module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Topology.UrysohnsLemma | {
"line": 462,
"column": 4
} | {
"line": 470,
"column": 23
} | [
{
"pp": "case refine_2\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : RegularSpace X\ninst✝ : LocallyCompactSpace X\ns t : Set X\nhs : IsCompact s\nh's : IsGδ s\nht : IsClosed[inst✝²] t\nhd : Disjoint s t\nU : ℕ → Set X\nU_open : ∀ (n : ℕ), IsOpen[inst✝²] (U n)\nhU : s = ⋂ n, U n\nm : Set X\nm_comp : IsCo... | apply Subset.antisymm (fun x hx ↦ by simp [g, fs _ hx, hu]) ?_
apply compl_subset_compl.1
intro x hx
obtain ⟨n, hn⟩ : ∃ n, x ∉ U n := by simpa [hU] using hx
have fnx : f n x = 0 := fm _ (by simp [hn])
have : g x < 1 := by
apply lt_of_lt_of_le ?_ hu.le
exact (S x).tsum_lt_tsum (i := n) (f... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UrysohnsLemma | {
"line": 462,
"column": 4
} | {
"line": 470,
"column": 23
} | [
{
"pp": "case refine_2\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : RegularSpace X\ninst✝ : LocallyCompactSpace X\ns t : Set X\nhs : IsCompact s\nh's : IsGδ s\nht : IsClosed[inst✝²] t\nhd : Disjoint s t\nU : ℕ → Set X\nU_open : ∀ (n : ℕ), IsOpen[inst✝²] (U n)\nhU : s = ⋂ n, U n\nm : Set X\nm_comp : IsCo... | apply Subset.antisymm (fun x hx ↦ by simp [g, fs _ hx, hu]) ?_
apply compl_subset_compl.1
intro x hx
obtain ⟨n, hn⟩ : ∃ n, x ∉ U n := by simpa [hU] using hx
have fnx : f n x = 0 := fm _ (by simp [hn])
have : g x < 1 := by
apply lt_of_lt_of_le ?_ hu.le
exact (S x).tsum_lt_tsum (i := n) (f... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 416,
"column": 2
} | {
"line": 419,
"column": 52
} | [
{
"pp": "case refine_2\nX : Type u_1\n𝕜 : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : CompactSpace X\ninst✝¹ : T2Space X\ninst✝ : RCLike 𝕜\nφ : ↑(characterSpace 𝕜 C(X, 𝕜))\n⊢ ∃ a, (continuousMapEval X 𝕜) a = φ",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedCo... | · obtain ⟨x, hx⟩ := (ideal_isMaximal_iff (RingHom.ker φ)).mp inferInstance
refine ⟨x, CharacterSpace.ext_ker <| Ideal.ext fun f => ?_⟩
simpa only [RingHom.mem_ker, continuousMapEval_apply_apply, mem_idealOfSet_compl_singleton,
RingHom.mem_ker] using SetLike.ext_iff.mp hx f | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Basic | {
"line": 394,
"column": 8
} | {
"line": 394,
"column": 29
} | [
{
"pp": "case a\nA : Type u_1\ninst✝ : NonUnitalCStarAlgebra A\nx y : A\nhxy : IsSelfAdjoint (↑y - ↑x) ∧ SpectrumRestricts (↑y - ↑x) ⇑ContinuousMap.realToNNReal\nhyx : IsSelfAdjoint (↑x - ↑y) ∧ SpectrumRestricts (↑x - ↑y) ⇑ContinuousMap.realToNNReal\n⊢ ↑(x - y) = ↑0",
"usedConstants": [
"AddGroup.toSu... | Unitization.inr_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 98,
"column": 4
} | {
"line": 98,
"column": 50
} | [
{
"pp": "case pos\nX : Type u_1\nR : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : MetricSpace R\ninst✝⁶ : IsTopologicalSemiring R\ninst✝⁵ : ContinuousStar R\ninst✝⁴ : Ring A\ninst✝³ : StarRing A\ninst✝² : TopologicalSpace A\ninst✝¹ : Algebra R A\ninst✝ : Continuou... | apply cfcHom_continuous _ |>.tendsto _ |>.comp | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 196,
"column": 6
} | {
"line": 196,
"column": 17
} | [
{
"pp": "A : Type u_1\ninst✝²⁵ : PartialOrder A\ninst✝²⁴ : NonUnitalRing A\ninst✝²³ : TopologicalSpace A\ninst✝²² : StarRing A\ninst✝²¹ : Module ℝ A\ninst✝²⁰ : SMulCommClass ℝ A A\ninst✝¹⁹ : IsScalarTower ℝ A A\ninst✝¹⁸ : StarOrderedRing A\ninst✝¹⁷ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst... | Prod.le_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 589,
"column": 6
} | {
"line": 589,
"column": 17
} | [
{
"pp": "A : Type u_1\ninst✝²² : PartialOrder A\ninst✝²¹ : Ring A\ninst✝²⁰ : StarRing A\ninst✝¹⁹ : TopologicalSpace A\ninst✝¹⁸ : StarOrderedRing A\ninst✝¹⁷ : Algebra ℝ A\ninst✝¹⁶ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹⁵ : NonnegSpectrumClass ℝ A\ninst✝¹⁴ : IsSemitopologicalRing A\ninst✝¹³ : T2S... | Prod.le_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 708,
"column": 71
} | {
"line": 708,
"column": 89
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : PartialOrder A\ninst✝⁸ : Ring A\ninst✝⁷ : StarRing A\ninst✝⁶ : TopologicalSpace A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : Algebra ℝ A\ninst✝³ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝² : NonnegSpectrumClass ℝ A\ninst✝¹ : IsSemitopologicalRing A\ninst✝ : T2Space A\na :... | one_div_mul_eq_div | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 739,
"column": 2
} | {
"line": 739,
"column": 22
} | [
{
"pp": "𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NonUnitalNormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : NormedSpace 𝕜 A\ninst✝³ : IsScalarTower 𝕜 A A\ninst✝² : SMulCommClass 𝕜 A A\ninst✝¹ : ContinuousStar A\ninst✝ : NonUnitalIsometricContinuousFunctionalCalculus 𝕜 A p\ns : Set ... | by_cases hs0 : 0 ∈ s | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.CStarAlgebra.ApproximateUnit | {
"line": 60,
"column": 2
} | {
"line": 65,
"column": 98
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na b : A\nhab : ↑a ≤ ↑b\nha : 0 ≤ ↑a\nhb : 0 ≤ ↑b\n⊢ cfc (fun x ↦ 1 - (1 + x)⁻¹) ↑a ≤ cfc (fun x ↦ 1 - (1 + x)⁻¹) ↑b",
"usedConstants": [
"NNReal.instTopologicalSpace",
"Iff.mpr",
"z... | have h_cfc_one_sub (c : A⁺¹) (hc : 0 ≤ c := by cfc_tac) :
cfc (fun x : ℝ≥0 ↦ 1 - (1 + x)⁻¹) c = 1 - cfc (·⁻¹ : ℝ≥0 → ℝ≥0) (1 + c) := by
rw [cfc_tsub _ _ _ (fun x _ ↦ by simp) (hg := by fun_prop (disch := intro _ _; positivity)),
cfc_const_one ℝ≥0 c, cfc_comp' (·⁻¹) (1 + ·) c ?_, cfc_add .., cfc_const_on... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 244,
"column": 30
} | {
"line": 244,
"column": 48
} | [
{
