (p ^ i) < i * natDegree p",
"tactic": "refine lt_of_le_of_ne natDegree_pow_le fun h => hp2 ?_"
},
{
"state_after": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (natDegree (p ^ i)) = 0\nhp1 : leadingCoeff p ^ i = 0\nhp2 : ¬p ^ i = 0\nh : natDegree (p ^ i) = i * natDegree p\n⊢ p ^ i = 0",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : leadingCoeff p ^ i = 0\nhp2 : ¬p ^ i = 0\nh : natDegree (p ^ i) = i * natDegree p\n⊢ p ^ i = 0",
"tactic": "rw [← h, hp1] at hi"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (natDegree (p ^ i)) = 0\nhp1 : leadingCoeff p ^ i = 0\nhp2 : ¬p ^ i = 0\nh : natDegree (p ^ i) = i * natDegree p\n⊢ p ^ i = 0",
"tactic": "exact leadingCoeff_eq_zero.mp hi"
},
{
"state_after": "case neg\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : ¬leadingCoeff p ^ i = 0\n⊢ coeff (p ^ i * p) (natDegree (p ^ i) + natDegree p) = leadingCoeff (p ^ i) * leadingCoeff p",
"state_before": "case neg\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : ¬leadingCoeff p ^ i = 0\n⊢ coeff (p ^ i * p) (i * natDegree p + natDegree p) = leadingCoeff p ^ i * leadingCoeff p",
"tactic": "rw [← natDegree_pow' hp1, ← leadingCoeff_pow' hp1]"
},
{
"state_after": "no goals",
"state_before": "case neg\nR : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np✝ q : R[X]\nι : Type ?u.697185\np : R[X]\ni : ℕ\nhi : coeff (p ^ i) (i * natDegree p) = leadingCoeff p ^ i\nhp1 : ¬leadingCoeff p ^ i = 0\n⊢ coeff (p ^ i * p) (natDegree (p ^ i) + natDegree p) = leadingCoeff (p ^ i) * leadingCoeff p",
"tactic": "exact coeff_mul_degree_add_degree _ _"
}
] | [
1065,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1046,
1
] |
Mathlib/GroupTheory/Perm/Cycle/Basic.lean | Equiv.Perm.mem_list_cycles_iff | [
{
"state_after": "ι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\n⊢ IsCycle σ → (σ ∈ l ↔ ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a)",
"state_before": "ι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\n⊢ σ ∈ l ↔ IsCycle σ ∧ ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a",
"tactic": "suffices σ.IsCycle → (σ ∈ l ↔ ∀ (a : α) (_ : σ a ≠ a), σ a = l.prod a) by\n exact ⟨fun hσ => ⟨h1 σ hσ, (this (h1 σ hσ)).mp hσ⟩, fun hσ => (this hσ.1).mpr hσ.2⟩"
},
{
"state_after": "ι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\n⊢ σ ∈ l ↔ ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a",
"state_before": "ι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\n⊢ IsCycle σ → (σ ∈ l ↔ ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a)",
"tactic": "intro h3"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nthis : IsCycle σ → (σ ∈ l ↔ ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a)\n⊢ σ ∈ l ↔ IsCycle σ ∧ ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a",
"tactic": "exact ⟨fun hσ => ⟨h1 σ hσ, (this (h1 σ hσ)).mp hσ⟩, fun hσ => (this hσ.1).mpr hσ.2⟩"
},
{
"state_after": "case intro\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\n⊢ σ ∈ l ↔ ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a",
"state_before": "ι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\n⊢ σ ∈ l ↔ ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a",
"tactic": "cases nonempty_fintype α"
},
{
"state_after": "case intro.mp\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\n⊢ σ ∈ l → ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\n\ncase intro.mpr\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\n⊢ (∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a) → σ ∈ l",
"state_before": "case intro\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\n⊢ σ ∈ l ↔ ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a",
"tactic": "constructor"
},
{
"state_after": "case intro.mp\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : σ ∈ l\na : α\nha : ↑σ a ≠ a\n⊢ ↑σ a = ↑(List.prod l) a",
"state_before": "case intro.mp\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\n⊢ σ ∈ l → ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a",
"tactic": "intro h a ha"
},
{
"state_after": "no goals",
"state_before": "case intro.mp\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : σ ∈ l\na : α\nha : ↑σ a ≠ a\n⊢ ↑σ a = ↑(List.prod l) a",
"tactic": "exact eq_on_support_mem_disjoint h h2 _ (mem_support.mpr ha)"
},
{
"state_after": "case intro.mpr\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\n⊢ σ ∈ l",
"state_before": "case intro.mpr\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\n⊢ (∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a) → σ ∈ l",
"tactic": "intro h"
},
{
"state_after": "case intro.mpr\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nhσl : support σ ⊆ support (List.prod l)\n⊢ σ ∈ l",
"state_before": "case intro.mpr\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\n⊢ σ ∈ l",
"tactic": "have hσl : σ.support ⊆ l.prod.support := by\n intro x hx\n rw [mem_support] at hx\n rwa [mem_support, ← h _ hx]"
},
{
"state_after": "case intro.mpr.intro.intro\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nhσl : support σ ⊆ support (List.prod l)\na : α\nha : ↑σ a ≠ a\n⊢ σ ∈ l",
"state_before": "case intro.mpr\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nhσl : support σ ⊆ support (List.prod l)\n⊢ σ ∈ l",
"tactic": "obtain ⟨a, ha, -⟩ := id h3"
},
{
"state_after": "case intro.mpr.intro.intro\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nhσl : support σ ⊆ support (List.prod l)\na : α\nha✝ : ↑σ a ≠ a\nha : a ∈ support σ\n⊢ σ ∈ l",
"state_before": "case intro.mpr.intro.intro\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nhσl : support σ ⊆ support (List.prod l)\na : α\nha : ↑σ a ≠ a\n⊢ σ ∈ l",
"tactic": "rw [← mem_support] at ha"
},
{
"state_after": "case intro.mpr.intro.intro.intro.intro\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nhσl : support σ ⊆ support (List.prod l)\na : α\nha✝ : ↑σ a ≠ a\nha : a ∈ support σ\nτ : Perm α\nhτ : τ ∈ l\nhτa : a ∈ support τ\n⊢ σ ∈ l",
"state_before": "case intro.mpr.intro.intro\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nhσl : support σ ⊆ support (List.prod l)\na : α\nha✝ : ↑σ a ≠ a\nha : a ∈ support σ\n⊢ σ ∈ l",
"tactic": "obtain ⟨τ, hτ, hτa⟩ := exists_mem_support_of_mem_support_prod (hσl ha)"
},
{
"state_after": "case intro.mpr.intro.intro.intro.intro\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nhσl : support σ ⊆ support (List.prod l)\na : α\nha✝ : ↑σ a ≠ a\nha : a ∈ support σ\nτ : Perm α\nhτ : τ ∈ l\nhτa : a ∈ support τ\nhτl : ∀ (x : α), x ∈ support τ → ↑τ x = ↑(List.prod l) x\n⊢ σ ∈ l",
"state_before": "case intro.mpr.intro.intro.intro.intro\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nhσl : support σ ⊆ support (List.prod l)\na : α\nha✝ : ↑σ a ≠ a\nha : a ∈ support σ\nτ : Perm α\nhτ : τ ∈ l\nhτa : a ∈ support τ\n⊢ σ ∈ l",
"tactic": "have hτl : ∀ x ∈ τ.support, τ x = l.prod x := eq_on_support_mem_disjoint hτ h2"
},
{
"state_after": "case intro.mpr.intro.intro.intro.intro\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nhσl : support σ ⊆ support (List.prod l)\na : α\nha✝ : ↑σ a ≠ a\nha : a ∈ support σ\nτ : Perm α\nhτ : τ ∈ l\nhτa : a ∈ support τ\nhτl : ∀ (x : α), x ∈ support τ → ↑τ x = ↑(List.prod l) x\nkey : ∀ (x : α), x ∈ support σ ∩ support τ → ↑σ x = ↑τ x\n⊢ σ ∈ l",
"state_before": "case intro.mpr.intro.intro.intro.intro\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nhσl : support σ ⊆ support (List.prod l)\na : α\nha✝ : ↑σ a ≠ a\nha : a ∈ support σ\nτ : Perm α\nhτ : τ ∈ l\nhτa : a ∈ support τ\nhτl : ∀ (x : α), x ∈ support τ → ↑τ x = ↑(List.prod l) x\n⊢ σ ∈ l",
"tactic": "have key : ∀ x ∈ σ.support ∩ τ.support, σ x = τ x := by\n intro x hx\n rw [h x (mem_support.mp (mem_of_mem_inter_left hx)), hτl x (mem_of_mem_inter_right hx)]"
},
{
"state_after": "case h.e'_4\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nhσl : support σ ⊆ support (List.prod l)\na : α\nha✝ : ↑σ a ≠ a\nha : a ∈ support σ\nτ : Perm α\nhτ : τ ∈ l\nhτa : a ∈ support τ\nhτl : ∀ (x : α), x ∈ support τ → ↑τ x = ↑(List.prod l) x\nkey : ∀ (x : α), x ∈ support σ ∩ support τ → ↑σ x = ↑τ x\n⊢ σ = τ",
"state_before": "case intro.mpr.intro.intro.intro.intro\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nhσl : support σ ⊆ support (List.prod l)\na : α\nha✝ : ↑σ a ≠ a\nha : a ∈ support σ\nτ : Perm α\nhτ : τ ∈ l\nhτa : a ∈ support τ\nhτl : ∀ (x : α), x ∈ support τ → ↑τ x = ↑(List.prod l) x\nkey : ∀ (x : α), x ∈ support σ ∩ support τ → ↑σ x = ↑τ x\n⊢ σ ∈ l",
"tactic": "convert hτ"
},
{
"state_after": "case h.e'_4\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nhσl : support σ ⊆ support (List.prod l)\na : α\nha✝ : ↑σ a ≠ a\nha : a ∈ support σ\nτ : Perm α\nhτ : τ ∈ l\nhτa : a ∈ support τ\nhτl : ∀ (x : α), x ∈ support τ → ↑τ x = ↑(List.prod l) x\nkey : ∀ (x : α), x ∈ support σ ∩ support τ → ↑σ x = ↑τ x\n⊢ ↑σ a = ↑τ a",
"state_before": "case h.e'_4\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nhσl : support σ ⊆ support (List.prod l)\na : α\nha✝ : ↑σ a ≠ a\nha : a ∈ support σ\nτ : Perm α\nhτ : τ ∈ l\nhτa : a ∈ support τ\nhτl : ∀ (x : α), x ∈ support τ → ↑τ x = ↑(List.prod l) x\nkey : ∀ (x : α), x ∈ support σ ∩ support τ → ↑σ x = ↑τ x\n⊢ σ = τ",
"tactic": "refine' h3.eq_on_support_inter_nonempty_congr (h1 _ hτ) key _ ha"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4\nι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nhσl : support σ ⊆ support (List.prod l)\na : α\nha✝ : ↑σ a ≠ a\nha : a ∈ support σ\nτ : Perm α\nhτ : τ ∈ l\nhτa : a ∈ support τ\nhτl : ∀ (x : α), x ∈ support τ → ↑τ x = ↑(List.prod l) x\nkey : ∀ (x : α), x ∈ support σ ∩ support τ → ↑σ x = ↑τ x\n⊢ ↑σ a = ↑τ a",
"tactic": "exact key a (mem_inter_of_mem ha hτa)"
},
{
"state_after": "ι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nx : α\nhx : x ∈ support σ\n⊢ x ∈ support (List.prod l)",
"state_before": "ι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\n⊢ support σ ⊆ support (List.prod l)",
"tactic": "intro x hx"
},
{
"state_after": "ι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nx : α\nhx : ↑σ x ≠ x\n⊢ x ∈ support (List.prod l)",
"state_before": "ι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nx : α\nhx : x ∈ support σ\n⊢ x ∈ support (List.prod l)",
"tactic": "rw [mem_support] at hx"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nx : α\nhx : ↑σ x ≠ x\n⊢ x ∈ support (List.prod l)",
"tactic": "rwa [mem_support, ← h _ hx]"
},
{
"state_after": "ι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nhσl : support σ ⊆ support (List.prod l)\na : α\nha✝ : ↑σ a ≠ a\nha : a ∈ support σ\nτ : Perm α\nhτ : τ ∈ l\nhτa : a ∈ support τ\nhτl : ∀ (x : α), x ∈ support τ → ↑τ x = ↑(List.prod l) x\nx : α\nhx : x ∈ support σ ∩ support τ\n⊢ ↑σ x = ↑τ x",
"state_before": "ι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nhσl : support σ ⊆ support (List.prod l)\na : α\nha✝ : ↑σ a ≠ a\nha : a ∈ support σ\nτ : Perm α\nhτ : τ ∈ l\nhτa : a ∈ support τ\nhτl : ∀ (x : α), x ∈ support τ → ↑τ x = ↑(List.prod l) x\n⊢ ∀ (x : α), x ∈ support σ ∩ support τ → ↑σ x = ↑τ x",
"tactic": "intro x hx"
},
{
"state_after": "no goals",
"state_before": "ι : Type ?u.2666441\nα✝ : Type ?u.2666444\nβ : Type ?u.2666447\ninst✝¹ : DecidableEq α✝\nα : Type u_1\ninst✝ : Finite α\nl : List (Perm α)\nh1 : ∀ (σ : Perm α), σ ∈ l → IsCycle σ\nh2 : List.Pairwise Disjoint l\nσ : Perm α\nh3 : IsCycle σ\nval✝ : Fintype α\nh : ∀ (a : α), ↑σ a ≠ a → ↑σ a = ↑(List.prod l) a\nhσl : support σ ⊆ support (List.prod l)\na : α\nha✝ : ↑σ a ≠ a\nha : a ∈ support σ\nτ : Perm α\nhτ : τ ∈ l\nhτa : a ∈ support τ\nhτl : ∀ (x : α), x ∈ support τ → ↑τ x = ↑(List.prod l) x\nx : α\nhx : x ∈ support σ ∩ support τ\n⊢ ↑σ x = ↑τ x",
"tactic": "rw [h x (mem_support.mp (mem_of_mem_inter_left hx)), hτl x (mem_of_mem_inter_right hx)]"
}
] | [
1310,
44
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1285,
1
] |
Mathlib/Combinatorics/SimpleGraph/Connectivity.lean | SimpleGraph.isBridge_iff | [] | [
2447,
93
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2446,
1
] |
Mathlib/Order/Filter/Partial.lean | Filter.rmap_rmap | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nr : Rel α β\ns : Rel β γ\nl : Filter α\n⊢ (rmap s (rmap r l)).sets = (rmap (Rel.comp r s) l).sets",
"tactic": "simp [rmap_sets, Set.preimage, Rel.core_comp]"
}
] | [
83,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
81,
1
] |
Mathlib/Analysis/LocallyConvex/Basic.lean | balanced_empty | [
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_1\n𝕝 : Type ?u.11887\nE : Type u_2\nι : Sort ?u.11893\nκ : ι → Sort ?u.11898\ninst✝¹ : SeminormedRing 𝕜\ninst✝ : SMul 𝕜 E\ns t u v A B : Set E\nx✝¹ : 𝕜\nx✝ : ‖x✝¹‖ ≤ 1\n⊢ x✝¹ • ∅ ⊆ ∅",
"tactic": "rw [smul_set_empty]"
}
] | [
175,
85
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
175,
1
] |
Mathlib/Topology/Inseparable.lean | SeparationQuotient.lift_comp_mk | [] | [
537,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
536,
1
] |
Mathlib/Order/Filter/SmallSets.lean | Filter.smallSets_principal | [] | [
110,
36
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
109,
1
] |
Mathlib/Data/Dfinsupp/NeLocus.lean | Dfinsupp.neLocus_eq_empty | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nN : α → Type u_2\ninst✝² : DecidableEq α\ninst✝¹ : (a : α) → DecidableEq (N a)\ninst✝ : (a : α) → Zero (N a)\nf✝ g✝ f g : Π₀ (a : α), N a\nh : f = g\n⊢ neLocus f f = ∅",
"tactic": "simp only [neLocus, Ne.def, eq_self_iff_true, not_true, Finset.filter_False]"
}
] | [
62,
97
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
59,
1
] |
Mathlib/Data/Set/Pointwise/Interval.lean | Set.preimage_const_add_Iio | [] | [
59,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
58,
1
] |
Mathlib/Analysis/SpecialFunctions/Pow/Complex.lean | Complex.cpow_sub | [
{
"state_after": "no goals",
"state_before": "x y z : ℂ\nhx : x ≠ 0\n⊢ x ^ (y - z) = x ^ y / x ^ z",
"tactic": "rw [sub_eq_add_neg, cpow_add _ _ hx, cpow_neg, div_eq_mul_inv]"
}
] | [
111,
65
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
110,
1
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Mathlib/Data/List/Forall2.lean | List.rel_filter | [
{
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"tactic": "dsimp [LiftFun] at hpq"
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{
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{
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"tactic": "have : q b := by rwa [← hpq h₁]"
},
{
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"tactic": "simp only [filter_cons_of_pos _ h, filter_cons_of_pos _ this, forall₂_cons, h₁, true_and_iff,\n rel_filter hpq h₂]"
},
{
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"tactic": "rwa [← hpq h₁]"
},
{
"state_after": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type ?u.148999\nδ : Type ?u.149002\nR S : α → β → Prop\nP : γ → δ → Prop\nRₐ : α → α → Prop\np : α → Bool\nq : β → Bool\nhpq : ∀ ⦃a : α⦄ ⦃b : β⦄, R a b → (p a = true ↔ q b = true)\na : α\nas : List α\nb : β\nbs : List β\nh₁ : R a b\nh₂ : Forall₂ R as bs\nh : ¬p a = true\nthis : ¬q b = true\n⊢ Forall₂ R (filter p (a :: as)) (filter q (b :: bs))",
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"tactic": "have : ¬q b := by rwa [← hpq h₁]"
},
{
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"tactic": "simp only [filter_cons_of_neg _ h, filter_cons_of_neg _ this, rel_filter hpq h₂]"
},
{
"state_after": "no goals",
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"tactic": "rwa [← hpq h₁]"
}
] | [
302,
87
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
291,
1
] |
Mathlib/Data/Finsupp/Lex.lean | Finsupp.lex_eq_invImage_dfinsupp_lex | [] | [
52,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
50,
1
] |
Mathlib/Algebra/Star/Subalgebra.lean | StarSubalgebra.mem_carrier | [] | [
82,
10
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
81,
1
] |
Mathlib/Topology/Semicontinuous.lean | upperSemicontinuous_sum | [] | [
987,
59
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
985,
1
] |
Mathlib/Data/Polynomial/Degree/Definitions.lean | Polynomial.natDegree_mem_support_of_nonzero | [
{
"state_after": "R : Type u\nS : Type v\na b c d : R\nn m : ℕ\ninst✝ : Semiring R\np q : R[X]\nι : Type ?u.515454\nH : p ≠ 0\n⊢ coeff p (natDegree p) ≠ 0",
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"tactic": "rw [mem_support_iff]"
},
{
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"tactic": "exact (not_congr leadingCoeff_eq_zero).mpr H"
}
] | [
677,
47
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
675,
1
] |
Mathlib/RingTheory/WittVector/Basic.lean | WittVector.mapFun.zsmul | [
{
"state_after": "no goals",
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"tactic": "map_fun_tac"
}
] | [
130,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
130,
1
] |
Mathlib/Data/Finset/Prod.lean | Finset.filter_product | [
{
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"tactic": "ext ⟨a, b⟩"
},
{
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"tactic": "simp [mem_filter, mem_product, decide_eq_true_eq, and_comm, and_left_comm, and_assoc]"
