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.Data.Matrix.PEquiv | {
"line": 165,
"column": 16
} | {
"line": 165,
"column": 24
} | [
{
"pp": "case neg\nn : Type u_4\nα : Type u_5\ninst✝¹ : DecidableEq n\ninst✝ : AddGroupWithOne α\ni j i✝ j✝ : n\nh✝⁴ : ¬i✝ = i\nh✝³ : ¬i✝ = j\nh✝² : j✝ ∉ some i✝\nh✝¹ : ¬i✝ = j✝\nh✝ : j✝ ∉ none\n⊢ 0 = 0 - 0 - 0 + 0 + 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"AddMonoid.toAddSemigrou... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Convex.Approximation | {
"line": 192,
"column": 56
} | {
"line": 192,
"column": 68
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_2\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace ℝ E\nhφc : Lower... | sSup_image', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Between | {
"line": 833,
"column": 4
} | {
"line": 833,
"column": 13
} | [
{
"pp": "case inl\nR : Type u_1\ninst✝² : Field R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx y z : R\nhxz✝ : x ≤ z\nhxz : x = z\n⊢ Wbtw R x y z ↔ x ≤ y ∧ y ≤ z",
"usedConstants": []
}
] | subst hxz | Lean.Elab.Tactic.evalSubst | Lean.Parser.Tactic.subst |
Mathlib.Analysis.Convex.Between | {
"line": 1073,
"column": 4
} | {
"line": 1073,
"column": 12
} | [
{
"pp": "case inl\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : y -ᵥ x = 0\n⊢ Wbtw R x y z ∨ Wbtw R x z y",
"usedConstants": [
"congrArg",
"A... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Convex.Between | {
"line": 1073,
"column": 4
} | {
"line": 1073,
"column": 12
} | [
{
"pp": "case inl\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : y -ᵥ x = 0\n⊢ Wbtw R x y z ∨ Wbtw R x z y",
"usedConstants": [
"congrArg",
"A... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Between | {
"line": 1073,
"column": 4
} | {
"line": 1073,
"column": 12
} | [
{
"pp": "case inl\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : y -ᵥ x = 0\n⊢ Wbtw R x y z ∨ Wbtw R x z y",
"usedConstants": [
"congrArg",
"A... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Between | {
"line": 1074,
"column": 4
} | {
"line": 1074,
"column": 12
} | [
{
"pp": "case inr.inl\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : z -ᵥ x = 0\n⊢ Wbtw R x y z ∨ Wbtw R x z y",
"usedConstants": [
"congrArg",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Convex.Between | {
"line": 1074,
"column": 4
} | {
"line": 1074,
"column": 12
} | [
{
"pp": "case inr.inl\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : z -ᵥ x = 0\n⊢ Wbtw R x y z ∨ Wbtw R x z y",
"usedConstants": [
"congrArg",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Between | {
"line": 1074,
"column": 4
} | {
"line": 1074,
"column": 12
} | [
{
"pp": "case inr.inl\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y z : P\nh : z -ᵥ x = 0\n⊢ Wbtw R x y z ∨ Wbtw R x z y",
"usedConstants": [
"congrArg",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Birkhoff | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 19
} | [
{
"pp": "case refine_2\nR : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Field R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\n⊢ {x | ∃ σ, Equiv.Perm.permMatrix R σ = x} ⊆ Set.extremePoints R ↑(doublyStochastic R n)",
"usedConstants": [
"Matrix"
]
}
] | rintro _ ⟨σ, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.Convex.Birkhoff | {
"line": 192,
"column": 4
} | {
"line": 192,
"column": 12
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Field R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nσ : Equiv.Perm n\nx₁ : Matrix n n R\nhx₁ hx₂ : x₁ ∈ ↑(doublyStochastic R n)\nhσ : Equiv.Perm.permMatrix R σ ∈ openSegment R x₁ x₁\nthis : ∀ (i j : n), x₁ i j ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Convex.Cone.Pointed | {
"line": 351,
"column": 11
} | {
"line": 351,
"column": 43
} | [
{
"pp": "R : Type u_1\nE : Type u_2\ninst✝⁴ : Ring R\ninst✝³ : LinearOrder R\ninst✝² : IsOrderedRing R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\nC : PointedCone R E\n⊢ C.lineal = sSup {S | ↑S ≤ C}",
"usedConstants": [
"Eq.mpr",
"Submodule",
"IsOrderedRing.toPosMulMono",
"IsOrdere... | gc_ofSubmodule_lineal.le_iff_le, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Convex.Cone.Dual | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 34
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : PartialOrder R\ninst✝⁹ : IsOrderedRing R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : ClosedIciTopology R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : TopologicalSpace M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ ... | ext; simp [forall_comm (α := M)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Cone.Dual | {
"line": 98,
"column": 2
} | {
"line": 98,
"column": 34
