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.Analysis.InnerProductSpace.Orthonormal | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 24
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nv w : ι → E\nhw : ∀ (i : ι), w i = v i ∨ w i = -v i\ni j : ι\nhi : w i = -v i\nhj : w j = -v j\nh : i = j\nhv : ⟪v i, v j⟫ = 1\n⊢ ⟪w i, w j⟫ = 1",
"usedConstan... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 231,
"column": 4
} | {
"line": 232,
"column": 24
} | [
{
"pp": "case neg\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nv w : ι → E\nhw : ∀ (i : ι), w i = v i ∨ w i = -v i\ni j : ι\nhi : w i = -v i\nhj : w j = -v j\nh : ¬i = j\nhv : ⟪v i, v j⟫ = 0\n⊢ ⟪w i, w j⟫ = 0",
"usedConsta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthonormal | {
"line": 259,
"column": 86
} | {
"line": 259,
"column": 97
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\ns : Set (Set E)\nhs : DirectedOn (fun x1 x2 ↦ x1 ⊆ x2) s\nh : ∀ a ∈ s, Orthonormal 𝕜 fun x ↦ ↑x\n⊢ ∀ (i : ↑s), Orthonormal 𝕜 fun x ↦ ↑x",
"usedConstants": [
"Eq.mpr",
"O... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Subspace | {
"line": 261,
"column": 35
} | {
"line": 261,
"column": 72
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nV : ι → Submodule 𝕜 E\nhV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ\nv : Π₀ (i : ι), ↥(V i)\nhv : ((DFinsupp.lsum ℕ) fun i ↦ (V i).subtype) v = 0\ni : ι\nt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Subspace | {
"line": 264,
"column": 6
} | {
"line": 264,
"column": 76
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_4\nV : ι → Submodule 𝕜 E\nhV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ\nv : Π₀ (i : ι), ↥(V i)\nhv : ((DFinsupp.lsum ℕ) fun i ↦ (V i).subtype) v = 0\ni : ι\n⊢... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Subspace | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nι : Type u_4\ninst✝ : DecidableEq ι\nV : ι → Submodule 𝕜 E\nhV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ\nhV_sum : IsInternal fun i ↦ V i\nα : ι → Type u_6\nv_family : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.LinearMap | {
"line": 393,
"column": 71
} | {
"line": 393,
"column": 82
} | [
{
"pp": "𝕜 : Type u_4\ninst✝⁴ : RCLike 𝕜\nF : Type u_8\nH : Type u_9\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace 𝕜 H\na c : F\nb d : H\nha : a ≠ 0\nhb : b ≠ 0\nh : ∀ (x : H), ⟪b, x⟫_𝕜 • a = ⟪d, x⟫_𝕜 • c\nh₂ : ∀ (x : F), ⟪a, x⟫_𝕜... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.ClosedSubmodule | {
"line": 99,
"column": 18
} | {
"line": 99,
"column": 29
} | [
{
"pp": "ι : Sort u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\nO : Type u_5\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : TopologicalSpace N\ninst✝³ : Module R N\ninst✝² : AddCommMonoid O\ninst✝¹ : TopologicalSpace O\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.ClosedSubmodule | {
"line": 116,
"column": 28
} | {
"line": 116,
"column": 39
} | [
{
"pp": "ι : Sort u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\nO : Type u_5\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : TopologicalSpace N\ninst✝³ : Module R N\ninst✝² : AddCommMonoid O\ninst✝¹ : TopologicalSpace O\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.ClosedSubmodule | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 80
} | [
{
"pp": "R : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : TopologicalSpace N\ninst✝² : Module R N\ninst✝¹ : ContinuousAdd N\ninst✝ : ContinuousConstSMul R N\nf : M →L[R] N\ns : ClosedSubm... | simp [map, Submodule.map_le_iff_le_comap]; simp [← toSubmodule_le_toSubmodule] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Module.ClosedSubmodule | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 80
} | [
{
"pp": "R : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝⁸ : Semiring R\ninst✝⁷ : AddCommMonoid M\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : Module R M\ninst✝⁴ : AddCommMonoid N\ninst✝³ : TopologicalSpace N\ninst✝² : Module R N\ninst✝¹ : ContinuousAdd N\ninst✝ : ContinuousConstSMul R N\nf : M →L[R] N\ns : ClosedSubm... | simp [map, Submodule.map_le_iff_le_comap]; simp [← toSubmodule_le_toSubmodule] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.ClosedSubmodule | {
"line": 315,
"column": 50
} | {
"line": 315,
"column": 61
} | [
{