}
] | [
149,
88
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
146,
1
] |
Mathlib/Algebra/ModEq.lean | AddCommGroup.modEq_iff_eq_mod_zmultiples | [
{
"state_after": "no goals",
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"tactic": "simp_rw [modEq_iff_eq_add_zsmul, QuotientAddGroup.eq_iff_sub_mem, AddSubgroup.mem_zmultiples_iff,\n eq_sub_iff_add_eq', eq_comm]"
}
] | [
302,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
300,
1
] |
Mathlib/Analysis/NormedSpace/OperatorNorm.lean | ContinuousLinearMap.prodMapL_apply | [] | [
1067,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1065,
1
] |
Mathlib/Topology/MetricSpace/PiNat.lean | PiCountable.dist_eq_tsum | [] | [
833,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
831,
1
] |
Mathlib/Order/Minimal.lean | IsAntichain.maximals_lowerClosure | [] | [
244,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
242,
1
] |
Mathlib/Order/WithBot.lean | WithTop.untop'_eq_untop'_iff | [] | [
724,
31
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
722,
1
] |
Mathlib/Algebra/Support.lean | Function.mulSupport_mul_inv | [
{
"state_after": "no goals",
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"tactic": "simp"
}
] | [
292,
61
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
291,
1
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Mathlib/Topology/Constructions.lean | Filter.Tendsto.update | [
{
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"tactic": "rcases eq_or_ne j i with (rfl | hj) <;> simp [*, hf.apply]"
}
] | [
1234,
91
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1231,
1
] |
Mathlib/Data/Set/Finite.lean | Set.Finite.inf_of_right | [] | [
757,
21
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
756,
1
] |
Mathlib/Data/Real/Irrational.lean | Irrational.of_div_int | [] | [
432,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
431,
1
] |
Mathlib/MeasureTheory/Group/Action.lean | MeasureTheory.map_smul | [] | [
101,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
100,
1
] |
Mathlib/Geometry/Manifold/SmoothManifoldWithCorners.lean | LocalHomeomorph.extend_symm_preimage_inter_range_eventuallyEq_aux | [
{
"state_after": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_1\nH : Type u_2\nE' : Type ?u.162184\nM' : Type ?u.162187\nH' : Type ?u.162190\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns✝ t s : Set M\nx : M\nhx : x ∈ f.source\n⊢ ↑(LocalEquiv.symm (extend f I)) ⁻¹' s ∩ range ↑I =ᶠ[𝓝 (↑(extend f I) x)]\n ↑(ModelWithCorners.symm I) ⁻¹' f.target ∩ (↑(LocalEquiv.symm (extend f I)) ⁻¹' s ∩ range ↑I)",
"state_before": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_1\nH : Type u_2\nE' : Type ?u.162184\nM' : Type ?u.162187\nH' : Type ?u.162190\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns✝ t s : Set M\nx : M\nhx : x ∈ f.source\n⊢ ↑(LocalEquiv.symm (extend f I)) ⁻¹' s ∩ range ↑I =ᶠ[𝓝 (↑(extend f I) x)]\n (extend f I).target ∩ ↑(LocalEquiv.symm (extend f I)) ⁻¹' s",
"tactic": "rw [f.extend_target, inter_assoc, inter_comm (range I)]"
},
{
"state_after": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_1\nH : Type u_2\nE' : Type ?u.162184\nM' : Type ?u.162187\nH' : Type ?u.162190\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns✝ t s : Set M\nx : M\nhx : x ∈ f.source\n⊢ ↑(ModelWithCorners.symm I) ⁻¹' f.target ∈ 𝓝 (↑(extend f I) x)",
"state_before": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_1\nH : Type u_2\nE' : Type ?u.162184\nM' : Type ?u.162187\nH' : Type ?u.162190\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns✝ t s : Set M\nx : M\nhx : x ∈ f.source\n⊢ univ ∩ (↑(LocalEquiv.symm (extend f I)) ⁻¹' s ∩ range ↑I) =ᶠ[𝓝 (↑(extend f I) x)]\n ↑(ModelWithCorners.symm I) ⁻¹' f.target ∩ (↑(LocalEquiv.symm (extend f I)) ⁻¹' s ∩ range ↑I)",
"tactic": "refine' (eventuallyEq_univ.mpr _).symm.inter EventuallyEq.rfl"
},
{
"state_after": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_1\nH : Type u_2\nE' : Type ?u.162184\nM' : Type ?u.162187\nH' : Type ?u.162190\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns✝ t s : Set M\nx : M\nhx : x ∈ f.source\n⊢ ↑(ModelWithCorners.symm I) (↑(extend f I) x) ∈ f.target",
"state_before": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_1\nH : Type u_2\nE' : Type ?u.162184\nM' : Type ?u.162187\nH' : Type ?u.162190\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns✝ t s : Set M\nx : M\nhx : x ∈ f.source\n⊢ ↑(ModelWithCorners.symm I) ⁻¹' f.target ∈ 𝓝 (↑(extend f I) x)",
"tactic": "refine' I.continuousAt_symm.preimage_mem_nhds (f.open_target.mem_nhds _)"
},
{
"state_after": "no goals",
"state_before": "𝕜 : Type u_4\nE : Type u_3\nM : Type u_1\nH : Type u_2\nE' : Type ?u.162184\nM' : Type ?u.162187\nH' : Type ?u.162190\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : TopologicalSpace H\ninst✝⁴ : TopologicalSpace M\nf f' : LocalHomeomorph M H\nI : ModelWithCorners 𝕜 E H\ninst✝³ : NormedAddCommGroup E'\ninst✝² : NormedSpace 𝕜 E'\ninst✝¹ : TopologicalSpace H'\ninst✝ : TopologicalSpace M'\nI' : ModelWithCorners 𝕜 E' H'\nx✝ : M\ns✝ t s : Set M\nx : M\nhx : x ∈ f.source\n⊢ ↑(ModelWithCorners.symm I) (↑(extend f I) x) ∈ f.target",
"tactic": "simp_rw [f.extend_coe, Function.comp_apply, I.left_inv, f.mapsTo hx]"
}
] | [
949,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
939,
1
] |
Mathlib/GroupTheory/MonoidLocalization.lean | Submonoid.LocalizationMap.sec_spec' | [
{
"state_after": "no goals",
"state_before": "M : Type u_1\ninst✝² : CommMonoid M\nS : Submonoid M\nN : Type u_2\ninst✝¹ : CommMonoid N\nP : Type ?u.309265\ninst✝ : CommMonoid P\nf : LocalizationMap S N\nz : N\n⊢ ↑(toMap f) (sec f z).fst = ↑(toMap f) ↑(sec f z).snd * z",
"tactic": "rw [mul_comm, sec_spec]"
}
] | [
610,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
609,
1
] |
Mathlib/Algebra/Order/LatticeGroup.lean | LatticeOrderedCommGroup.neg_of_inv_le_one | [] | [
500,
24
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
499,
1
] |
Mathlib/CategoryTheory/Limits/Shapes/BinaryProducts.lean | CategoryTheory.Limits.WalkingPair.swap_symm_apply_tt | [] | [
76,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
75,
1
] |
Std/Data/Int/Lemmas.lean | Int.zero_mul | [] | [
408,
89
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
408,
19
] |
Mathlib/MeasureTheory/Measure/Lebesgue/EqHaar.lean | MeasureTheory.Measure.tendsto_add_haar_inter_smul_zero_of_density_zero_aux1 | [
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 0)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\n⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 0)",
"tactic": "have A : Tendsto (fun r : ℝ => μ (s ∩ ({x} + r • t)) / μ (closedBall x r)) (𝓝[>] 0) (𝓝 0) := by\n apply\n tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds h\n (eventually_of_forall fun b => zero_le _)\n filter_upwards [self_mem_nhdsWithin]\n rintro r (rpos : 0 < r)\n apply mul_le_mul_right' (measure_mono (inter_subset_inter_right _ _)) _\n intro y hy\n have : y - x ∈ r • closedBall (0 : E) 1 := by\n apply smul_set_mono t_bound\n simpa [neg_add_eq_sub] using hy\n simpa only [smul_closedBall _ _ zero_le_one, Real.norm_of_nonneg rpos.le,\n mem_closedBall_iff_norm, mul_one, sub_zero, smul_zero]"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\n⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 0)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 0)",
"tactic": "have B :\n Tendsto (fun r : ℝ => μ (closedBall x r) / μ ({x} + r • u)) (𝓝[>] 0)\n (𝓝 (μ (closedBall x 1) / μ ({x} + u))) := by\n apply tendsto_const_nhds.congr' _\n filter_upwards [self_mem_nhdsWithin]\n rintro r (rpos : 0 < r)\n have : closedBall x r = {x} + r • closedBall (0 : E) 1 := by\n simp only [_root_.smul_closedBall, Real.norm_of_nonneg rpos.le, zero_le_one, add_zero,\n mul_one, singleton_add_closedBall, smul_zero]\n simp only [this, add_haar_singleton_add_smul_div_singleton_add_smul μ rpos.ne']\n simp only [add_haar_closedBall_center, image_add_left, measure_preimage_add, singleton_add]"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\nC :\n Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)))\n (𝓝[Ioi 0] 0) (𝓝 (0 * (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u))))\n⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 0)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\n⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 0)",
"tactic": "have C : Tendsto (fun r : ℝ =>\n μ (s ∩ ({x} + r • t)) / μ (closedBall x r) * (μ (closedBall x r) / μ ({x} + r • u)))\n (𝓝[>] 0) (𝓝 (0 * (μ (closedBall x 1) / μ ({x} + u)))) := by\n apply ENNReal.Tendsto.mul A _ B (Or.inr ENNReal.zero_ne_top)\n simp only [ne_eq, not_true, singleton_add, image_add_left, measure_preimage_add, false_or,\n ENNReal.div_eq_top, h'u, false_or_iff, not_and, and_false_iff]\n intro aux\n exact (measure_closedBall_lt_top.ne aux).elim"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\nC :\n Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)))\n (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 0)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\nC :\n Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)))\n (𝓝[Ioi 0] 0) (𝓝 (0 * (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u))))\n⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 0)",
"tactic": "simp only [zero_mul] at C"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\nC :\n Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)))\n (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u))) =ᶠ[𝓝[Ioi 0] 0]\n fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • u)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\nC :\n Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)))\n (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 0)",
"tactic": "apply C.congr' _"
},
{
"state_after": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\nC :\n Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)))\n (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ ∀ (a : ℝ),\n a ∈ Ioi 0 →\n ↑↑μ (s ∩ ({x} + a • t)) / ↑↑μ (closedBall x a) * (↑↑μ (closedBall x a) / ↑↑μ ({x} + a • u)) =\n ↑↑μ (s ∩ ({x} + a • t)) / ↑↑μ ({x} + a • u)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\nC :\n Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)))\n (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u))) =ᶠ[𝓝[Ioi 0] 0]\n fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • u)",
"tactic": "filter_upwards [self_mem_nhdsWithin]"
},
{
"state_after": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\nC :\n Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)))\n (𝓝[Ioi 0] 0) (𝓝 0)\nr : ℝ\nrpos : 0 < r\n⊢ ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) =\n ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • u)",
"state_before": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\nC :\n Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)))\n (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ ∀ (a : ℝ),\n a ∈ Ioi 0 →\n ↑↑μ (s ∩ ({x} + a • t)) / ↑↑μ (closedBall x a) * (↑↑μ (closedBall x a) / ↑↑μ ({x} + a • u)) =\n ↑↑μ (s ∩ ({x} + a • t)) / ↑↑μ ({x} + a • u)",
"tactic": "rintro r (rpos : 0 < r)"
},
{
"state_after": "no goals",
"state_before": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\nC :\n Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)))\n (𝓝[Ioi 0] 0) (𝓝 0)\nr : ℝ\nrpos : 0 < r\n⊢ ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) =\n ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • u)",
"tactic": "calc\n μ (s ∩ ({x} + r • t)) / μ (closedBall x r) * (μ (closedBall x r) / μ ({x} + r • u)) =\n μ (closedBall x r) * (μ (closedBall x r))⁻¹ * (μ (s ∩ ({x} + r • t)) / μ ({x} + r • u)) :=\n by simp only [div_eq_mul_inv]; ring\n _ = μ (s ∩ ({x} + r • t)) / μ ({x} + r • u) := by\n rw [ENNReal.mul_inv_cancel (measure_closedBall_pos μ x rpos).ne'\n measure_closedBall_lt_top.ne,\n one_mul]"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\n⊢ ∀ᶠ (b : ℝ) in 𝓝[Ioi 0] 0,\n ↑↑μ (s ∩ ({x} + b • t)) / ↑↑μ (closedBall x b) ≤ ↑↑μ (s ∩ closedBall x b) / ↑↑μ (closedBall x b)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\n⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)",
"tactic": "apply\n tendsto_of_tendsto_of_tendsto_of_le_of_le' tendsto_const_nhds h\n (eventually_of_forall fun b => zero_le _)"
},
{
"state_after": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\n⊢ ∀ (a : ℝ),\n a ∈ Ioi 0 → ↑↑μ (s ∩ ({x} + a • t)) / ↑↑μ (closedBall x a) ≤ ↑↑μ (s ∩ closedBall x a) / ↑↑μ (closedBall x a)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\n⊢ ∀ᶠ (b : ℝ) in 𝓝[Ioi 0] 0,\n ↑↑μ (s ∩ ({x} + b • t)) / ↑↑μ (closedBall x b) ≤ ↑↑μ (s ∩ closedBall x b) / ↑↑μ (closedBall x b)",
"tactic": "filter_upwards [self_mem_nhdsWithin]"
},
{
"state_after": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nr : ℝ\nrpos : 0 < r\n⊢ ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) ≤ ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)",
"state_before": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\n⊢ ∀ (a : ℝ),\n a ∈ Ioi 0 → ↑↑μ (s ∩ ({x} + a • t)) / ↑↑μ (closedBall x a) ≤ ↑↑μ (s ∩ closedBall x a) / ↑↑μ (closedBall x a)",
"tactic": "rintro r (rpos : 0 < r)"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nr : ℝ\nrpos : 0 < r\n⊢ {x} + r • t ⊆ closedBall x r",
"state_before": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nr : ℝ\nrpos : 0 < r\n⊢ ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) ≤ ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)",
"tactic": "apply mul_le_mul_right' (measure_mono (inter_subset_inter_right _ _)) _"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nr : ℝ\nrpos : 0 < r\ny : E\nhy : y ∈ {x} + r • t\n⊢ y ∈ closedBall x r",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nr : ℝ\nrpos : 0 < r\n⊢ {x} + r • t ⊆ closedBall x r",
"tactic": "intro y hy"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nr : ℝ\nrpos : 0 < r\ny : E\nhy : y ∈ {x} + r • t\nthis : y - x ∈ r • closedBall 0 1\n⊢ y ∈ closedBall x r",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nr : ℝ\nrpos : 0 < r\ny : E\nhy : y ∈ {x} + r • t\n⊢ y ∈ closedBall x r",
"tactic": "have : y - x ∈ r • closedBall (0 : E) 1 := by\n apply smul_set_mono t_bound\n simpa [neg_add_eq_sub] using hy"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nr : ℝ\nrpos : 0 < r\ny : E\nhy : y ∈ {x} + r • t\nthis : y - x ∈ r • closedBall 0 1\n⊢ y ∈ closedBall x r",
"tactic": "simpa only [smul_closedBall _ _ zero_le_one, Real.norm_of_nonneg rpos.le,\n mem_closedBall_iff_norm, mul_one, sub_zero, smul_zero]"
},
{
"state_after": "case a\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nr : ℝ\nrpos : 0 < r\ny : E\nhy : y ∈ {x} + r • t\n⊢ y - x ∈ r • t",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nr : ℝ\nrpos : 0 < r\ny : E\nhy : y ∈ {x} + r • t\n⊢ y - x ∈ r • closedBall 0 1",
"tactic": "apply smul_set_mono t_bound"
},
{
"state_after": "no goals",
"state_before": "case a\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nr : ℝ\nrpos : 0 < r\ny : E\nhy : y ∈ {x} + r • t\n⊢ y - x ∈ r • t",
"tactic": "simpa [neg_add_eq_sub] using hy"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ (fun x_1 => ↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)) =ᶠ[𝓝[Ioi 0] 0] fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))",
"tactic": "apply tendsto_const_nhds.congr' _"
},
{
"state_after": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ ∀ (a : ℝ), a ∈ Ioi 0 → ↑↑μ (closedBall x 1) / ↑↑μ ({x} + u) = ↑↑μ (closedBall x a) / ↑↑μ ({x} + a • u)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ (fun x_1 => ↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)) =ᶠ[𝓝[Ioi 0] 0] fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)",
"tactic": "filter_upwards [self_mem_nhdsWithin]"
},
{
"state_after": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nr : ℝ\nrpos : 0 < r\n⊢ ↑↑μ (closedBall x 1) / ↑↑μ ({x} + u) = ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)",
"state_before": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\n⊢ ∀ (a : ℝ), a ∈ Ioi 0 → ↑↑μ (closedBall x 1) / ↑↑μ ({x} + u) = ↑↑μ (closedBall x a) / ↑↑μ ({x} + a • u)",
"tactic": "rintro r (rpos : 0 < r)"
},
{
"state_after": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nr : ℝ\nrpos : 0 < r\nthis : closedBall x r = {x} + r • closedBall 0 1\n⊢ ↑↑μ (closedBall x 1) / ↑↑μ ({x} + u) = ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)",
"state_before": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nr : ℝ\nrpos : 0 < r\n⊢ ↑↑μ (closedBall x 1) / ↑↑μ ({x} + u) = ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)",
"tactic": "have : closedBall x r = {x} + r • closedBall (0 : E) 1 := by\n simp only [_root_.smul_closedBall, Real.norm_of_nonneg rpos.le, zero_le_one, add_zero,\n mul_one, singleton_add_closedBall, smul_zero]"
},
{
