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.TensorProduct | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 90
} | [
{
"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\nx y : E ⊗[𝕜] F\n⊢ x = y ↔ ∀ (a : E) (b : F), inner 𝕜 (a ⊗ₜ[𝕜] b) x = inner 𝕜 (a ⊗ₜ[𝕜] b) y",
"usedConsta... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 200,
"column": 2
} | {
"line": 200,
"column": 90
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : InnerProductSpace 𝕜 G\nx y : E ⊗[𝕜] F ⊗[𝕜] G\n⊢ x = y ↔ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 354,
"column": 30
} | {
"line": 354,
"column": 41
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : Nontrivial E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Rayleigh | {
"line": 355,
"column": 66
} | {
"line": 355,
"column": 90
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : RCLike 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : Nontrivial E\nhT : T.IsSymmetric\nthis : ProperSpace E\nT' : ↥(selfAdjoint (E →L[𝕜] E)) := hT.toSelfAdjoint\nx : E\nhx : x ≠ 0\nH₁ :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 72,
"column": 29
} | {
"line": 72,
"column": 40
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖\nhK : ∀ K > 0, ∃ x, ‖(T - μ • 1) x‖ < K * ‖x‖\nC : 𝕜\nhC : 1 < ‖C‖\nε : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.TensorProduct | {
"line": 449,
"column": 2
} | {
"line": 449,
"column": 90
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : InnerProductSpace 𝕜 G\nx y : E ⊗[𝕜] (F ⊗[𝕜] G)\n⊢ x = y ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 88,
"column": 46
} | {
"line": 88,
"column": 57
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖\nc : ℝ\nhc₀ : c > 0\nhc : ∀ ε > 0, ∃ x, ‖x‖ ≤ 1 ∧ c ≤ ‖x‖ ∧ ‖(T - μ • 1) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 90,
"column": 6
} | {
"line": 90,
"column": 29
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖\nc : ℝ\nhc₀ : c > 0\nhc : ∀ ε > 0, ∃ x, ‖x‖ ≤ 1 ∧ c ≤ ‖x‖ ∧ ‖(T - μ • 1) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 99,
"column": 4
} | {
"line": 99,
"column": 35
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖\nc : ℝ\nhc₀ : c > 0\nhc : ∀ ε > 0, ∃ x, ‖x‖ ≤ 1 ∧ c ≤ ‖x‖ ∧ ‖(T - μ • 1) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 79
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh : ∀ c > 0, ∃ x, ‖(T - μ • 1) x‖ < c * ‖x‖\nc : ℝ\nhc₀ : c > 0\nhc : ∀ ε > 0, ∃ x, ‖x‖ ≤ 1 ∧ c ≤ ‖x‖ ∧ ‖(T - μ • 1) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 549,
"column": 69
} | {
"line": 555,
"column": 14
} | [
{
"pp": "E : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝² : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : InnerProductSpace ℝ F'\nu : E → F'\nhu : ContDiff ... | by
rw [mul_assoc, ← lintegral_const_mul γ]
· gcongr
simp_rw [← mul_assoc]
exact enorm_fderiv_norm_rpow_le (hu.differentiable one_ne_zero) h1γ
dsimp [enorm]
fun_prop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 144,
"column": 6
} | {
"line": 144,
"column": 17
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nS : End 𝕜 X\nhS_not_surj : ¬Function.Surjective ⇑S\nhS_anti : Topology.IsClosedEmbedding ⇑S\nc : 𝕜\nhc : 1 < ‖c‖\nR : ℝ\nhR : ‖c‖ < R\nV : ℕ → Submodule 𝕜 X := fun n ↦ (LinearMa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 146,
"column": 6
} | {
"line": 147,
"column": 34
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nS : End 𝕜 X\nhS_not_surj : ¬Function.Surjective ⇑S\nhS_anti : Topology.IsClosedEmbedding ⇑S\nc : 𝕜\nhc : 1 < ‖c‖\nR : ℝ\nhR : ‖c‖ < R\nV : ℕ → Submodule 𝕜 X := fun n ↦ (LinearMa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 151,
"column": 21
} | {
"line": 151,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nS : End 𝕜 X\nhS_not_surj : ¬Function.Surjective ⇑S\nhS_anti : Topology.IsClosedEmbedding ⇑S\nc : 𝕜\nhc : 1 < ‖c‖\nR : ℝ\nhR : ‖c‖ < R\nV : ℕ → Submodule 𝕜 X := fun n ↦ (LinearMa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Charpoly | {
"line": 52,
"column": 52
} | {
"line": 52,
"column": 63
} | [
{
"pp": "K : Type u_3\nV : Type u_4\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nf : End K V\nh : (charpoly f).Splits\n⊢ ((toMatrix (Free.chooseBasis K V) (Free.chooseBasis K V)) f).charpoly.Splits",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Eigenspace.Charpoly | {