"pp": "ι : Sort u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\nO : Type u_5\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : TopologicalSpace N\ninst✝³ : Module R N\ninst✝² : AddCommMonoid O\ninst✝¹ : TopologicalSpace O\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Module.ClosedSubmodule | {
"line": 316,
"column": 56
} | {
"line": 316,
"column": 67
} | [
{
"pp": "ι : Sort u_1\nR : Type u_2\nM : Type u_3\nN : Type u_4\nO : Type u_5\ninst✝⁹ : Semiring R\ninst✝⁸ : AddCommMonoid M\ninst✝⁷ : TopologicalSpace M\ninst✝⁶ : Module R M\ninst✝⁵ : AddCommMonoid N\ninst✝⁴ : TopologicalSpace N\ninst✝³ : Module R N\ninst✝² : AddCommMonoid O\ninst✝¹ : TopologicalSpace O\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.RCLike.Lemmas | {
"line": 48,
"column": 6
} | {
"line": 48,
"column": 30
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nthis : Module.rank ℝ ↥(Submodule.span ℝ {1, I}) ≤ ↑(#{1, I})\n⊢ Module.rank ℝ ↥(Submodule.span ℝ {1, I}) ≤ ↑(#{1, I})",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Submodule",
"Real... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 73,
"column": 39
} | {
"line": 73,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx y : E\n⊢ (starRingEnd 𝕜) ⟪T x, y⟫ = ⟪T y, x⟫",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Inner.inner",
... | hT x y, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 91,
"column": 33
} | {
"line": 91,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT S : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nhS : S.IsSymmetric\nx y : E\n⊢ ⟪T x, y⟫ + ⟪S x, y⟫ = ⟪x, (T + S) y⟫",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",... | hT x y, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 36
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_3\nT : ι → E →ₗ[𝕜] E\ns : Finset ι\nhT : ∀ i ∈ s, (T i).IsSymmetric\nx✝¹ x✝ : E\n⊢ ⟪(∑ i ∈ s, T i) x✝¹, x✝⟫ = ⟪x✝¹, (∑ i ∈ s, T i) x✝⟫",
"usedConstants": [
"Eq.mpr",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 101,
"column": 33
} | {
"line": 101,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT S : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nhS : S.IsSymmetric\nx y : E\n⊢ ⟪T x, y⟫ - ⟪S x, y⟫ = ⟪x, (T - S) y⟫",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
... | hT x y, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Orthogonal | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 13
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\nx✝ : E\n⊢ x✝ ∈ Kᗮ ↔ x✝ ∈ ⨅ v, (↑((innerSL 𝕜) ↑v)).ker",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.toNor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthogonal | {
"line": 132,
"column": 2
} | {
"line": 135,
"column": 5
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nK : Submodule 𝕜 E\nf : E ≃ₗᵢ[𝕜] F\n⊢ map (↑f.toLinearEquiv) Kᗮ = (map (↑f.toLinearEquiv) K)ᗮ",
"usedConstan... | refine (map_orthogonal K f.toLinearIsometry).trans ?_
have : f.toLinearIsometry.range = ⊤ := f.range
rw [this, inf_top_eq]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Orthogonal | {
"line": 132,
"column": 2
} | {
"line": 135,
"column": 5
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nK : Submodule 𝕜 E\nf : E ≃ₗᵢ[𝕜] F\n⊢ map (↑f.toLinearEquiv) Kᗮ = (map (↑f.toLinearEquiv) K)ᗮ",
"usedConstan... | refine (map_orthogonal K f.toLinearIsometry).trans ?_
have : f.toLinearIsometry.range = ⊤ := f.range
rw [this, inf_top_eq]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 223,
"column": 4
} | {
"line": 223,
"column": 55
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nF : Type u_3\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nT : E →ₗ[𝕜] E\nf : E ≃ₗᵢ[𝕜] F\nh : (↑f.toLinearEquiv ∘ₗ T ∘ₗ ↑f.symm.toLinearEquiv).IsSy... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 238,
"column": 39
} | {
"line": 238,
"column": 50
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E ≃ₗ[𝕜] E\nhT : (↑T).IsSymmetric\nx y : E\n⊢ ⟪↑T.symm x, y⟫ = ⟪x, ↑T.symm y⟫",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"LinearEquiv.symm",
"Inner.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthogonal | {
"line": 194,
"column": 6
} | {
"line": 195,
"column": 33
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\n⊢ K = ⊥ → Kᗮ = ⊤",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Submodule",
"AddCommGroup.toAddCommMonoid",
"NormedSpace.toModule",