"state_after": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nr : ℝ\nrpos : 0 < r\nthis : closedBall x r = {x} + r • closedBall 0 1\n⊢ ↑↑μ (closedBall x 1) / ↑↑μ ({x} + u) = ↑↑μ (closedBall 0 1) / ↑↑μ u",
"state_before": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nr : ℝ\nrpos : 0 < r\nthis : closedBall x r = {x} + r • closedBall 0 1\n⊢ ↑↑μ (closedBall x 1) / ↑↑μ ({x} + u) = ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)",
"tactic": "simp only [this, add_haar_singleton_add_smul_div_singleton_add_smul μ rpos.ne']"
},
{
"state_after": "no goals",
"state_before": "case h\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nr : ℝ\nrpos : 0 < r\nthis : closedBall x r = {x} + r • closedBall 0 1\n⊢ ↑↑μ (closedBall x 1) / ↑↑μ ({x} + u) = ↑↑μ (closedBall 0 1) / ↑↑μ u",
"tactic": "simp only [add_haar_closedBall_center, image_add_left, measure_preimage_add, singleton_add]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nr : ℝ\nrpos : 0 < r\n⊢ closedBall x r = {x} + r • closedBall 0 1",
"tactic": "simp only [_root_.smul_closedBall, Real.norm_of_nonneg rpos.le, zero_le_one, add_zero,\n mul_one, singleton_add_closedBall, smul_zero]"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\n⊢ 0 ≠ 0 ∨ ↑↑μ (closedBall x 1) / ↑↑μ ({x} + u) ≠ ⊤",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\n⊢ Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)))\n (𝓝[Ioi 0] 0) (𝓝 (0 * (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u))))",
"tactic": "apply ENNReal.Tendsto.mul A _ B (Or.inr ENNReal.zero_ne_top)"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\n⊢ ↑↑μ (closedBall x 1) = ⊤ → ¬¬↑↑μ u = ⊤",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\n⊢ 0 ≠ 0 ∨ ↑↑μ (closedBall x 1) / ↑↑μ ({x} + u) ≠ ⊤",
"tactic": "simp only [ne_eq, not_true, singleton_add, image_add_left, measure_preimage_add, false_or,\n ENNReal.div_eq_top, h'u, false_or_iff, not_and, and_false_iff]"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\naux : ↑↑μ (closedBall x 1) = ⊤\n⊢ ¬¬↑↑μ u = ⊤",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\n⊢ ↑↑μ (closedBall x 1) = ⊤ → ¬¬↑↑μ u = ⊤",
"tactic": "intro aux"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\naux : ↑↑μ (closedBall x 1) = ⊤\n⊢ ¬¬↑↑μ u = ⊤",
"tactic": "exact (measure_closedBall_lt_top.ne aux).elim"
},
{
"state_after": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\nC :\n Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)))\n (𝓝[Ioi 0] 0) (𝓝 0)\nr : ℝ\nrpos : 0 < r\n⊢ ↑↑μ (s ∩ ({x} + r • t)) * (↑↑μ (closedBall x r))⁻¹ * (↑↑μ (closedBall x r) * (↑↑μ ({x} + r • u))⁻¹) =\n ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r))⁻¹ * (↑↑μ (s ∩ ({x} + r • t)) * (↑↑μ ({x} + r • u))⁻¹)",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\nC :\n Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)))\n (𝓝[Ioi 0] 0) (𝓝 0)\nr : ℝ\nrpos : 0 < r\n⊢ ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) =\n ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r))⁻¹ * (↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • u))",
"tactic": "simp only [div_eq_mul_inv]"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\nC :\n Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)))\n (𝓝[Ioi 0] 0) (𝓝 0)\nr : ℝ\nrpos : 0 < r\n⊢ ↑↑μ (s ∩ ({x} + r • t)) * (↑↑μ (closedBall x r))⁻¹ * (↑↑μ (closedBall x r) * (↑↑μ ({x} + r • u))⁻¹) =\n ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r))⁻¹ * (↑↑μ (s ∩ ({x} + r • t)) * (↑↑μ ({x} + r • u))⁻¹)",
"tactic": "ring"
},
{
"state_after": "no goals",
"state_before": "E : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\ninst✝⁶ : MeasurableSpace E\ninst✝⁵ : BorelSpace E\ninst✝⁴ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝³ : IsAddHaarMeasure μ\nF : Type ?u.2104866\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : CompleteSpace F\ns✝ s : Set E\nx : E\nh : Tendsto (fun r => ↑↑μ (s ∩ closedBall x r) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nt u : Set E\nh'u : ↑↑μ u ≠ 0\nt_bound : t ⊆ closedBall 0 1\nA : Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r)) (𝓝[Ioi 0] 0) (𝓝 0)\nB : Tendsto (fun r => ↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)) (𝓝[Ioi 0] 0) (𝓝 (↑↑μ (closedBall x 1) / ↑↑μ ({x} + u)))\nC :\n Tendsto (fun r => ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r) / ↑↑μ ({x} + r • u)))\n (𝓝[Ioi 0] 0) (𝓝 0)\nr : ℝ\nrpos : 0 < r\n⊢ ↑↑μ (closedBall x r) * (↑↑μ (closedBall x r))⁻¹ * (↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • u)) =\n ↑↑μ (s ∩ ({x} + r • t)) / ↑↑μ ({x} + r • u)",
"tactic": "rw [ENNReal.mul_inv_cancel (measure_closedBall_pos μ x rpos).ne'\n measure_closedBall_lt_top.ne,\n one_mul]"
}
] | [
657,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
608,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean | MeasureTheory.Integrable.inf | [
{
"state_after": "α : Type u_2\nβ✝ : Type ?u.953675\nγ : Type ?u.953678\nδ : Type ?u.953681\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β✝\ninst✝¹ : NormedAddCommGroup γ\nβ : Type u_1\ninst✝ : NormedLatticeAddCommGroup β\nf g : α → β\nhf : Memℒp f 1\nhg : Memℒp g 1\n⊢ Memℒp (f ⊓ g) 1",
"state_before": "α : Type u_2\nβ✝ : Type ?u.953675\nγ : Type ?u.953678\nδ : Type ?u.953681\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β✝\ninst✝¹ : NormedAddCommGroup γ\nβ : Type u_1\ninst✝ : NormedLatticeAddCommGroup β\nf g : α → β\nhf : Integrable f\nhg : Integrable g\n⊢ Integrable (f ⊓ g)",
"tactic": "rw [← memℒp_one_iff_integrable] at hf hg⊢"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ✝ : Type ?u.953675\nγ : Type ?u.953678\nδ : Type ?u.953681\nm : MeasurableSpace α\nμ ν : Measure α\ninst✝³ : MeasurableSpace δ\ninst✝² : NormedAddCommGroup β✝\ninst✝¹ : NormedAddCommGroup γ\nβ : Type u_1\ninst✝ : NormedLatticeAddCommGroup β\nf g : α → β\nhf : Memℒp f 1\nhg : Memℒp g 1\n⊢ Memℒp (f ⊓ g) 1",
"tactic": "exact hf.inf hg"
}
] | [
699,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
696,
1
] |
Mathlib/Analysis/InnerProductSpace/l2Space.lean | lp.summable_inner | [
{
"state_after": "case refine'_1\nι : Type u_1\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\nE : Type ?u.47553\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nf g : { x // x ∈ lp G 2 }\n⊢ Real.IsConjugateExponent (ENNReal.toReal 2) (ENNReal.toReal 2)\n\ncase refine'_2\nι : Type u_1\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\nE : Type ?u.47553\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nf g : { x // x ∈ lp G 2 }\n⊢ ∀ (i : ι), ‖inner (↑f i) (↑g i)‖ ≤ (fun i => ‖↑f i‖ * ‖↑g i‖) i",
"state_before": "ι : Type u_1\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\nE : Type ?u.47553\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nf g : { x // x ∈ lp G 2 }\n⊢ Summable fun i => inner (↑f i) (↑g i)",
"tactic": "refine' summable_of_norm_bounded (fun i => ‖f i‖ * ‖g i‖) (lp.summable_mul _ f g) _"
},
{
"state_after": "case refine'_2\nι : Type u_1\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\nE : Type ?u.47553\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nf g : { x // x ∈ lp G 2 }\ni : ι\n⊢ ‖inner (↑f i) (↑g i)‖ ≤ (fun i => ‖↑f i‖ * ‖↑g i‖) i",
"state_before": "case refine'_2\nι : Type u_1\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\nE : Type ?u.47553\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nf g : { x // x ∈ lp G 2 }\n⊢ ∀ (i : ι), ‖inner (↑f i) (↑g i)‖ ≤ (fun i => ‖↑f i‖ * ‖↑g i‖) i",
"tactic": "intro i"
},
{
"state_after": "no goals",
"state_before": "case refine'_2\nι : Type u_1\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\nE : Type ?u.47553\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nf g : { x // x ∈ lp G 2 }\ni : ι\n⊢ ‖inner (↑f i) (↑g i)‖ ≤ (fun i => ‖↑f i‖ * ‖↑g i‖) i",
"tactic": "exact norm_inner_le_norm (𝕜 := 𝕜) _ _"
},
{
"state_after": "no goals",
"state_before": "case refine'_1\nι : Type u_1\n𝕜 : Type u_3\ninst✝⁴ : IsROrC 𝕜\nE : Type ?u.47553\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ncplt : CompleteSpace E\nG : ι → Type u_2\ninst✝¹ : (i : ι) → NormedAddCommGroup (G i)\ninst✝ : (i : ι) → InnerProductSpace 𝕜 (G i)\nf g : { x // x ∈ lp G 2 }\n⊢ Real.IsConjugateExponent (ENNReal.toReal 2) (ENNReal.toReal 2)",
"tactic": "rw [Real.isConjugateExponent_iff] <;> norm_num"
}
] | [
122,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
116,
1
] |
Mathlib/LinearAlgebra/QuotientPi.lean | Submodule.piQuotientLift_mk | [
{
"state_after": "ι : Type u_1\nR : Type u_2\ninst✝⁸ : CommRing R\nMs : ι → Type u_3\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type ?u.16262\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\nq : Submodule R N\nf : (i : ι) → Ms i →ₗ[R] N\nhf : ∀ (i : ι), p i ≤ comap (f i) q\nx : (i : ι) → Ms i\n⊢ (Finset.sum Finset.univ fun d =>\n ↑(comp (mapQ (p d) q (f d) (_ : p d ≤ comap (f d) q)) (proj d)) fun i => Quotient.mk (x i)) =\n Finset.sum Finset.univ fun x_1 => ↑(mkQ q) (↑(comp (f x_1) (proj x_1)) x)",
"state_before": "ι : Type u_1\nR : Type u_2\ninst✝⁸ : CommRing R\nMs : ι → Type u_3\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type ?u.16262\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\nq : Submodule R N\nf : (i : ι) → Ms i →ₗ[R] N\nhf : ∀ (i : ι), p i ≤ comap (f i) q\nx : (i : ι) → Ms i\n⊢ (↑(piQuotientLift p q f hf) fun i => Quotient.mk (x i)) = Quotient.mk (↑(↑(lsum R (fun i => Ms i) R) f) x)",
"tactic": "rw [piQuotientLift, lsum_apply, sum_apply, ← mkQ_apply, lsum_apply, sum_apply, _root_.map_sum]"
},
{
"state_after": "no goals",
"state_before": "ι : Type u_1\nR : Type u_2\ninst✝⁸ : CommRing R\nMs : ι → Type u_3\ninst✝⁷ : (i : ι) → AddCommGroup (Ms i)\ninst✝⁶ : (i : ι) → Module R (Ms i)\nN : Type u_4\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nNs : ι → Type ?u.16262\ninst✝³ : (i : ι) → AddCommGroup (Ns i)\ninst✝² : (i : ι) → Module R (Ns i)\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\np : (i : ι) → Submodule R (Ms i)\nq : Submodule R N\nf : (i : ι) → Ms i →ₗ[R] N\nhf : ∀ (i : ι), p i ≤ comap (f i) q\nx : (i : ι) → Ms i\n⊢ (Finset.sum Finset.univ fun d =>\n ↑(comp (mapQ (p d) q (f d) (_ : p d ≤ comap (f d) q)) (proj d)) fun i => Quotient.mk (x i)) =\n Finset.sum Finset.univ fun x_1 => ↑(mkQ q) (↑(comp (f x_1) (proj x_1)) x)",
"tactic": "simp only [coe_proj, mapQ_apply, mkQ_apply, comp_apply]"
}
] | [
52,
58
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
48,
1
] |
Mathlib/Analysis/Calculus/BumpFunctionInner.lean | expNegInvGlue.differentiable_polynomial_eval_inv_mul | [] | [
142,
60
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
140,
1
] |
Mathlib/MeasureTheory/Function/L1Space.lean | MeasureTheory.lintegral_nnnorm_add_left | [] | [
92,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
90,
1
] |
Mathlib/AlgebraicGeometry/StructureSheaf.lean | AlgebraicGeometry.StructureSheaf.toBasicOpen_surjective | [
{
"state_after": "R : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"state_before": "R : Type u\ninst✝ : CommRing R\nf : R\n⊢ Function.Surjective ↑(toBasicOpen R f)",
"tactic": "intro s"
},
{
"state_after": "R : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"state_before": "R : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"tactic": "let ι : Type u := PrimeSpectrum.basicOpen f"
},
{
"state_after": "R : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\na' h' : { x // x ∈ PrimeSpectrum.basicOpen f } → R\niDh' : (x : { x // x ∈ PrimeSpectrum.basicOpen f }) → PrimeSpectrum.basicOpen (h' x) ⟶ PrimeSpectrum.basicOpen f\nhxDh' : ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }), ↑x ∈ PrimeSpectrum.basicOpen (h' x)\ns_eq' :\n ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }),\n const R (a' x) (h' x) (PrimeSpectrum.basicOpen (h' x))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h' x) → y ∈ PrimeSpectrum.basicOpen (h' x)) =\n ↑((structureSheaf R).val.map (iDh' x).op) s\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"state_before": "R : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"tactic": "choose a' h' iDh' hxDh' s_eq' using locally_const_basicOpen R (PrimeSpectrum.basicOpen f) s"
},
{
"state_after": "case intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\na' h' : { x // x ∈ PrimeSpectrum.basicOpen f } → R\niDh' : (x : { x // x ∈ PrimeSpectrum.basicOpen f }) → PrimeSpectrum.basicOpen (h' x) ⟶ PrimeSpectrum.basicOpen f\nhxDh' : ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }), ↑x ∈ PrimeSpectrum.basicOpen (h' x)\ns_eq' :\n ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }),\n const R (a' x) (h' x) (PrimeSpectrum.basicOpen (h' x))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h' x) → y ∈ PrimeSpectrum.basicOpen (h' x)) =\n ↑((structureSheaf R).val.map (iDh' x).op) s\nt : Finset ι\nht_cover' : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h' i))\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"state_before": "R : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\na' h' : { x // x ∈ PrimeSpectrum.basicOpen f } → R\niDh' : (x : { x // x ∈ PrimeSpectrum.basicOpen f }) → PrimeSpectrum.basicOpen (h' x) ⟶ PrimeSpectrum.basicOpen f\nhxDh' : ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }), ↑x ∈ PrimeSpectrum.basicOpen (h' x)\ns_eq' :\n ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }),\n const R (a' x) (h' x) (PrimeSpectrum.basicOpen (h' x))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h' x) → y ∈ PrimeSpectrum.basicOpen (h' x)) =\n ↑((structureSheaf R).val.map (iDh' x).op) s\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"tactic": "obtain ⟨t, ht_cover'⟩ :=\n (PrimeSpectrum.isCompact_basicOpen f).elim_finite_subcover\n (fun i : ι => PrimeSpectrum.basicOpen (h' i)) (fun i => PrimeSpectrum.isOpen_basicOpen)\n fun x hx => by rw [Set.mem_iUnion]; exact ⟨⟨x, hx⟩, hxDh' ⟨x, hx⟩⟩"
},
{
"state_after": "case intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\na' h' : { x // x ∈ PrimeSpectrum.basicOpen f } → R\niDh' : (x : { x // x ∈ PrimeSpectrum.basicOpen f }) → PrimeSpectrum.basicOpen (h' x) ⟶ PrimeSpectrum.basicOpen f\nhxDh' : ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }), ↑x ∈ PrimeSpectrum.basicOpen (h' x)\ns_eq' :\n ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }),\n const R (a' x) (h' x) (PrimeSpectrum.basicOpen (h' x))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h' x) → y ∈ PrimeSpectrum.basicOpen (h' x)) =\n ↑((structureSheaf R).val.map (iDh' x).op) s\nt : Finset ι\nht_cover' :\n PrimeSpectrum.basicOpen f ≤ ⨆ (i : { x // x ∈ PrimeSpectrum.basicOpen f }) (_ : i ∈ t), PrimeSpectrum.basicOpen (h' i)\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"state_before": "case intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\na' h' : { x // x ∈ PrimeSpectrum.basicOpen f } → R\niDh' : (x : { x // x ∈ PrimeSpectrum.basicOpen f }) → PrimeSpectrum.basicOpen (h' x) ⟶ PrimeSpectrum.basicOpen f\nhxDh' : ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }), ↑x ∈ PrimeSpectrum.basicOpen (h' x)\ns_eq' :\n ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }),\n const R (a' x) (h' x) (PrimeSpectrum.basicOpen (h' x))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h' x) → y ∈ PrimeSpectrum.basicOpen (h' x)) =\n ↑((structureSheaf R).val.map (iDh' x).op) s\nt : Finset ι\nht_cover' : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h' i))\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"tactic": "simp only [← Opens.coe_iSup, SetLike.coe_subset_coe] at ht_cover'"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\na' h' : { x // x ∈ PrimeSpectrum.basicOpen f } → R\niDh' : (x : { x // x ∈ PrimeSpectrum.basicOpen f }) → PrimeSpectrum.basicOpen (h' x) ⟶ PrimeSpectrum.basicOpen f\nhxDh' : ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }), ↑x ∈ PrimeSpectrum.basicOpen (h' x)\ns_eq' :\n ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }),\n const R (a' x) (h' x) (PrimeSpectrum.basicOpen (h' x))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h' x) → y ∈ PrimeSpectrum.basicOpen (h' x)) =\n ↑((structureSheaf R).val.map (iDh' x).op) s\nt : Finset ι\nht_cover' :\n PrimeSpectrum.basicOpen f ≤ ⨆ (i : { x // x ∈ PrimeSpectrum.basicOpen f }) (_ : i ∈ t), PrimeSpectrum.basicOpen (h' i)\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nht_cover : PrimeSpectrum.basicOpen f ≤ ⨆ (i : ι) (_ : i ∈ t), PrimeSpectrum.basicOpen (h i)\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"state_before": "case intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\na' h' : { x // x ∈ PrimeSpectrum.basicOpen f } → R\niDh' : (x : { x // x ∈ PrimeSpectrum.basicOpen f }) → PrimeSpectrum.basicOpen (h' x) ⟶ PrimeSpectrum.basicOpen f\nhxDh' : ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }), ↑x ∈ PrimeSpectrum.basicOpen (h' x)\ns_eq' :\n ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }),\n const R (a' x) (h' x) (PrimeSpectrum.basicOpen (h' x))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h' x) → y ∈ PrimeSpectrum.basicOpen (h' x)) =\n ↑((structureSheaf R).val.map (iDh' x).op) s\nt : Finset ι\nht_cover' :\n PrimeSpectrum.basicOpen f ≤ ⨆ (i : { x // x ∈ PrimeSpectrum.basicOpen f }) (_ : i ∈ t), PrimeSpectrum.basicOpen (h' i)\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"tactic": "obtain ⟨a, h, iDh, ht_cover, ah_ha, s_eq⟩ :=\n normalize_finite_fraction_representation R (PrimeSpectrum.basicOpen f)\n s t a' h' iDh' ht_cover' s_eq'"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nht_cover : PrimeSpectrum.basicOpen f ≤ ⨆ (i : ι) (_ : i ∈ t), PrimeSpectrum.basicOpen (h i)\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\na' h' : { x // x ∈ PrimeSpectrum.basicOpen f } → R\niDh' : (x : { x // x ∈ PrimeSpectrum.basicOpen f }) → PrimeSpectrum.basicOpen (h' x) ⟶ PrimeSpectrum.basicOpen f\nhxDh' : ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }), ↑x ∈ PrimeSpectrum.basicOpen (h' x)\ns_eq' :\n ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }),\n const R (a' x) (h' x) (PrimeSpectrum.basicOpen (h' x))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h' x) → y ∈ PrimeSpectrum.basicOpen (h' x)) =\n ↑((structureSheaf R).val.map (iDh' x).op) s\nt : Finset ι\nht_cover' :\n PrimeSpectrum.basicOpen f ≤ ⨆ (i : { x // x ∈ PrimeSpectrum.basicOpen f }) (_ : i ∈ t), PrimeSpectrum.basicOpen (h' i)\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nht_cover : PrimeSpectrum.basicOpen f ≤ ⨆ (i : ι) (_ : i ∈ t), PrimeSpectrum.basicOpen (h i)\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"tactic": "clear s_eq' iDh' hxDh' ht_cover' a' h'"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι), ↑(⨆ (_ : i ∈ t), PrimeSpectrum.basicOpen (h i))\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nht_cover : PrimeSpectrum.basicOpen f ≤ ⨆ (i : ι) (_ : i ∈ t), PrimeSpectrum.basicOpen (h i)\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"tactic": "rw [← SetLike.coe_subset_coe, Opens.coe_iSup] at ht_cover"