"line": 58,
"column": 53
} | {
"line": 58,
"column": 64
} | [
{
"pp": "K : Type u_3\nV : Type u_4\ninst✝³ : Field K\ninst✝² : AddCommGroup V\ninst✝¹ : Module K V\ninst✝ : Module.Finite K V\nf : End K V\nh : (charpoly f).Splits\nb : Basis (Free.ChooseBasisIndex K V) K V := Free.chooseBasis K V\n⊢ ((toMatrix (Free.chooseBasis K V) (Free.chooseBasis K V)) f).charpoly.Splits"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 189,
"column": 4
} | {
"line": 189,
"column": 44
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\ninst✝ : CompleteSpace X\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh₁ : ¬HasEigenvalue (↑T) μ\nS : X →L[𝕜] X := T - μ • 1\nK : NNReal\nhK : AntilipschitzWith... | rw [iterate_succ', LinearMap.range_comp] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Fin.Tuple.Sort | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 40
} | [
{
"pp": "n : ℕ\nα : Type u_1\ninst✝ : LinearOrder α\nf : Fin n → α\nx✝ : ↥(graph f)\nx : α\ni : Fin n\nh : (x, i) ∈ graph f\n⊢ (fun i ↦ ⟨(f i, i), ⋯⟩) ((fun p ↦ (↑p).2) ⟨(x, i), h⟩) = ⟨(x, i), h⟩",
"usedConstants": [
"Eq.mpr",
"Tuple.graphEquiv₁._proof_4",
"Finset.univ",
"LinearOrder... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Normed.Operator.Compact.FredholmAlternative | {
"line": 214,
"column": 55
} | {
"line": 214,
"column": 85
} | [
{
"pp": "𝕜 : Type u_1\nX : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedAddCommGroup X\ninst✝¹ : NormedSpace 𝕜 X\nT : X →L[𝕜] X\nμ : 𝕜\ninst✝ : CompleteSpace X\nhT : IsCompactOperator ⇑T\nhμ : μ ≠ 0\nh₁ : ¬HasEigenvalue (↑T) μ\nS : X →L[𝕜] X := T - μ • 1\nK✝ : NNReal\nhK✝ : AntilipschitzWi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 573,
"column": 80
} | {
"line": 583,
"column": 16
} | [
{
"pp": "E : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MeasurableSpace E\ninst✝⁴ : BorelSpace E\ninst✝³ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝² : μ.IsAddHaarMeasure\nF' : Type u_5\ninst✝¹ : NormedAddCommGroup F'\ninst✝ : InnerProductSpace ℝ F'\nu : E → F'\nhu : ContDiff ... | by
suffices (C : ℝ) * γ = eLpNormLESNormFDerivOfEqInnerConst μ p by
rw [eLpNorm_nnreal_eq_lintegral h0p]
congr
norm_cast at this ⊢
simp_rw [eLpNormLESNormFDerivOfEqInnerConst, γ]
refold_let n n' C
rw [NNReal.coe_mul, NNReal.coe_mk, Real.coe_toNNReal', mul_eq_mul_left_iff,... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Fin.Tuple.Sort | {
"line": 200,
"column": 4
} | {
"line": 200,
"column": 15
} | [
{
"pp": "n : ℕ\nσ : Equiv.Perm (Fin n)\n⊢ Monotone (⇑σ ∘ ⇑σ⁻¹)",
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"Equiv.Perm.instInv",
"congrArg",
"PartialOrder.toPreorder",
"Monotone",
"Function.comp",
"SemilatticeInf.toPartialOrder",
"DistribLattic... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Fin.Tuple.Sort | {
"line": 202,
"column": 22
} | {
"line": 202,
"column": 33
} | [
{
"pp": "n : ℕ\nσ : Equiv.Perm (Fin n)\ni✝ j✝ : Fin n\nhij : i✝ < j✝\nh : σ (σ⁻¹ i✝) = σ (σ⁻¹ j✝)\n⊢ i✝ = j✝",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Hermitian | {
"line": 51,
"column": 2
} | {
"line": 51,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_3\nA : Matrix n n 𝕜\ninst✝⁴ : RCLike 𝕜\ninst✝³ : Fintype n\ninst✝² : DecidableEq n\nE : Type u_4\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nb : OrthonormalBasis n 𝕜 E\nthis : FiniteDimensional 𝕜 E\nh : ∀ (x y : E), inner 𝕜 (((toLin b.toBasis b.toBasis... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Hermitian | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_3\nA : Matrix n n 𝕜\ninst✝¹ : RCLike 𝕜\ninst✝ : Fintype n\nhA : A.IsHermitian\nx : n → 𝕜\n⊢ im (star x ⬝ᵥ A *ᵥ x) = 0",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Pi.instStarForall",
"Real",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.FunctionalSpaces.SobolevInequality | {
"line": 691,
"column": 10
} | {
"line": 691,
"column": 21
} | [
{
"pp": "case h.e'_4.h.e'_6.h\nF : Type u_3\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\nE : Type u_4\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\ninst✝² : FiniteDimensional ℝ E\nμ : Measure E\ninst✝¹ : μ.IsAddHaarMeasure\ninst✝ : Fini... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 73,