... | rintro rfl
exact bot_orthogonal_eq_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Orthogonal | {
"line": 194,
"column": 6
} | {
"line": 195,
"column": 33
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\n⊢ K = ⊥ → Kᗮ = ⊤",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Submodule",
"AddCommGroup.toAddCommMonoid",
"NormedSpace.toModule",
... | rintro rfl
exact bot_orthogonal_eq_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 285,
"column": 44
} | {
"line": 285,
"column": 55
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nS T : E →ₗ[𝕜] E\nhS : S.IsSymmetric\nhT : T.IsSymmetric\nh : S.range ≤ T.range\nv : E\nhv : T v = 0\n⊢ ∃ y, T y = S (S v)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthogonal | {
"line": 370,
"column": 2
} | {
"line": 370,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nf : E ≃ₗᵢ[𝕜] F\nU V : Submodule 𝕜 E\nh : Submodule.map (↑↑↑f) U ⟂ Submodule.map (↑↑↑f) V\nhf : ∀ (p : Submodule... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 336,
"column": 4
} | {
"line": 336,
"column": 50
} | [
{
"pp": "case h.mp\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx : E\n⊢ x ∈ T.rangeᗮ → x ∈ T.ker",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Submodule",
"Ri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 372,
"column": 2
} | {
"line": 373,
"column": 33
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : f.IsSymmetric\nα : 𝕜\nh : (α • f).IsSymmetric\nhf' : ¬∀ (x v : E), ⟪v, f x⟫ = 0\n⊢ IsSelfAdjoint α",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Orthogonal | {
"line": 447,
"column": 2
} | {
"line": 447,
"column": 13
} | [
{
"pp": "𝕜 : Type u_4\nE : Type u_5\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nK : ClosedSubmodule 𝕜 E\nx : E\n⊢ x ∈ K ⊓ Kᗮ → x ∈ ⊥",
"usedConstants": [
"ClosedSubmodule.mem_inf._simp_1",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Inner.i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Reflection | {
"line": 156,
"column": 4
} | {
"line": 156,
"column": 26
} | [
{
"pp": "case a\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nv w : F\nh : ‖v‖ = ‖w‖\nR : F ≃ₗᵢ[ℝ] F := ⋯\nthis : R v + R v = w + w\n⊢ (fun x ↦ 2 • x) (R v) = (fun x ↦ 2 • x) w",
"usedConstants": [
"LinearIsometryEquiv.instEquivLike",
"Eq.mpr",
"InnerProductS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Submodule | {
"line": 63,
"column": 6
} | {
"line": 63,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\ny : E\nhy : y ∈ K\nz : E\nhz : z ∈ Kᗮ\nhv : y + z ∈ Kᗮᗮ\nhyz : ⟪z, y⟫ = 0\n⊢ z = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Submodule | {
"line": 71,
"column": 14
} | {
"line": 71,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nK₀ K₁ : Submodule 𝕜 E\ninst✝¹ : K₀.HasOrthogonalProjection\ninst✝ : K₁.HasOrthogonalProjection\nh : K₀ᗮ ≤ K₁ᗮ\n⊢ K₁ ≤ K₀",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Submodule | {
"line": 103,
"column": 14
} | {
"line": 103,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nK L : Submodule 𝕜 E\ninst✝¹ : K.HasOrthogonalProjection\ninst✝ : L.HasOrthogonalProjection\nh : Kᗮ = Lᗮ\n⊢ K = L",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Submodule | {
"line": 130,
"column": 4
} | {
"line": 130,
"column": 85
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_4\ninst✝² : Preorder ι\nU : ι → Submodule 𝕜 E\ninst✝¹ : ∀ (i : ι), (U i).HasOrthogonalProjection\ninst✝ : (⨆ i, U i).topologicalClosure.HasOrthogonalProjection\nhU : Monotone U\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Submodule | {
"line": 228,
"column": 14
} | {
"line": 228,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nK₁ K₂ : ClosedSubmodule 𝕜 E\ninst✝¹ : (↑K₁).HasOrthogonalProjection\ninst✝ : (↑K₂).HasOrthogonalProjection\nh : K₁ᗮ = K₂ᗮ\n⊢ K₁ = K₂",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Submodule | {
"line": 238,
"column": 2
} | {
"line": 238,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nK₁ K₂ : ClosedSubmodule 𝕜 E\n⊢ K₁ᗮ ⊔ K₂ᗮ = (K₁ ⊓ K₂)ᗮ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 332,
"column": 2
} | {
"line": 332,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : RCLike 𝕜\nE : Type u_4\nE' : Type u_5\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup E'\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace 𝕜 E'\nf : E ≃ₗᵢ[𝕜] E'\np : Submodule 𝕜 E\ninst✝ : p.HasOrthogonalProjection\nx : E'\n⊢ (map (↑f.toLinearEquiv) p)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 384,