},
{
"state_after": "case ht_cover\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι), ↑(⨆ (_ : i ∈ t), PrimeSpectrum.basicOpen (h i))\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\n⊢ ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\n\ncase intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι), ↑(⨆ (_ : i ∈ t), PrimeSpectrum.basicOpen (h i))\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"tactic": "replace ht_cover : (PrimeSpectrum.basicOpen f : Set <| PrimeSpectrum R) ⊆\n ⋃ (i : ι) (x : i ∈ t), (PrimeSpectrum.basicOpen (h i) : Set _)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"state_before": "case ht_cover\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι), ↑(⨆ (_ : i ∈ t), PrimeSpectrum.basicOpen (h i))\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\n⊢ ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\n\ncase intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"tactic": ". convert ht_cover using 2\n exact funext fun j => by rw [Opens.coe_iSup]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nhn : f ^ n ∈ Ideal.span (h '' ↑t)\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"state_before": "case intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"tactic": "obtain ⟨n, hn⟩ : f ∈ (Ideal.span (h '' ↑t)).radical := by\n rw [← PrimeSpectrum.vanishingIdeal_zeroLocus_eq_radical, PrimeSpectrum.zeroLocus_span]\n replace ht_cover : (PrimeSpectrum.zeroLocus {f})ᶜ ⊆\n ⋃ (i : ι) (x : i ∈ t), (PrimeSpectrum.zeroLocus {h i})ᶜ\n . convert ht_cover\n . rw [PrimeSpectrum.basicOpen_eq_zeroLocus_compl]\n . simp only [Opens.iSup_mk, Opens.carrier_eq_coe, PrimeSpectrum.basicOpen_eq_zeroLocus_compl]\n rw [Set.compl_subset_comm] at ht_cover\n simp_rw [Set.compl_iUnion, compl_compl, ← PrimeSpectrum.zeroLocus_iUnion,\n ← Finset.set_biUnion_coe, ← Set.image_eq_iUnion] at ht_cover\n apply PrimeSpectrum.vanishingIdeal_anti_mono ht_cover\n exact PrimeSpectrum.subset_vanishingIdeal_zeroLocus {f} (Set.mem_singleton f)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nhn : f * f ^ n ∈ Ideal.span (h '' ↑t)\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nhn : f ^ n ∈ Ideal.span (h '' ↑t)\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"tactic": "replace hn := Ideal.mul_mem_left _ f hn"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nhn : ∃ l, l ∈ Finsupp.supported R R ↑t ∧ ↑(Finsupp.total ι R R h) l = f ^ (n + 1)\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nhn : f * f ^ n ∈ Ideal.span (h '' ↑t)\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"tactic": "erw [← pow_succ, Finsupp.mem_span_image_iff_total] at hn"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ↑(Finsupp.total ι R R h) b = f ^ (n + 1)\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nhn : ∃ l, l ∈ Finsupp.supported R R ↑t ∧ ↑(Finsupp.total ι R R h) l = f ^ (n + 1)\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"tactic": "rcases hn with ⟨b, b_supp, hb⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i • h i = f ^ (n + 1)\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ↑(Finsupp.total ι R R h) b = f ^ (n + 1)\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"tactic": "rw [Finsupp.total_apply_of_mem_supported R b_supp] at hb"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i • h i = f ^ (n + 1)\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"tactic": "dsimp at hb"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\n⊢ ↑(toBasicOpen R f)\n (IsLocalization.mk' (Localization.Away f) (∑ i in t, ↑b i * a i)\n { val := f ^ (n + 1), property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = f ^ (n + 1)) }) =\n s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\n⊢ ∃ a, ↑(toBasicOpen R f) a = s",
"tactic": "use\n IsLocalization.mk' (Localization.Away f) (∑ i : ι in t, b i * a i)\n (⟨f ^ (n + 1), n + 1, rfl⟩ : Submonoid.powers _)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\n⊢ const R (∑ i in t, ↑b i * a i)\n (↑{ val := f ^ (n + 1), property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = f ^ (n + 1)) })\n (PrimeSpectrum.basicOpen f)\n (_ :\n ∀ (x : ↑(PrimeSpectrum.Top R)),\n x ∈ PrimeSpectrum.basicOpen f →\n ↑{ val := f ^ (n + 1), property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = f ^ (n + 1)) } ∈\n Ideal.primeCompl x.asIdeal) =\n s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\n⊢ ↑(toBasicOpen R f)\n (IsLocalization.mk' (Localization.Away f) (∑ i in t, ↑b i * a i)\n { val := f ^ (n + 1), property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = f ^ (n + 1)) }) =\n s",
"tactic": "rw [toBasicOpen_mk']"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\n⊢ const R (∑ i in t, ↑b i * a i)\n (↑{ val := f ^ (n + 1), property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = f ^ (n + 1)) })\n (PrimeSpectrum.basicOpen f)\n (_ :\n ∀ (x : ↑(PrimeSpectrum.Top R)),\n x ∈ PrimeSpectrum.basicOpen f →\n ↑{ val := f ^ (n + 1), property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = f ^ (n + 1)) } ∈\n Ideal.primeCompl x.asIdeal) =\n s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\n⊢ const R (∑ i in t, ↑b i * a i)\n (↑{ val := f ^ (n + 1), property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = f ^ (n + 1)) })\n (PrimeSpectrum.basicOpen f)\n (_ :\n ∀ (x : ↑(PrimeSpectrum.Top R)),\n x ∈ PrimeSpectrum.basicOpen f →\n ↑{ val := f ^ (n + 1), property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = f ^ (n + 1)) } ∈\n Ideal.primeCompl x.asIdeal) =\n s",
"tactic": "let tt := ((t : Set (PrimeSpectrum.basicOpen f)) : Type u)"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.hcover\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\n⊢ PrimeSpectrum.basicOpen f ≤ ⨆ (i : tt), PrimeSpectrum.basicOpen (h ↑i)\n\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\n⊢ ∀ (i : tt),\n (forget CommRingCat).map ((structureSheaf R).val.map (iDh ↑i).op)\n (const R (∑ i in t, ↑b i * a i)\n (↑{ val := f ^ (n + 1), property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = f ^ (n + 1)) })\n (PrimeSpectrum.basicOpen f)\n (_ :\n ∀ (x : ↑(PrimeSpectrum.Top R)),\n x ∈ PrimeSpectrum.basicOpen f →\n ↑{ val := f ^ (n + 1), property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = f ^ (n + 1)) } ∈\n Ideal.primeCompl x.asIdeal)) =\n (forget CommRingCat).map ((structureSheaf R).val.map (iDh ↑i).op) s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\n⊢ const R (∑ i in t, ↑b i * a i)\n (↑{ val := f ^ (n + 1), property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = f ^ (n + 1)) })\n (PrimeSpectrum.basicOpen f)\n (_ :\n ∀ (x : ↑(PrimeSpectrum.Top R)),\n x ∈ PrimeSpectrum.basicOpen f →\n ↑{ val := f ^ (n + 1), property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = f ^ (n + 1)) } ∈\n Ideal.primeCompl x.asIdeal) =\n s",
"tactic": "apply\n (structureSheaf R).eq_of_locally_eq' (fun i : tt => PrimeSpectrum.basicOpen (h i))\n (PrimeSpectrum.basicOpen f) fun i : tt => iDh i"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\n⊢ (forget CommRingCat).map ((structureSheaf R).val.map (iDh ↑{ val := i, property := hi }).op)\n (const R (∑ i in t, ↑b i * a i)\n (↑{ val := f ^ (n + 1), property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = f ^ (n + 1)) })\n (PrimeSpectrum.basicOpen f)\n (_ :\n ∀ (x : ↑(PrimeSpectrum.Top R)),\n x ∈ PrimeSpectrum.basicOpen f →\n ↑{ val := f ^ (n + 1), property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = f ^ (n + 1)) } ∈\n Ideal.primeCompl x.asIdeal)) =\n (forget CommRingCat).map ((structureSheaf R).val.map (iDh ↑{ val := i, property := hi }).op) s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\n⊢ ∀ (i : tt),\n (forget CommRingCat).map ((structureSheaf R).val.map (iDh ↑i).op)\n (const R (∑ i in t, ↑b i * a i)\n (↑{ val := f ^ (n + 1), property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = f ^ (n + 1)) })\n (PrimeSpectrum.basicOpen f)\n (_ :\n ∀ (x : ↑(PrimeSpectrum.Top R)),\n x ∈ PrimeSpectrum.basicOpen f →\n ↑{ val := f ^ (n + 1), property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = f ^ (n + 1)) } ∈\n Ideal.primeCompl x.asIdeal)) =\n (forget CommRingCat).map ((structureSheaf R).val.map (iDh ↑i).op) s",
"tactic": "rintro ⟨i, hi⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\n⊢ ↑((structureSheaf R).val.map (iDh i).op)\n (const R (∑ i in t, ↑b i * a i) (f ^ (n + 1)) (PrimeSpectrum.basicOpen f)\n (_ :\n ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ PrimeSpectrum.basicOpen f → f ^ (n + 1) ∈ Ideal.primeCompl x.asIdeal)) =\n ↑((structureSheaf R).val.map (iDh i).op) s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\n⊢ (forget CommRingCat).map ((structureSheaf R).val.map (iDh ↑{ val := i, property := hi }).op)\n (const R (∑ i in t, ↑b i * a i)\n (↑{ val := f ^ (n + 1), property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = f ^ (n + 1)) })\n (PrimeSpectrum.basicOpen f)\n (_ :\n ∀ (x : ↑(PrimeSpectrum.Top R)),\n x ∈ PrimeSpectrum.basicOpen f →\n ↑{ val := f ^ (n + 1), property := (_ : ∃ y, (fun x x_1 => x ^ x_1) f y = f ^ (n + 1)) } ∈\n Ideal.primeCompl x.asIdeal)) =\n (forget CommRingCat).map ((structureSheaf R).val.map (iDh ↑{ val := i, property := hi }).op) s",
"tactic": "dsimp"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\n⊢ ↑((structureSheaf R).val.map (iDh i).op)\n (const R (∑ i in t, ↑b i * a i) (f ^ (n + 1)) (PrimeSpectrum.basicOpen f)\n (_ :\n ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ PrimeSpectrum.basicOpen f → f ^ (n + 1) ∈ Ideal.primeCompl x.asIdeal)) =\n ↑((structureSheaf R).val.map (iDh i).op) s",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\n⊢ ↑((structureSheaf R).val.map (iDh i).op)\n (const R (∑ i in t, ↑b i * a i) (f ^ (n + 1)) (PrimeSpectrum.basicOpen f)\n (_ :\n ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ PrimeSpectrum.basicOpen f → f ^ (n + 1) ∈ Ideal.primeCompl x.asIdeal)) =\n ↑((structureSheaf R).val.map (iDh i).op) s",
"tactic": "change (structureSheaf R).1.map _ _ = (structureSheaf R).1.map _ _"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\n⊢ const R (∑ i in t, ↑b i * a i) (f ^ (n + 1)) (PrimeSpectrum.basicOpen (h i))\n ?intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.hv =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\n\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.hv\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\n⊢ ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ PrimeSpectrum.basicOpen (h i) → f ^ (n + 1) ∈ Ideal.primeCompl x.asIdeal",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\n⊢ ↑((structureSheaf R).val.map (iDh i).op)\n (const R (∑ i in t, ↑b i * a i) (f ^ (n + 1)) (PrimeSpectrum.basicOpen f)\n (_ :\n ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ PrimeSpectrum.basicOpen f → f ^ (n + 1) ∈ Ideal.primeCompl x.asIdeal)) =\n ↑((structureSheaf R).val.map (iDh i).op) s",
"tactic": "rw [s_eq i hi, res_const]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.hv\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\n⊢ ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ PrimeSpectrum.basicOpen (h i) → f ^ (n + 1) ∈ Ideal.primeCompl x.asIdeal\n\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\n⊢ const R (∑ i in t, ↑b i * a i) (f ^ (n + 1)) (PrimeSpectrum.basicOpen (h i))\n ?intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.hv =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\n⊢ const R (∑ i in t, ↑b i * a i) (f ^ (n + 1)) (PrimeSpectrum.basicOpen (h i))\n ?intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.hv =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\n\ncase intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.hv\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\n⊢ ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ PrimeSpectrum.basicOpen (h i) → f ^ (n + 1) ∈ Ideal.primeCompl x.asIdeal",
"tactic": "swap"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.h\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\n⊢ (∑ i in t, ↑b i * a i) * h i = a i * f ^ (n + 1)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\n⊢ const R (∑ i in t, ↑b i * a i) (f ^ (n + 1)) (PrimeSpectrum.basicOpen (h i))\n (_ :\n ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (f ^ (n + 1))) =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))",
"tactic": "apply const_ext"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.h\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\n⊢ ∑ x in t, ↑b x * a x * h i = ∑ x in t, a i * (↑b x * h x)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.h\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\n⊢ (∑ i in t, ↑b i * a i) * h i = a i * f ^ (n + 1)",
"tactic": "rw [← hb, Finset.sum_mul, Finset.mul_sum]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.h\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\n⊢ ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }), x ∈ t → ↑b x * a x * h i = a i * (↑b x * h x)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.h\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\n⊢ ∑ x in t, ↑b x * a x * h i = ∑ x in t, a i * (↑b x * h x)",
"tactic": "apply Finset.sum_congr rfl"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.h\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\nj : { x // x ∈ PrimeSpectrum.basicOpen f }\nhj : j ∈ t\n⊢ ↑b j * a j * h i = a i * (↑b j * h j)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.h\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\n⊢ ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }), x ∈ t → ↑b x * a x * h i = a i * (↑b x * h x)",
"tactic": "intro j hj"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.h\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\nj : { x // x ∈ PrimeSpectrum.basicOpen f }\nhj : j ∈ t\n⊢ ↑b j * (h j * a i) = a i * (↑b j * h j)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.h\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\nj : { x // x ∈ PrimeSpectrum.basicOpen f }\nhj : j ∈ t\n⊢ ↑b j * a j * h i = a i * (↑b j * h j)",
"tactic": "rw [mul_assoc, ah_ha j hj i hi]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.h\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\nj : { x // x ∈ PrimeSpectrum.basicOpen f }\nhj : j ∈ t\n⊢ ↑b j * (h j * a i) = a i * (↑b j * h j)",
"tactic": "ring"
},
{
"state_after": "R : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\na' h' : { x // x ∈ PrimeSpectrum.basicOpen f } → R\niDh' : (x : { x // x ∈ PrimeSpectrum.basicOpen f }) → PrimeSpectrum.basicOpen (h' x) ⟶ PrimeSpectrum.basicOpen f\nhxDh' : ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }), ↑x ∈ PrimeSpectrum.basicOpen (h' x)\ns_eq' :\n ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }),\n const R (a' x) (h' x) (PrimeSpectrum.basicOpen (h' x))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h' x) → y ∈ PrimeSpectrum.basicOpen (h' x)) =\n ↑((structureSheaf R).val.map (iDh' x).op) s\nx : PrimeSpectrum R\nhx : x ∈ ↑(PrimeSpectrum.basicOpen f)\n⊢ ∃ i, x ∈ (fun i => ↑(PrimeSpectrum.basicOpen (h' i))) i",