"column": 2
} | {
"line": 73,
"column": 13
} | [
{
"pp": "n : Type un\nn₀ : Type un₀\nR : Type uR\ninst✝ : Semiring R\nm m₀ : Type um\nA : Matrix m n R\nr : m₀ → m\nc : n₀ → n\n⊢ (A.submatrix r c).cRank ≤ A.cRank",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 81,
"column": 32
} | {
"line": 81,
"column": 43
} | [
{
"pp": "m : Type um\nn : Type un\nR : Type uR\ninst✝² : Semiring R\ninst✝¹ : StrongRankCondition R\ninst✝ : Fintype n\nA : Matrix m n R\n⊢ #↑(range Aᵀ) ≤ ↑(Fintype.card n)",
"usedConstants": [
"Eq.mpr",
"Cardinal",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Cardinal.mk",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 96,
"column": 2
} | {
"line": 96,
"column": 13
} | [
{
"pp": "m : Type um\nm₀ : Type um₀\nn : Type un\nn₀ : Type un₀\nR : Type uR\ninst✝ : Semiring R\nA : Matrix m n R\nr : m₀ → m\nc : n₀ → n\n⊢ (A.submatrix r c).eRank ≤ A.eRank",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 100,
"column": 33
} | {
"line": 100,
"column": 44
} | [
{
"pp": "m : Type um\nn : Type un\nR : Type uR\ninst✝¹ : Semiring R\ninst✝ : StrongRankCondition R\nA : Matrix m n R\nthis :\n ∀ {m : Type um} {n : Type un} {R : Type uR} [inst : Semiring R] [StrongRankCondition R] (A : Matrix m n R),\n Finite n → A.eRank ≤ ENat.card n\nhfin : ¬Finite n\n⊢ Infinite ?m.33",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 108,
"column": 33
} | {
"line": 108,
"column": 44
} | [
{
"pp": "m : Type um\nn : Type un\nR : Type uR\ninst✝¹ : Semiring R\ninst✝ : StrongRankCondition R\nA : Matrix m n R\nthis :\n ∀ {m : Type um} {n : Type un} {R : Type uR} [inst : Semiring R] [StrongRankCondition R] (A : Matrix m n R),\n Finite m → A.eRank ≤ ENat.card m\nhfin : ¬Finite m\n⊢ Infinite ?m.34",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 34
} | [
{
"pp": "m : Type um\nm₀ : Type um₀\nn : Type un\nn₀ : Type un₀\nR : Type uR\ninst✝² : Fintype n\ninst✝¹ : CommRing R\ninst✝ : Fintype n₀\nA : Matrix m n R\nem : m₀ ≃ m\nen : n₀ ≃ n\n⊢ (A.submatrix ⇑em ⇑en).rank = A.rank",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 259,
"column": 10
} | {
"line": 259,
"column": 21
} | [
{
"pp": "m : Type um\nm₀ : Type um₀\nn₀ : Type un₀\nR : Type uR\ninst✝ : CommRing R\nn : Type un\nA : Matrix m n R\nem : m₀ ≃ m\nen : n₀ ≃ n\n⊢ lift.{um₀, max uR um} A.cRank ≤ lift.{um, max uR um₀} (A.submatrix ⇑em ⇑en).cRank",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 265,
"column": 2
} | {
"line": 265,
"column": 37
} | [
{
"pp": "m : Type um\nn₀ : Type un₀\nR : Type uR\ninst✝ : CommRing R\nm₀ : Type um\nn : Type un\nA : Matrix m n R\nem : m₀ ≃ m\nen : n₀ ≃ n\n⊢ (A.submatrix ⇑em ⇑en).cRank = A.cRank",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 279,
"column": 2
} | {
"line": 279,
"column": 37
} | [
{
"pp": "m : Type um\nm₀ : Type um₀\nn₀ : Type un₀\nR : Type uR\ninst✝ : CommRing R\nn : Type un\nA : Matrix m n R\nem : m₀ ≃ m\nen : n₀ ≃ n\n⊢ (A.submatrix ⇑em ⇑en).eRank = A.eRank",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 403,
"column": 2
} | {
"line": 403,
"column": 68
} | [
{
"pp": "m : Type um\nn : Type un\nR : Type uR\ninst✝⁵ : Fintype n\ninst✝⁴ : Fintype m\ninst✝³ : Field R\ninst✝² : PartialOrder R\ninst✝¹ : StarRing R\ninst✝ : StarOrderedRing R\nA : Matrix m n R\n⊢ (A * Aᴴ).rank = A.rank",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Spectrum | {
"line": 75,
"column": 2
} | {
"line": 76,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nn : Type u_2\ninst✝¹ : Fintype n\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.IsHermitian\nj : n\n⊢ A *ᵥ (hA.eigenvectorBasis j).ofLp = hA.eigenvalues j • (hA.eigenvectorBasis j).ofLp",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.HermitianFunctionalCalculus | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 29
} | [
{
"pp": "n : Type u_1\n𝕜 : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.IsHermitian\nh0 : FiniteDimensional ℝ C(↑(spectrum ℝ A), ℝ)\nf : C(↑(spectrum ℝ A), ℝ)\nhf : ↑hA.eigenvectorUnitary * diagonal (RCLike.ofReal ∘ ⇑f ∘ fun i ↦ ⟨hA.eigenvalues i, ⋯⟩) = 0\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Matrix.Rank | {
"line": 445,
"column": 2
} | {
"line": 445,
"column": 56
} | [
{