"column": 2
} | {
"line": 385,
"column": 9
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nhK : K ≠ ⊥\nx : E\nhxK : x ∈ K\nhx_ne_zero : x ≠ 0\n⊢ 1 ≤ ‖K.orthogonalProjection‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 396,
"column": 4
} | {
"line": 396,
"column": 15
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv w : E\nthis : (𝕜 ∙ v).starProjection (↑‖v‖ ^ 2 • w) = ⟪v, w⟫ • v\n⊢ ↑(‖v‖ ^ 2) • (𝕜 ∙ v).starProjection w = ⟪v, w⟫ • v",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"I... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 150,
"column": 12
} | {
"line": 150,
"column": 23
} | [
{
"pp": "case h₁\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜' F\nf : E →SL[σ] F\nσ' : 𝕜' →+... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 150,
"column": 12
} | {
"line": 150,
"column": 28
} | [
{
"pp": "case h₁\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜' F\nf : E →SL[σ] F\nσ' : 𝕜' →+... | simpa using dinv | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 150,
"column": 12
} | {
"line": 150,
"column": 28
} | [
{
"pp": "case h₁\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜' F\nf : E →SL[σ] F\nσ' : 𝕜' →+... | simpa using dinv | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 150,
"column": 12
} | {
"line": 150,
"column": 28
} | [
{
"pp": "case h₁\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜' F\nf : E →SL[σ] F\nσ' : 𝕜' →+... | simpa using dinv | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 467,
"column": 2
} | {
"line": 467,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nU V : Submodule 𝕜 E\ninst✝¹ : U.HasOrthogonalProjection\ninst✝ : V.HasOrthogonalProjection\nh : ∀ (x : E), U.orthogonalProjection ↑(V.orthogonalProjection x) = 0\nx : ↥V\n⊢ U.orthogonalProj... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 485,
"column": 4
} | {
"line": 485,
"column": 43
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nU V : Submodule 𝕜 E\ninst✝¹ : U.HasOrthogonalProjection\ninst✝ : V.HasOrthogonalProjection\nh : U ≤ V\nx : E\n⊢ U.orthogonalProjection x = U.orthogonalProjection (V.starProjection x)",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 546,
"column": 2
} | {
"line": 546,
"column": 76
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nU : Submodule 𝕜 E\ninst✝ : U.HasOrthogonalProjection\nv : E\nh : ‖U.starProjection v‖ = ‖v‖\n⊢ v ∈ U",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 153,
"column": 4
} | {
"line": 154,
"column": 33
} | [
{
"pp": "case zero.h\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ F\ninst✝ : FiniteDimensional ℝ F\nφ : F ≃ₗᵢ[ℝ] F\nhn : finrank ℝ ↥(↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).kerᗮ ≤ 0\nthis✝ : (↑(ContinuousLinearMap.id ℝ F - ↑↑φ)).ker = ⊤\nx : F\nthis : ↑(ContinuousLinearMap.id ℝ F - ↑↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 605,
"column": 2
} | {
"line": 605,
"column": 13
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\np : E →ₗ[𝕜] E\nhp : p.IsSymmetricProjection\nthis : p.range.HasOrthogonalProjection\nx : E\n⊢ x - p x ∈ p.ker",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNorm... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 318,
"column": 36
} | {
"line": 318,
"column": 68
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁷ : RCLike 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : InnerProductSpace ℝ F\nK : Submodule 𝕜 E\nι : Type u_4\ninst✝² : DecidableEq ι\ninst✝¹ : Fintype ι\nV : ι → Submodule 𝕜 E\ninst✝ : ∀ (... | rw [DFinsupp.sum_eq_sum_fintype] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 352,
"column": 8
} | {
"line": 352,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhev : e ∈ v\nthis : e ∈ span 𝕜 v ⊓ (span �... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 371,
"column": 8
} | {
"line": 371,
"column": 24
} | [
{
"pp": "case mp.refine_2.inl\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : e ∉ v\nh_end :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 374,
"column": 8
} | {
"line": 374,
"column": 24
} | [
{
"pp": "case mp.refine_2.inr.inl\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := (↑‖x‖)⁻¹ • x\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : e ∉ v\nh_e... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 378,