"state_before": "R : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\na' h' : { x // x ∈ PrimeSpectrum.basicOpen f } → R\niDh' : (x : { x // x ∈ PrimeSpectrum.basicOpen f }) → PrimeSpectrum.basicOpen (h' x) ⟶ PrimeSpectrum.basicOpen f\nhxDh' : ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }), ↑x ∈ PrimeSpectrum.basicOpen (h' x)\ns_eq' :\n ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }),\n const R (a' x) (h' x) (PrimeSpectrum.basicOpen (h' x))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h' x) → y ∈ PrimeSpectrum.basicOpen (h' x)) =\n ↑((structureSheaf R).val.map (iDh' x).op) s\nx : PrimeSpectrum R\nhx : x ∈ ↑(PrimeSpectrum.basicOpen f)\n⊢ x ∈ ⋃ (i : ι), (fun i => ↑(PrimeSpectrum.basicOpen (h' i))) i",
"tactic": "rw [Set.mem_iUnion]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\na' h' : { x // x ∈ PrimeSpectrum.basicOpen f } → R\niDh' : (x : { x // x ∈ PrimeSpectrum.basicOpen f }) → PrimeSpectrum.basicOpen (h' x) ⟶ PrimeSpectrum.basicOpen f\nhxDh' : ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }), ↑x ∈ PrimeSpectrum.basicOpen (h' x)\ns_eq' :\n ∀ (x : { x // x ∈ PrimeSpectrum.basicOpen f }),\n const R (a' x) (h' x) (PrimeSpectrum.basicOpen (h' x))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h' x) → y ∈ PrimeSpectrum.basicOpen (h' x)) =\n ↑((structureSheaf R).val.map (iDh' x).op) s\nx : PrimeSpectrum R\nhx : x ∈ ↑(PrimeSpectrum.basicOpen f)\n⊢ ∃ i, x ∈ (fun i => ↑(PrimeSpectrum.basicOpen (h' i))) i",
"tactic": "exact ⟨⟨x, hx⟩, hxDh' ⟨x, hx⟩⟩"
},
{
"state_after": "case h.e'_4.h.h.e'_3.h\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι), ↑(⨆ (_ : i ∈ t), PrimeSpectrum.basicOpen (h i))\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\ne_1✝¹ : Set (PrimeSpectrum R) = Set ↑(PrimeSpectrum.Top R)\ne_1✝ : PrimeSpectrum R = ↑(PrimeSpectrum.Top R)\n⊢ (fun i => ⋃ (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))) = fun i => ↑(⨆ (_ : i ∈ t), PrimeSpectrum.basicOpen (h i))",
"state_before": "case ht_cover\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι), ↑(⨆ (_ : i ∈ t), PrimeSpectrum.basicOpen (h i))\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\n⊢ ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))",
"tactic": "convert ht_cover using 2"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4.h.h.e'_3.h\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι), ↑(⨆ (_ : i ∈ t), PrimeSpectrum.basicOpen (h i))\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\ne_1✝¹ : Set (PrimeSpectrum R) = Set ↑(PrimeSpectrum.Top R)\ne_1✝ : PrimeSpectrum R = ↑(PrimeSpectrum.Top R)\n⊢ (fun i => ⋃ (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))) = fun i => ↑(⨆ (_ : i ∈ t), PrimeSpectrum.basicOpen (h i))",
"tactic": "exact funext fun j => by rw [Opens.coe_iSup]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι), ↑(⨆ (_ : i ∈ t), PrimeSpectrum.basicOpen (h i))\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\ne_1✝¹ : Set (PrimeSpectrum R) = Set ↑(PrimeSpectrum.Top R)\ne_1✝ : PrimeSpectrum R = ↑(PrimeSpectrum.Top R)\nj : ι\n⊢ (⋃ (_ : j ∈ t), ↑(PrimeSpectrum.basicOpen (h j))) = ↑(⨆ (_ : j ∈ t), PrimeSpectrum.basicOpen (h j))",
"tactic": "rw [Opens.coe_iSup]"
},
{
"state_after": "R : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\n⊢ f ∈ PrimeSpectrum.vanishingIdeal (PrimeSpectrum.zeroLocus (h '' ↑t))",
"state_before": "R : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\n⊢ f ∈ Ideal.radical (Ideal.span (h '' ↑t))",
"tactic": "rw [← PrimeSpectrum.vanishingIdeal_zeroLocus_eq_radical, PrimeSpectrum.zeroLocus_span]"
},
{
"state_after": "case ht_cover\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\n⊢ PrimeSpectrum.zeroLocus {f}ᶜ ⊆ ⋃ (i : ι) (_ : i ∈ t), PrimeSpectrum.zeroLocus {h i}ᶜ\n\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : PrimeSpectrum.zeroLocus {f}ᶜ ⊆ ⋃ (i : ι) (_ : i ∈ t), PrimeSpectrum.zeroLocus {h i}ᶜ\n⊢ f ∈ PrimeSpectrum.vanishingIdeal (PrimeSpectrum.zeroLocus (h '' ↑t))",
"state_before": "R : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\n⊢ f ∈ PrimeSpectrum.vanishingIdeal (PrimeSpectrum.zeroLocus (h '' ↑t))",
"tactic": "replace ht_cover : (PrimeSpectrum.zeroLocus {f})ᶜ ⊆\n ⋃ (i : ι) (x : i ∈ t), (PrimeSpectrum.zeroLocus {h i})ᶜ"
},
{
"state_after": "R : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : PrimeSpectrum.zeroLocus {f}ᶜ ⊆ ⋃ (i : ι) (_ : i ∈ t), PrimeSpectrum.zeroLocus {h i}ᶜ\n⊢ f ∈ PrimeSpectrum.vanishingIdeal (PrimeSpectrum.zeroLocus (h '' ↑t))",
"state_before": "case ht_cover\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\n⊢ PrimeSpectrum.zeroLocus {f}ᶜ ⊆ ⋃ (i : ι) (_ : i ∈ t), PrimeSpectrum.zeroLocus {h i}ᶜ\n\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : PrimeSpectrum.zeroLocus {f}ᶜ ⊆ ⋃ (i : ι) (_ : i ∈ t), PrimeSpectrum.zeroLocus {h i}ᶜ\n⊢ f ∈ PrimeSpectrum.vanishingIdeal (PrimeSpectrum.zeroLocus (h '' ↑t))",
"tactic": ". convert ht_cover\n . rw [PrimeSpectrum.basicOpen_eq_zeroLocus_compl]\n . simp only [Opens.iSup_mk, Opens.carrier_eq_coe, PrimeSpectrum.basicOpen_eq_zeroLocus_compl]"
},
{
"state_after": "R : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : (⋃ (i : ι) (_ : i ∈ t), PrimeSpectrum.zeroLocus {h i}ᶜ)ᶜ ⊆ PrimeSpectrum.zeroLocus {f}\n⊢ f ∈ PrimeSpectrum.vanishingIdeal (PrimeSpectrum.zeroLocus (h '' ↑t))",
"state_before": "R : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : PrimeSpectrum.zeroLocus {f}ᶜ ⊆ ⋃ (i : ι) (_ : i ∈ t), PrimeSpectrum.zeroLocus {h i}ᶜ\n⊢ f ∈ PrimeSpectrum.vanishingIdeal (PrimeSpectrum.zeroLocus (h '' ↑t))",
"tactic": "rw [Set.compl_subset_comm] at ht_cover"
},
{
"state_after": "R : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : PrimeSpectrum.zeroLocus ((fun i => h i) '' ↑t) ⊆ PrimeSpectrum.zeroLocus {f}\n⊢ f ∈ PrimeSpectrum.vanishingIdeal (PrimeSpectrum.zeroLocus (h '' ↑t))",
"state_before": "R : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : (⋃ (i : ι) (_ : i ∈ t), PrimeSpectrum.zeroLocus {h i}ᶜ)ᶜ ⊆ PrimeSpectrum.zeroLocus {f}\n⊢ f ∈ PrimeSpectrum.vanishingIdeal (PrimeSpectrum.zeroLocus (h '' ↑t))",
"tactic": "simp_rw [Set.compl_iUnion, compl_compl, ← PrimeSpectrum.zeroLocus_iUnion,\n ← Finset.set_biUnion_coe, ← Set.image_eq_iUnion] at ht_cover"
},
{
"state_after": "case a\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : PrimeSpectrum.zeroLocus ((fun i => h i) '' ↑t) ⊆ PrimeSpectrum.zeroLocus {f}\n⊢ f ∈ PrimeSpectrum.vanishingIdeal (PrimeSpectrum.zeroLocus {f})",
"state_before": "R : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : PrimeSpectrum.zeroLocus ((fun i => h i) '' ↑t) ⊆ PrimeSpectrum.zeroLocus {f}\n⊢ f ∈ PrimeSpectrum.vanishingIdeal (PrimeSpectrum.zeroLocus (h '' ↑t))",
"tactic": "apply PrimeSpectrum.vanishingIdeal_anti_mono ht_cover"
},
{
"state_after": "no goals",
"state_before": "case a\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : PrimeSpectrum.zeroLocus ((fun i => h i) '' ↑t) ⊆ PrimeSpectrum.zeroLocus {f}\n⊢ f ∈ PrimeSpectrum.vanishingIdeal (PrimeSpectrum.zeroLocus {f})",
"tactic": "exact PrimeSpectrum.subset_vanishingIdeal_zeroLocus {f} (Set.mem_singleton f)"
},
{
"state_after": "case h.e'_3\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\n⊢ PrimeSpectrum.zeroLocus {f}ᶜ = ↑(PrimeSpectrum.basicOpen f)\n\ncase h.e'_4.h.e'_3.h.f\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nx✝¹ : ι\nx✝ : x✝¹ ∈ t\n⊢ PrimeSpectrum.zeroLocus {h x✝¹}ᶜ = ↑(PrimeSpectrum.basicOpen (h x✝¹))",
"state_before": "case ht_cover\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\n⊢ PrimeSpectrum.zeroLocus {f}ᶜ ⊆ ⋃ (i : ι) (_ : i ∈ t), PrimeSpectrum.zeroLocus {h i}ᶜ",
"tactic": "convert ht_cover"
},
{
"state_after": "case h.e'_4.h.e'_3.h.f\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nx✝¹ : ι\nx✝ : x✝¹ ∈ t\n⊢ PrimeSpectrum.zeroLocus {h x✝¹}ᶜ = ↑(PrimeSpectrum.basicOpen (h x✝¹))",
"state_before": "case h.e'_3\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\n⊢ PrimeSpectrum.zeroLocus {f}ᶜ = ↑(PrimeSpectrum.basicOpen f)\n\ncase h.e'_4.h.e'_3.h.f\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nx✝¹ : ι\nx✝ : x✝¹ ∈ t\n⊢ PrimeSpectrum.zeroLocus {h x✝¹}ᶜ = ↑(PrimeSpectrum.basicOpen (h x✝¹))",
"tactic": ". rw [PrimeSpectrum.basicOpen_eq_zeroLocus_compl]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4.h.e'_3.h.f\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nx✝¹ : ι\nx✝ : x✝¹ ∈ t\n⊢ PrimeSpectrum.zeroLocus {h x✝¹}ᶜ = ↑(PrimeSpectrum.basicOpen (h x✝¹))",
"tactic": ". simp only [Opens.iSup_mk, Opens.carrier_eq_coe, PrimeSpectrum.basicOpen_eq_zeroLocus_compl]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_3\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\n⊢ PrimeSpectrum.zeroLocus {f}ᶜ = ↑(PrimeSpectrum.basicOpen f)",
"tactic": "rw [PrimeSpectrum.basicOpen_eq_zeroLocus_compl]"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4.h.e'_3.h.f\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nx✝¹ : ι\nx✝ : x✝¹ ∈ t\n⊢ PrimeSpectrum.zeroLocus {h x✝¹}ᶜ = ↑(PrimeSpectrum.basicOpen (h x✝¹))",
"tactic": "simp only [Opens.iSup_mk, Opens.carrier_eq_coe, PrimeSpectrum.basicOpen_eq_zeroLocus_compl]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.hcover\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\nx : ↑(PrimeSpectrum.Top R)\nhx : x ∈ ↑(PrimeSpectrum.basicOpen f)\n⊢ x ∈ ↑(⨆ (i : tt), PrimeSpectrum.basicOpen (h ↑i))",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.hcover\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\n⊢ PrimeSpectrum.basicOpen f ≤ ⨆ (i : tt), PrimeSpectrum.basicOpen (h ↑i)",
"tactic": "intro x hx"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.hcover\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\nx : ↑(PrimeSpectrum.Top R)\nhx : x ∈ ↑(PrimeSpectrum.basicOpen f)\n⊢ ∃ i, x ∈ PrimeSpectrum.basicOpen (h ↑i)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.hcover\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\nx : ↑(PrimeSpectrum.Top R)\nhx : x ∈ ↑(PrimeSpectrum.basicOpen f)\n⊢ x ∈ ↑(⨆ (i : tt), PrimeSpectrum.basicOpen (h ↑i))",
"tactic": "erw [TopologicalSpace.Opens.mem_iSup]"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.hcover\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\nx : ↑(PrimeSpectrum.Top R)\nhx : x ∈ ↑(PrimeSpectrum.basicOpen f)\nthis : x ∈ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\n⊢ ∃ i, x ∈ PrimeSpectrum.basicOpen (h ↑i)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.hcover\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\nx : ↑(PrimeSpectrum.Top R)\nhx : x ∈ ↑(PrimeSpectrum.basicOpen f)\n⊢ ∃ i, x ∈ PrimeSpectrum.basicOpen (h ↑i)",
"tactic": "have := ht_cover hx"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.hcover\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\nx : ↑(PrimeSpectrum.Top R)\nhx : x ∈ ↑(PrimeSpectrum.basicOpen f)\nthis : ∃ i j, x ∈ ↑(PrimeSpectrum.basicOpen (h i))\n⊢ ∃ i, x ∈ PrimeSpectrum.basicOpen (h ↑i)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.hcover\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\nx : ↑(PrimeSpectrum.Top R)\nhx : x ∈ ↑(PrimeSpectrum.basicOpen f)\nthis : x ∈ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\n⊢ ∃ i, x ∈ PrimeSpectrum.basicOpen (h ↑i)",
"tactic": "rw [← Finset.set_biUnion_coe, Set.mem_iUnion₂] at this"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.hcover.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\nx : ↑(PrimeSpectrum.Top R)\nhx : x ∈ ↑(PrimeSpectrum.basicOpen f)\ni : ι\ni_mem : i ∈ ↑t\nx_mem : x ∈ ↑(PrimeSpectrum.basicOpen (h i))\n⊢ ∃ i, x ∈ PrimeSpectrum.basicOpen (h ↑i)",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.hcover\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\nx : ↑(PrimeSpectrum.Top R)\nhx : x ∈ ↑(PrimeSpectrum.basicOpen f)\nthis : ∃ i j, x ∈ ↑(PrimeSpectrum.basicOpen (h i))\n⊢ ∃ i, x ∈ PrimeSpectrum.basicOpen (h ↑i)",
"tactic": "rcases this with ⟨i, i_mem, x_mem⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.hcover.intro.intro\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\nx : ↑(PrimeSpectrum.Top R)\nhx : x ∈ ↑(PrimeSpectrum.basicOpen f)\ni : ι\ni_mem : i ∈ ↑t\nx_mem : x ∈ ↑(PrimeSpectrum.basicOpen (h i))\n⊢ ∃ i, x ∈ PrimeSpectrum.basicOpen (h ↑i)",
"tactic": "refine ⟨⟨i, i_mem⟩, x_mem⟩"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.hv\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\ny : ↑(PrimeSpectrum.Top R)\nhy : y ∈ PrimeSpectrum.basicOpen (h i)\n⊢ f ^ (n + 1) ∈ Ideal.primeCompl y.asIdeal",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.hv\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\n⊢ ∀ (x : ↑(PrimeSpectrum.Top R)), x ∈ PrimeSpectrum.basicOpen (h i) → f ^ (n + 1) ∈ Ideal.primeCompl x.asIdeal",
"tactic": "intro y hy"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.hv\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\ny : ↑(PrimeSpectrum.Top R)\nhy : y ∈ PrimeSpectrum.basicOpen (h i)\n⊢ y ∈ PrimeSpectrum.basicOpen (f ^ (n + 1))",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.hv\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\ny : ↑(PrimeSpectrum.Top R)\nhy : y ∈ PrimeSpectrum.basicOpen (h i)\n⊢ f ^ (n + 1) ∈ Ideal.primeCompl y.asIdeal",
"tactic": "change y ∈ PrimeSpectrum.basicOpen (f ^ (n + 1))"
},
{
"state_after": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.hv\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\ny : ↑(PrimeSpectrum.Top R)\nhy : y ∈ PrimeSpectrum.basicOpen (h i)\n⊢ y ∈ PrimeSpectrum.basicOpen f",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.hv\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\ny : ↑(PrimeSpectrum.Top R)\nhy : y ∈ PrimeSpectrum.basicOpen (h i)\n⊢ y ∈ PrimeSpectrum.basicOpen (f ^ (n + 1))",
"tactic": "rw [PrimeSpectrum.basicOpen_pow f (n + 1) (by linarith)]"
},
{
"state_after": "no goals",
"state_before": "case intro.intro.intro.intro.intro.intro.intro.intro.intro.h.mk.hv\nR : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\ny : ↑(PrimeSpectrum.Top R)\nhy : y ∈ PrimeSpectrum.basicOpen (h i)\n⊢ y ∈ PrimeSpectrum.basicOpen f",
"tactic": "exact (leOfHom (iDh i) : _) hy"
},
{
"state_after": "no goals",
"state_before": "R : Type u\ninst✝ : CommRing R\nf : R\ns : ↑((structureSheaf R).val.obj (PrimeSpectrum.basicOpen f).op)\nι : Type u := { x // x ∈ PrimeSpectrum.basicOpen f }\nt : Finset ι\na h : ι → R\niDh : (i : ι) → PrimeSpectrum.basicOpen (h i) ⟶ PrimeSpectrum.basicOpen f\nah_ha : ∀ (i : ι), i ∈ t → ∀ (j : ι), j ∈ t → a i * h j = h i * a j\ns_eq :\n ∀ (i : ι),\n i ∈ t →\n ↑((structureSheaf R).val.map (iDh i).op) s =\n const R (a i) (h i) (PrimeSpectrum.basicOpen (h i))\n (_ : ∀ (y : ↑(PrimeSpectrum.Top R)), y ∈ PrimeSpectrum.basicOpen (h i) → y ∈ PrimeSpectrum.basicOpen (h i))\nht_cover : ↑(PrimeSpectrum.basicOpen f) ⊆ ⋃ (i : ι) (_ : i ∈ t), ↑(PrimeSpectrum.basicOpen (h i))\nn : ℕ\nb : ι →₀ R\nb_supp : b ∈ Finsupp.supported R R ↑t\nhb : ∑ i in t, ↑b i * h i = f ^ (n + 1)\ntt : Type u := ↑↑t\ni : ι\nhi : i ∈ ↑t\ny : ↑(PrimeSpectrum.Top R)\nhy : y ∈ PrimeSpectrum.basicOpen (h i)\n⊢ 0 < n + 1",
"tactic": "linarith"
}
] | [
912,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
826,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineEquiv.lean | AffineEquiv.constVAdd_add | [] | [
497,
22
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
495,
1
] |
Mathlib/RingTheory/Polynomial/Basic.lean | Polynomial.sup_aeval_range_eq_top_of_coprime | [
{
"state_after": "R : Type u\nS : Type ?u.696957\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nhpq : IsCoprime p q\n⊢ ⊤ ≤ LinearMap.range (↑(aeval f) p) ⊔ LinearMap.range (↑(aeval f) q)",
"state_before": "R : Type u\nS : Type ?u.696957\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nhpq : IsCoprime p q\n⊢ LinearMap.range (↑(aeval f) p) ⊔ LinearMap.range (↑(aeval f) q) = ⊤",
"tactic": "rw [eq_top_iff]"
},
{
"state_after": "R : Type u\nS : Type ?u.696957\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nhpq : IsCoprime p q\nv : M\na✝ : v ∈ ⊤\n⊢ v ∈ LinearMap.range (↑(aeval f) p) ⊔ LinearMap.range (↑(aeval f) q)",
"state_before": "R : Type u\nS : Type ?u.696957\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nhpq : IsCoprime p q\n⊢ ⊤ ≤ LinearMap.range (↑(aeval f) p) ⊔ LinearMap.range (↑(aeval f) q)",
"tactic": "intro v _"
},
{
"state_after": "R : Type u\nS : Type ?u.696957\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nhpq : IsCoprime p q\nv : M\na✝ : v ∈ ⊤\n⊢ ∃ y, y ∈ LinearMap.range (↑(aeval f) p) ∧ ∃ z, z ∈ LinearMap.range (↑(aeval f) q) ∧ y + z = v",
"state_before": "R : Type u\nS : Type ?u.696957\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nhpq : IsCoprime p q\nv : M\na✝ : v ∈ ⊤\n⊢ v ∈ LinearMap.range (↑(aeval f) p) ⊔ LinearMap.range (↑(aeval f) q)",
"tactic": "rw [Submodule.mem_sup]"
},
{
"state_after": "case intro.intro\nR : Type u\nS : Type ?u.696957\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nv : M\na✝ : v ∈ ⊤\np' q' : R[X]\nhpq' : p' * p + q' * q = 1\n⊢ ∃ y, y ∈ LinearMap.range (↑(aeval f) p) ∧ ∃ z, z ∈ LinearMap.range (↑(aeval f) q) ∧ y + z = v",