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : PartialOrder R\ninst✝⁹ : IsOrderedRing R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : ClosedIciTopology R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : TopologicalSpace M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ ... | ext; simp [forall_comm (α := M)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Cone.Dual | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 34
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : PartialOrder R\ninst✝⁹ : IsOrderedRing R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : ClosedIciTopology R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : TopologicalSpace M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ ... | ext; simp [forall_comm (α := M)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Cone.Dual | {
"line": 101,
"column": 2
} | {
"line": 101,
"column": 34
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\ninst✝¹¹ : CommRing R\ninst✝¹⁰ : PartialOrder R\ninst✝⁹ : IsOrderedRing R\ninst✝⁸ : TopologicalSpace R\ninst✝⁷ : ClosedIciTopology R\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : Module R M\ninst✝⁴ : TopologicalSpace M\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ ... | ext; simp [forall_comm (α := M)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Convex.Cone.Dual | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommSemiring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : IsOrderedRing R\nM : Type u_2\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\np : M →ₗ[R] N →ₗ[R] R\nι : Sort u_4\nf : ι → Set M\n⊢ dual p (⋃ i, f i) = ⨅ i, dual p (f i)"... | ext; simp [forall_comm (α := M)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Convex.Cone.Dual | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommSemiring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : IsOrderedRing R\nM : Type u_2\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\np : M →ₗ[R] N →ₗ[R] R\nι : Sort u_4\nf : ι → Set M\n⊢ dual p (⋃ i, f i) = ⨅ i, dual p (f i)"... | ext; simp [forall_comm (α := M)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Convex.Cone.Dual | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommSemiring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : IsOrderedRing R\nM : Type u_2\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\np : M →ₗ[R] N →ₗ[R] R\nS : Set (Set M)\n⊢ dual p (⋃₀ S) = sInf (dual p '' S)",
"usedCons... | ext; simp [forall_comm (α := M)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Convex.Cone.Dual | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 34
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommSemiring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : IsOrderedRing R\nM : Type u_2\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\np : M →ₗ[R] N →ₗ[R] R\nS : Set (Set M)\n⊢ dual p (⋃₀ S) = sInf (dual p '' S)",
"usedCons... | ext; simp [forall_comm (α := M)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Convex.Cone.Dual | {
"line": 109,
"column": 23
} | {
"line": 109,
"column": 98
} | [
{
"pp": "case smul\nR : Type u_1\ninst✝⁶ : CommSemiring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : IsOrderedRing R\nM : Type u_2\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\np : M →ₗ[R] N →ₗ[R] R\ns : Set M\nx : N\nhx : x ∈ dual p s\ny✝ : M\nt : R≥0\ny : ... | rw [map_smul_of_tower, Nonneg.mk_smul, smul_apply]; exact mul_nonneg t.2 hy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Convex.Cone.Dual | {
"line": 109,
"column": 23
} | {
"line": 109,
"column": 98
} | [
{
"pp": "case smul\nR : Type u_1\ninst✝⁶ : CommSemiring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : IsOrderedRing R\nM : Type u_2\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\np : M →ₗ[R] N →ₗ[R] R\ns : Set M\nx : N\nhx : x ∈ dual p s\ny✝ : M\nt : R≥0\ny : ... | rw [map_smul_of_tower, Nonneg.mk_smul, smul_apply]; exact mul_nonneg t.2 hy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Intrinsic | {
"line": 168,
"column": 76
} | {
"line": 170,
"column": 16
} | [
{
"pp": "𝕜 : Type u_1\nV : Type u_2\nP : Type u_5\ninst✝⁴ : Ring 𝕜\ninst✝³ : AddCommGroup V\ninst✝² : Module 𝕜 V\ninst✝¹ : TopologicalSpace P\ninst✝ : AddTorsor V P\ns : Set P\n⊢ intrinsicInterior 𝕜 s ∪ intrinsicFrontier 𝕜 s = intrinsicClosure 𝕜 s",
"usedConstants": [
"frontier",
"intrinsi... | by
simp [intrinsicClosure, intrinsicInterior, intrinsicFrontier, closure_eq_interior_union_frontier,
image_union] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Join | {
"line": 72,
"column": 34
} | {
"line": 72,
"column": 44
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_3\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns₁ s₂ t : Set E\n⊢ ⋃ x, ⋃ (_ : x ∈ s₁ ∨ x ∈ s₂), ⋃ y ∈ t, segment 𝕜 x y =\n (⋃ x ∈ s₁, ⋃ y ∈ t, segment 𝕜 x y) ∪ ⋃ x ∈ s₂, ⋃ y ∈ t, segment 𝕜 x y",