"pp": "A : Type u_1\ninst✝⁷ : NonUnitalCStarAlgebra A\ninst✝⁶ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁵ : Fintype ι\ninst✝⁴ : (i : ι) → NormedAddCommGroup (E i)\ninst✝³ : (i : ι) → Module ℂ (E i)\ninst✝² : (i : ι) → SMul A (E i)\ninst✝¹ : (i : ι) → CStarModule A (E i)\ninst✝ : StarOrderedRing A... | by simp [smul_sum] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 512,
"column": 53
} | {
"line": 512,
"column": 76
} | [
{
"pp": "case h\nA : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nf : ℝ → ℝ\ns : Set A\nhf : ConvexOn ℝ (inr '' s) (cfc f)\ninrl : A →ₗ[ℝ] Unitization ℂ A := inrHom ℝ ℂ A\nhf₀ : f 0 = 0\nh₁ : inr ∘ cfcₙ f = fun x ↦ ↑(cfcₙ f x)\nx✝ : A\n⊢ ↑(cfcₙ f x✝) = cfc f ↑x... | real_cfcₙ_eq_cfc_inr .. | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 274,
"column": 6
} | {
"line": 274,
"column": 17
} | [
{
"pp": "A : Type u_1\ninst✝⁷ : NonUnitalCStarAlgebra A\ninst✝⁶ : PartialOrder A\nι : Type u_2\nE : ι → Type u_3\ninst✝⁵ : Fintype ι\ninst✝⁴ : (i : ι) → NormedAddCommGroup (E i)\ninst✝³ : (i : ι) → Module ℂ (E i)\ninst✝² : (i : ι) → SMul A (E i)\ninst✝¹ : (i : ι) → CStarModule A (E i)\ninst✝ : StarOrderedRing A... | pi_norm_sq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ApproximateUnit | {
"line": 177,
"column": 2
} | {
"line": 177,
"column": 80
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nl : Filter A\nh : ∀ (m : A), 0 ≤ m → ‖m‖ < 1 → Tendsto (fun x ↦ x * m) l (𝓝 m)\nn : ℕ\nc : Fin n → ℂ\nx : Fin n → ↑({x | 0 ≤ x} ∩ ball 0 1)\ni : Fin n\nx✝ : i ∈ Finset.univ\n⊢ Tendsto (fun x_1 ↦ c i • (... | exact tendsto_const_nhds.smul <| h (x i) (x i).2.1 <| by simpa using (x i).2.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Matrix.Normed | {
"line": 638,
"column": 6
} | {
"line": 638,
"column": 24
} | [
{
"pp": "l : Type u_2\nm : Type u_3\nn : Type u_4\nα : Type u_5\ninst✝³ : Fintype l\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : RCLike α\nA : Matrix l m α\nB : Matrix m n α\n⊢ (∑ x, ∑ x_1, ‖∑ j, A x j * B j x_1‖₊ ^ 2) ^ (1 / 2) ≤\n (∑ i, ∑ j, ‖A i j‖₊ ^ 2) ^ (1 / 2) * (∑ i, ∑ j, ‖B i j‖₊ ^ 2) ^ (1 / 2)"... | ← NNReal.mul_rpow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.CompletelyPositiveMap | {
"line": 99,
"column": 2
} | {
"line": 99,
"column": 14
} | [
{
"pp": "F : Type u_1\nA₁ : Type u_2\nA₂ : Type u_3\ninst✝⁷ : NonUnitalCStarAlgebra A₁\ninst✝⁶ : NonUnitalCStarAlgebra A₂\ninst✝⁵ : PartialOrder A₁\ninst✝⁴ : PartialOrder A₂\ninst✝³ : StarOrderedRing A₁\ninst✝² : StarOrderedRing A₂\ninst✝¹ : FunLike F A₁ A₂\ninst✝ : LinearMapClass F ℂ A₁ A₂\nh : ∀ (φ : F) (k : ... | intro φ a ha | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 77,
"column": 57
} | {
"line": 79,
"column": 5
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nA : Type u_5\nM : CStarMatrix m n A\n⊢ M.map id = M",
"usedConstants": [
"id",
"Eq.refl",
"CStarMatrix.map",
"CStarMatrix.ext"
]
}
] | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 471,
"column": 20
} | {
"line": 471,
"column": 53
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\nS : Type u_4\nA : Type u_5\nB : Type u_6\ninst✝⁵ : Unique n\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid A\ninst✝² : Mul A\ninst✝¹ : Star A\ninst✝ : Module R A\nx✝ : A\n⊢ (fun x y ↦ star x✝) = star fun x y ↦ x✝",
"usedConstants": [
"CStarMatrix.instSt... | ext; simp [star_eq_conjTranspose] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 471,
"column": 20
} | {
"line": 471,
"column": 53
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nR : Type u_3\nS : Type u_4\nA : Type u_5\nB : Type u_6\ninst✝⁵ : Unique n\ninst✝⁴ : Semiring R\ninst✝³ : AddCommMonoid A\ninst✝² : Mul A\ninst✝¹ : Star A\ninst✝ : Module R A\nx✝ : A\n⊢ (fun x y ↦ star x✝) = star fun x y ↦ x✝",
"usedConstants": [
"CStarMatrix.instSt... | ext; simp [star_eq_conjTranspose] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Dual | {
"line": 241,
"column": 2
} | {
"line": 243,
"column": 40
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\ninst✝² : Fintype ι\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nb : OrthonormalBasis ι ℝ E\nL : StrongDual ℝ E\n⊢ ‖L‖ ^ 2 = ∑ i, L (b i) ^ 2",
"usedConstants": [
"LinearIsometryEquiv.instEquivLike",
"Norm.norm",
"Eq.mpr",
"InnerP... | have := b.toBasis.finiteDimensional_of_finite
simp_rw [← (InnerProductSpace.toDual ℝ E).symm.norm_map, ← b.sum_sq_inner_left,
InnerProductSpace.toDual_symm_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Dual | {
"line": 241,
"column": 2
} | {
"line": 243,
"column": 40
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\ninst✝² : Fintype ι\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nb : OrthonormalBasis ι ℝ E\nL : StrongDual ℝ E\n⊢ ‖L‖ ^ 2 = ∑ i, L (b i) ^ 2",
"usedConstants": [
"LinearIsometryEquiv.instEquivLike",
"Norm.norm",
"Eq.mpr",
"InnerP... | have := b.toBasis.finiteDimensional_of_finite
simp_rw [← (InnerProductSpace.toDual ℝ E).symm.norm_map, ← b.sum_sq_inner_left,
InnerProductSpace.toDual_symm_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Extreme | {
"line": 205,
"column": 4
} | {
"line": 205,
"column": 35
} | [
{
"pp": "case h.refine_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : Semiring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns : Set E\nt : Set F\nx : E\ny : F\nhx : (x, y) ∈ s ×ˢ t\nh : ∀ ⦃x₁ : E × F⦄, x₁ ∈ s ×ˢ t → ∀ ⦃x₂ : E ... | exact mem_image_of_mem _ hx_fst | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.CStarAlgebra.Hom | {
"line": 37,
"column": 2
} | {
"line": 41,
"column": 46
} | [
{
"pp": "F : Type u_1\nA : Type u_2\nB : Type u_3\ninst✝⁴ : CStarAlgebra A\ninst✝³ : CStarAlgebra B\ninst✝² : FunLike F A B\ninst✝¹ : AlgHomClass F ℂ A B\ninst✝ : StarHomClass F A B\na : A\nha : IsSelfAdjoint a\nφ : F\nhφ : Function.Injective ⇑φ\nh_spec : spectrum ℝ ((StarAlgHom.restrictScalars ℝ ↑φ) a) ⊆ spect... | suffices φ (cfc f a) = 0 by
rw [map_eq_zero_iff φ hφ, ← cfc_zero ℝ a, cfc_eq_cfc_iff_eqOn] at this