"state_before": "R : Type u\nS : Type ?u.696957\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nhpq : IsCoprime p q\nv : M\na✝ : v ∈ ⊤\n⊢ ∃ y, y ∈ LinearMap.range (↑(aeval f) p) ∧ ∃ z, z ∈ LinearMap.range (↑(aeval f) q) ∧ y + z = v",
"tactic": "rcases hpq with ⟨p', q', hpq'⟩"
},
{
"state_after": "case intro.intro\nR : Type u\nS : Type ?u.696957\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nv : M\na✝ : v ∈ ⊤\np' q' : R[X]\nhpq' : p' * p + q' * q = 1\n⊢ ↑(↑(aeval f) (p * p')) v ∈ LinearMap.range (↑(aeval f) p) ∧\n ∃ z, z ∈ LinearMap.range (↑(aeval f) q) ∧ ↑(↑(aeval f) (p * p')) v + z = v",
"state_before": "case intro.intro\nR : Type u\nS : Type ?u.696957\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nv : M\na✝ : v ∈ ⊤\np' q' : R[X]\nhpq' : p' * p + q' * q = 1\n⊢ ∃ y, y ∈ LinearMap.range (↑(aeval f) p) ∧ ∃ z, z ∈ LinearMap.range (↑(aeval f) q) ∧ y + z = v",
"tactic": "use aeval f (p * p') v"
},
{
"state_after": "case intro.intro\nR : Type u\nS : Type ?u.696957\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nv : M\na✝ : v ∈ ⊤\np' q' : R[X]\nhpq' : p' * p + q' * q = 1\n⊢ ∃ z, z ∈ LinearMap.range (↑(aeval f) q) ∧ ↑(↑(aeval f) (p * p')) v + z = v",
"state_before": "case intro.intro\nR : Type u\nS : Type ?u.696957\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nv : M\na✝ : v ∈ ⊤\np' q' : R[X]\nhpq' : p' * p + q' * q = 1\n⊢ ↑(↑(aeval f) (p * p')) v ∈ LinearMap.range (↑(aeval f) p) ∧\n ∃ z, z ∈ LinearMap.range (↑(aeval f) q) ∧ ↑(↑(aeval f) (p * p')) v + z = v",
"tactic": "use LinearMap.mem_range.2 ⟨aeval f p' v, by simp only [LinearMap.mul_apply, aeval_mul]⟩"
},
{
"state_after": "case intro.intro\nR : Type u\nS : Type ?u.696957\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nv : M\na✝ : v ∈ ⊤\np' q' : R[X]\nhpq' : p' * p + q' * q = 1\n⊢ ↑(↑(aeval f) (q * q')) v ∈ LinearMap.range (↑(aeval f) q) ∧ ↑(↑(aeval f) (p * p')) v + ↑(↑(aeval f) (q * q')) v = v",
"state_before": "case intro.intro\nR : Type u\nS : Type ?u.696957\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nv : M\na✝ : v ∈ ⊤\np' q' : R[X]\nhpq' : p' * p + q' * q = 1\n⊢ ∃ z, z ∈ LinearMap.range (↑(aeval f) q) ∧ ↑(↑(aeval f) (p * p')) v + z = v",
"tactic": "use aeval f (q * q') v"
},
{
"state_after": "case intro.intro\nR : Type u\nS : Type ?u.696957\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nv : M\na✝ : v ∈ ⊤\np' q' : R[X]\nhpq' : p' * p + q' * q = 1\n⊢ ↑(↑(aeval f) (p * p')) v + ↑(↑(aeval f) (q * q')) v = v",
"state_before": "case intro.intro\nR : Type u\nS : Type ?u.696957\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nv : M\na✝ : v ∈ ⊤\np' q' : R[X]\nhpq' : p' * p + q' * q = 1\n⊢ ↑(↑(aeval f) (q * q')) v ∈ LinearMap.range (↑(aeval f) q) ∧ ↑(↑(aeval f) (p * p')) v + ↑(↑(aeval f) (q * q')) v = v",
"tactic": "use LinearMap.mem_range.2 ⟨aeval f q' v, by simp only [LinearMap.mul_apply, aeval_mul]⟩"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nR : Type u\nS : Type ?u.696957\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nv : M\na✝ : v ∈ ⊤\np' q' : R[X]\nhpq' : p' * p + q' * q = 1\n⊢ ↑(↑(aeval f) (p * p')) v + ↑(↑(aeval f) (q * q')) v = v",
"tactic": "simpa only [mul_comm p p', mul_comm q q', aeval_one, aeval_add] using\n congr_arg (fun p : R[X] => aeval f p v) hpq'"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type ?u.696957\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nv : M\na✝ : v ∈ ⊤\np' q' : R[X]\nhpq' : p' * p + q' * q = 1\n⊢ ↑(↑(aeval f) p) (↑(↑(aeval f) p') v) = ↑(↑(aeval f) (p * p')) v",
"tactic": "simp only [LinearMap.mul_apply, aeval_mul]"
},
{
"state_after": "no goals",
"state_before": "R : Type u\nS : Type ?u.696957\nσ : Type v\nM : Type w\ninst✝³ : CommRing R\ninst✝² : CommRing S\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nf : M →ₗ[R] M\np q : R[X]\nv : M\na✝ : v ∈ ⊤\np' q' : R[X]\nhpq' : p' * p + q' * q = 1\n⊢ ↑(↑(aeval f) q) (↑(↑(aeval f) q') v) = ↑(↑(aeval f) (q * q')) v",
"tactic": "simp only [LinearMap.mul_apply, aeval_mul]"
}
] | [
1017,
49
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1006,
1
] |
Mathlib/Algebra/BigOperators/Basic.lean | Multiset.disjoint_finset_sum_left | [
{
"state_after": "case h.e'_2.a\nι : Type ?u.896168\nβ✝ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β✝\nβ : Type u_1\ni : Finset β\nf : β → Multiset α\na : Multiset α\n⊢ (∀ (b : β), b ∈ i → Disjoint (f b) a) ↔ ∀ (b : Multiset α), b ∈ map f i.val → Disjoint b a",
"state_before": "ι : Type ?u.896168\nβ✝ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β✝\nβ : Type u_1\ni : Finset β\nf : β → Multiset α\na : Multiset α\n⊢ Disjoint (Finset.sum i f) a ↔ ∀ (b : β), b ∈ i → Disjoint (f b) a",
"tactic": "convert @disjoint_sum_left _ a (map f i.val)"
},
{
"state_after": "no goals",
"state_before": "case h.e'_2.a\nι : Type ?u.896168\nβ✝ : Type u\nα : Type v\nγ : Type w\ns s₁ s₂ : Finset α\na✝ : α\nf✝ g : α → β✝\nβ : Type u_1\ni : Finset β\nf : β → Multiset α\na : Multiset α\n⊢ (∀ (b : β), b ∈ i → Disjoint (f b) a) ↔ ∀ (b : Multiset α), b ∈ map f i.val → Disjoint b a",
"tactic": "simp [and_congr_left_iff, iff_self_iff]"
}
] | [
2082,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2079,
1
] |
src/lean/Init/Core.lean | mt | [] | [
517,
23
] | d5348dfac847a56a4595fb6230fd0708dcb4e7e9 | https://github.com/leanprover/lean4 | [
516,
1
] |
Mathlib/Topology/Instances/EReal.lean | EReal.continuousAt_add_coe_top | [] | [
195,
95
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
192,
1
] |
Mathlib/MeasureTheory/Function/LpSeminorm.lean | MeasureTheory.snorm_const_lt_top_iff | [
{
"state_after": "α : Type u_1\nE : Type ?u.1263233\nF : Type u_2\nG : Type ?u.1263239\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : F\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhp : 0 < ENNReal.toReal p\n⊢ snorm (fun x => c) p μ < ⊤ ↔ c = 0 ∨ ↑↑μ Set.univ < ⊤",
"state_before": "α : Type u_1\nE : Type ?u.1263233\nF : Type u_2\nG : Type ?u.1263239\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : F\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\n⊢ snorm (fun x => c) p μ < ⊤ ↔ c = 0 ∨ ↑↑μ Set.univ < ⊤",
"tactic": "have hp : 0 < p.toReal := ENNReal.toReal_pos hp_ne_zero hp_ne_top"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.1263233\nF : Type u_2\nG : Type ?u.1263239\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : F\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhp : 0 < ENNReal.toReal p\nhμ : μ = 0\n⊢ snorm (fun x => c) p μ < ⊤ ↔ c = 0 ∨ ↑↑μ Set.univ < ⊤\n\ncase neg\nα : Type u_1\nE : Type ?u.1263233\nF : Type u_2\nG : Type ?u.1263239\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : F\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhp : 0 < ENNReal.toReal p\nhμ : ¬μ = 0\n⊢ snorm (fun x => c) p μ < ⊤ ↔ c = 0 ∨ ↑↑μ Set.univ < ⊤",
"state_before": "α : Type u_1\nE : Type ?u.1263233\nF : Type u_2\nG : Type ?u.1263239\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : F\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhp : 0 < ENNReal.toReal p\n⊢ snorm (fun x => c) p μ < ⊤ ↔ c = 0 ∨ ↑↑μ Set.univ < ⊤",
"tactic": "by_cases hμ : μ = 0"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.1263233\nF : Type u_2\nG : Type ?u.1263239\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : F\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhp : 0 < ENNReal.toReal p\nhμ : ¬μ = 0\nhc : c = 0\n⊢ snorm (fun x => c) p μ < ⊤ ↔ c = 0 ∨ ↑↑μ Set.univ < ⊤\n\ncase neg\nα : Type u_1\nE : Type ?u.1263233\nF : Type u_2\nG : Type ?u.1263239\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : F\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhp : 0 < ENNReal.toReal p\nhμ : ¬μ = 0\nhc : ¬c = 0\n⊢ snorm (fun x => c) p μ < ⊤ ↔ c = 0 ∨ ↑↑μ Set.univ < ⊤",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.1263233\nF : Type u_2\nG : Type ?u.1263239\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : F\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhp : 0 < ENNReal.toReal p\nhμ : ¬μ = 0\n⊢ snorm (fun x => c) p μ < ⊤ ↔ c = 0 ∨ ↑↑μ Set.univ < ⊤",
"tactic": "by_cases hc : c = 0"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type ?u.1263233\nF : Type u_2\nG : Type ?u.1263239\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : F\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhp : 0 < ENNReal.toReal p\nhμ : ¬μ = 0\nhc : ¬c = 0\n⊢ ↑‖c‖₊ * ↑↑μ Set.univ ^ (1 / ENNReal.toReal p) < ⊤ ↔ c = 0 ∨ ↑↑μ Set.univ < ⊤",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.1263233\nF : Type u_2\nG : Type ?u.1263239\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : F\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhp : 0 < ENNReal.toReal p\nhμ : ¬μ = 0\nhc : ¬c = 0\n⊢ snorm (fun x => c) p μ < ⊤ ↔ c = 0 ∨ ↑↑μ Set.univ < ⊤",
"tactic": "rw [snorm_const' c hp_ne_zero hp_ne_top]"
},
{
"state_after": "case pos\nα : Type u_1\nE : Type ?u.1263233\nF : Type u_2\nG : Type ?u.1263239\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : F\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhp : 0 < ENNReal.toReal p\nhμ : ¬μ = 0\nhc : ¬c = 0\nhμ_top : ↑↑μ Set.univ = ⊤\n⊢ ↑‖c‖₊ * ↑↑μ Set.univ ^ (1 / ENNReal.toReal p) < ⊤ ↔ c = 0 ∨ ↑↑μ Set.univ < ⊤\n\ncase neg\nα : Type u_1\nE : Type ?u.1263233\nF : Type u_2\nG : Type ?u.1263239\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : F\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhp : 0 < ENNReal.toReal p\nhμ : ¬μ = 0\nhc : ¬c = 0\nhμ_top : ¬↑↑μ Set.univ = ⊤\n⊢ ↑‖c‖₊ * ↑↑μ Set.univ ^ (1 / ENNReal.toReal p) < ⊤ ↔ c = 0 ∨ ↑↑μ Set.univ < ⊤",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.1263233\nF : Type u_2\nG : Type ?u.1263239\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : F\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhp : 0 < ENNReal.toReal p\nhμ : ¬μ = 0\nhc : ¬c = 0\n⊢ ↑‖c‖₊ * ↑↑μ Set.univ ^ (1 / ENNReal.toReal p) < ⊤ ↔ c = 0 ∨ ↑↑μ Set.univ < ⊤",
"tactic": "by_cases hμ_top : μ Set.univ = ∞"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type ?u.1263233\nF : Type u_2\nG : Type ?u.1263239\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : F\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhp : 0 < ENNReal.toReal p\nhμ : ¬μ = 0\nhc : ¬c = 0\nhμ_top : ¬↑↑μ Set.univ = ⊤\n⊢ ↑‖c‖₊ < ⊤ ∧ ↑↑μ Set.univ ^ (1 / ENNReal.toReal p) < ⊤ ∨ ↑‖c‖₊ = 0 ∨ ↑↑μ Set.univ ^ (1 / ENNReal.toReal p) = 0 ↔\n c = 0 ∨ ↑↑μ Set.univ < ⊤",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.1263233\nF : Type u_2\nG : Type ?u.1263239\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : F\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhp : 0 < ENNReal.toReal p\nhμ : ¬μ = 0\nhc : ¬c = 0\nhμ_top : ¬↑↑μ Set.univ = ⊤\n⊢ ↑‖c‖₊ * ↑↑μ Set.univ ^ (1 / ENNReal.toReal p) < ⊤ ↔ c = 0 ∨ ↑↑μ Set.univ < ⊤",
"tactic": "rw [ENNReal.mul_lt_top_iff]"
},
{
"state_after": "case neg\nα : Type u_1\nE : Type ?u.1263233\nF : Type u_2\nG : Type ?u.1263239\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : F\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhp : 0 < ENNReal.toReal p\nhμ : ¬μ = 0\nhc : ¬c = 0\nhμ_top : ¬↑↑μ Set.univ = ⊤\n⊢ ↑↑μ Set.univ ^ (ENNReal.toReal p)⁻¹ < ⊤",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.1263233\nF : Type u_2\nG : Type ?u.1263239\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : F\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhp : 0 < ENNReal.toReal p\nhμ : ¬μ = 0\nhc : ¬c = 0\nhμ_top : ¬↑↑μ Set.univ = ⊤\n⊢ ↑‖c‖₊ < ⊤ ∧ ↑↑μ Set.univ ^ (1 / ENNReal.toReal p) < ⊤ ∨ ↑‖c‖₊ = 0 ∨ ↑↑μ Set.univ ^ (1 / ENNReal.toReal p) = 0 ↔\n c = 0 ∨ ↑↑μ Set.univ < ⊤",
"tactic": "simp only [true_and_iff, one_div, ENNReal.rpow_eq_zero_iff, hμ, false_or_iff, or_false_iff,\n ENNReal.coe_lt_top, nnnorm_eq_zero, ENNReal.coe_eq_zero,\n MeasureTheory.Measure.measure_univ_eq_zero, hp, inv_lt_zero, hc, and_false_iff, false_and_iff,\n _root_.inv_pos, or_self_iff, hμ_top, Ne.lt_top hμ_top, iff_true_iff]"
},
{
"state_after": "no goals",
"state_before": "case neg\nα : Type u_1\nE : Type ?u.1263233\nF : Type u_2\nG : Type ?u.1263239\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : F\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhp : 0 < ENNReal.toReal p\nhμ : ¬μ = 0\nhc : ¬c = 0\nhμ_top : ¬↑↑μ Set.univ = ⊤\n⊢ ↑↑μ Set.univ ^ (ENNReal.toReal p)⁻¹ < ⊤",
"tactic": "exact ENNReal.rpow_lt_top_of_nonneg (inv_nonneg.mpr hp.le) hμ_top"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type ?u.1263233\nF : Type u_2\nG : Type ?u.1263239\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : F\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhp : 0 < ENNReal.toReal p\nhμ : μ = 0\n⊢ snorm (fun x => c) p μ < ⊤ ↔ c = 0 ∨ ↑↑μ Set.univ < ⊤",
"tactic": "simp only [hμ, Measure.coe_zero, Pi.zero_apply, or_true_iff, WithTop.zero_lt_top,\n snorm_measure_zero]"
},
{
"state_after": "no goals",
"state_before": "case pos\nα : Type u_1\nE : Type ?u.1263233\nF : Type u_2\nG : Type ?u.1263239\nm m0 : MeasurableSpace α\np✝ : ℝ≥0∞\nq : ℝ\nμ ν : Measure α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\np : ℝ≥0∞\nc : F\nhp_ne_zero : p ≠ 0\nhp_ne_top : p ≠ ⊤\nhp : 0 < ENNReal.toReal p\nhμ : ¬μ = 0\nhc : c = 0\n⊢ snorm (fun x => c) p μ < ⊤ ↔ c = 0 ∨ ↑↑μ Set.univ < ⊤",
"tactic": "simp only [hc, true_or_iff, eq_self_iff_true, WithTop.zero_lt_top, snorm_zero']"
},
{
"state_after": "no goals",
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319,
68
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303,
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Mathlib/Topology/Constructions.lean | ContinuousAt.fst' | [] | [
353,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
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Mathlib/LinearAlgebra/Matrix/NonsingularInverse.lean | Matrix.inv_submatrix_equiv | [
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"tactic": "rw [← invOf_eq_nonsing_inv, ← invOf_eq_nonsing_inv, invOf_submatrix_equiv_eq A]"
},
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"tactic": "simp_rw [nonsing_inv_eq_ring_inverse, Ring.inverse_non_unit _ h, Ring.inverse_non_unit _ this,\n submatrix_zero, Pi.zero_apply]"
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687,
37
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Mathlib/Logic/Basic.lean | ite_ne_right_iff | [
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60
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Mathlib/LinearAlgebra/Dfinsupp.lean | Dfinsupp.sum_mapRange_index.linearMap | [
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227,
77
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Mathlib/Analysis/InnerProductSpace/Adjoint.lean | LinearMap.eq_adjoint_iff_basis_left | [
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74
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Mathlib/Data/Multiset/Basic.lean | Multiset.foldl_induction' | [
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1451,
94
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Mathlib/Algebra/Algebra/Basic.lean | NoZeroSMulDivisors.iff_algebraMap_injective | [] | [
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101
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Mathlib/Analysis/InnerProductSpace/PiL2.lean | EuclideanSpace.norm_single | [] | [
286,
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Mathlib/CategoryTheory/Sites/InducedTopology.lean | CategoryTheory.over_forget_locallyCoverDense | [
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},
{
"state_after": "case h.e'_4.h\nC : Type u_1\ninst✝² : Category C\nD : Type ?u.13308\ninst✝¹ : Category D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst✝ : Category A\nX : C\nY : Over X\nT : ↑(GrothendieckTopology.sieves J ((Over.forget X).obj Y))\nZ : C\n⊢ ∀ (f : Z ⟶ (Over.forget X).obj Y),\n (Sieve.functorPushforward (Over.forget X) (Sieve.functorPullback (Over.forget X) ↑T)).arrows f ↔ (↑T).arrows f",
"state_before": "case h.e'_4\nC : Type u_1\ninst✝² : Category C\nD : Type ?u.13308\ninst✝¹ : Category D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst✝ : Category A\nX : C\nY : Over X\nT : ↑(GrothendieckTopology.sieves J ((Over.forget X).obj Y))\n⊢ Sieve.functorPushforward (Over.forget X) (Sieve.functorPullback (Over.forget X) ↑T) = ↑T",
"tactic": "ext Z"
},
{
"state_after": "case h.e'_4.h\nC : Type u_1\ninst✝² : Category C\nD : Type ?u.13308\ninst✝¹ : Category D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst✝ : Category A\nX : C\nY : Over X\nT : ↑(GrothendieckTopology.sieves J ((Over.forget X).obj Y))\nZ : C\nf : Z ⟶ (Over.forget X).obj Y\n⊢ (Sieve.functorPushforward (Over.forget X) (Sieve.functorPullback (Over.forget X) ↑T)).arrows f ↔ (↑T).arrows f",
"state_before": "case h.e'_4.h\nC : Type u_1\ninst✝² : Category C\nD : Type ?u.13308\ninst✝¹ : Category D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst✝ : Category A\nX : C\nY : Over X\nT : ↑(GrothendieckTopology.sieves J ((Over.forget X).obj Y))\nZ : C\n⊢ ∀ (f : Z ⟶ (Over.forget X).obj Y),\n (Sieve.functorPushforward (Over.forget X) (Sieve.functorPullback (Over.forget X) ↑T)).arrows f ↔ (↑T).arrows f",
"tactic": "intro f"
},
{