"usedConstants": [
... | iUnion_or, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Convex.Intrinsic | {
"line": 272,
"column": 2
} | {
"line": 272,
"column": 57
} | [
{
"pp": "case h\n𝕜 : Type u_1\nV : Type u_2\nP : Type u_5\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : CompleteSpace 𝕜\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace 𝕜 V\ninst✝² : FiniteDimensional 𝕜 V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns : Set P\nx : P\n⊢ (∃ y,\n (∀ (o : Set... | refine ⟨?_, fun h => ⟨⟨x, _⟩, ?_, Subtype.coe_mk _ ?_⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Convex.ConvexSpace.Prod | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 12
} | [
{
"pp": "I : Type u_1\nR : Type u_2\ninst✝⁴ : Semiring R\ninst✝³ : PartialOrder R\ninst✝² : IsStrictOrderedRing R\nι : Type u_3\nX : Type u_4\ninst✝¹ : Zero X\ninst✝ : ConvexSpace R X\ni✝ : ι\nw : StdSimplex R (ι →₀ X)\ni : ι\nhi : i ∉ w.weights.support.biUnion support\n⊢ (iConvexComb w fun x ↦ x i) = 0",
"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Geometry.Convex.ConvexSpace.AffineSpace | {
"line": 91,
"column": 2
} | {
"line": 99,
"column": 32
} | [
{
"pp": "case neg\nR : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AffineSpace V P\nf : StdSimplex R (StdSimplex R P)\nb : P\nhL :\n (Finset.affineCombination R (StdSimplex.map convexCombin... | · refine Finset.sum_congr ?_ (fun _ _ => rfl)
ext p
simp only [Finsupp.mem_support_iff, ne_eq]
constructor
· intro hp
exact (mul_pos ((f.nonneg d).lt_of_ne' hd) ((d.nonneg p).lt_of_ne' hp)).ne'
· intro hp hp'
simp only [Finsupp.coe_smul, Pi.smul_apply, smul_eq_mul, hp', mul_zero,
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Convex.Set | {
"line": 131,
"column": 4
} | {
"line": 131,
"column": 12
} | [
{
"pp": "case neg\nR : Type u_3\nX : Type u_5\nY : Type u_6\ninst✝⁴ : Semiring R\ninst✝³ : PartialOrder R\ninst✝² : IsStrictOrderedRing R\ninst✝¹ : ConvexSpace R X\ninst✝ : ConvexSpace R Y\nf : X → Y\ns : Set X\nhf : IsAffineMap R f\nhs : IsConvexSet R s\nw : StdSimplex R Y\nhw : ↑w.weights.support ⊆ f '' s\nu ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Convex.Quasiconvex | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 58
} | [
{
"pp": "case h\n𝕜 : Type u_1\nE : Type u_2\nβ : Type u_3\ninst✝⁴ : Semiring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddCommMonoid E\ninst✝¹ : LinearOrder β\ninst✝ : SMul 𝕜 E\ns : Set E\nf : E → β\n⊢ ((∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 ≤ a → 0 ≤ b → a + b = 1 → f (a • x + b • y) ≤ ... | simp_rw [← forall_and, ← Icc_min_max, mem_Icc, and_comm] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 186,
"column": 4
} | {
"line": 186,
"column": 57
} | [
{
"pp": "case ha\nX : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\ni : ↑(Set.Icc 0 1)\nx y : X × X\n⊢ ↑i * dist x.1 y.1 + (1 - ↑i) * dist x.2 y.2 ≤ max (dist x.1 y.1) (dist x.2 y.2)",
"usedConstants": [
"Real.instIsOrderedRing",
"le_refl",
"Real.parti... | nth_grw 1 [le_max_left (dist x.1 y.1) (dist x.2 y.2)] | Mathlib.Tactic._aux_Mathlib_Tactic_GRewrite_Elab___macroRules_Mathlib_Tactic_tacticNth_grw______1 | Mathlib.Tactic.tacticNth_grw_____ |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 215,
"column": 67
} | {
"line": 215,
"column": 75
} | [
{
"pp": "X : Type u_2\ninst✝³ : ConvexSpace ℝ X\ninst✝² : MetricSpace X\ninst✝¹ : IsConvexDist X\nT : Type u_3\ninst✝ : TopologicalSpace T\nf : T → ℝ\nhf : Continuous f\nhf0 : ∀ (t : T), 0 ≤ f t\nhf1 : ∀ (t : T), f t ≤ 1\nx y : T → X\nhx : ContinuousOn x (f ⁻¹' {0}ᶜ)\nhy : ContinuousOn y (f ⁻¹' {1}ᶜ)\nhx' : Bor... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 216,
"column": 53
} | {
"line": 216,
"column": 61
} | [
{
"pp": "X : Type u_2\ninst✝³ : ConvexSpace ℝ X\ninst✝² : MetricSpace X\ninst✝¹ : IsConvexDist X\nT : Type u_3\ninst✝ : TopologicalSpace T\nf : T → ℝ\nhf : Continuous f\nhf0 : ∀ (t : T), 0 ≤ f t\nhf1 : ∀ (t : T), f t ≤ 1\nx y : T → X\nhx : ContinuousOn x (f ⁻¹' {0}ᶜ)\nhy : ContinuousOn y (f ⁻¹' {1}ᶜ)\nhx' : Bor... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Convex.SimplicialComplex.AffineIndependentUnion | {
"line": 77,
"column": 4
} | {
"line": 79,
"column": 90
} | [
{
"pp": "case a.hs\n𝕜 : Type ?u.15364\nE : Type ?u.15649\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : DecidableEq E\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nabstract : PreAbstractSimplicialComplex E\nindep : AffineIndependent 𝕜 Subtype.val\ns t : Finset E\nhs : ... | · apply indep.mono
simp only [Finset.coe_union]
exact Set.union_subset (Set.subset_biUnion_of_mem hs) (Set.subset_biUnion_of_mem ht) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.StrictConvexBetween | {
"line": 66,
"column": 18
} | {
"line": 66,
"column": 29
} | [
{