"pp": "m : Type um\nn : Type un\nR : Type uR\ninst✝⁴ : Fintype n\ninst✝³ : Field R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : Fintype m\nA : Matrix m n R\n⊢ (A * Aᵀ).rank = A.rank",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.PosDef | {
"line": 50,
"column": 40
} | {
"line": 50,
"column": 51
} | [
{
"pp": "n : Type u_2\n𝕜 : Type u_3\ninst✝² : Fintype n\ninst✝¹ : RCLike 𝕜\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.PosSemidef\ni : n\nx✝ : i ∈ Finset.univ\n⊢ 0 ≤ ↑(⋯.eigenvalues i)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"Real.instLE",
"Real",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.PosDef | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 13
} | [
{
"pp": "case h0\nn : Type u_2\n𝕜 : Type u_3\ninst✝² : Fintype n\ninst✝¹ : RCLike 𝕜\nA : Matrix n n 𝕜\ninst✝ : DecidableEq n\nhA : A.PosDef\ni : n\na✝ : i ∈ Finset.univ\n⊢ 0 < ↑(⋯.eigenvalues i)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"Real",
"Preorder.toLT",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.PosDef | {
"line": 124,
"column": 6
} | {
"line": 124,
"column": 50
} | [
{
"pp": "m : Type u_1\nn : Type u_2\n𝕜 : Type u_3\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : RCLike 𝕜\nA M : Matrix n n 𝕜\nhM : M.PosDef\nx : n → 𝕜\nhx : M *ᵥ x ⬝ᵥ star x = 0\nh : x ≠ 0\n⊢ False",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 59
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ : 𝕜\nhμ : HasEigenvalue T μ\nv : E\nhv₁ : T v = μ • v\nhv₂ : v ≠ 0\n⊢ (starRingEnd 𝕜) μ = μ",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 14
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv : v ∈ eigenspace T μ\nw : E\nhw : w ∈ eigenspace T ν\nhv' : v = 0\n⊢ ⟪((fun μ ↦ (eigenspace T μ).subtypeₗᵢ) μ) ⟨... | simp [hv'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 14
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv : v ∈ eigenspace T μ\nw : E\nhw : w ∈ eigenspace T ν\nhv' : v = 0\n⊢ ⟪((fun μ ↦ (eigenspace T μ).subtypeₗᵢ) μ) ⟨... | simp [hv'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 14
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv : v ∈ eigenspace T μ\nw : E\nhw : w ∈ eigenspace T ν\nhv' : v = 0\n⊢ ⟪((fun μ ↦ (eigenspace T μ).subtypeₗᵢ) μ) ⟨... | simp [hv'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 109,
"column": 2
} | {
"line": 109,
"column": 60
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nμ ν : 𝕜\nhμν : μ ≠ ν\nv : E\nhv✝ : v ∈ eigenspace T μ\nhv : T v = μ • v\nw : E\nhw✝ : w ∈ eigenspace T ν\nhw : T w = ν • w\nhv' : ¬v = 0\nH : (s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 197,
"column": 4
} | {
"line": 197,
"column": 97
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nhT : T.IsSymmetric\nv : E\nμ : Eigenvalues T\nthis :\n ∀ (w : PiLp 2 fun μ ↦ ↥(eigenspace T (↑T 1 μ))),\n T (hT.diagonalization.symm w) = ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 268,
"column": 26
} | {
"line": 268,
"column": 37
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nn : ℕ\nhT : T.IsSymmetric\nhn : finrank 𝕜 E = n\ni : Fin n\nv : E := (hT.unsortedEigenvectorBasis hn) i\nμ : 𝕜 := ↑T (DirectSum.IsInternal.s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 20
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nn : ℕ\nhT : T.IsSymmetric\nhn : finrank 𝕜 E = n\ni : Fin n\nv : E := (hT.unsortedEigenvectorBasis hn) i\nμ : 𝕜 := ↑T (DirectSum.IsInternal.s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 340,
"column": 4
} | {
"line": 340,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nn : ℕ\nhT : T.IsSymmetric\nhn : finrank 𝕜 E = n\nv : E\ni : Fin n\nthis :\n ∀ (w : EuclideanSpace 𝕜 (Fin n)),\n T ((hT.eigenvectorBasis ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 70,
"column": 20
} | {
"line": 70,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA : Matrix n n 𝕜\nh₁ : A.PosSemidef\nh₂ : (-A).PosSemidef\ni✝ j i : n\n⊢ A i i ≤ 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 70,
"column": 52
} | {
"line": 70,
"column": 63
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA : Matrix n n 𝕜\nh₁ : A.PosSemidef\nh₂ : (-A).PosSemidef\ni✝ j i : n\n⊢ 0 ≤ A i i",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 13
} | [
{