"state_after": "case h.e'_4.h.mp\nC : Type u_1\ninst✝² : Category C\nD : Type ?u.13308\ninst✝¹ : Category D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst✝ : Category A\nX : C\nY : Over X\nT : ↑(GrothendieckTopology.sieves J ((Over.forget X).obj Y))\nZ : C\nf : Z ⟶ (Over.forget X).obj Y\n⊢ (Sieve.functorPushforward (Over.forget X) (Sieve.functorPullback (Over.forget X) ↑T)).arrows f → (↑T).arrows f\n\ncase h.e'_4.h.mpr\nC : Type u_1\ninst✝² : Category C\nD : Type ?u.13308\ninst✝¹ : Category D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst✝ : Category A\nX : C\nY : Over X\nT : ↑(GrothendieckTopology.sieves J ((Over.forget X).obj Y))\nZ : C\nf : Z ⟶ (Over.forget X).obj Y\n⊢ (↑T).arrows f → (Sieve.functorPushforward (Over.forget X) (Sieve.functorPullback (Over.forget X) ↑T)).arrows f",
"state_before": "case h.e'_4.h\nC : Type u_1\ninst✝² : Category C\nD : Type ?u.13308\ninst✝¹ : Category D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst✝ : Category A\nX : C\nY : Over X\nT : ↑(GrothendieckTopology.sieves J ((Over.forget X).obj Y))\nZ : C\nf : Z ⟶ (Over.forget X).obj Y\n⊢ (Sieve.functorPushforward (Over.forget X) (Sieve.functorPullback (Over.forget X) ↑T)).arrows f ↔ (↑T).arrows f",
"tactic": "constructor"
},
{
"state_after": "case h.e'_4.h.mp.intro.intro.intro.intro\nC : Type u_1\ninst✝² : Category C\nD : Type ?u.13308\ninst✝¹ : Category D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst✝ : Category A\nX : C\nY : Over X\nT : ↑(GrothendieckTopology.sieves J ((Over.forget X).obj Y))\nZ : C\nw✝¹ : Over X\nw✝ : w✝¹ ⟶ Y\ng' : Z ⟶ (Over.forget X).obj w✝¹\nhg : (Sieve.functorPullback (Over.forget X) ↑T).arrows w✝\n⊢ (↑T).arrows (g' ≫ (Over.forget X).map w✝)",
"state_before": "case h.e'_4.h.mp\nC : Type u_1\ninst✝² : Category C\nD : Type ?u.13308\ninst✝¹ : Category D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst✝ : Category A\nX : C\nY : Over X\nT : ↑(GrothendieckTopology.sieves J ((Over.forget X).obj Y))\nZ : C\nf : Z ⟶ (Over.forget X).obj Y\n⊢ (Sieve.functorPushforward (Over.forget X) (Sieve.functorPullback (Over.forget X) ↑T)).arrows f → (↑T).arrows f",
"tactic": "rintro ⟨_, _, g', hg, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4.h.mp.intro.intro.intro.intro\nC : Type u_1\ninst✝² : Category C\nD : Type ?u.13308\ninst✝¹ : Category D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst✝ : Category A\nX : C\nY : Over X\nT : ↑(GrothendieckTopology.sieves J ((Over.forget X).obj Y))\nZ : C\nw✝¹ : Over X\nw✝ : w✝¹ ⟶ Y\ng' : Z ⟶ (Over.forget X).obj w✝¹\nhg : (Sieve.functorPullback (Over.forget X) ↑T).arrows w✝\n⊢ (↑T).arrows (g' ≫ (Over.forget X).map w✝)",
"tactic": "exact T.val.downward_closed hg g'"
},
{
"state_after": "case h.e'_4.h.mpr\nC : Type u_1\ninst✝² : Category C\nD : Type ?u.13308\ninst✝¹ : Category D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst✝ : Category A\nX : C\nY : Over X\nT : ↑(GrothendieckTopology.sieves J ((Over.forget X).obj Y))\nZ : C\nf : Z ⟶ (Over.forget X).obj Y\nhf : (↑T).arrows f\n⊢ (Sieve.functorPushforward (Over.forget X) (Sieve.functorPullback (Over.forget X) ↑T)).arrows f",
"state_before": "case h.e'_4.h.mpr\nC : Type u_1\ninst✝² : Category C\nD : Type ?u.13308\ninst✝¹ : Category D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst✝ : Category A\nX : C\nY : Over X\nT : ↑(GrothendieckTopology.sieves J ((Over.forget X).obj Y))\nZ : C\nf : Z ⟶ (Over.forget X).obj Y\n⊢ (↑T).arrows f → (Sieve.functorPushforward (Over.forget X) (Sieve.functorPullback (Over.forget X) ↑T)).arrows f",
"tactic": "intro hf"
},
{
"state_after": "no goals",
"state_before": "case h.e'_4.h.mpr\nC : Type u_1\ninst✝² : Category C\nD : Type ?u.13308\ninst✝¹ : Category D\nG : C ⥤ D\nJ : GrothendieckTopology C\nK : GrothendieckTopology D\nA : Type v\ninst✝ : Category A\nX : C\nY : Over X\nT : ↑(GrothendieckTopology.sieves J ((Over.forget X).obj Y))\nZ : C\nf : Z ⟶ (Over.forget X).obj Y\nhf : (↑T).arrows f\n⊢ (Sieve.functorPushforward (Over.forget X) (Sieve.functorPullback (Over.forget X) ↑T)).arrows f",
"tactic": "exact ⟨Over.mk (f ≫ Y.hom), Over.homMk f, 𝟙 _, hf, (Category.id_comp _).symm⟩"
}
] | [
143,
82
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
134,
1
] |
Mathlib/Topology/ContinuousFunction/CocompactMap.lean | CocompactMap.isCompact_preimage | [
{
"state_after": "case intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.10402\nδ : Type ?u.10405\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ninst✝ : T2Space β\nf : CocompactMap α β\ns : Set β\nhs : IsCompact s\nt : Set ?m.10680\nht : IsCompact t\nhts : ?m.10705ᶜ ⊆ t\n⊢ IsCompact (↑f ⁻¹' s)",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.10402\nδ : Type ?u.10405\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ninst✝ : T2Space β\nf : CocompactMap α β\ns : Set β\nhs : IsCompact s\n⊢ IsCompact (↑f ⁻¹' s)",
"tactic": "obtain ⟨t, ht, hts⟩ :=\n mem_cocompact'.mp\n (by\n simpa only [preimage_image_preimage, preimage_compl] using\n mem_map.mp\n (cocompact_tendsto f <|\n mem_cocompact.mpr ⟨s, hs, compl_subset_compl.mpr (image_preimage_subset f _)⟩))"
},
{
"state_after": "no goals",
"state_before": "case intro.intro\nα : Type u_2\nβ : Type u_1\nγ : Type ?u.10402\nδ : Type ?u.10405\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ninst✝ : T2Space β\nf : CocompactMap α β\ns : Set β\nhs : IsCompact s\nt : Set ?m.10680\nht : IsCompact t\nhts : ?m.10705ᶜ ⊆ t\n⊢ IsCompact (↑f ⁻¹' s)",
"tactic": "exact\n isCompact_of_isClosed_subset ht (hs.isClosed.preimage <| map_continuous f) (by simpa using hts)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.10402\nδ : Type ?u.10405\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ninst✝ : T2Space β\nf : CocompactMap α β\ns : Set β\nhs : IsCompact s\n⊢ ?m.10705 ∈ cocompact α",
"tactic": "simpa only [preimage_image_preimage, preimage_compl] using\n mem_map.mp\n (cocompact_tendsto f <|\n mem_cocompact.mpr ⟨s, hs, compl_subset_compl.mpr (image_preimage_subset f _)⟩)"
},
{
"state_after": "no goals",
"state_before": "α : Type u_2\nβ : Type u_1\nγ : Type ?u.10402\nδ : Type ?u.10405\ninst✝⁴ : TopologicalSpace α\ninst✝³ : TopologicalSpace β\ninst✝² : TopologicalSpace γ\ninst✝¹ : TopologicalSpace δ\ninst✝ : T2Space β\nf : CocompactMap α β\ns : Set β\nhs : IsCompact s\nt : Set α\nht : IsCompact t\nhts : (↑f ⁻¹' s)ᶜᶜ ⊆ t\n⊢ ↑f ⁻¹' s ⊆ t",
"tactic": "simpa using hts"
}
] | [
194,
100
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
184,
1
] |
Mathlib/GroupTheory/Subgroup/Basic.lean | Subgroup.center_toSubmonoid | [] | [
2075,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2074,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/InverseDeriv.lean | Real.differentiableWithinAt_arcsin_Iic | [
{
"state_after": "x : ℝ\nh : DifferentiableWithinAt ℝ arcsin (Iic x) x\n⊢ x ≠ 1",
"state_before": "x : ℝ\n⊢ DifferentiableWithinAt ℝ arcsin (Iic x) x ↔ x ≠ 1",
"tactic": "refine' ⟨fun h => _, fun h => (hasDerivWithinAt_arcsin_Iic h).differentiableWithinAt⟩"
},
{
"state_after": "x : ℝ\nh : DifferentiableWithinAt ℝ arcsin (Neg.neg '' Ici (-x)) (- -x)\n⊢ x ≠ 1",
"state_before": "x : ℝ\nh : DifferentiableWithinAt ℝ arcsin (Iic x) x\n⊢ x ≠ 1",
"tactic": "rw [← neg_neg x, ← image_neg_Ici] at h"
},
{
"state_after": "x : ℝ\nh : DifferentiableWithinAt ℝ arcsin (Neg.neg '' Ici (-x)) (- -x)\nthis : DifferentiableWithinAt ℝ (fun y => -(arcsin ∘ Neg.neg) y) (Ici (-x)) (-x)\n⊢ x ≠ 1",
"state_before": "x : ℝ\nh : DifferentiableWithinAt ℝ arcsin (Neg.neg '' Ici (-x)) (- -x)\n⊢ x ≠ 1",
"tactic": "have := (h.comp (-x) differentiableWithinAt_id.neg (mapsTo_image _ _)).neg"
}
] | [
101,
64
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
96,
1
] |
Mathlib/LinearAlgebra/Matrix/BilinearForm.lean | BilinForm.toMatrix_compRight | [
{
"state_after": "no goals",
"state_before": "R : Type ?u.1541084\nM : Type ?u.1541087\ninst✝²⁰ : Semiring R\ninst✝¹⁹ : AddCommMonoid M\ninst✝¹⁸ : Module R M\nR₁ : Type ?u.1541123\nM₁ : Type ?u.1541126\ninst✝¹⁷ : Ring R₁\ninst✝¹⁶ : AddCommGroup M₁\ninst✝¹⁵ : Module R₁ M₁\nR₂ : Type u_1\nM₂ : Type u_2\ninst✝¹⁴ : CommSemiring R₂\ninst✝¹³ : AddCommMonoid M₂\ninst✝¹² : Module R₂ M₂\nR₃ : Type ?u.1541925\nM₃ : Type ?u.1541928\ninst✝¹¹ : CommRing R₃\ninst✝¹⁰ : AddCommGroup M₃\ninst✝⁹ : Module R₃ M₃\nV : Type ?u.1542516\nK : Type ?u.1542519\ninst✝⁸ : Field K\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module K V\nB✝ : BilinForm R M\nB₁ : BilinForm R₁ M₁\nB₂ : BilinForm R₂ M₂\nn : Type u_3\no : Type ?u.1543736\ninst✝⁵ : Fintype n\ninst✝⁴ : Fintype o\ninst✝³ : DecidableEq n\nb : Basis n R₂ M₂\nM₂' : Type ?u.1543895\ninst✝² : AddCommMonoid M₂'\ninst✝¹ : Module R₂ M₂'\nc : Basis o R₂ M₂'\ninst✝ : DecidableEq o\nB : BilinForm R₂ M₂\nf : M₂ →ₗ[R₂] M₂\n⊢ ↑(toMatrix b) (compRight B f) = ↑(toMatrix b) B ⬝ ↑(LinearMap.toMatrix b b) f",
"tactic": "simp only [BilinForm.compRight, BilinForm.toMatrix_comp b b, toMatrix_id, transpose_one,\n Matrix.one_mul]"
}
] | [
395,
20
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
392,
1
] |
Mathlib/Algebra/GroupPower/Order.lean | one_le_sq_iff | [] | [
593,
51
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
592,
1
] |
Mathlib/Topology/FiberBundle/Trivialization.lean | Pretrivialization.target_inter_preimage_symm_source_eq | [
{
"state_after": "no goals",
"state_before": "ι : Type ?u.10841\nB : Type u_1\nF : Type u_2\nE : B → Type ?u.10852\nZ : Type u_3\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nproj : Z → B\ne✝ : Pretrivialization F proj\nx : Z\ne f : Pretrivialization F proj\n⊢ f.target ∩ ↑(LocalEquiv.symm f.toLocalEquiv) ⁻¹' e.source = (e.baseSet ∩ f.baseSet) ×ˢ univ",
"tactic": "rw [inter_comm, f.target_eq, e.source_eq, f.preimage_symm_proj_inter]"
}
] | [
203,
72
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
201,
1
] |
Mathlib/Algebra/Algebra/Operations.lean | Submodule.one_eq_span | [
{
"state_after": "case h\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n⊢ ∀ (x : A), x ∈ 1 ↔ x ∈ span R {1}",
"state_before": "ι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n⊢ 1 = span R {1}",
"tactic": "apply Submodule.ext"
},
{
"state_after": "case h\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n a : A\n⊢ a ∈ 1 ↔ a ∈ span R {1}",
"state_before": "case h\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n : A\n⊢ ∀ (x : A), x ∈ 1 ↔ x ∈ span R {1}",
"tactic": "intro a"
},
{
"state_after": "no goals",
"state_before": "case h\nι : Sort uι\nR : Type u\ninst✝² : CommSemiring R\nA : Type v\ninst✝¹ : Semiring A\ninst✝ : Algebra R A\nS T : Set A\nM N P Q : Submodule R A\nm n a : A\n⊢ a ∈ 1 ↔ a ∈ span R {1}",
"tactic": "simp only [mem_one, mem_span_singleton, Algebra.smul_def, mul_one]"
}
] | [
114,
69
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
111,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/Deriv.lean | HasStrictDerivAt.cos | [] | [
799,
46
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
797,
1
] |
Mathlib/Data/Set/Intervals/Basic.lean | Set.Iic_union_Ioc | [
{
"state_after": "case inl\nα : Type u_1\nβ : Type ?u.101649\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhcd : c ≤ d\nh : c < b\n⊢ Iic b ∪ Ioc c d = Iic (max b d)\n\ncase inr\nα : Type u_1\nβ : Type ?u.101649\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhcd : d ≤ c\nh : d < b\n⊢ Iic b ∪ Ioc c d = Iic (max b d)",
"state_before": "α : Type u_1\nβ : Type ?u.101649\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nh : min c d < b\n⊢ Iic b ∪ Ioc c d = Iic (max b d)",
"tactic": "cases' le_total c d with hcd hcd <;> simp [hcd] at h"
},
{
"state_after": "no goals",
"state_before": "case inl\nα : Type u_1\nβ : Type ?u.101649\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhcd : c ≤ d\nh : c < b\n⊢ Iic b ∪ Ioc c d = Iic (max b d)",
"tactic": "exact Iic_union_Ioc' h"
},
{
"state_after": "case inr\nα : Type u_1\nβ : Type ?u.101649\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhcd : d ≤ c\nh : d < b\n⊢ Iic b ∪ Ioc c d = Iic (max d b)",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.101649\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhcd : d ≤ c\nh : d < b\n⊢ Iic b ∪ Ioc c d = Iic (max b d)",
"tactic": "rw [max_comm]"
},
{
"state_after": "no goals",
"state_before": "case inr\nα : Type u_1\nβ : Type ?u.101649\ninst✝ : LinearOrder α\na a₁ a₂ b b₁ b₂ c d : α\nhcd : d ≤ c\nh : d < b\n⊢ Iic b ∪ Ioc c d = Iic (max d b)",
"tactic": "simp [*, max_eq_right_of_lt h]"
}
] | [
1421,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1417,
1
] |
Mathlib/LinearAlgebra/Matrix/ToLin.lean | Matrix.toLin'_one | [] | [
358,
23
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
357,
1
] |
Mathlib/SetTheory/Ordinal/Arithmetic.lean | Ordinal.IsLimit.one_lt | [
{
"state_after": "no goals",
"state_before": "α : Type ?u.93011\nβ : Type ?u.93014\nγ : Type ?u.93017\nr : α → α → Prop\ns : β → β → Prop\nt : γ → γ → Prop\no : Ordinal\nh : IsLimit o\n⊢ 1 < o",
"tactic": "simpa only [succ_zero] using h.2 _ h.pos"
}
] | [
292,
43
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
291,
1
] |
Mathlib/Order/LiminfLimsup.lean | Filter.bliminf_sup_le_and_aux_right | [] | [
916,
40
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
915,
1
] |
Mathlib/GroupTheory/GroupAction/Basic.lean | MulAction.smul_orbit | [] | [
228,
70
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
224,
1
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Mathlib/CategoryTheory/Functor/EpiMono.lean | CategoryTheory.Functor.isSplitEpi_iff | [
{
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{
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"tactic": "intro h"
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{
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"tactic": "exact IsSplitEpi.mk' ((splitEpiEquiv F f).invFun h.exists_splitEpi.some)"
},
{
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"tactic": "intro h"
},
{
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"tactic": "exact IsSplitEpi.mk' ((splitEpiEquiv F f).toFun h.exists_splitEpi.some)"
}
] | [
249,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
244,
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Mathlib/NumberTheory/Liouville/LiouvilleWith.lean | LiouvilleWith.sub_nat | [] | [
287,
18
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
286,
1
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Mathlib/Algebra/Algebra/Unitization.lean | Unitization.fst_inr | [] | [
123,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
122,
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Mathlib/Analysis/SpecialFunctions/Exponential.lean | hasStrictFDerivAt_exp_smul_const_of_mem_ball' | [
{
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"tactic": "let ⟨_, _⟩ := analyticAt_exp_of_mem_ball (t • x) htx"
},
{
"state_after": "case h.e'_10\n𝕂 : Type u_3\n𝕊 : Type u_2\n𝔸 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕂\ninst✝⁸ : CharZero 𝕂\ninst✝⁷ : NormedCommRing 𝕊\ninst✝⁶ : NormedRing 𝔸\ninst✝⁵ : NormedSpace 𝕂 𝕊\ninst✝⁴ : NormedAlgebra 𝕂 𝔸\ninst✝³ : Algebra 𝕊 𝔸\ninst✝² : ContinuousSMul 𝕊 𝔸\ninst✝¹ : IsScalarTower 𝕂 𝕊 𝔸\ninst✝ : CompleteSpace 𝔸\nx : 𝔸\nt : 𝕊\nhtx : t • x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nw✝ : FormalMultilinearSeries 𝕂 𝔸 𝔸\nh✝ : HasFPowerSeriesAt (exp 𝕂) w✝ (t • x)\n⊢ ContinuousLinearMap.smulRight (ContinuousLinearMap.smulRight 1 x) (exp 𝕂 (t • x)) =\n exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x",
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"tactic": "convert hasStrictFDerivAt_exp_smul_const_of_mem_ball 𝕂 _ _ htx using 1"
},
{
"state_after": "case h.e'_10.h\n𝕂 : Type u_3\n𝕊 : Type u_2\n𝔸 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕂\ninst✝⁸ : CharZero 𝕂\ninst✝⁷ : NormedCommRing 𝕊\ninst✝⁶ : NormedRing 𝔸\ninst✝⁵ : NormedSpace 𝕂 𝕊\ninst✝⁴ : NormedAlgebra 𝕂 𝔸\ninst✝³ : Algebra 𝕊 𝔸\ninst✝² : ContinuousSMul 𝕊 𝔸\ninst✝¹ : IsScalarTower 𝕂 𝕊 𝔸\ninst✝ : CompleteSpace 𝔸\nx : 𝔸\nt : 𝕊\nhtx : t • x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nw✝ : FormalMultilinearSeries 𝕂 𝔸 𝔸\nh✝ : HasFPowerSeriesAt (exp 𝕂) w✝ (t • x)\nt' : 𝕊\n⊢ ↑(ContinuousLinearMap.smulRight (ContinuousLinearMap.smulRight 1 x) (exp 𝕂 (t • x))) t' =\n ↑(exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) t'",
"state_before": "case h.e'_10\n𝕂 : Type u_3\n𝕊 : Type u_2\n𝔸 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕂\ninst✝⁸ : CharZero 𝕂\ninst✝⁷ : NormedCommRing 𝕊\ninst✝⁶ : NormedRing 𝔸\ninst✝⁵ : NormedSpace 𝕂 𝕊\ninst✝⁴ : NormedAlgebra 𝕂 𝔸\ninst✝³ : Algebra 𝕊 𝔸\ninst✝² : ContinuousSMul 𝕊 𝔸\ninst✝¹ : IsScalarTower 𝕂 𝕊 𝔸\ninst✝ : CompleteSpace 𝔸\nx : 𝔸\nt : 𝕊\nhtx : t • x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nw✝ : FormalMultilinearSeries 𝕂 𝔸 𝔸\nh✝ : HasFPowerSeriesAt (exp 𝕂) w✝ (t • x)\n⊢ ContinuousLinearMap.smulRight (ContinuousLinearMap.smulRight 1 x) (exp 𝕂 (t • x)) =\n exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x",
"tactic": "ext t'"
},
{