exact zero_ne_one <| calc
0 = f x := (this hx).symm
_ = 1 := h_eqOn_x <| Set.mem_singleton x | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Topology.OpenPartialHomeomorph.Constructions | {
"line": 202,
"column": 9
} | {
"line": 204,
"column": 30
} | [
{
"pp": "X : Type u_1\nX' : Type u_2\nY : Type u_3\nY' : Type u_4\nZ : Type u_5\nZ' : Type u_6\ninst✝⁷ : TopologicalSpace X\ninst✝⁶ : TopologicalSpace X'\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : TopologicalSpace Y'\ninst✝³ : TopologicalSpace Z\ninst✝² : TopologicalSpace Z'\ne✝ e e' : OpenPartialHomeomorph X Y\nin... | by
rw [e.open_source.inter_frontier_eq]
exact eqOn_empty _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.OpenPartialHomeomorph.Constructions | {
"line": 310,
"column": 4
} | {
"line": 311,
"column": 66
} | [
{
"pp": "X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\nU : Opens X\nhU : Nonempty ↥U\ny : Y\nhy : y ∈ (e.subtypeRestr hU).target\n⊢ ↑e ((Subtype.val ∘ ↑(e.subtypeRestr hU).symm) y) = y",
"usedConstants": [
"Eq.mpr",
"OpenParti... | change restrict _ e _ = _
rw [← e.subtypeRestr_coe hU, (e.subtypeRestr hU).right_inv hy] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.OpenPartialHomeomorph.Constructions | {
"line": 310,
"column": 4
} | {
"line": 311,
"column": 66
} | [
{
"pp": "X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\nU : Opens X\nhU : Nonempty ↥U\ny : Y\nhy : y ∈ (e.subtypeRestr hU).target\n⊢ ↑e ((Subtype.val ∘ ↑(e.subtypeRestr hU).symm) y) = y",
"usedConstants": [
"Eq.mpr",
"OpenParti... | change restrict _ e _ = _
rw [← e.subtypeRestr_coe hU, (e.subtypeRestr hU).right_inv hy] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.OpenPartialHomeomorph.Constructions | {
"line": 309,
"column": 2
} | {
"line": 313,
"column": 39
} | [
{
"pp": "X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\nU : Opens X\nhU : Nonempty ↥U\ny : Y\nhy : y ∈ (e.subtypeRestr hU).target\n⊢ (Subtype.val ∘ ↑(e.subtypeRestr hU).symm) y = ↑e.symm y",
"usedConstants": [
"Eq.mpr",
"OpenPa... | rw [e.eq_symm_apply _ hy.1]
· change restrict _ e _ = _
rw [← e.subtypeRestr_coe hU, (e.subtypeRestr hU).right_inv hy]
· have := OpenPartialHomeomorph.map_target _ hy
rwa [e.subtypeRestr_source] at this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.OpenPartialHomeomorph.Constructions | {
"line": 309,
"column": 2
} | {
"line": 313,
"column": 39
} | [
{
"pp": "X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\nU : Opens X\nhU : Nonempty ↥U\ny : Y\nhy : y ∈ (e.subtypeRestr hU).target\n⊢ (Subtype.val ∘ ↑(e.subtypeRestr hU).symm) y = ↑e.symm y",
"usedConstants": [
"Eq.mpr",
"OpenPa... | rw [e.eq_symm_apply _ hy.1]
· change restrict _ e _ = _
rw [← e.subtypeRestr_coe hU, (e.subtypeRestr hU).right_inv hy]
· have := OpenPartialHomeomorph.map_target _ hy
rwa [e.subtypeRestr_source] at this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.OpenPartialHomeomorph.Constructions | {
"line": 321,
"column": 36
} | {
"line": 332,
"column": 62
} | [
{
"pp": "X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\nU V : Opens X\nhU : Nonempty ↥U\nhV : Nonempty ↥V\nhUV : U ≤ V\n⊢ EqOn (↑(e.subtypeRestr hV).symm) (inclusion hUV ∘ ↑(e.subtypeRestr hU).symm) (e.subtypeRestr hU).target",
"usedConsta... | by
set i := Set.inclusion hUV
intro y hy
dsimp [OpenPartialHomeomorph.subtypeRestr_def] at hy ⊢
have hyV : e.symm y ∈ (V.openPartialHomeomorphSubtypeCoe hV).target := by
rw [Opens.openPartialHomeomorphSubtypeCoe_target] at hy ⊢
exact hUV hy.2
refine (V.openPartialHomeomorphSubtypeCoe hV).injOn ?_ triv... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 276,
"column": 27
} | {
"line": 278,
"column": 48
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ne✝ : Pretrivialization F proj\nx✝ : Z\ne' : Pretrivialization F TotalSpace.proj\nb : B\ny : E b\ne : Pretrivialization F proj\ns : Set B\ninst✝ : Nonempty (↑s → F → ↑(proj... | exact if h : (x.1.1, x.2) ∈ e.target then ⟨e.invFun (x.1, x.2), by
simpa only [mem_preimage, ← e.proj_toFun _ (e.map_target' h), e.right_inv' h] using x.1.2⟩
else Classical.arbitrary (s → F → _) x.1 x.2 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 276,
"column": 27
} | {
"line": 278,
"column": 48
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ne✝ : Pretrivialization F proj\nx✝ : Z\ne' : Pretrivialization F TotalSpace.proj\nb : B\ny : E b\ne : Pretrivialization F proj\ns : Set B\ninst✝ : Nonempty (↑s → F → ↑(proj... | exact if h : (x.1.1, x.2) ∈ e.target then ⟨e.invFun (x.1, x.2), by
simpa only [mem_preimage, ← e.proj_toFun _ (e.map_target' h), e.right_inv' h] using x.1.2⟩
else Classical.arbitrary (s → F → _) x.1 x.2 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 276,
"column": 27
} | {
"line": 278,
"column": 48
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ne✝ : Pretrivialization F proj\nx✝ : Z\ne' : Pretrivialization F TotalSpace.proj\nb : B\ny : E b\ne : Pretrivialization F proj\ns : Set B\ninst✝ : Nonempty (↑s → F → ↑(proj... | exact if h : (x.1.1, x.2) ∈ e.target then ⟨e.invFun (x.1, x.2), by
simpa only [mem_preimage, ← e.proj_toFun _ (e.map_target' h), e.right_inv' h] using x.1.2⟩
else Classical.arbitrary (s → F → _) x.1 x.2 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 285,
"column": 4
} | {
"line": 285,
"column": 19
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ne✝ : Pretrivialization F proj\nx✝ : Z\ne' : Pretrivialization F TotalSpace.proj\nb : B\ny : E b\ne : Pretrivialization F proj\ns : Set B\ninst✝ : Nonempty (↑s → F → ↑(proj... | rw [dif_pos hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 291,
"column": 30
} | {
"line": 291,
"column": 45