"pp": "case a\n𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA : Matrix n n 𝕜\nh₁ : A.PosSemidef\nh₂ : (-A).PosSemidef\ni j : n\nhdiag : True\nh1 : 0 ≤ A i j * star (A i j) + A i j * star (A i j)\nh2 : A i j * star (A i j) + A i j * star (A i j) ≤ 0\n⊢ A i j = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 412,
"column": 38
} | {
"line": 412,
"column": 85
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nμ : ℝ\nT : E →ₗ[𝕜] E\nhμ : HasEigenvalue T ↑μ\nhnn : ∀ (x : E), 0 ≤ RCLike.re ⟪x, T x⟫\nv : E\nhv₁ : v ∈ (genEigenspace T ↑μ) 1\nhv₂ : v ≠ 0\n⊢ 0 < ‖v‖ ^ 2",
"usedConstants": [
"Ad... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA B : Matrix n n 𝕜\nh₁ : A ≤ B\nh₂ : B ≤ A\n⊢ A = B",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 81,
"column": 9
} | {
"line": 81,
"column": 47
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝ : RCLike 𝕜\nA B : Matrix n n 𝕜\nh₁ : A ≤ B\nh₂ : B ≤ A\n⊢ (-(B - A)).PosSemidef",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 421,
"column": 38
} | {
"line": 421,
"column": 85
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : RCLike 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nμ : ℝ\nT : E →ₗ[𝕜] E\nhμ : HasEigenvalue T ↑μ\nhnn : ∀ (x : E), 0 < RCLike.re ⟪x, T x⟫\nv : E\nhv₁ : v ∈ (genEigenspace T ↑μ) 1\nhv₂ : v ≠ 0\n⊢ 0 < ‖v‖ ^ 2",
"usedConstants": [
"_p... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 150,
"column": 2
} | {
"line": 150,
"column": 41
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nhA : A.PosSemidef\nx : n → 𝕜\n⊢ (((toLinearMap₂' 𝕜) A) (star x)) x = 0 ↔ A *ᵥ x = 0",
"usedConstants": [
"Eq.mpr",
"Pi.Function.module",
"Pi.instStarForall",
"Alg... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 176,
"column": 4
} | {
"line": 176,
"column": 44
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix n n 𝕜\nx✝ : A.IsHermitian ∧ spectrum 𝕜 A ⊆ {a | 0 ≤ a}\nh1 : A.IsHermitian\nh2 : spectrum 𝕜 A ⊆ {a | 0 ≤ a}\ni : n\n⊢ 0 i ≤ h1.eigenvalues i",
"usedConstants": [
"Real.inst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 83
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : Finite n\nm : Type u_3\ninst✝ : Finite m\nthis✝ : Fintype n\nthis : Fintype m\na : Matrix n n 𝕜\nhx : (star a * a).PosSemidef\nb : Matrix m m 𝕜\nhy : (star b * b).PosSemidef\n⊢ (kroneckerMap (fun x1 x2 ↦ x1 * x2) (star a * a) (star b * b)).Pos... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Spectrum | {
"line": 459,
"column": 2
} | {
"line": 459,
"column": 17
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nT : E →L[𝕜] E\nhT : IsCompactOperator ⇑T\nhT' : (↑T).IsSymmetric\nS : ↥(⨆ μ, eigenspace (↑T) μ)ᗮ →L[𝕜] ↥(⨆ μ, eigenspace (↑T) μ)ᗮ := T.restrict ⋯\nhS_compact : IsC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 249,
"column": 2
} | {
"line": 250,
"column": 73
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nι : Type u_3\nA B : Matrix ι ι 𝕜\nhA : A.PosSemidef\nhB : B.PosSemidef\nx : ι →₀ 𝕜\nhAB : ((A ⊙ B).submatrix Subtype.val Subtype.val).PosSemidef\n⊢ 0 ≤ x.sum fun i xi ↦ x.sum fun j xj ↦ star xi * (A ⊙ B) i j * xj",
"usedConstants": [
"NormedCommRing.toNorme... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 271,
"column": 2
} | {
"line": 271,
"column": 46
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : RCLike 𝕜\nι : Type u_3\nA B : Matrix ι ι 𝕜\nhA : A.PosDef\nhB : B.PosDef\nx : ι →₀ 𝕜\nhx : x ≠ 0\nhAB : ((A ⊙ B).submatrix Subtype.val Subtype.val).PosDef\n⊢ Finsupp.subtypeDomain (fun x_1 ↦ x_1 ∈ x.support) x ≠ 0",
"usedConstants": [
"NormedCommRing.toNormedRing",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Matrix.Order | {
"line": 329,
"column": 6
} | {
"line": 329,
"column": 31
} | [
{
"pp": "𝕜 : Type u_1\nn : Type u_2\ninst✝¹ : RCLike 𝕜\ninst✝ : Fintype n\nx : Matrix n n 𝕜\ny : (Matrix n n 𝕜)ˣ\nhM : (star ↑y * ↑y).PosDef\n__spread✝⁻⁰ : PreInnerProductSpace.Core 𝕜 (Matrix n n 𝕜) := ⋯.matrixPreInnerProductSpace\nhx : ↑y * xᴴ = 0\n⊢ x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.GramMatrix | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 49
} | [
{
"pp": "E : Type u_1\nn : Type u_2\n𝕜 : Type u_4\ninst✝³ : RCLike 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : Fintype n\nv : n → E\nx y : n → 𝕜\n⊢ star x ⬝ᵥ gram 𝕜 v *ᵥ y = ⟪∑ i, x i • v i, ∑ i, y i • v i⟫_𝕜",
"usedConstants": [