"column": 8
} | {
"line": 378,
"column": 19
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv : Set E\nhv : Orthonormal 𝕜 Subtype.val\nx : E\nhx' : x ∈ (span 𝕜 v)ᗮ\nhx : x ≠ 0\ne : E := ⋯\nhe : ‖e‖ = 1\nhe' : e ∈ (span 𝕜 v)ᗮ\nhe'' : e ∉ v\nh_end : ∀ a ∈ v, ⟪a, e⟫_𝕜 = 0\na : E\nh... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.UnitaryGroup | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 36
} | [
{
"pp": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα : Type v\ninst✝¹ : CommRing α\ninst✝ : StarRing α\nA : Matrix n n α\nhA : A * star A = 1\n⊢ star A * A = 1",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Matrix.instMulOneOfFintypeOfDecidableEqOfAddCommMonoid",
"NonU... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.UnitaryGroup | {
"line": 83,
"column": 4
} | {
"line": 83,
"column": 37
} | [
{
"pp": "case left\nn : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα : Type v\ninst✝¹ : CommRing α\ninst✝ : StarRing α\nA : Matrix n n α\nhA : A ∈ unitaryGroup n α\n⊢ star A.det * A.det = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.UnitaryGroup | {
"line": 84,
"column": 4
} | {
"line": 84,
"column": 37
} | [
{
"pp": "case right\nn : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα : Type v\ninst✝¹ : CommRing α\ninst✝ : StarRing α\nA : Matrix n n α\nhA : A ∈ unitaryGroup n α\n⊢ A.det * star A.det = 1",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.UnitaryGroup | {
"line": 267,
"column": 24
} | {
"line": 267,
"column": 35
} | [
{
"pp": "n : Type u\ninst✝³ : DecidableEq n\ninst✝² : Fintype n\nα : Type v\ninst✝¹ : CommRing α\ninst✝ : StarRing α\nA✝ : Matrix n n α\nA : ↥(specialUnitaryGroup n α)\n⊢ star ↑A ∈ ↑(unitaryGroup n α)",
"usedConstants": [
"Eq.mpr",
"Matrix.instStar",
"SetLike.mem_coe._simp_1",
"NonUn... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 629,
"column": 17
} | {
"line": 629,
"column": 47
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹¹ : NontriviallyNormedField 𝕜\ninst✝¹⁰ : NontriviallyNormedField 𝕜'\nE : Type u_3\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nσ : 𝕜 →+* 𝕜'\nσ' : 𝕜' →+* 𝕜\ninst✝⁷ : RingHomInvPair σ σ'\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : Normed... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 436,
"column": 4
} | {
"line": 436,
"column": 83
} | [
{
"pp": "case inr\np : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : WithLp p (α × β)\nh : 1 ≤ p.toReal\npos : 0 < p.toReal\nnonneg : 0 ≤ 1 / p.toReal\n⊢ edist x y ≤ ↑(2 ^ (1 / p).toReal) * edist x.ofLp y.ofLp",
"usedConstants": [
... | have cancel : p.toReal * (1 / p.toReal) = 1 := mul_div_cancel₀ 1 (ne_of_gt pos) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 632,
"column": 2
} | {
"line": 632,
"column": 43
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nx y : WithLp p (α × β)\n⊢ nndist x.fst y.fst ≤ nndist x y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 636,
"column": 2
} | {
"line": 636,
"column": 43
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : PseudoMetricSpace α\ninst✝ : PseudoMetricSpace β\nx y : WithLp p (α × β)\n⊢ nndist x.snd y.snd ≤ nndist x y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 667,
"column": 18
} | {
"line": 667,
"column": 61
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : WithLp ∞ (α × β)\n⊢ edist x.ofLp y.ofLp ≤ edist x y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 669,
"column": 6
} | {
"line": 670,
"column": 22
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nx y : WithLp ∞ (α × β)\n⊢ edist x y ≤ edist x.ofLp y.ofLp",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 694,
"column": 2
} | {
"line": 694,
"column": 13
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : WithLp p (α × β)\n⊢ ‖x.fst‖ₑ ≤ ‖x‖ₑ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 698,
"column": 2
} | {
"line": 698,
"column": 13
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : WithLp p (α × β)\n⊢ ‖x.snd‖ₑ ≤ ‖x‖ₑ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 702,
"column": 2
} | {
"line": 702,
"column": 13
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : WithLp p (α × β)\n⊢ ‖x.fst‖₊ ≤ ‖x‖₊",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 706,
"column": 2
} | {
"line": 706,
"column": 13