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝² : TopologicalSpace B\ninst✝¹ : TopologicalSpace F\nproj : Z → B\ne✝ : Pretrivialization F proj\nx✝ : Z\ne' : Pretrivialization F TotalSpace.proj\nb : B\ny : E b\ne : Pretrivialization F proj\ns : Set B\ninst✝ : Nonempty (↑s → F → ↑(proj... | rw [dif_pos hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.FiberBundle.Basic | {
"line": 530,
"column": 2
} | {
"line": 530,
"column": 40
} | [
{
"pp": "ι : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝¹ : TopologicalSpace B\ninst✝ : TopologicalSpace F\nZ : FiberBundleCore ι B F\ni : ι\np : B × F\n⊢ p ∈ (Z.localTrivAsPartialEquiv i).target ↔ p.1 ∈ Z.baseSet i",
"usedConstants": [
"FiberBundleCore.indexAt",
"Set.instSProd",
"Eq.mpr",... | rw [localTrivAsPartialEquiv, mem_prod] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 937,
"column": 4
} | {
"line": 943,
"column": 38
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace F\nproj : Z → B\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace (TotalSpace F E)\ne✝ : Trivialization F proj\nx : Z\ne'✝ : Trivialization F TotalSpace.proj\nb : B\ny : E b\ne e' : T... | rintro p (hp | hp')
· change (e.source.piecewise e e' p).1 = proj p
rw [piecewise_eq_of_mem, e.coe_fst] <;> exact hp
· change (e.source.piecewise e e' p).1 = proj p
rw [piecewise_eq_of_notMem, e'.coe_fst hp']
simp only [source_eq] at hp' ⊢
exact fun h => H.le_bot ⟨h, hp'⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 937,
"column": 4
} | {
"line": 943,
"column": 38
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝³ : TopologicalSpace B\ninst✝² : TopologicalSpace F\nproj : Z → B\ninst✝¹ : TopologicalSpace Z\ninst✝ : TopologicalSpace (TotalSpace F E)\ne✝ : Trivialization F proj\nx : Z\ne'✝ : Trivialization F TotalSpace.proj\nb : B\ny : E b\ne e' : T... | rintro p (hp | hp')
· change (e.source.piecewise e e' p).1 = proj p
rw [piecewise_eq_of_mem, e.coe_fst] <;> exact hp
· change (e.source.piecewise e e' p).1 = proj p
rw [piecewise_eq_of_notMem, e'.coe_fst hp']
simp only [source_eq] at hp' ⊢
exact fun h => H.le_bot ⟨h, hp'⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 986,
"column": 2
} | {
"line": 986,
"column": 44
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nZ : Type u_4\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace F\nproj : Z → B\ninst✝² : TopologicalSpace Z\nT : Trivialization F proj\nι : Type u_5\ninst✝¹ : TopologicalSpace ι\ninst✝ : LocallyCompactPair ι ↑T.baseSet\nγ : C(ι, ↑T.baseSet)\ni : ι\ne : ↑T.source\nh : p... | simp [clift, liftCM, ← h, lift_self, this] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.Unitary.Span | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 60
} | [
{
"pp": "A : Type u_1\ninst✝ : CStarAlgebra A\nu : Fin 4 → ↥(unitary A)\nc : Fin 4 → ℂ\nh : ∀ (i : Fin 4), ‖c i‖ ≤ ‖∑ i, c i • ↑(u i)‖ / 2\n⊢ ∑ i, c i • ↑(u i) ∈ span ℂ ↑(unitary A)",
"usedConstants": [
"Submodule",
"CStarAlgebra.toNonUnitalCStarAlgebra",
"Submodule.addSubmonoidClass",
... | exact sum_mem fun i _ ↦ smul_mem _ _ (subset_span (u i).2) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.AbsolutelyMonotone | {
"line": 83,
"column": 2
} | {
"line": 84,
"column": 70
} | [
{
"pp": "f : ℝ → ℝ\ns : Set ℝ\nhs : UniqueDiffOn ℝ s\nn : ℕ\nx : ℝ\nhx : x ∈ s\np : ℝ → FormalMultilinearSeries ℝ ℝ ℝ\nhp : HasFTaylorSeriesUpToOn ∞ f p s\nhp_nn : ∀ (n : ℕ) ⦃x : ℝ⦄, x ∈ s → 0 ≤ (p x n) fun x ↦ 1\n⊢ 0 ≤ iteratedDerivWithin n f s x",
"usedConstants": [
"Real",
"Semiring.toModule"... | have heq : p x n = iteratedFDerivWithin ℝ n f s x :=
hp.eq_iteratedFDerivWithin_of_uniqueDiffOn (mod_cast le_top) hs hx | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 113,
"column": 47
} | {
"line": 119,
"column": 16
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\ns₁ s₂ : AffineSubspace k P\ninst✝¹ : FiniteDimensional k ↥s₁.direction\ninst✝ : FiniteDimensional k ↥s₂.direction\n⊢ FiniteDimensional k ↥(s₁ ⊔ s₂).di... | by
rcases eq_bot_or_nonempty s₁ with rfl | ⟨p₁, hp₁⟩
· rwa [bot_sup_eq]
rcases eq_bot_or_nonempty s₂ with rfl | ⟨p₂, hp₂⟩
· rwa [sup_bot_eq]
rw [AffineSubspace.direction_sup hp₁ hp₂]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 295,
"column": 4
} | {
"line": 295,
"column": 44
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : CharZero k\ns : Simplex k P n\ni : Fin (n + 1)\n⊢ (↑n + 1) • (s.centroid -ᵥ s.points i) = ∑ x, (s.points x -ᵥ s.points i)",
"u... | smul_centroid_vsub_point_eq_sum_vsub s i | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 132,
"column": 56
} | {
"line": 145,
"column": 86
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : DecidableEq P\np : ι → P\nhi : AffineIndependent k p\ns : Finset ι\nn : ℕ\nhc : #s = n + 1\n⊢ finrank k ↥(vectorSpan k ↑(Finset.image p s)) = ... | by
classical
have hi' := hi.range.mono (Set.image_subset_range p ↑s)
have hc' : #(s.image p) = n + 1 := by rwa [s.card_image_of_injective hi.injective]
have hn : (s.image p).Nonempty := by simp [hc', ← Finset.card_pos]
rcases hn with ⟨p₁, hp₁⟩
have hp₁' : p₁ ∈ p '' s := by simpa using hp₁
rw [affineIndepe... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 303,
"column": 49
} | {
"line": 303,
"column": 91
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : CharZero k\ns : Simplex k P n\ni j : Fin (n + 1)\n⊢ ((↑n)⁻¹ • ∑ x, (s.points x -ᵥ s.points i) +ᵥ s.points i) -ᵥ s.faceOppositeCent... | faceOppositeCentroid_eq_sum_vsub_vadd s j, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 211,
"column": 2
} | {
"line": 213,
"column": 69
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : Fintype ι\np : ι → P\nn : ℕ\nhc : Fintype.card ι = n + 1\ni₁ : ι\n⊢ AffineIndependent k p ↔ finrank k ↥(vectorSpan k (Set.range p)) = n",
... | rw [affineIndependent_iff_linearIndependent_vsub _ _ i₁,
linearIndependent_iff_card_eq_finrank_span, eq_comm,
vectorSpan_range_eq_span_range_vsub_right_ne k p i₁, Set.finrank] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 55