"state_after": "case h.e'_10.h\n𝕂 : Type u_3\n𝕊 : Type u_2\n𝔸 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕂\ninst✝⁸ : CharZero 𝕂\ninst✝⁷ : NormedCommRing 𝕊\ninst✝⁶ : NormedRing 𝔸\ninst✝⁵ : NormedSpace 𝕂 𝕊\ninst✝⁴ : NormedAlgebra 𝕂 𝔸\ninst✝³ : Algebra 𝕊 𝔸\ninst✝² : ContinuousSMul 𝕊 𝔸\ninst✝¹ : IsScalarTower 𝕂 𝕊 𝔸\ninst✝ : CompleteSpace 𝔸\nx : 𝔸\nt : 𝕊\nhtx : t • x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nw✝ : FormalMultilinearSeries 𝕂 𝔸 𝔸\nh✝ : HasFPowerSeriesAt (exp 𝕂) w✝ (t • x)\nt' : 𝕊\n⊢ Commute (t' • x) (exp 𝕂 (t • x))",
"state_before": "case h.e'_10.h\n𝕂 : Type u_3\n𝕊 : Type u_2\n𝔸 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕂\ninst✝⁸ : CharZero 𝕂\ninst✝⁷ : NormedCommRing 𝕊\ninst✝⁶ : NormedRing 𝔸\ninst✝⁵ : NormedSpace 𝕂 𝕊\ninst✝⁴ : NormedAlgebra 𝕂 𝔸\ninst✝³ : Algebra 𝕊 𝔸\ninst✝² : ContinuousSMul 𝕊 𝔸\ninst✝¹ : IsScalarTower 𝕂 𝕊 𝔸\ninst✝ : CompleteSpace 𝔸\nx : 𝔸\nt : 𝕊\nhtx : t • x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nw✝ : FormalMultilinearSeries 𝕂 𝔸 𝔸\nh✝ : HasFPowerSeriesAt (exp 𝕂) w✝ (t • x)\nt' : 𝕊\n⊢ ↑(ContinuousLinearMap.smulRight (ContinuousLinearMap.smulRight 1 x) (exp 𝕂 (t • x))) t' =\n ↑(exp 𝕂 (t • x) • ContinuousLinearMap.smulRight 1 x) t'",
"tactic": "show Commute (t' • x) (exp 𝕂 (t • x))"
},
{
"state_after": "no goals",
"state_before": "case h.e'_10.h\n𝕂 : Type u_3\n𝕊 : Type u_2\n𝔸 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕂\ninst✝⁸ : CharZero 𝕂\ninst✝⁷ : NormedCommRing 𝕊\ninst✝⁶ : NormedRing 𝔸\ninst✝⁵ : NormedSpace 𝕂 𝕊\ninst✝⁴ : NormedAlgebra 𝕂 𝔸\ninst✝³ : Algebra 𝕊 𝔸\ninst✝² : ContinuousSMul 𝕊 𝔸\ninst✝¹ : IsScalarTower 𝕂 𝕊 𝔸\ninst✝ : CompleteSpace 𝔸\nx : 𝔸\nt : 𝕊\nhtx : t • x ∈ EMetric.ball 0 (FormalMultilinearSeries.radius (expSeries 𝕂 𝔸))\nw✝ : FormalMultilinearSeries 𝕂 𝔸 𝔸\nh✝ : HasFPowerSeriesAt (exp 𝕂) w✝ (t • x)\nt' : 𝕊\n⊢ Commute (t' • x) (exp 𝕂 (t • x))",
"tactic": "exact (((Commute.refl x).smul_left t').smul_right t).exp_right 𝕂"
}
] | [
338,
67
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
330,
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Mathlib/Order/Filter/Basic.lean | Filter.push_pull | [
{
"state_after": "case a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.286447\nι : Sort x\nf : α → β\nF : Filter α\nG : Filter β\n⊢ map f (F ⊓ comap f G) ≤ map f F ⊓ G\n\ncase a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.286447\nι : Sort x\nf : α → β\nF : Filter α\nG : Filter β\n⊢ map f F ⊓ G ≤ map f (F ⊓ comap f G)",
"state_before": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.286447\nι : Sort x\nf : α → β\nF : Filter α\nG : Filter β\n⊢ map f (F ⊓ comap f G) = map f F ⊓ G",
"tactic": "apply le_antisymm"
},
{
"state_after": "no goals",
"state_before": "case a\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.286447\nι : Sort x\nf : α → β\nF : Filter α\nG : Filter β\n⊢ map f (F ⊓ comap f G) ≤ map f F ⊓ G",
"tactic": "calc\n map f (F ⊓ comap f G) ≤ map f F ⊓ (map f <| comap f G) := map_inf_le\n _ ≤ map f F ⊓ G := inf_le_inf_left (map f F) map_comap_le"
},
{
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"tactic": "rintro U ⟨V, V_in, W, ⟨Z, Z_in, hZ⟩, h⟩"
},
{
"state_after": "case a.intro.intro.intro.intro.intro.intro\nα : Type u\nβ : Type v\nγ : Type w\nδ : Type ?u.286447\nι : Sort x\nf : α → β\nF : Filter α\nG : Filter β\nU : Set β\nV : Set α\nV_in : V ∈ F\nW : Set α\nh : f ⁻¹' U = V ∩ W\nZ : Set β\nZ_in : Z ∈ G\nhZ : f ⁻¹' Z ⊆ W\n⊢ f '' V ∩ Z ⊆ U",
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"tactic": "apply mem_inf_of_inter (image_mem_map V_in) Z_in"
},
{
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"tactic": "calc\n f '' V ∩ Z = f '' (V ∩ f ⁻¹' Z) := by rw [image_inter_preimage]\n _ ⊆ f '' (V ∩ W) := image_subset _ (inter_subset_inter_right _ ‹_›)\n _ = f '' (f ⁻¹' U) := by rw [h]\n _ ⊆ U := image_preimage_subset f U"
},
{
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},
{
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"tactic": "rw [h]"
}
] | [
2577,
41
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
2564,
11
] |
Mathlib/Order/UpperLower/Basic.lean | lowerClosure_union | [] | [
1439,
35
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1438,
1
] |
Mathlib/Data/MvPolynomial/Basic.lean | MvPolynomial.algHom_ext' | [] | [
489,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
483,
1
] |
Mathlib/LinearAlgebra/AffineSpace/AffineMap.lean | AffineMap.ext_iff | [] | [
150,
28
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
149,
1
] |
Mathlib/CategoryTheory/Monoidal/OfHasFiniteProducts.lean | CategoryTheory.monoidalOfHasFiniteProducts.tensorObj | [] | [
94,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
93,
1
] |
Mathlib/Data/Set/Image.lean | Subtype.exists_set_subtype | [
{
"state_after": "no goals",
"state_before": "α : Type u_1\nt : Set α\np : Set α → Prop\n⊢ (∃ s, p (val '' s)) ↔ ∃ s, s ⊆ t ∧ p s",
"tactic": "rw [← exists_subset_range_and_iff, range_coe]"
}
] | [
1458,
48
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1456,
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] |
Mathlib/GroupTheory/SpecificGroups/Alternating.lean | Equiv.Perm.IsThreeCycle.mem_alternatingGroup | [] | [
86,
34
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
84,
1
] |
Mathlib/Data/Int/Log.lean | Int.lt_zpow_succ_log_self | [
{
"state_after": "case inl\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : r ≤ 0\n⊢ r < ↑b ^ (log b r + 1)\n\ncase inr\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\n⊢ r < ↑b ^ (log b r + 1)",
"state_before": "R : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\n⊢ r < ↑b ^ (log b r + 1)",
"tactic": "cases' le_or_lt r 0 with hr hr"
},
{
"state_after": "case inr.inl\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : 1 ≤ r\n⊢ r < ↑b ^ (log b r + 1)\n\ncase inr.inr\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : r < 1\n⊢ r < ↑b ^ (log b r + 1)",
"state_before": "case inr\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\n⊢ r < ↑b ^ (log b r + 1)",
"tactic": "cases' le_or_lt 1 r with hr1 hr1"
},
{
"state_after": "case inl\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : r ≤ 0\n⊢ r < ↑b",
"state_before": "case inl\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : r ≤ 0\n⊢ r < ↑b ^ (log b r + 1)",
"tactic": "rw [log_of_right_le_zero _ hr, zero_add, zpow_one]"
},
{
"state_after": "no goals",
"state_before": "case inl\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : r ≤ 0\n⊢ r < ↑b",
"tactic": "exact hr.trans_lt (zero_lt_one.trans_le <| by exact_mod_cast hb.le)"
},
{
"state_after": "no goals",
"state_before": "R : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : r ≤ 0\n⊢ 1 ≤ ↑b",
"tactic": "exact_mod_cast hb.le"
},
{
"state_after": "case inr.inl\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : 1 ≤ r\n⊢ r < ↑b ^ (↑(Nat.log b ⌊r⌋₊) + 1)",
"state_before": "case inr.inl\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : 1 ≤ r\n⊢ r < ↑b ^ (log b r + 1)",
"tactic": "rw [log_of_one_le_right _ hr1]"
},
{
"state_after": "case inr.inl\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : 1 ≤ r\n⊢ r < ↑(b ^ Nat.succ (Nat.log b ⌊r⌋₊))",
"state_before": "case inr.inl\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : 1 ≤ r\n⊢ r < ↑b ^ (↑(Nat.log b ⌊r⌋₊) + 1)",
"tactic": "rw [Int.ofNat_add_one_out, zpow_ofNat, ← Nat.cast_pow]"
},
{
"state_after": "case inr.inl.h\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : 1 ≤ r\n⊢ ⌊r⌋₊ < b ^ Nat.succ (Nat.log b ⌊r⌋₊)",
"state_before": "case inr.inl\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : 1 ≤ r\n⊢ r < ↑(b ^ Nat.succ (Nat.log b ⌊r⌋₊))",
"tactic": "apply Nat.lt_of_floor_lt"
},
{
"state_after": "no goals",
"state_before": "case inr.inl.h\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : 1 ≤ r\n⊢ ⌊r⌋₊ < b ^ Nat.succ (Nat.log b ⌊r⌋₊)",
"tactic": "exact Nat.lt_pow_succ_log_self hb _"
},
{
"state_after": "case inr.inr\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : r < 1\n⊢ r < ↑b ^ (-↑(Nat.clog b ⌈r⁻¹⌉₊) + 1)",
"state_before": "case inr.inr\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : r < 1\n⊢ r < ↑b ^ (log b r + 1)",
"tactic": "rw [log_of_right_le_one _ hr1.le]"
},
{
"state_after": "case inr.inr\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : r < 1\nhcri : 1 < r⁻¹\n⊢ r < ↑b ^ (-↑(Nat.clog b ⌈r⁻¹⌉₊) + 1)",
"state_before": "case inr.inr\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : r < 1\n⊢ r < ↑b ^ (-↑(Nat.clog b ⌈r⁻¹⌉₊) + 1)",
"tactic": "have hcri : 1 < r⁻¹ := one_lt_inv hr hr1"
},
{
"state_after": "case inr.inr\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : r < 1\nhcri : 1 < r⁻¹\nthis : 1 ≤ Nat.clog b ⌈r⁻¹⌉₊\n⊢ r < ↑b ^ (-↑(Nat.clog b ⌈r⁻¹⌉₊) + 1)",
"state_before": "case inr.inr\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : r < 1\nhcri : 1 < r⁻¹\n⊢ r < ↑b ^ (-↑(Nat.clog b ⌈r⁻¹⌉₊) + 1)",
"tactic": "have : 1 ≤ Nat.clog b ⌈r⁻¹⌉₊ :=\n Nat.succ_le_of_lt (Nat.clog_pos hb <| Nat.one_lt_cast.1 <| hcri.trans_le (Nat.le_ceil _))"
},
{
"state_after": "case inr.inr\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : r < 1\nhcri : 1 < r⁻¹\nthis : 1 ≤ Nat.clog b ⌈r⁻¹⌉₊\n⊢ ↑(b ^ (Nat.clog b ⌈r⁻¹⌉₊ - 1)) < r⁻¹",
"state_before": "case inr.inr\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : r < 1\nhcri : 1 < r⁻¹\nthis : 1 ≤ Nat.clog b ⌈r⁻¹⌉₊\n⊢ r < ↑b ^ (-↑(Nat.clog b ⌈r⁻¹⌉₊) + 1)",
"tactic": "rw [neg_add_eq_sub, ← neg_sub, ← Int.ofNat_one, ← Int.ofNat_sub this, zpow_neg, zpow_ofNat,\n lt_inv hr (pow_pos (Nat.cast_pos.mpr <| zero_lt_one.trans hb) _), ← Nat.cast_pow]"
},
{
"state_after": "case inr.inr\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : r < 1\nhcri : 1 < r⁻¹\nthis : 1 ≤ Nat.clog b ⌈r⁻¹⌉₊\n⊢ b ^ (Nat.clog b ⌈r⁻¹⌉₊ - 1) < ⌈r⁻¹⌉₊",
"state_before": "case inr.inr\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : r < 1\nhcri : 1 < r⁻¹\nthis : 1 ≤ Nat.clog b ⌈r⁻¹⌉₊\n⊢ ↑(b ^ (Nat.clog b ⌈r⁻¹⌉₊ - 1)) < r⁻¹",
"tactic": "refine' Nat.lt_ceil.1 _"
},
{
"state_after": "no goals",
"state_before": "case inr.inr\nR : Type u_1\ninst✝¹ : LinearOrderedSemifield R\ninst✝ : FloorSemiring R\nb : ℕ\nhb : 1 < b\nr : R\nhr : 0 < r\nhr1 : r < 1\nhcri : 1 < r⁻¹\nthis : 1 ≤ Nat.clog b ⌈r⁻¹⌉₊\n⊢ b ^ (Nat.clog b ⌈r⁻¹⌉₊ - 1) < ⌈r⁻¹⌉₊",
"tactic": "exact Nat.pow_pred_clog_lt_self hb <| Nat.one_lt_cast.1 <| hcri.trans_le <| Nat.le_ceil _"
}
] | [
121,
94
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
105,
1
] |
Mathlib/Algebra/GroupPower/Order.lean | pow_mono | [
{
"state_after": "β : Type ?u.218801\nA : Type ?u.218804\nG : Type ?u.218807\nM : Type ?u.218810\nR : Type u_1\ninst✝ : OrderedSemiring R\na x y : R\nn✝ m : ℕ\nh : 1 ≤ a\nn : ℕ\n⊢ a ^ n ≤ a * a ^ n",
"state_before": "β : Type ?u.218801\nA : Type ?u.218804\nG : Type ?u.218807\nM : Type ?u.218810\nR : Type u_1\ninst✝ : OrderedSemiring R\na x y : R\nn✝ m : ℕ\nh : 1 ≤ a\nn : ℕ\n⊢ a ^ n ≤ a ^ (n + 1)",
"tactic": "rw [pow_succ]"
},
{
"state_after": "no goals",
"state_before": "β : Type ?u.218801\nA : Type ?u.218804\nG : Type ?u.218807\nM : Type ?u.218810\nR : Type u_1\ninst✝ : OrderedSemiring R\na x y : R\nn✝ m : ℕ\nh : 1 ≤ a\nn : ℕ\n⊢ a ^ n ≤ a * a ^ n",
"tactic": "exact le_mul_of_one_le_left (pow_nonneg (zero_le_one.trans h) _) h"
}
] | [
447,
71
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
444,
1
] |
Mathlib/Topology/Algebra/InfiniteSum/Basic.lean | Function.Surjective.summable_iff_of_hasSum_iff | [] | [
306,
42
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
303,
1
] |
Mathlib/LinearAlgebra/Dual.lean | Subspace.dualLift_rightInverse | [] | [
1044,
29
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
1042,
1
] |
Mathlib/Analysis/SpecialFunctions/Trigonometric/ComplexDeriv.lean | Complex.hasDerivAt_tan | [] | [
37,
38
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
36,
1
] |
Mathlib/Data/Set/Intervals/ProjIcc.lean | Set.IccExtend_val | [] | [
139,
33
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
138,
1
] |
Mathlib/Analysis/InnerProductSpace/Orthogonal.lean | Submodule.IsOrtho.ge | [] | [
324,
9
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
323,
1
] |
Mathlib/Algebra/CharP/Basic.lean | CharP.char_is_prime | [] | [
570,
76
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
569,
1
] |
Mathlib/Algebra/Group/UniqueProds.lean | UniqueMul.set_subsingleton | [
{
"state_after": "case mk.mk\nG : Type u_1\nH : Type ?u.1533\ninst✝¹ : Mul G\ninst✝ : Mul H\nA✝ B✝ : Finset G\na0✝ b0✝ : G\nA B : Finset G\na0 b0 : G\nh : UniqueMul A B a0 b0\nx1 y1 : G\nhx : x1 ∈ A ∧ y1 ∈ B ∧ x1 * y1 = a0 * b0\nx2 y2 : G\nhy : x2 ∈ A ∧ y2 ∈ B ∧ x2 * y2 = a0 * b0\n⊢ (x1, y1) = (x2, y2)",
"state_before": "G : Type u_1\nH : Type ?u.1533\ninst✝¹ : Mul G\ninst✝ : Mul H\nA✝ B✝ : Finset G\na0✝ b0✝ : G\nA B : Finset G\na0 b0 : G\nh : UniqueMul A B a0 b0\n⊢ Set.Subsingleton {ab | ab.fst ∈ A ∧ ab.snd ∈ B ∧ ab.fst * ab.snd = a0 * b0}",
"tactic": "rintro ⟨x1, y1⟩ (hx : x1 ∈ A ∧ y1 ∈ B ∧ x1 * y1 = a0 * b0) ⟨x2, y2⟩\n (hy : x2 ∈ A ∧ y2 ∈ B ∧ x2 * y2 = a0 * b0)"
},
{
"state_after": "case mk.mk.intro\nG : Type u_1\nH : Type ?u.1533\ninst✝¹ : Mul G\ninst✝ : Mul H\nA✝ B✝ : Finset G\na0 b0 : G\nA B : Finset G\nx1 y1 x2 y2 : G\nh : UniqueMul A B x1 y1\nhx : x1 ∈ A ∧ y1 ∈ B ∧ x1 * y1 = x1 * y1\nhy : x2 ∈ A ∧ y2 ∈ B ∧ x2 * y2 = x1 * y1\n⊢ (x1, y1) = (x2, y2)",
"state_before": "case mk.mk\nG : Type u_1\nH : Type ?u.1533\ninst✝¹ : Mul G\ninst✝ : Mul H\nA✝ B✝ : Finset G\na0✝ b0✝ : G\nA B : Finset G\na0 b0 : G\nh : UniqueMul A B a0 b0\nx1 y1 : G\nhx : x1 ∈ A ∧ y1 ∈ B ∧ x1 * y1 = a0 * b0\nx2 y2 : G\nhy : x2 ∈ A ∧ y2 ∈ B ∧ x2 * y2 = a0 * b0\n⊢ (x1, y1) = (x2, y2)",
"tactic": "rcases h hx.1 hx.2.1 hx.2.2 with ⟨rfl, rfl⟩"
},
{
"state_after": "case mk.mk.intro.intro\nG : Type u_1\nH : Type ?u.1533\ninst✝¹ : Mul G\ninst✝ : Mul H\nA✝ B✝ : Finset G\na0 b0 : G\nA B : Finset G\nx2 y2 : G\nh : UniqueMul A B x2 y2\nhx hy : x2 ∈ A ∧ y2 ∈ B ∧ x2 * y2 = x2 * y2\n⊢ (x2, y2) = (x2, y2)",
"state_before": "case mk.mk.intro\nG : Type u_1\nH : Type ?u.1533\ninst✝¹ : Mul G\ninst✝ : Mul H\nA✝ B✝ : Finset G\na0 b0 : G\nA B : Finset G\nx1 y1 x2 y2 : G\nh : UniqueMul A B x1 y1\nhx : x1 ∈ A ∧ y1 ∈ B ∧ x1 * y1 = x1 * y1\nhy : x2 ∈ A ∧ y2 ∈ B ∧ x2 * y2 = x1 * y1\n⊢ (x1, y1) = (x2, y2)",
"tactic": "rcases h hy.1 hy.2.1 hy.2.2 with ⟨rfl, rfl⟩"
},
{
"state_after": "no goals",
"state_before": "case mk.mk.intro.intro\nG : Type u_1\nH : Type ?u.1533\ninst✝¹ : Mul G\ninst✝ : Mul H\nA✝ B✝ : Finset G\na0 b0 : G\nA B : Finset G\nx2 y2 : G\nh : UniqueMul A B x2 y2\nhx hy : x2 ∈ A ∧ y2 ∈ B ∧ x2 * y2 = x2 * y2\n⊢ (x2, y2) = (x2, y2)",
"tactic": "rfl"
}
] | [
79,
6
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
73,
1
] |
Std/Data/Nat/Gcd.lean | Nat.gcd_eq_right_iff_dvd | [
{
"state_after": "m n : Nat\n⊢ m ∣ n ↔ gcd m n = m",
"state_before": "m n : Nat\n⊢ m ∣ n ↔ gcd n m = m",
"tactic": "rw [gcd_comm]"
},
{
"state_after": "no goals",
"state_before": "m n : Nat\n⊢ m ∣ n ↔ gcd m n = m",
"tactic": "exact gcd_eq_left_iff_dvd"
}
] | [
68,
43
] | e68aa8f5fe47aad78987df45f99094afbcb5e936 | https://github.com/leanprover/std4 | [
67,
1
] |
Mathlib/Analysis/NormedSpace/Extr.lean | IsMaxFilter.norm_add_self | [] | [
52,
45
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
50,
1
] |
Mathlib/CategoryTheory/Subobject/FactorThru.lean | CategoryTheory.Subobject.factorThru_comp_arrow | [
{
"state_after": "case h\nC : Type u₁\ninst✝¹ : Category C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝ : Category D\nX Y : C\nP : Subobject Y\nf : X ⟶ underlying.obj P\nh : Factors P (f ≫ arrow P)\n⊢ factorThru P (f ≫ arrow P) h ≫ arrow P = f ≫ arrow P",
"state_before": "C : Type u₁\ninst✝¹ : Category C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝ : Category D\nX Y : C\nP : Subobject Y\nf : X ⟶ underlying.obj P\nh : Factors P (f ≫ arrow P)\n⊢ factorThru P (f ≫ arrow P) h = f",
"tactic": "ext"
},
{
"state_after": "no goals",
"state_before": "case h\nC : Type u₁\ninst✝¹ : Category C\nX✝ Y✝ Z : C\nD : Type u₂\ninst✝ : Category D\nX Y : C\nP : Subobject Y\nf : X ⟶ underlying.obj P\nh : Factors P (f ≫ arrow P)\n⊢ factorThru P (f ≫ arrow P) h ≫ arrow P = f ≫ arrow P",
"tactic": "simp"
}
] | [
146,
7
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
143,
1
] |
Mathlib/Combinatorics/Pigeonhole.lean | Finset.exists_lt_sum_fiber_of_maps_to_of_nsmul_lt_sum | [
{
"state_after": "no goals",
"state_before": "α : Type u\nβ : Type v\nM : Type w\ninst✝¹ : DecidableEq β\ns : Finset α\nt : Finset β\nf : α → β\nw : α → M\nb : M\nn : ℕ\ninst✝ : LinearOrderedCancelAddCommMonoid M\nhf : ∀ (a : α), a ∈ s → f a ∈ t\nhb : card t • b < ∑ x in s, w x\n⊢ ∑ i in t, b < ∑ i in t, ∑ x in filter (fun x => f x = i) s, w x",
"tactic": "simpa only [sum_fiberwise_of_maps_to hf, sum_const]"
}
] | [
122,
80
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
120,
1
] |
Mathlib/Order/Hom/Bounded.lean | BoundedOrderHom.copy_eq | [] | [
615,
17
] | 5a919533f110b7d76410134a237ee374f24eaaad | https://github.com/leanprover-community/mathlib4 | [
614,
1
] |
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