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : WithLp p (α × β)\n⊢ ‖x.snd‖₊ ≤ ‖x‖₊",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 710,
"column": 2
} | {
"line": 710,
"column": 13
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : WithLp p (α × β)\n⊢ ‖x.fst‖ ≤ ‖x‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 714,
"column": 2
} | {
"line": 714,
"column": 13
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : WithLp p (α × β)\n⊢ ‖x.snd‖ ≤ ‖x‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 813,
"column": 4
} | {
"line": 813,
"column": 37
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : WithLp 2 (α × β)\nthis : ‖x‖₊ ^ 2 = ‖x.fst‖₊ ^ 2 + ‖x.snd‖₊ ^ 2\n⊢ ‖x‖ ^ 2 = ‖x.fst‖ ^ 2 + ‖x.snd‖ ^ 2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 839,
"column": 2
} | {
"line": 844,
"column": 91
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : α\n⊢ ‖toLp p (x, 0)‖₊ = ‖x‖₊",
"usedConstants": [
"ENNReal.coe_ne_top._simp_1",
"WithLp",
"GroupWithZero.toMonoidWithZero",
"ENNReal.instIsOrdere... | induction p generalizing hp with
| top =>
simp [prod_nnnorm_eq_sup]
| coe p =>
have hp0 : (p : ℝ) ≠ 0 := mod_cast (zero_lt_one.trans_le <| Fact.out (p := 1 ≤ (p : ℝ≥0∞))).ne'
simp [prod_nnnorm_eq_add, NNReal.zero_rpow hp0, ← NNReal.rpow_mul, mul_inv_cancel₀ hp0] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 839,
"column": 2
} | {
"line": 844,
"column": 91
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : α\n⊢ ‖toLp p (x, 0)‖₊ = ‖x‖₊",
"usedConstants": [
"ENNReal.coe_ne_top._simp_1",
"WithLp",
"GroupWithZero.toMonoidWithZero",
"ENNReal.instIsOrdere... | induction p generalizing hp with
| top =>
simp [prod_nnnorm_eq_sup]
| coe p =>
have hp0 : (p : ℝ) ≠ 0 := mod_cast (zero_lt_one.trans_le <| Fact.out (p := 1 ≤ (p : ℝ≥0∞))).ne'
simp [prod_nnnorm_eq_add, NNReal.zero_rpow hp0, ← NNReal.rpow_mul, mul_inv_cancel₀ hp0] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 839,
"column": 2
} | {
"line": 844,
"column": 91
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\nx : α\n⊢ ‖toLp p (x, 0)‖₊ = ‖x‖₊",
"usedConstants": [
"ENNReal.coe_ne_top._simp_1",
"WithLp",
"GroupWithZero.toMonoidWithZero",
"ENNReal.instIsOrdere... | induction p generalizing hp with
| top =>
simp [prod_nnnorm_eq_sup]
| coe p =>
have hp0 : (p : ℝ) ≠ 0 := mod_cast (zero_lt_one.trans_le <| Fact.out (p := 1 ≤ (p : ℝ≥0∞))).ne'
simp [prod_nnnorm_eq_add, NNReal.zero_rpow hp0, ← NNReal.rpow_mul, mul_inv_cancel₀ hp0] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 847,
"column": 2
} | {
"line": 852,
"column": 91
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\ny : β\n⊢ ‖toLp p (0, y)‖₊ = ‖y‖₊",
"usedConstants": [
"ENNReal.coe_ne_top._simp_1",
"WithLp",
"GroupWithZero.toMonoidWithZero",
"ENNReal.instIsOrdere... | induction p generalizing hp with
| top =>
simp [prod_nnnorm_eq_sup]
| coe p =>
have hp0 : (p : ℝ) ≠ 0 := mod_cast (zero_lt_one.trans_le <| Fact.out (p := 1 ≤ (p : ℝ≥0∞))).ne'
simp [prod_nnnorm_eq_add, NNReal.zero_rpow hp0, ← NNReal.rpow_mul, mul_inv_cancel₀ hp0] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 847,
"column": 2
} | {
"line": 852,
"column": 91
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\ny : β\n⊢ ‖toLp p (0, y)‖₊ = ‖y‖₊",
"usedConstants": [
"ENNReal.coe_ne_top._simp_1",
"WithLp",
"GroupWithZero.toMonoidWithZero",
"ENNReal.instIsOrdere... | induction p generalizing hp with
| top =>
simp [prod_nnnorm_eq_sup]
| coe p =>
have hp0 : (p : ℝ) ≠ 0 := mod_cast (zero_lt_one.trans_le <| Fact.out (p := 1 ≤ (p : ℝ≥0∞))).ne'
simp [prod_nnnorm_eq_add, NNReal.zero_rpow hp0, ← NNReal.rpow_mul, mul_inv_cancel₀ hp0] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 847,
"column": 2
} | {
"line": 852,
"column": 91
} | [
{
"pp": "p : ℝ≥0∞\nα : Type u_2\nβ : Type u_3\nhp : Fact (1 ≤ p)\ninst✝¹ : SeminormedAddCommGroup α\ninst✝ : SeminormedAddCommGroup β\ny : β\n⊢ ‖toLp p (0, y)‖₊ = ‖y‖₊",
"usedConstants": [
"ENNReal.coe_ne_top._simp_1",
"WithLp",
"GroupWithZero.toMonoidWithZero",
"ENNReal.instIsOrdere... | induction p generalizing hp with
| top =>
simp [prod_nnnorm_eq_sup]
| coe p =>
have hp0 : (p : ℝ) ≠ 0 := mod_cast (zero_lt_one.trans_le <| Fact.out (p := 1 ≤ (p : ℝ≥0∞))).ne'
simp [prod_nnnorm_eq_add, NNReal.zero_rpow hp0, ← NNReal.rpow_mul, mul_inv_cancel₀ hp0] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 205,