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nA : E →L[𝕜] F\nx : F\n⊢ ‖(adjointAux A) x‖ ≤ ... | rw [adjointAux_apply, LinearIsometryEquiv.norm_map] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 55
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nA : E →L[𝕜] F\nx : E\n⊢ ‖(adjointAux (adjoint... | rw [adjointAux_apply, LinearIsometryEquiv.norm_map] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 577,
"column": 2
} | {
"line": 580,
"column": 53
} | [
{
"pp": "case h\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : CharZero k\ns : Simplex k P n\nhmem1 : s.medial.points 0 ∈ affineSpan k (Set.range s.medial.points)\nhmem2 : s.medial.poin... | suffices (∃ a b, (n : k)⁻¹ • (s.points b -ᵥ s.points a) = v) ↔
∃ a b, -((n : k)⁻¹ • (s.points a -ᵥ s.points b)) = v by
simpa [Set.mem_vsub, Set.mem_smul_set, medial_points,
faceOppositeCentroid_vsub_faceOppositeCentroid] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.Normed.Affine.AddTorsorBases | {
"line": 105,
"column": 20
} | {
"line": 105,
"column": 30
} | [
{
"pp": "case neg\nV : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Se... | exact hf p | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 788,
"column": 2
} | {
"line": 788,
"column": 39
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\nh : finrank k V = 2\nthis : FiniteDimensional k V\n⊢ Coplanar k s",
"usedConstants": [
"Eq.mpr",
"Submodule",
"vectorSpan",
"c... | rw [coplanar_iff_finrank_le_two, ← h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Group.Integral | {
"line": 94,
"column": 28
} | {
"line": 94,
"column": 48
} | [
{
"pp": "G : Type u_4\nE : Type u_5\ninst✝⁵ : MeasurableSpace G\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nμ : Measure G\ninst✝² : Group G\ninst✝¹ : MeasurableMul G\ninst✝ : μ.IsMulLeftInvariant\nf : G → E\ng : G\nh_mul : MeasurableEmbedding fun x ↦ g * x\n⊢ ∫ (y : G), f y ∂map (fun x ↦ g * x) μ ... | map_mul_left_eq_self | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 434,
"column": 2
} | {
"line": 435,
"column": 32
} | [
{
"pp": "α : Type u_3\nE : α → Type u_4\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\ninst✝ : Nonempty α\nf : ↥(lp E ∞)\n⊢ IsLUB (Set.range fun i ↦ ‖↑f i‖) ‖f‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"iSup",
"lp.instNormSubtype... | rw [lp.norm_eq_ciSup]
exact isLUB_ciSup (lp.memℓp f) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 434,
"column": 2
} | {
"line": 435,
"column": 32
} | [
{
"pp": "α : Type u_3\nE : α → Type u_4\ninst✝¹ : (i : α) → NormedAddCommGroup (E i)\ninst✝ : Nonempty α\nf : ↥(lp E ∞)\n⊢ IsLUB (Set.range fun i ↦ ‖↑f i‖) ‖f‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"congrArg",
"iSup",
"lp.instNormSubtype... | rw [lp.norm_eq_ciSup]
exact isLUB_ciSup (lp.memℓp f) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 498,
"column": 4
} | {
"line": 498,
"column": 85
} | [
{
"pp": "case inr.inl.inr\nα : Type u_3\nE : α → Type u_4\ninst✝ : (i : α) → NormedAddCommGroup (E i)\nf : ↥(lp E ∞)\nh : ‖f‖ = 0\n_i : Nonempty α\n⊢ f = 0",
"usedConstants": [
"lp.isLUB_norm",
"Norm.norm",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
"lp.ins... | have H : IsLUB (Set.range fun i => ‖f i‖) 0 := by simpa [h] using lp.isLUB_norm f | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.UniformLimitsDeriv | {
"line": 291,
"column": 27
} | {
"line": 291,
"column": 48
} | [
{
"pp": "case pos\nι : Type u_1\nl : Filter ι\nE : Type u_5\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_6\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_7\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\nhf' : Te... | simp [hyz', hqpos.le] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.UniformLimitsDeriv | {
"line": 291,
"column": 27
} | {
"line": 291,
"column": 48
} | [
{
"pp": "case pos\nι : Type u_1\nl : Filter ι\nE : Type u_5\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_6\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_7\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\nhf' : Te... | simp [hyz', hqpos.le] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.UniformLimitsDeriv | {
"line": 291,
"column": 27
} | {
"line": 291,
"column": 48
} | [
{
"pp": "case pos\nι : Type u_1\nl : Filter ι\nE : Type u_5\ninst✝⁴ : NormedAddCommGroup E\n𝕜 : Type u_6\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\nG : Type u_7\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ι → E → G\ng : E → G\nf' : ι → E → E →L[𝕜] G\ng' : E → E →L[𝕜] G\nx : E\nhf' : Te... | simp [hyz', hqpos.le] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.lpSpace | {
"line": 710,
"column": 8
} | {
"line": 710,
"column": 26
} | [
{
"pp": "case inr.inr\n𝕜 : Type u_1\nα : Type u_3\nE : α → Type u_4\np : ℝ≥0∞\ninst✝³ : (i : α) → NormedAddCommGroup (E i)\ninst✝² : NormedRing 𝕜\ninst✝¹ : (i : α) → Module 𝕜 (E i)\ninst✝ : ∀ (i : α), IsBoundedSMul 𝕜 (E i)\nhp✝ : p ≠ 0\nc : 𝕜\nf : ↥(lp E p)\nhp : 0 < p.toReal\ninst : NNNorm ↥(lp E p)\ncoe_... | ← NNReal.mul_rpow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.EuclideanDist | {
"line": 119,
"column": 2
} | {
"line": 119,
"column": 23
} | [
{
"pp": "F : Type u_2\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\nG : Type u_3\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace ℝ G\ninst✝ : FiniteDimensional ℝ G\nf g : F → G\nn : ℕ∞\nhf : ContDiff ℝ (↑n) f\nhg : ContDiff ℝ (↑n) g\nh : ∀ (x : F), f x ≠ g x\n⊢ ContDiff ℝ ↑n fun x ↦ Dist.dist (... | apply ContDiff.dist ℝ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.MeasureTheory.Measure.EverywherePos | {
"line": 293,
"column": 4
} | {
"line": 296,
"column": 26
} | [
{