"pp": "case neg\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : StrictConvexSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\np p₁ p₂ p₃ : P\nr : ℝ\nh : Collinear ℝ {p₁, p₂, p₃}\nhp₁ : dist p₁ p = r\nhp₂ : dist p₂ p ≤ r\nhp₃ : dist p₃ p = r\nh... | lt_max_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.StrictConvexBetween | {
"line": 72,
"column": 18
} | {
"line": 72,
"column": 29
} | [
{
"pp": "case neg\nV : Type u_1\nP : Type u_2\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : StrictConvexSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\np p₁ p₂ p₃ : P\nr : ℝ\nh : Collinear ℝ {p₁, p₂, p₃}\nhp₁ : dist p₁ p = r\nhp₂ : dist p₂ p ≤ r\nhp₃ : dist p₃ p = r\nh... | lt_max_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Side | {
"line": 236,
"column": 2
} | {
"line": 239,
"column": 76
} | [
{
"pp": "case mp\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v ∈ s.direction\n⊢ s.WSameSide (v +ᵥ x) y → s.WSameSide x y"... | · rintro ⟨p₁, hp₁, p₂, hp₂, h⟩
refine
⟨-v +ᵥ p₁, AffineSubspace.vadd_mem_of_mem_direction (Submodule.neg_mem _ hv) hp₁, p₂, hp₂, ?_⟩
rwa [vsub_vadd_eq_vsub_sub, sub_neg_eq_add, add_comm, ← vadd_vsub_assoc] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Side | {
"line": 259,
"column": 2
} | {
"line": 262,
"column": 76
} | [
{
"pp": "case mp\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : CommRing R\ninst✝⁴ : PartialOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ns : AffineSubspace R P\nx y : P\nv : V\nhv : v ∈ s.direction\n⊢ s.WOppSide (v +ᵥ x) y → s.WOppSide x y",
... | · rintro ⟨p₁, hp₁, p₂, hp₂, h⟩
refine
⟨-v +ᵥ p₁, AffineSubspace.vadd_mem_of_mem_direction (Submodule.neg_mem _ hv) hp₁, p₂, hp₂, ?_⟩
rwa [vsub_vadd_eq_vsub_sub, sub_neg_eq_add, add_comm, ← vadd_vsub_assoc] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Side | {
"line": 540,
"column": 4
} | {
"line": 540,
"column": 67
} | [
{
"pp": "case mpr.inr\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ns : AffineSubspace R P\nx y : P\nh : y ∈ s\n⊢ s.WSameSide x y ∧ s.WOppSide x y",
"usedConstants": [
... | · exact ⟨wSameSide_of_right_mem x h, wOppSide_of_right_mem x h⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Visible | {
"line": 81,
"column": 4
} | {
"line": 86,
"column": 50
} | [
{
"pp": "case inl\n𝕜 : Type u_1\nV : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup V\ninst✝ : Module 𝕜 V\ns : Set V\nx : V\nι : Type u_4\nt : Finset ι\na : ι → V\nw : ι → 𝕜\nhw₀ : ∀ i ∈ t, 0 ≤ w i\nhw₁ : ∑ i ∈ t, w i = 1\nha : ∀ i ∈ t, a i ∈ s\nh... | convert! hw
rw [← one_smul 𝕜 (a i), ← hwi, eq_comm]
rw [← hwi, ← sub_eq_zero, ← sum_erase_eq_sub hi,
sum_eq_zero_iff_of_nonneg fun j hj ↦ hw₀ _ <| erase_subset _ _ hj] at hw₁
refine sum_eq_single _ (fun j hj hji ↦ ?_) (by simp [hi])
rw [hw₁ _ <| mem_erase.2 ⟨hji, hj⟩, zero_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Visible | {
"line": 81,
"column": 4
} | {
"line": 86,
"column": 50
} | [
{
"pp": "case inl\n𝕜 : Type u_1\nV : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup V\ninst✝ : Module 𝕜 V\ns : Set V\nx : V\nι : Type u_4\nt : Finset ι\na : ι → V\nw : ι → 𝕜\nhw₀ : ∀ i ∈ t, 0 ≤ w i\nhw₁ : ∑ i ∈ t, w i = 1\nha : ∀ i ∈ t, a i ∈ s\nh... | convert! hw
rw [← one_smul 𝕜 (a i), ← hwi, eq_comm]
rw [← hwi, ← sub_eq_zero, ← sum_erase_eq_sub hi,
sum_eq_zero_iff_of_nonneg fun j hj ↦ hw₀ _ <| erase_subset _ _ hj] at hw₁
refine sum_eq_single _ (fun j hj hji ↦ ?_) (by simp [hi])
rw [hw₁ _ <| mem_erase.2 ⟨hji, hj⟩, zero_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Visible | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 72
} | [
{
"pp": "V : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module ℝ V\ns : Set V\ny : V\ninst✝² : TopologicalSpace V\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : ContinuousSMul ℝ V\nhs : IsClosed[inst✝²] s\nhy : y ∈ s\nx : V\nt : Set ℝ := Set.Ici 0 ∩ ⇑(lineMap x y) ⁻¹' s\nht₁ : 1 ∈ t\nht : BddBelow t\nδ : ℝ := sInf... | replace hδ₀ : 0 < δ := hδ₀.lt_of_ne' <| by rintro hδ₀; simp [hδ₀] at h | Lean.Elab.Tactic.evalReplace | Lean.Parser.Tactic.replace |
Mathlib.Analysis.Convex.Side | {
"line": 839,
"column": 37
} | {
"line": 839,
"column": 45
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex R P n\nw₁ w₂ : Fin (n + 1) → R\nhw₁ : ∑ j, w₁ j = 1\nhw₂ : ∑ j, w₂ j = 1\ni : ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Convex.Side | {
"line": 840,
"column": 2
} | {
"line": 841,
"column": 72
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AffineSpace V P\nn : ℕ\ninst✝ : NeZero n\ns : Simplex R P n\nw₁ w₂ : Fin (n + 1) → R\nhw₁ : ∑ j, w₁ j = 1\nhw₂ : ∑ j, w₂ j = 1\ni : ... | refine ⟨?_, (s.affineCombination_mem_affineSpan_faceOpposite_iff hw₁).not.2 h0,
(s.affineCombination_mem_affineSpan_faceOpposite_iff hw₂).not.2 h0'⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Convex.StrictCombination | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 12