"column": 2
} | {
"line": 207,
"column": 9
} | [
{
"pp": "p : ℝ≥0∞\n𝕜 : Type u_1\ninst✝³ : Semiring 𝕜\nη : Type u_5\nιs : η → Type u_6\nMs : η → Type u_7\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module 𝕜 (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent 𝕜 (v i)\n⊢ LinearIndependent 𝕜 fun ji ↦ s... | suffices LinearIndependent 𝕜 ((WithLp.linearEquiv p 𝕜 _).symm.toLinearMap ∘
fun ji : Σ j, ιs j ↦ Pi.single ji.1 (v ji.1 ji.2)) by
simpa | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 407,
"column": 6
} | {
"line": 415,
"column": 36
} | [
{
"pp": "case inr\np : ℝ≥0∞\n𝕜 : Type u_1\nι : Type u_2\nα : ι → Type u_3\nβ : ι → Type u_4\ninst✝³ : Fact (1 ≤ p)\ninst✝² : (i : ι) → PseudoMetricSpace (α i)\ninst✝¹ : (i : ι) → PseudoEMetricSpace (β i)\ninst✝ : Fintype ι\nf g h : PiLp p β\nhp : 1 ≤ p.toReal\n⊢ (∑ i, edist (f.ofLp i) (h.ofLp i) ^ p.toReal) ^ ... | calc
(∑ i, edist (f i) (h i) ^ p.toReal) ^ (1 / p.toReal) ≤
(∑ i, (edist (f i) (g i) + edist (g i) (h i)) ^ p.toReal) ^ (1 / p.toReal) := by
gcongr
apply edist_triangle
_ ≤
(∑ i, edist (f i) (g i) ^ p.toReal) ^ (1 / p.toReal) +
(∑ i, edist (g i) ... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 1051,
"column": 4
} | {
"line": 1051,
"column": 46
} | [
{
"pp": "case refine_1\np : ℝ≥0∞\nhp : Fact (1 ≤ p)\nα : Type u_4\nβ : Type u_5\ninst✝⁶ : SeminormedAddCommGroup α\ninst✝⁵ : SeminormedAddCommGroup β\nR : Type u_6\ninst✝⁴ : SeminormedRing R\ninst✝³ : Module R α\ninst✝² : Module R β\ninst✝¹ : IsBoundedSMul R α\ninst✝ : IsBoundedSMul R β\nthis : PseudoMetricSpac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 1052,
"column": 4
} | {
"line": 1052,
"column": 46
} | [
{
"pp": "case refine_2\np : ℝ≥0∞\nhp : Fact (1 ≤ p)\nα : Type u_4\nβ : Type u_5\ninst✝⁶ : SeminormedAddCommGroup α\ninst✝⁵ : SeminormedAddCommGroup β\nR : Type u_6\ninst✝⁴ : SeminormedRing R\ninst✝³ : Module R α\ninst✝² : Module R β\ninst✝¹ : IsBoundedSMul R α\ninst✝ : IsBoundedSMul R β\nthis : PseudoMetricSpac... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 468,
"column": 4
} | {
"line": 468,
"column": 36
} | [
{
"pp": "case inl\nι : Type u_2\nβ : ι → Type u_4\ninst✝² : (i : ι) → PseudoEMetricSpace (β i)\ninst✝¹ : Fintype ι\ni : ι\ninst✝ : Fact (1 ≤ ∞)\nx y : PiLp ∞ β\n⊢ edist (x.ofLp i) (y.ofLp i) ≤ edist x y",
"usedConstants": [
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"PiLp",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 493,
"column": 4
} | {
"line": 493,
"column": 83
} | [
{
"pp": "case inr\np : ℝ≥0∞\nι : Type u_2\nβ : ι → Type u_4\ninst✝² : Fact (1 ≤ p)\ninst✝¹ : (i : ι) → PseudoEMetricSpace (β i)\ninst✝ : Fintype ι\nx y : WithLp p ((i : ι) → β i)\nh : 1 ≤ p.toReal\npos : 0 < p.toReal\nnonneg : 0 ≤ 1 / p.toReal\n⊢ edist x y ≤ ↑(↑(Fintype.card ι) ^ (1 / p).toReal) * edist x.ofLp ... | have cancel : p.toReal * (1 / p.toReal) = 1 := mul_div_cancel₀ 1 (ne_of_gt pos) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Haar.OfBasis | {
"line": 87,
"column": 6
} | {
"line": 87,
"column": 47
} | [
{
"pp": "case h.refine_1\nι : Type u_1\nι' : Type u_2\nE : Type u_3\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nv : ι → E\ne : ι' ≃ ι\nK : (ι' → ℝ) ≃ (ι → ℝ) := Equiv.piCongrLeft' (fun _a ↦ ℝ) e\nx : ι' → ℝ\nh : (∀ (i : ι), 0 ≤ x (e.symm i)) ∧ ∀ (i : ι), x (e.symm i) ≤... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 645,
"column": 2
} | {
"line": 645,
"column": 43
} | [
{
"pp": "p : ℝ≥0∞\nι : Type u_2\nβ : ι → Type u_4\nhp : Fact (1 ≤ p)\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoMetricSpace (β i)\nx y : PiLp p β\ni : ι\n⊢ nndist (x.ofLp i) (y.ofLp i) ≤ nndist x y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 672,
"column": 18
} | {
"line": 672,
"column": 61
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_4\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nx y : WithLp ∞ ((i : ι) → β i)\n⊢ edist x.ofLp y.ofLp ≤ edist x y",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 673,
"column": 8
} | {
"line": 674,
"column": 20
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_4\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nx y : WithLp ∞ ((i : ι) → β i)\n⊢ edist x y ≤ edist x.ofLp y.ofLp",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Haar.OfBasis | {
"line": 88,
"column": 6
} | {
"line": 88,
"column": 47
} | [
{