"InnerProductSpace.toNormedSpace... | trans ∑ i, ∑ j, conj (x i) * y j * ⟪v i, v j⟫_𝕜 | Batteries.Tactic._aux_Batteries_Tactic_Trans___elabRules_Batteries_Tactic_tacticTrans____1 | Batteries.Tactic.tacticTrans___ |
Mathlib.Analysis.InnerProductSpace.JointEigenspace | {
"line": 95,
"column": 2
} | {
"line": 96,
"column": 9
} | [
{
"pp": "case e_p\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nα : 𝕜\nA B : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nhB : B.IsSymmetric\nhAB : Commute A B\n⊢ ⨆ i, (genEigenspace (B.restrict ⋯) i) 1 = ⊤",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.JointEigenspace | {
"line": 104,
"column": 2
} | {
"line": 104,
"column": 64
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nA B : E →ₗ[𝕜] E\ninst✝ : FiniteDimensional 𝕜 E\nhA : A.IsSymmetric\nhB : B.IsSymmetric\nhAB : Commute A B\n⊢ ⨆ α, ⨆ γ, eigenspace A α ⊓ eigenspace B γ = ⊤",
"usedConstants": [
"E... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.LaxMilgram | {
"line": 63,
"column": 23
} | {
"line": 63,
"column": 34
} | [
{
"pp": "V : Type u\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : CompleteSpace V\nB : V →L[ℝ] V →L[ℝ] ℝ\nC : ℝ\nC_ge_0 : 0 < C\ncoercivity : ∀ (u : V), C * ‖u‖ * ‖u‖ ≤ (B u) u\nv : V\nh : ¬0 < ‖v‖\n⊢ v = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.LaxMilgram | {
"line": 72,
"column": 2
} | {
"line": 72,
"column": 13
} | [
{
"pp": "V : Type u\ninst✝² : NormedAddCommGroup V\ninst✝¹ : InnerProductSpace ℝ V\ninst✝ : CompleteSpace V\nB : V →L[ℝ] V →L[ℝ] ℝ\ncoercive : IsCoercive B\nC : ℝ\nC_pos : 0 < C\nbelow_bound : ∀ (v : V), C * ‖v‖ ≤ ‖(continuousLinearMapOfBilin B) v‖\n⊢ ∀ (x : V), C⁻¹⁻¹ * ‖x‖ ≤ ‖(continuousLinearMapOfBilin B) x‖"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.BirkhoffSum.Basic | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 27
} | [
{
"pp": "α : Type u_1\nM : Type u_2\ninst✝ : AddCommMonoid M\nf : α → α\ng g' : α → M\nn : ℕ\nx : α\n⊢ birkhoffSum f (g + g') n x = birkhoffSum f g n x + birkhoffSum f g' n x",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Membership.mem",
"id",
"Nat.iterate",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 13
} | [
{
"pp": "A : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : No... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 138,
"column": 2
} | {
"line": 138,
"column": 24
} | [
{
"pp": "A : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : No... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 142,
"column": 2
} | {
"line": 142,
"column": 24
} | [
{
"pp": "A : Type u_2\ninst✝¹¹ : NonUnitalRing A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Module ℝ A\ninst✝⁷ : SMulCommClass ℝ A A\ninst✝⁶ : IsScalarTower ℝ A A\ninst✝⁵ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁴ : PartialOrder A\ninst✝³ : StarOrderedRing A\ninst✝² : No... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 154,
"column": 49
} | {
"line": 154,
"column": 60
} | [
{
"pp": "A : Type u_2\ninst✝¹⁶ : NonUnitalRing A\ninst✝¹⁵ : StarRing A\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : Module ℝ A\ninst✝¹² : SMulCommClass ℝ A A\ninst✝¹¹ : IsScalarTower ℝ A A\ninst✝¹⁰ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁹ : PartialOrder A\ninst✝⁸ : StarOrderedRing A\ninst✝⁷... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Dynamics.BirkhoffSum.NormedSpace | {
"line": 123,
"column": 6
} | {
"line": 123,
"column": 17
} | [
{
"pp": "case hab.h\n𝕜 : Type u_1\nX : Type u_2\nE : Type u_3\ninst✝³ : PseudoEMetricSpace X\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : X → X\ng : X → E\nhf : LipschitzWith 1 f\nhg : UniformContinuous g\nε : ℝ\nhε : 0 < ε\nδ : ℝ≥0∞\nhδ₀ : 0 < δ\nhδε : ∀ (x y : X), (x, y) ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 204,
"column": 55
} | {
"line": 204,
"column": 66
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹⁶ : RCLike 𝕜\ninst✝¹⁵ : NonUnitalRing A\ninst✝¹⁴ : TopologicalSpace A\ninst✝¹³ : Module 𝕜 A\ninst✝¹² : StarRing A\ninst✝¹¹ : PartialOrder A\ninst✝¹⁰ : StarOrderedRing A\ninst✝⁹ : IsScalarTower 𝕜 A A\ninst✝⁸ : SMulCommClass 𝕜 A A\ninst✝⁷ : NonUnitalCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 242,
"column": 2
} | {
"line": 242,
"column": 46
} | [
{