} | [
{
"pp": "case inl\nV : Type u_2\nι : Type u_4\ninst✝² : NormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\ninst✝ : StrictConvexSpace ℝ V\nt : Finset ι\nw : ι → ℝ\np : V\nz : ι → V\nh0 : ∀ i ∈ t, 0 ≤ w i\ni j : ι\nhi : i ∈ t\nhj : j ∈ t\nhij : z i ≠ z j\nhi0 : w i ≠ 0\nhj0 : w j ≠ 0\nhz : ∀ i ∈ t, z i ∈ closedBall ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Convex.StrictCombination | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 12
} | [
{
"pp": "case inl\nV : Type u_2\nι : Type u_4\ninst✝² : NormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\ninst✝ : StrictConvexSpace ℝ V\nt : Finset ι\nw : ι → ℝ\np : V\nz : ι → V\nh0 : ∀ i ∈ t, 0 ≤ w i\ni j : ι\nhi : i ∈ t\nhj : j ∈ t\nhij : z i ≠ z j\nhi0 : w i ≠ 0\nhj0 : w j ≠ 0\nhz : ∀ i ∈ t, z i ∈ closedBall ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.StrictCombination | {
"line": 96,
"column": 4
} | {
"line": 96,
"column": 12
} | [
{
"pp": "case inl\nV : Type u_2\nι : Type u_4\ninst✝² : NormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\ninst✝ : StrictConvexSpace ℝ V\nt : Finset ι\nw : ι → ℝ\np : V\nz : ι → V\nh0 : ∀ i ∈ t, 0 ≤ w i\ni j : ι\nhi : i ∈ t\nhj : j ∈ t\nhij : z i ≠ z j\nhi0 : w i ≠ 0\nhj0 : w j ≠ 0\nhz : ∀ i ∈ t, z i ∈ closedBall ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.StrictCombination | {
"line": 137,
"column": 4
} | {
"line": 137,
"column": 12
} | [
{
"pp": "V : Type u_2\nP : Type u_3\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : NormedSpace ℝ V\ninst✝² : StrictConvexSpace ℝ V\ninst✝¹ : PseudoMetricSpace P\ninst✝ : NormedAddTorsor V P\nn : ℕ\ns : Simplex ℝ P n\nr : ℝ\np₀ : P\nhr : ∀ (i : Fin (n + 1)), dist (s.points i) p₀ ≤ r\nw : Fin (n + 1) → ℝ\nhw : ∑ i, w i... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Normed.Group.ZeroAtInfty | {
"line": 35,
"column": 54
} | {
"line": 35,
"column": 62
} | [
{
"pp": "E : Type u_1\nF : Type u_2\n𝓕 : Type u_3\ninst✝³ : SeminormedAddGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : FunLike 𝓕 E F\ninst✝ : ZeroAtInftyContinuousMapClass 𝓕 E F\nf : 𝓕\nε : ℝ\nhε : 0 < ε\nh : (fun x ↦ ‖f x‖) ⁻¹' Metric.ball 0 ε ∈ cocompact E\nr : ℝ\nhr : (Metric.closedBall 0 r)ᶜ ⊆ (f... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Normed.Group.ZeroAtInfty | {
"line": 35,
"column": 54
} | {
"line": 35,
"column": 62
} | [
{
"pp": "E : Type u_1\nF : Type u_2\n𝓕 : Type u_3\ninst✝³ : SeminormedAddGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : FunLike 𝓕 E F\ninst✝ : ZeroAtInftyContinuousMapClass 𝓕 E F\nf : 𝓕\nε : ℝ\nhε : 0 < ε\nh : (fun x ↦ ‖f x‖) ⁻¹' Metric.ball 0 ε ∈ cocompact E\nr : ℝ\nhr : (Metric.closedBall 0 r)ᶜ ⊆ (f... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.ZeroAtInfty | {
"line": 35,
"column": 54
} | {
"line": 35,
"column": 62
} | [
{
"pp": "E : Type u_1\nF : Type u_2\n𝓕 : Type u_3\ninst✝³ : SeminormedAddGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : FunLike 𝓕 E F\ninst✝ : ZeroAtInftyContinuousMapClass 𝓕 E F\nf : 𝓕\nε : ℝ\nhε : 0 < ε\nh : (fun x ↦ ‖f x‖) ⁻¹' Metric.ball 0 ε ∈ cocompact E\nr : ℝ\nhr : (Metric.closedBall 0 r)ᶜ ⊆ (f... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.ZeroAtInfty | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 10
} | [
{
"pp": "case h.a\nE : Type u_1\nF : Type u_2\n𝓕 : Type u_3\ninst✝³ : SeminormedAddGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : FunLike 𝓕 E F\ninst✝ : ZeroAtInftyContinuousMapClass 𝓕 E F\nf : 𝓕\nε : ℝ\nhε : 0 < ε\nh : (fun x ↦ ‖f x‖) ⁻¹' Metric.ball 0 ε ∈ cocompact E\nr : ℝ\nhr : (Metric.closedBall ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.MeasureTheory.Function.L2Space | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 54
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup F\nf : ↥(Lp F 2 μ)\nh_two : ENNReal.ofReal 2 = 2\n⊢ eLpNorm (fun x ↦ ‖↑↑f x‖ ^ 2) 1 μ < ∞",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Mul... | rw [eLpNorm_norm_rpow f zero_lt_two, one_mul, h_two] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.L2Space | {
"line": 145,
"column": 6
} | {
"line": 145,
"column": 40
} | [
{
"pp": "α : Type u_1\nE : Type u_2\n𝕜 : Type u_4\ninst✝³ : RCLike 𝕜\ninst✝² : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nf : ↥(Lp E 2 μ)\n⊢ ∫ (x : α), ‖↑↑f x‖ ^ 2 ∂μ = (∫⁻ (a : α), ↑(‖↑↑f a‖₊ ^ 2) ∂μ).toReal",
"usedConstants": [
"Norm.norm",
... | integral_eq_lintegral_of_nonneg_ae | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.L2Space | {
"line": 227,
"column": 2
} | {
"line": 227,
"column": 72
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_4\ninst✝¹ : RCLike 𝕜\ninst✝ : MeasurableSpace α\nμ : Measure α\ns : Set α\nhs : MeasurableSet s\nhμs : μ s ≠ ∞\nf : ↥(Lp 𝕜 2 μ)\n⊢ ⟪indicatorConstLp 2 hs hμs 1, f⟫ = ∫ (x : α) in s, ↑↑f x ∂μ",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