"pp": "case refine_1\nG : Type u_2\ninst✝⁸ : Group G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : IsTopologicalGroup G\ninst✝⁵ : LocallyCompactSpace G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsMulLeftInvariant\ninst✝¹ : IsFiniteMeasureOnCompacts μ\ninst✝ : μ.InnerRegularCompact... | obtain ⟨⟨f, f_cont⟩, Lf, -, f_comp, -⟩ : ∃ f : C(G, ℝ), L = f ⁻¹' {1} ∧ EqOn f 0 ∅
∧ HasCompactSupport f ∧ ∀ x, f x ∈ Icc (0 : ℝ) 1 :=
exists_continuous_one_zero_of_isCompact_of_isGδ L_comp L_Gδ isClosed_empty
(disjoint_empty L) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Convolution | {
"line": 440,
"column": 7
} | {
"line": 440,
"column": 79
} | [
{
"pp": "case h\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedAddCommGroup E'\ninst✝⁸ : NormedAddCommGroup F\nf : G → E\ng : G → E'\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : NormedSpace 𝕜 E'\ninst✝⁴ : NormedS... | simp only [Pi.smul_apply, convolution_def, ← integral_smul, L.map_smul₂] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | {
"line": 239,
"column": 4
} | {
"line": 244,
"column": 44
} | [
{
"pp": "case refine_3.refine_2\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nA : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball 0 1)\nf : E → ℝ\nf_support : support f = ball 0 1\nf_smooth : ContDiff ℝ ∞ f\nf_range : range f ⊆ Icc 0 1\n... | · have I1 : x ∉ support f := by rwa [f_support]
have I2 : -x ∉ support f := by
rw [f_support]
simpa using hx
simp only [mem_support, Classical.not_not] at I1 I2
simp only [I1, I2, add_zero, zero_div] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 13
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nD : ℝ\n⊢ w D = fun x ↦ ((∫ (x : E), u x ∂μ) * |D| ^ finrank ℝ E)⁻¹ • u (D⁻¹ • x)",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
... | ext1 x; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 13
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nD : ℝ\n⊢ w D = fun x ↦ ((∫ (x : E), u x ∂μ) * |D| ^ finrank ℝ E)⁻¹ • u (D⁻¹ • x)",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
... | ext1 x; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | {
"line": 326,
"column": 47
} | {
"line": 326,
"column": 76
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nD : ℝ\nDpos : 0 < D\n⊢ IsCompact (closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (ball 0 D))",
"usedConstants": [
"Eq.mpr",
... | closure_ball (0 : E) Dpos.ne' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.BumpFunction.FiniteDimension | {
"line": 513,
"column": 8
} | {
"line": 513,
"column": 31
} | [
{
"pp": "E✝ : Type u_1\ninst✝⁵ : NormedAddCommGroup E✝\ninst✝⁴ : NormedSpace ℝ E✝\ninst✝³ : FiniteDimensional ℝ E✝\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nthis✝¹ : MeasurableSpace E := borel E\nthis✝ : BorelSpace E\nIR : ∀ (R : ℝ), 1 < R → 0 < (R - ... | simp only [hR, if_true] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.ContDiff.Convolution | {
"line": 273,
"column": 4
} | {
"line": 273,
"column": 21
} | [
{
"pp": "case h\n𝕜 : Type u𝕜\nG : Type uG\nE : Type uE\nE' : Type uE'\nF : Type uF\nP : Type uP\ninst✝¹³ : NormedAddCommGroup E\ninst✝¹² : NormedAddCommGroup E'\ninst✝¹¹ : NormedAddCommGroup F\nf : G → E\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : NormedSpace 𝕜 E\ninst✝⁸ : NormedSpace 𝕜 E'\ninst✝⁷ : NormedSpace ℝ F\nins... | exact Z.comp x Z' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.BumpFunction.Normed | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 31
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : HasContDiffBump E\ninst✝ : MeasurableSpace E\nμ : Measure E\nf : ContDiffBump 0\nx : E\n⊢ f.normed μ (-x) = f.normed μ x",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NegZeroClass.... | simp_rw [f.normed_def, f.neg] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Calculus.BumpFunction.Normed | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 31
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : HasContDiffBump E\ninst✝ : MeasurableSpace E\nμ : Measure E\nf : ContDiffBump 0\nx : E\n⊢ f.normed μ (-x) = f.normed μ x",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NegZeroClass.... | simp_rw [f.normed_def, f.neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.BumpFunction.Normed | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 31
} | [
{
"pp": "E : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : HasContDiffBump E\ninst✝ : MeasurableSpace E\nμ : Measure E\nf : ContDiffBump 0\nx : E\n⊢ f.normed μ (-x) = f.normed μ x",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NegZeroClass.... | simp_rw [f.normed_def, f.neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 644,
"column": 10
} | {
"line": 645,
"column": 52
} | [
{
"pp": "case hm\nα : Type u_1\ninst✝⁴ : MetricSpace α\ninst✝³ : SecondCountableTopology α\ninst✝² : MeasurableSpace α\ninst✝¹ : OpensMeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nN : ℕ\nτ : ℝ\nhτ : 1 < τ\nhN : IsEmpty (SatelliteConfig α N τ)\ns : Set α\nr : α → ℝ\nrpos : ∀ x ∈ s, 0 < r x\nrle : ... | intro b _
apply omeas.inter measurableSet_closedBall | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Covering.Besicovitch | {
"line": 644,
"column": 10
} | {
"line": 645,
"column": 52
} | [
{
"pp": "case hm\nα : Type u_1\ninst✝⁴ : MetricSpace α\ninst✝³ : SecondCountableTopology α\ninst✝² : MeasurableSpace α\ninst✝¹ : OpensMeasurableSpace α\nμ : Measure α\ninst✝ : IsFiniteMeasure μ\nN : ℕ\nτ : ℝ\nhτ : 1 < τ\nhN : IsEmpty (SatelliteConfig α N τ)\ns : Set α\nr : α → ℝ\nrpos : ∀ x ∈ s, 0 < r x\nrle : ... | intro b _
apply omeas.inter measurableSet_closedBall | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 808,
"column": 6
} | {
"line": 808,
"column": 38
} | [
{