"pp": "case h.refine_2\nι : Type u_1\nι' : Type u_2\nE : Type u_3\ninst✝³ : Fintype ι\ninst✝² : Fintype ι'\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nv : ι → E\ne : ι' ≃ ι\nK : (ι' → ℝ) ≃ (ι → ℝ) := Equiv.piCongrLeft' (fun _a ↦ ℝ) e\nx : ι' → ℝ\nh : (∀ (i : ι), 0 ≤ x (e.symm i)) ∧ ∀ (i : ι), x (e.symm i) ≤... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 683,
"column": 4
} | {
"line": 688,
"column": 57
} | [
{
"pp": "case inr\np : ℝ≥0∞\n𝕜 : Type u_1\nι : Type u_2\nα : ι → Type u_3\nβ : ι → Type u_4\nhp : Fact (1 ≤ p)\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx y : PiLp p β\nh : 1 ≤ p.toReal\n⊢ dist x y = ‖-x + y‖",
"usedConstants": [
"WithLp",
"PiLp.instNorm",
"Real... | · have : p ≠ ∞ := by
intro hp
rw [hp, ENNReal.toReal_top] at h
linarith
simp only [dist_eq_sum (zero_lt_one.trans_le h), norm_eq_sum (zero_lt_one.trans_le h),
dist_eq_norm, add_apply, neg_apply, norm_neg_add] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 702,
"column": 2
} | {
"line": 702,
"column": 13
} | [
{
"pp": "p : ℝ≥0∞\nι : Type u_2\nβ : ι → Type u_4\nhp : Fact (1 ≤ p)\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx : PiLp p β\ni : ι\n⊢ ‖x.ofLp i‖ₑ ≤ ‖x‖ₑ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 706,
"column": 2
} | {
"line": 706,
"column": 13
} | [
{
"pp": "p : ℝ≥0∞\nι : Type u_2\nβ : ι → Type u_4\nhp : Fact (1 ≤ p)\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx : PiLp p β\ni : ι\n⊢ ‖x.ofLp i‖₊ ≤ ‖x‖₊",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 710,
"column": 2
} | {
"line": 710,
"column": 13
} | [
{
"pp": "p : ℝ≥0∞\nι : Type u_2\nβ : ι → Type u_4\nhp : Fact (1 ≤ p)\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx : PiLp p β\ni : ι\n⊢ ‖x.ofLp i‖ ≤ ‖x‖",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Haar.OfBasis | {
"line": 108,
"column": 4
} | {
"line": 108,
"column": 25
} | [
{
"pp": "ι : Type u_1\ninst✝ : Fintype ι\nb : OrthonormalBasis ι ℝ ℝ\ne : ι ≃ Fin 1\nB : parallelepiped ⇑(b.reindex e) = parallelepiped ⇑b\nF : ℝ → Fin 1 → ℝ := fun t _i ↦ t\n⊢ Icc 0 1 = F '' Icc 0 1",
"usedConstants": [
"Set.Subset.antisymm",
"Real",
"Pi.preorder",
"Real.instZero",
... | apply Subset.antisymm | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 788,
"column": 4
} | {
"line": 788,
"column": 37
} | [
{
"pp": "ι : Type u_2\ninst✝¹ : Fintype ι\nβ : ι → Type u_5\ninst✝ : (i : ι) → SeminormedAddCommGroup (β i)\nx : PiLp 2 β\nthis : ‖x‖₊ ^ 2 = ∑ i, ‖x.ofLp i‖₊ ^ 2\n⊢ ‖x‖ ^ 2 = ∑ i, ‖x.ofLp i‖ ^ 2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 28
} | [
{
"pp": "a r : ℝ\nhr : 0 ≤ r\n⊢ volume.real (ball a r) = 2 * r",
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"MeasureTheory.Measure",
"HMul.hMul",
"ENNReal.ofReal",
"congrArg",
"Com... | simp [measureReal_def, hr] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 28
} | [
{
"pp": "a r : ℝ\nhr : 0 ≤ r\n⊢ volume.real (ball a r) = 2 * r",
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"MeasureTheory.Measure",
"HMul.hMul",
"ENNReal.ofReal",
"congrArg",
"Com... | simp [measureReal_def, hr] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 28
} | [
{
"pp": "a r : ℝ\nhr : 0 ≤ r\n⊢ volume.real (ball a r) = 2 * r",
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"MeasureTheory.Measure",
"HMul.hMul",
"ENNReal.ofReal",
"congrArg",
"Com... | simp [measureReal_def, hr] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 28
} | [
{
"pp": "a r : ℝ\nhr : 0 ≤ r\n⊢ volume.real (closedBall a r) = 2 * r",
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"MeasureTheory.Measure",
"HMul.hMul",
"Real.volume_closedBall",
"ENNReal... | simp [measureReal_def, hr] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 28
} | [
{
"pp": "a r : ℝ\nhr : 0 ≤ r\n⊢ volume.real (closedBall a r) = 2 * r",
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"MeasureTheory.Measure",
"HMul.hMul",
"Real.volume_closedBall",
"ENNReal... | simp [measureReal_def, hr] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 28
} | [
{
"pp": "a r : ℝ\nhr : 0 ≤ r\n⊢ volume.real (closedBall a r) = 2 * r",
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"MeasureTheory.Measure",
"HMul.hMul",
"Real.volume_closedBall",
"ENNReal... | simp [measureReal_def, hr] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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