... | rw [L2.inner_indicatorConstLp_eq_inner_setIntegral 𝕜 hs hμs (1 : 𝕜) f] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 80,
"column": 2
} | {
"line": 80,
"column": 79
} | [
{
"pp": "case h\nE : Type u_5\nF : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nhf_temperate : HasTemperateGrowth f\nN : ℕ\nk : ℕ → ℕ\nhk : ∀ (n : ℕ), iteratedFDeriv ℝ n f =O[⊤] fun x ↦ (1 + ‖x‖) ^ k n\nn : ℕ\nhn : n ≤ N\nx... | rw [Real.norm_of_nonneg (by positivity), Real.norm_of_nonneg (by positivity)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Distribution.TemperateGrowth | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 18
} | [
{
"pp": "D : Type u_4\nE : Type u_5\nF : Type u_6\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedAddCommGroup D\ninst✝ : NormedSpace ℝ D\ng : E → F\nf : D → E\nt : Set E\nht : Set.range f ⊆ t\nht' : UniqueDiffOn ℝ t\nhg₁ : ContDi... | intro i hi hi' | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.MeasureTheory.Function.ContinuousMapDense | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : NormalSpace α\ninst✝⁴ : MeasurableSpace α\ninst✝³ : BorelSpace α\nE : Type u_2\ninst✝² : NormedAddCommGroup E\nμ : Measure α\np : ℝ≥0∞\ninst✝¹ : NormedSpace ℝ E\ninst✝ : μ.OuterRegular\nhp : p ≠ ∞\ns u : Set α\ns_closed : IsClosed[inst✝⁶] s\nu_open : ... | have hsv : s ⊆ v := subset_inter hsu sV | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Distribution.SchwartzSpace.Deriv | {
"line": 204,
"column": 4
} | {
"line": 204,
"column": 38
} | [
{
"pp": "case succ\nE : Type u_5\nF : Type u_8\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : NormedSpace ℝ E\nf : 𝓢(E, F)\nn : ℕ\nIH : ∀ (m : Fin n → E), tsupport ⇑(∂^{m} f) ⊆ tsupport ⇑f\nm : Fin (n + 1) → E\n⊢ tsupport ⇑(∂^{m} f) ⊆ tsupport ⇑f",
"usedCon... | rw [iteratedLineDerivOp_succ_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 265,
"column": 2
} | {
"line": 265,
"column": 15
} | [
{
"pp": "case h.e'_2.h.e'_9.h\nT : ℝ\nhT : Fact (0 < T)\np : ℝ≥0∞\ninst✝ : Fact (1 ≤ p)\nhp : p ≠ ∞\ne_3✝ : Complex.instSemiring = NormedField.toNormedCommRing.toSemiring\ne_6✝ : Lp.instModule ≍ Lp.instModule\n⊢ span ℂ (range (fourierLp p)) = Submodule.map (↑(toLp p haarAddCircle ℂ)) (span ℂ (range fourier))",
... | rw [map_span] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Distribution.ContDiffMapSupportedIn | {
"line": 946,
"column": 2
} | {
"line": 946,
"column": 93
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝¹³ : NontriviallyNormedField 𝕜\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace ℝ E\nn : ℕ∞\nK : Compacts E\nm : MeasurableSpace E\ninst✝¹⁰ : OpensMeasurableSpace E\nF₁ : Type u_5\nF₂ : Type u_6\nF₃ : Type u_7\ninst✝⁹ : NormedAddCommGroup F₁\ninst✝⁸ : NormedSpac... | rw [integralAgainstBilinLM_eq_integral hφ, setIntegral_eq_integral_of_forall_compl_eq_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 121,
"column": 2
} | {
"line": 123,
"column": 97
} | [
{
"pp": "case h.e'_12.h.h\nV : Type u_1\nW : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : NormedAddCommGroup W\ninst✝ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nv : V\nw : W\nha : HasFDerivAt (fun w' ↦ (L v) w') (L v) w\ne_8✝ : addCommGroup = instNormedAddCommGroup.toAddCommGroup\n... | simp only [neg_mul, ContinuousLinearMap.coe_smul', ContinuousLinearMap.coe_comp', Pi.smul_apply,
Function.comp_apply, ofRealCLM_apply, smul_eq_mul, ContinuousLinearMap.comp_neg,
ContinuousLinearMap.neg_apply, ContinuousLinearMap.toSpanSingleton_apply, real_smul, neg_inj] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Measure.Lebesgue.Integral | {
"line": 94,
"column": 6
} | {
"line": 94,
"column": 18
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nc : ℝ\nf : ℝ → E\n⊢ ∫ (x : ℝ) in Ioi c, f (-x) = ∫ (x : ℝ) in Iic (-c), f x",
"usedConstants": [
"Eq.mpr",
"Real",
"Set.Ioi",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"congrArg",
"Measu... | ← neg_neg c, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 391,
"column": 6
} | {
"line": 391,
"column": 36
} | [
{
"pp": "a : ℂ\nr : ℝ\nha : 0 < a.re\nhr : 0 < r\naux : (1 / ↑r) ^ a = 1 / ↑r * (1 / ↑r) ^ (a - 1)\n⊢ 1 / ↑r * (1 / ↑r) ^ (a - 1) * ∫ (t : ℝ) in Ioi 0, ↑t ^ (a - 1) * cexp (-↑t) = (1 / ↑r) ^ a * Gamma a",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"InnerProductSpace.t... | rw [aux, Gamma_eq_integral ha] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Fourier.FourierTransformDeriv | {
"line": 513,
"column": 2
} | {
"line": 515,
"column": 37
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℂ E\nV : Type u_2\nW : Type u_3\ninst✝⁶ : NormedAddCommGroup V\ninst✝⁵ : NormedSpace ℝ V\ninst✝⁴ : NormedAddCommGroup W\ninst✝³ : NormedSpace ℝ W\nL : V →L[ℝ] W →L[ℝ] ℝ\nf : V → E\ninst✝² : MeasurableSpace V\ninst✝¹ : BorelSpac... | · have : fourierIntegral 𝐞 μ L.toLinearMap₁₂ f = 0 := by