"pp": "A : Type u_2\ninst✝¹⁰ : Ring A\ninst✝⁹ : StarRing A\ninst✝⁸ : TopologicalSpace A\ninst✝⁷ : Algebra ℝ A\ninst✝⁶ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁵ : PartialOrder A\ninst✝⁴ : StarOrderedRing A\ninst✝³ : NonnegSpectrumClass ℝ A\ninst✝² : IsTopologicalRing A\ninst✝¹ : T2Space A\ninst✝... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Abs | {
"line": 246,
"column": 2
} | {
"line": 246,
"column": 13
} | [
{
"pp": "A : Type u_2\ninst✝¹¹ : Ring A\ninst✝¹⁰ : StarRing A\ninst✝⁹ : TopologicalSpace A\ninst✝⁸ : Algebra ℝ A\ninst✝⁷ : ContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝⁶ : PartialOrder A\ninst✝⁵ : StarOrderedRing A\ninst✝⁴ : NonnegSpectrumClass ℝ A\ninst✝³ : IsTopologicalRing A\ninst✝² : T2Space A\ninst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.MeanErgodic | {
"line": 59,
"column": 4
} | {
"line": 60,
"column": 11
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(eqLocus f 1)\nhg_proj : ∀ (x : ↥(eqLocus f 1)), g ↑x = x\nhg_ker : ↑(↑g).ker ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.MeanErgodic | {
"line": 72,
"column": 6
} | {
"line": 72,
"column": 21
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(eqLocus f 1)\nhg_proj : ∀ (x : ↥(eqLocus f 1)), g ↑x = x\nhg_ker : ↑(↑g).ker ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.MeanErgodic | {
"line": 74,
"column": 2
} | {
"line": 75,
"column": 9
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : E →ₗ[𝕜] E\nhf : LipschitzWith 1 ⇑f\ng : E →L[𝕜] ↥(eqLocus f 1)\nhg_proj : ∀ (x : ↥(eqLocus f 1)), g ↑x = x\nhg_ker : ↑(↑g).ker ⊆ closure[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] ↑... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.MeanErgodic | {
"line": 105,
"column": 6
} | {
"line": 105,
"column": 17
} | [
{
"pp": "case hg_ker.refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nf : E →L[𝕜] E\nhf : ‖f‖ ≤ 1\nx : E\nhx : x ∈ (↑f - 1).rangeᗮ\n⊢ ‖f x‖ ≤ ‖1 x‖",
"usedConstants": [
"Norm.norm",
"InnerProductS... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.MeanErgodic | {
"line": 107,
"column": 8
} | {
"line": 107,
"column": 75
} | [
{
"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\nf : E →L[𝕜] E\nhf : ‖f‖ ≤ 1\nx : E\nhx : x ∈ (↑f - 1).rangeᗮ\n⊢ ∀ (y : E), ⟪f y, x⟫ = ⟪y, x⟫",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.SingularValues | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 51
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝⁶ : RCLike 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\ninst✝³ : FiniteDimensional 𝕜 E\nF : Type u_3\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace 𝕜 F\ninst✝ : FiniteDimensional 𝕜 F\nT : E →ₗ[𝕜] F\ni j : ℕ\nhij : i ≤ j\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.OfNorm | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 28
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : InnerProductSpaceable E\nr : ℚ\nx y : E\na b : 𝕜\n⊢ (fun r ↦ inner_ 𝕜 (r • x) y) (a + b) = (fun r ↦ inner_ 𝕜 (r • x) y) a + (fun r ↦ inner_ 𝕜 (r • x) y) b",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.OfNorm | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 33
} | [
{
"pp": "𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : InnerProductSpaceable E\nr : ℚ\nx y : E\nhom : 𝕜 →ₗ[ℚ] 𝕜 := (AddMonoidHom.mk' (fun r ↦ inner_ 𝕜 (r • x) y) ⋯).toRatLinearMap\n⊢ inner_ 𝕜 (↑r • x) y = (starRingEnd 𝕜) ↑r * inner_ 𝕜 x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.OfNorm | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 15
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : RCLike 𝕜\nE : Type u_2\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : InnerProductSpaceable E\nhI : I = 0\n⊢ innerProp' E 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.StandardSubspace | {
"line": 110,
"column": 4
} | {
"line": 110,
"column": 40
} | [
{
"pp": "case mp\nH : Type u_1\ninst✝ : NormedAddCommGroup H\nipc : InnerProductSpace ℂ H\nx : H\nS : ClosedSubmodule ℝ H\nh : ∀ (u : H), -(I • u) ∈ S → inner ℝ u x = 0\ny : H\nhy : y ∈ S\nhiy : y ∈ S → inner ℝ (I • y) x = 0\n⊢ ⟪y, x⟫.im = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.StandardSubspace | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 33
} | [
{