"pp": "G : Type u_1\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : Group G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\ninst✝² : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝¹ : μ.IsHaarMeasure\ninst✝ : μ'.IsHaarMeasure\ns : Set G\nhs : MeasurableSet s\nh's : μ.IsEverywherePos s... | refine ⟨t, t_mem, fun u hu ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.MeasureTheory.Covering.BesicovitchVectorSpace | {
"line": 463,
"column": 2
} | {
"line": 463,
"column": 47
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nN : ℕ\nτ : ℝ\na : SatelliteConfig E N τ\nlastc : a.c (last N) = 0\nlastr : a.r (last N) = 1\nhτ : 1 ≤ τ\nδ : ℝ\nhδ1 : τ ≤ 1 + δ / 4\nhδ2 : δ ≤ 1\nc' : Fin N.succ → E := fun i ↦ if ‖a.c i‖ ≤ 2 then a.c i else (2 / ‖a.c i‖) • a.c i\nno... | wlog hij : ‖a.c i‖ ≤ ‖a.c j‖ generalizing i j | Mathlib.Tactic._aux_Mathlib_Tactic_WLOG___elabRules_Mathlib_Tactic_wlog_1 | Mathlib.Tactic.wlog |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 865,
"column": 4
} | {
"line": 865,
"column": 81
} | [
{
"pp": "case a\nG : Type u_1\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : Group G\ninst✝⁸ : IsTopologicalGroup G\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : BorelSpace G\ninst✝⁵ : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝⁴ : μ.IsHaarMeasure\ninst✝³ : IsFiniteMeasureOnCompacts μ'\ninst✝² : μ'.IsMulLeftInvariant\ninst... | exact (measurableSet_toMeasurable _ _).inter (measurableSet_toMeasurable _ _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 865,
"column": 4
} | {
"line": 865,
"column": 81
} | [
{
"pp": "case a\nG : Type u_1\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : Group G\ninst✝⁸ : IsTopologicalGroup G\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : BorelSpace G\ninst✝⁵ : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝⁴ : μ.IsHaarMeasure\ninst✝³ : IsFiniteMeasureOnCompacts μ'\ninst✝² : μ'.IsMulLeftInvariant\ninst... | exact (measurableSet_toMeasurable _ _).inter (measurableSet_toMeasurable _ _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 865,
"column": 4
} | {
"line": 865,
"column": 81
} | [
{
"pp": "case a\nG : Type u_1\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : Group G\ninst✝⁸ : IsTopologicalGroup G\ninst✝⁷ : MeasurableSpace G\ninst✝⁶ : BorelSpace G\ninst✝⁵ : LocallyCompactSpace G\nμ' μ : Measure G\ninst✝⁴ : μ.IsHaarMeasure\ninst✝³ : IsFiniteMeasureOnCompacts μ'\ninst✝² : μ'.IsMulLeftInvariant\ninst... | exact (measurableSet_toMeasurable _ _).inter (measurableSet_toMeasurable _ _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.Unique | {
"line": 966,
"column": 2
} | {
"line": 966,
"column": 74
} | [
{
"pp": "case inv_eq_self\nG : Type u_1\ninst✝⁷ : CommGroup G\ninst✝⁶ : TopologicalSpace G\ninst✝⁵ : IsTopologicalGroup G\ninst✝⁴ : MeasurableSpace G\ninst✝³ : BorelSpace G\nμ : Measure G\ninst✝² : μ.IsHaarMeasure\ninst✝¹ : LocallyCompactSpace G\ninst✝ : μ.Regular\nc : ℝ≥0∞ := ↑(μ.inv.haarScalarFactor μ)\n⊢ μ.i... | have hc : μ.inv = c • μ := isMulLeftInvariant_eq_smul_of_regular μ.inv μ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.MetricSpace.Holder | {
"line": 425,
"column": 4
} | {
"line": 425,
"column": 81
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝⁴ : PseudoMetricSpace X\ninst✝³ : SeminormedAddCommGroup Y\nC r : ℝ≥0\nf : X → Y\nα : Type u_4\ninst✝² : SeminormedRing α\ninst✝¹ : Module α Y\ninst✝ : NormSMulClass α Y\na : α\nha : ‖a‖₊ ≠ 0\n⊢ (∀ (x y : X), ↑‖a‖₊ * edist (f x) (f y) ≤ ↑‖a‖₊ * (↑C * edist x y ^ ↑r)) ↔\... | ENNReal.mul_le_mul_iff_right (ENNReal.coe_ne_zero.mpr ha) ENNReal.coe_ne_top, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Calculus.FDeriv.Symmetric | {
"line": 110,
"column": 20
} | {
"line": 110,
"column": 24
} | [
{
"pp": "case h\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns t : Set E\nf : E → F\nx : E\nh : t ∈ 𝓝[s] x\nhf : ContDiffWithinAt 𝕜 2 f t x\nhs : UniqueDiffOn ... | h''y | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Calculus.FDeriv.Symmetric | {
"line": 166,
"column": 2
} | {
"line": 168,
"column": 56
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nhs : UniqueDiffOn 𝕜 s\nhx : x ∈ s\n⊢ IsSymmSndFDerivWithinAt 𝕜 f s x ↔\n C... | simp_rw [IsSymmSndFDerivWithinAt, ContinuousMultilinearMap.ext_iff, Fin.forall_fin_succ_pi,
Fin.forall_fin_zero_pi]
simp [iteratedFDerivWithin_two_apply f hs hx, eq_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Symmetric | {
"line": 166,
"column": 2
} | {
"line": 168,
"column": 56
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nhs : UniqueDiffOn 𝕜 s\nhx : x ∈ s\n⊢ IsSymmSndFDerivWithinAt 𝕜 f s x ↔\n C... | simp_rw [IsSymmSndFDerivWithinAt, ContinuousMultilinearMap.ext_iff, Fin.forall_fin_succ_pi,
Fin.forall_fin_zero_pi]
simp [iteratedFDerivWithin_two_apply f hs hx, eq_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.DifferentialForm.VectorField | {
"line": 177,
"column": 30
} | {
"line": 177,
"column": 44
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nx : E\nhsx : UniqueDiffWithinAt 𝕜 s x\nn : ℕ\nω : E → E [⋀^Fin (n + 1)]→L[𝕜] F\nV : Fin (n + 1 ... | simpa using hj | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.DifferentialForm.VectorField | {
"line": 177,
"column": 30
} | {
"line": 177,
"column": 44
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nx : E\nhsx : UniqueDiffWithinAt 𝕜 s x\nn : ℕ\nω : E → E [⋀^Fin (n + 1)]→L[𝕜] F\nV : Fin (n + 1 ... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.DifferentialForm.VectorField | {
"line": 177,
"column": 30
} | {
"line": 177,
"column": 44
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nx : E\nhsx : UniqueDiffWithinAt 𝕜 s x\nn : ℕ\nω : E → E [⋀^Fin (n + 1)]→L[𝕜] F\nV : Fin (n + 1 ... | simpa using hj | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Norm | {
"line": 97,
"column": 23
} | {
"line": 97,
"column": 24