ext w; simp [fourierIntegral, integral, h'f]
simpa [this] using contDiff_const | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.PolarCoord | {
"line": 270,
"column": 35
} | {
"line": 270,
"column": 95
} | [
{
"pp": "ι : Type u_1\ninst✝² : Fintype ι\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : (ι → ℝ × ℝ) → E\n⊢ (∫ (p : ι → ℝ × ℝ) in Set.univ.pi fun x ↦ polarCoord.target, (∏ i, (p i).1) • f fun i ↦ ↑polarCoord.symm (p i)) =\n ∫ (x : ι → ℝ × ℝ) in Set.univ, f x",
"usedConstants":... | ← setIntegral_congr_set pi_polarCoord_symm_target_ae_eq_univ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 173,
"column": 2
} | {
"line": 189,
"column": 85
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\n⊢ ∫ (r : ℝ) in Ioi 0, ↑r * cexp (-b * ↑r ^ 2) = (2 * b)⁻¹",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"IsModuleTopology.toContinuousSMul",
"NormedCommRing.toNormedRing",
"AddG... | have hb' : b ≠ 0 := by contrapose! hb; rw [hb, zero_re]
have A : ∀ x : ℂ, HasDerivAt (fun x => -(2 * b)⁻¹ * cexp (-b * x ^ 2))
(x * cexp (-b * x ^ 2)) x := by
intro x
convert! ((hasDerivAt_pow 2 x).const_mul (-b)).cexp.const_mul (-(2 * b)⁻¹) using 1
field
have B : Tendsto (fun y : ℝ ↦ -(2 * b)⁻¹ * c... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 173,
"column": 2
} | {
"line": 189,
"column": 85
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\n⊢ ∫ (r : ℝ) in Ioi 0, ↑r * cexp (-b * ↑r ^ 2) = (2 * b)⁻¹",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"IsModuleTopology.toContinuousSMul",
"NormedCommRing.toNormedRing",
"AddG... | have hb' : b ≠ 0 := by contrapose! hb; rw [hb, zero_re]
have A : ∀ x : ℂ, HasDerivAt (fun x => -(2 * b)⁻¹ * cexp (-b * x ^ 2))
(x * cexp (-b * x ^ 2)) x := by
intro x
convert! ((hasDerivAt_pow 2 x).const_mul (-b)).cexp.const_mul (-(2 * b)⁻¹) using 1
field
have B : Tendsto (fun y : ℝ ↦ -(2 * b)⁻¹ * c... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 323,
"column": 48
} | {
"line": 323,
"column": 59
} | [
{
"pp": "case h.e'_3\nb : ℝ\nhb : 0 < b\n⊢ ↑((π / b) ^ (1 / 2)) / ↑2 = ↑(π / b) ^ (1 / 2) / 2",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"instHDiv",
"Real.pi",
"congrArg",
"Real.instDivInvMonoid",
"Complex.ofReal_cpow",
"Nat.instAtLeastTwoHAd... | ofReal_cpow | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Fourier.Inversion | {
"line": 119,
"column": 21
} | {
"line": 119,
"column": 32
} | [
{
"pp": "case h'φ\nV : Type u_1\nE : Type u_2\ninst✝⁷ : NormedAddCommGroup V\ninst✝⁶ : InnerProductSpace ℝ V\ninst✝⁵ : MeasurableSpace V\ninst✝⁴ : BorelSpace V\ninst✝³ : FiniteDimensional ℝ V\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\nf : V → E\ninst✝ : CompleteSpace E\nhf : Integrable f volume\n... | ← pow_one π | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gaussian.FourierTransform | {
"line": 314,
"column": 4
} | {
"line": 314,
"column": 22
} | [
{
"pp": "case h.e'_2.h.e'_7.h.h.e'_1.h.e'_6\nb : ℂ\nι : Type u_2\ninst✝ : Fintype ι\nhb : 0 < b.re\nc : ℂ\nw : EuclideanSpace ℝ ι\ny : ι → ℝ\n⊢ ∑ i, c * (↑(y i) * ↑(w.ofLp i)) = ∑ x, c * (↑(w.ofLp x) * ↑(y x))",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Sem... | simp_rw [mul_comm] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Distribution.FourierMultiplier | {
"line": 111,
"column": 2
} | {
"line": 112,
"column": 12
} | [
{
"pp": "E : Type u_3\nF : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ E\ninst✝⁴ : NormedSpace ℂ F\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : BorelSpace E\ninst✝ : CompleteSpace F\nf : 𝓢(E, F)\nι : Type := Fin (Module.finrank ℝ... | have : ∀ i (hi : i ∈ Finset.univ), (inner ℝ · (b i) ^ 2).HasTemperateGrowth := by
fun_prop | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Distribution.FourierMultiplier | {
"line": 220,
"column": 2
} | {
"line": 220,
"column": 66
} | [
{
"pp": "case e_f.h.h\nE : Type u_3\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace ℝ E\ninst✝³ : NormedSpace ℂ F\ninst✝² : FiniteDimensional ℝ E\ninst✝¹ : MeasurableSpace E\ninst✝ : BorelSpace E\nf : 𝓢'(E, F)\nι : Type := Fin (Module.finrank ℝ E)\nb : Or... | simp_rw [lineDeriv_eq_fourierMultiplierCLM, map_smul, smul_smul] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 79,
"column": 39
} | {
"line": 79,
"column": 47
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : RCLike 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℂ E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℂ 𝕜 E\nV : Type u_3\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\nin... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 82,
"column": 89
} | {
"line": 82,
"column": 97