"pp": "case mpr\nH : Type u_1\ninst✝ : NormedAddCommGroup H\nipc : InnerProductSpace ℂ H\nx : H\nS : ClosedSubmodule ℝ H\nh : ∀ y ∈ S, ⟪y, x⟫.im = 0\nu✝ : H\nhy : -(I • u✝) ∈ S\nhiy : ⟪-(I • u✝), x⟫.im = 0\n⊢ inner ℝ u✝ x = 0",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.StandardSubspace | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 46
} | [
{
"pp": "case carrier.h.mp\nH : Type u_1\ninst✝ : NormedAddCommGroup H\nipc : InnerProductSpace ℂ H\nS : ClosedSubmodule ℝ H\nx : H\nh : ∀ (a : ℝ), a • (scalarSMulCLE H UnitI).symm x ∈ S.mulI\n⊢ x ∈ S",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.StandardSubspace | {
"line": 140,
"column": 4
} | {
"line": 140,
"column": 46
} | [
{
"pp": "case carrier.h.mpr\nH : Type u_1\ninst✝ : NormedAddCommGroup H\nipc : InnerProductSpace ℂ H\nS : ClosedSubmodule ℝ H\nx : H\nh : ∀ (a : ℝ), a • x ∈ S\n⊢ x ∈ S.mulI.mulI",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Units.val",
"Eq.mpr",
"InnerProductSpace.toNormedSpa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.StandardSubspace | {
"line": 161,
"column": 2
} | {
"line": 161,
"column": 46
} | [
{
"pp": "H : Type u_1\ninst✝ : NormedAddCommGroup H\nipc : InnerProductSpace ℂ H\nS T : ClosedSubmodule ℝ H\n⊢ (S.mulI ⊔ T.mulI)ᗮ = S.mulIᗮ ⊓ T.mulIᗮ",
"usedConstants": [
"InnerProductSpace.toNormedSpace",
"Real",
"NormedSpace.toIsBoundedSMul",
"UniformContinuousConstSMul.to_continuo... | exact Eq.symm (inf_orthogonal S.mulI T.mulI) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.StarOrder | {
"line": 48,
"column": 6
} | {
"line": 49,
"column": 13
} | [
{
"pp": "𝕜 : Type u_1\nH : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup H\ninst✝³ : InnerProductSpace 𝕜 H\ninst✝² : CompleteSpace H\ninst✝¹ : Algebra ℝ (H →L[𝕜] H)\ninst✝ : IsScalarTower ℝ 𝕜 (H →L[𝕜] H)\nf : H →L[𝕜] H\nhf : f.IsPositive\nc✝ : ℝ\nc : ℝ := -c✝\nhc : 0 < c\nx : H\n⊢ re ⟪((algebr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.StandardSubspace | {
"line": 206,
"column": 21
} | {
"line": 206,
"column": 57
} | [
{
"pp": "H : Type u_1\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℂ H\nS : StandardSubspace H\n⊢ S.toClosedSubmodule.mulI ⊓ S.toClosedSubmodule.mulI.mulI = ⊥",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"ClosedSubmodule.mulI",
"congrAr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.StandardSubspace | {
"line": 207,
"column": 17
} | {
"line": 207,
"column": 53
} | [
{
"pp": "H : Type u_1\ninst✝¹ : NormedAddCommGroup H\ninst✝ : InnerProductSpace ℂ H\nS : StandardSubspace H\n⊢ S.toClosedSubmodule.mulI ⊔ S.toClosedSubmodule.mulI.mulI = ⊤",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"NormedSpace.toIsBoundedSMul",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.StarOrder | {
"line": 74,
"column": 8
} | {
"line": 74,
"column": 19
} | [
{
"pp": "case mpr.mem\n𝕜 : Type u_1\nH : Type u_2\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup H\ninst✝⁴ : InnerProductSpace 𝕜 H\ninst✝³ : CompleteSpace H\ninst✝² : Algebra ℝ (H →L[𝕜] H)\ninst✝¹ : IsScalarTower ℝ 𝕜 (H →L[𝕜] H)\ninst✝ : ContinuousFunctionalCalculus ℝ (H →L[𝕜] H) IsSelfAdjoint\nf✝ p f :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 25
} | [
{
"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\nT : ι → E →ₗ[𝕜] E\ns : Finset ι\nhT : ∀ i ∈ s, (T i).IsPositive\nx✝ : E\n⊢ 0 ≤ re ⟪(∑ i ∈ s, T i) x✝, x✝⟫",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 151,
"column": 23
} | {
"line": 151,
"column": 48
} | [
{
"pp": "𝕜 : 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.IsPositive\nhT' : T ≠ 0\nα : 𝕜\nh : (α • T).IsPositive\n⊢ ?m.48",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 152,
"column": 13
} | {
"line": 152,
"column": 43
} | [
{
"pp": "𝕜 : 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.IsPositive\nhT' : T ≠ 0\nα : 𝕜\nh : (α • T).IsPositive\nx : E\nhx : 0 < ⟪x, T x⟫\n⊢ ?m.78",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.InnerProductSpace.Positive | {
"line": 158,
"column": 2
} | {
"line": 160,
"column": 11
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : FiniteDimensional 𝕜 E\nT : E →ₗ[𝕜] E\nn : ℕ\nhT : T.IsPositive\nhn : Module.finrank 𝕜 E = n\ni : Fin n\n⊢ 0 ≤ ⋯.eigenvalues hn i",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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