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : StrongDual ℝ E\nx : E\nt : ℝ\nht : t ≠ 0\nh : HasStrictFDerivAt (fun x ↦ ‖x‖) f x\nh1 : HasStrictFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)\nh2 : HasStrictFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) x\n| Has... | 3 | Lean.Elab.Tactic.Conv.evalEnter | null |
Mathlib.Analysis.Calculus.FDeriv.Norm | {
"line": 121,
"column": 23
} | {
"line": 121,
"column": 24
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : StrongDual ℝ E\nx : E\nt : ℝ\nht : t ≠ 0\nh : HasFDerivAt (fun x ↦ ‖x‖) f x\nh1 : HasFDerivAt (fun y ↦ t⁻¹ • y) (t⁻¹ • ContinuousLinearMap.id ℝ E) (t • x)\nh2 : HasFDerivAt (fun y ↦ |t| * ‖y‖) (|t| • f) x\n| HasFDerivAt (fun y ↦ ... | 3 | Lean.Elab.Tactic.Conv.evalEnter | null |
Mathlib.Analysis.Calculus.Gradient.Basic | {
"line": 271,
"column": 6
} | {
"line": 271,
"column": 34
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : F → 𝕜\nx y : F\nh : DifferentiableAt 𝕜 f x\n⊢ ⟪∇ f x, y⟫ = (fderiv 𝕜 f x) y",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
... | h.hasGradientAt.fderiv_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Gradient.Basic | {
"line": 279,
"column": 25
} | {
"line": 279,
"column": 53
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : F → 𝕜\nx y : F\nh : DifferentiableAt 𝕜 f y\n⊢ (starRingEnd 𝕜) ⟪∇ f y, x⟫ = (starRingEnd 𝕜) ((fderiv 𝕜 f y) x)",
"usedConstants": [
"NormedCommRing... | h.hasGradientAt.fderiv_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 31
} | [
{
"pp": "case refine_4\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x... | · exact hu.mono (by simp [b]) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 120,
"column": 6
} | {
"line": 120,
"column": 31
} | [
{
"pp": "case inr\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf : MonotoneOn f s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\na : Set ℝ := {x | x ∈ s ∧ 𝓝[s ∩ Ioi x] x = ⊥} ∪ {x | x ∈ s ∧ 𝓝[s ∩ Iio x] x = ⊥}\na_count : a.Countable\ns₁ : Set ℝ := s \\ a\nhs₁ : MeasurableSet s₁\nu : Set ℝ := {c | ∃ x y, x... | · exact ⟨p, ps₁, h'p, hp⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 352,
"column": 4
} | {
"line": 352,
"column": 32
} | [
{
"pp": "case inl.hf\na b : ℝ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ f' x\nM : MonotoneOn f [[a, b]]\nhab : a ≤ b\n⊢ Contin... | · rwa [uIcc_of_le hab] at hf | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 677,
"column": 4
} | {
"line": 677,
"column": 66
} | [
{
"pp": "ι : Type u_1\nE : Type u_2\nμ : Measure ℝ\nl : Filter ι\ninst✝² : l.IsCountablyGenerated\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nb : ι → ℝ\nf : ℝ → E\nhb : Tendsto b l atTop\na : ℝ\nha : IntegrableOn f (Ioi a) μ\n⊢ ∀ᶠ (i : ι) in l, ∫ (x : ℝ) in Ioi a, f x ∂μ - ∫ (x : ℝ) in a..b i, f x ... | filter_upwards [hb.eventually_mem (Ici_mem_atTop a)] with i hi | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Function.JacobianOneDim | {
"line": 285,
"column": 4
} | {
"line": 286,
"column": 79
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\ns : Set ℝ\nf f' : ℝ → ℝ\nhs : MeasurableSet s\nhf' : ∀ x ∈ s, HasDerivWithinAt f (f' x) s x\nhf : MonotoneOn f s\ng : ℝ → F\nH : IntegrableOn g (f '' s) volume\nH' : IntegrableOn (fun x ↦ f' x • g (f x)) s volume\na b c : Set ℝ\nh_un... | have : ∫ x in b, f' x • g (f x) = 0 :=
setIntegral_eq_zero_of_forall_eq_zero (fun x hx ↦ by simp [deriv_b x hx]) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 382,
"column": 4
} | {
"line": 382,
"column": 32
} | [
{
"pp": "case inl.hf\na b : ℝ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ f' x\nM : MonotoneOn f [[a, b]]\nhab : a ≤ b\n⊢ Contin... | · rwa [uIcc_of_le hab] at hf | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntervalIntegral.IntegrationByParts | {
"line": 414,
"column": 4
} | {
"line": 414,
"column": 32
} | [
{
"pp": "case inl.hf\na b : ℝ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf f' : ℝ → ℝ\ng : ℝ → E\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt f (f' x) x\nhf' : ∀ x ∈ Ioo (min a b) (max a b), f' x ≤ 0\nM : AntitoneOn f [[a, b]]\nhab : a ≤ b\n⊢ Contin... | · rwa [uIcc_of_le hab] at hf | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 795,
"column": 6
} | {
"line": 795,
"column": 17
} | [
{
"pp": "case inl\nE : Type u_1\nf f' : ℝ → E\na : ℝ\nm : E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhcont : ContinuousWithinAt f (Ici a) a\nhderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Ioi a) volume\nhf : Tendsto f atTop (𝓝 m)\nhx : a ∈ Ici a\... | exact hcont | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 795,
"column": 6
} | {
"line": 795,
"column": 17
} | [
{
"pp": "case inl\nE : Type u_1\nf f' : ℝ → E\na : ℝ\nm : E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhcont : ContinuousWithinAt f (Ici a) a\nhderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Ioi a) volume\nhf : Tendsto f atTop (𝓝 m)\nhx : a ∈ Ici a\... | exact hcont | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 795,
"column": 6
} | {
"line": 795,
"column": 17
} | [
{
"pp": "case inl\nE : Type u_1\nf f' : ℝ → E\na : ℝ\nm : E\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nhcont : ContinuousWithinAt f (Ici a) a\nhderiv : ∀ x ∈ Ioi a, HasDerivAt f (f' x) x\nf'int : IntegrableOn f' (Ioi a) volume\nhf : Tendsto f atTop (𝓝 m)\nhx : a ∈ Ici a\... | exact hcont | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntegralEqImproper | {
"line": 841,
"column": 6
} | {
"line": 841,
"column": 17
} | [
{
"pp": "case inl\ng g' : ℝ → ℝ\na l : ℝ\nhcont : ContinuousWithinAt g (Ici a) a\nhderiv : ∀ x ∈ Ioi a, HasDerivAt g (g' x) x\ng'pos : ∀ x ∈ Ioi a, 0 ≤ g' x\nhg : Tendsto g atTop (𝓝 l)\nhx : a ∈ Ici a\n⊢ ContinuousWithinAt g (Ici a) a",
"usedConstants": []
}
] | exact hcont | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
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