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : RCLike 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℂ E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SMulCommClass ℂ 𝕜 E\nV : Type u_3\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\nin... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Distribution.SchwartzSpace.Fourier | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 44
} | [
{
"pp": "case h\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℂ E\nV : Type u_3\ninst✝⁴ : NormedAddCommGroup V\ninst✝³ : InnerProductSpace ℝ V\ninst✝² : FiniteDimensional ℝ V\ninst✝¹ : MeasurableSpace V\ninst✝ : BorelSpace V\nf : 𝓢(V, E)\nx : V\n⊢ (𝓕⁻ f) x = 𝓕⁻ (⇑f) x",
"usedConstant... | exact (fourierInv_eq_fourier_neg f x).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Fourier.BoundedContinuousFunctionChar | {
"line": 157,
"column": 67
} | {
"line": 162,
"column": 38
} | [
{
"pp": "V : Type u_1\nW : Type u_2\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module ℝ V\ninst✝³ : TopologicalSpace V\ninst✝² : AddCommGroup W\ninst✝¹ : Module ℝ W\ninst✝ : TopologicalSpace W\ne : AddChar ℝ Circle\nL : V →ₗ[ℝ] W →ₗ[ℝ] ℝ\nhe : Continuous ⇑e\nhL : Continuous fun p ↦ (L p.1) p.2\nw : W\n⊢ char he hL w ∈ ... | by
rw [mem_charPoly]
refine ⟨AddMonoidAlgebra.single w 1, ?_⟩
ext v
simp only [char_apply, AddMonoidAlgebra.single]
rw [Finset.sum_eq_single w] <;> simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Distribution.Sobolev | {
"line": 331,
"column": 53
} | {
"line": 332,
"column": 50
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : InnerProductSpace ℝ E\ninst✝⁴ : FiniteDimensional ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : InnerProductSpace ℂ F\ninst✝ : CompleteSpace F\ns : ℝ\nf : 𝓢'(E, F)\nhf : MemSobolev s 2... | by
rw [← Real.rpow_mul (by positivity)]; simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 283,
"column": 4
} | {
"line": 285,
"column": 40
} | [
{
"pp": "d : Type u_1\ninst✝ : Fintype d\nf : ↥(Lp ℂ 2 volume)\ni : d → ℤ\n⊢ ∫ (t : UnitAddTorus d), (starRingEnd ℂ) (↑↑(mFourierLp 2 i) t) * ↑↑f t = mFourierCoeff (↑↑f) i",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"MeasureTheory.ae",
"Eq.mpr",
"InnerProductSpace.toNorm... | apply integral_congr_ae
filter_upwards [coeFn_mFourierLp 2 i] with _ ht
rw [ht, ← mFourier_neg, smul_eq_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 283,
"column": 4
} | {
"line": 285,
"column": 40
} | [
{
"pp": "d : Type u_1\ninst✝ : Fintype d\nf : ↥(Lp ℂ 2 volume)\ni : d → ℤ\n⊢ ∫ (t : UnitAddTorus d), (starRingEnd ℂ) (↑↑(mFourierLp 2 i) t) * ↑↑f t = mFourierCoeff (↑↑f) i",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"MeasureTheory.ae",
"Eq.mpr",
"InnerProductSpace.toNorm... | apply integral_congr_ae
filter_upwards [coeFn_mFourierLp 2 i] with _ ht
rw [ht, ← mFourier_neg, smul_eq_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.FiniteAbelian.Orthogonality | {
"line": 68,
"column": 91
} | {
"line": 69,
"column": 60
} | [
{
"pp": "G : Type u_1\nR : Type u_3\ninst✝² : AddCommGroup G\ninst✝¹ : RCLike R\nψ₁ ψ₂ : AddChar G R\ninst✝ : Fintype G\n⊢ wInner cWeight ⇑ψ₁ ⇑ψ₂ = 0 ↔ ψ₁ ≠ ψ₂",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Ne.ite_eq_right_iff",
... | by
rw [wInner_cWeight_eq_boole, one_ne_zero.ite_eq_right_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.GroupTheory.FiniteAbelian.Basic | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 72
} | [
{
"pp": "case pos\nι : Type\ninst✝ : DecidableEq ι\np n : ι → ℕ\ni : ι\nx : ZMod (p i ^ n i)\nh : n i = 0\n⊢ (directSumNeZeroMulHom p n) (0 x) = (DirectSum.of (fun i ↦ ZMod (p i ^ n i)) i) x",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAs... | · simp [(ZMod.subsingleton_iff.2 <| by rw [h, pow_zero]).elim x 0] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.GroupTheory.FiniteAbelian.Basic | {
"line": 159,
"column": 2
} | {
"line": 159,
"column": 36
} | [
{
"pp": "G : Type u_1\ninst✝¹ : AddCommGroup G\ninst✝ : Finite G\nι : Type\nhι : Fintype ι\np : ι → ℕ\nhp : ∀ (i : ι), Nat.Prime (p i)\nn : ι → ℕ\ne : G ≃+ ⨁ (i : ι), ZMod (p i ^ n i)\ni : ι\nhi : n i ≠ 0\n⊢ 1 < (fun i ↦ p ↑i ^ n ↑i) ⟨i, hi⟩",
"usedConstants": [
"IsOrderedRing.toPosMulMono",
"Na... | exact one_lt_pow₀ (hp _).one_lt hi | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 64,
"column": 22
} | {
"line": 64,
"column": 52
} | [
{
"pp": "n : ℕ\ninst✝ : NeZero n\n⊢ ∀ ⦃a₁ a₂ : ZMod n⦄, zmod n a₁ = zmod n a₂ → a₁ = a₂",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Function.Surjective.forall",
"ZMod.commRing",
"AddGroupWithOne.toAddMonoidWithOne",
"AddChar",
"id",
"AddChar.zmod",
"DivI... | ZMod.intCast_surjective.forall | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Distribution.TemperedDistribution | {
"line": 354,
"column": 4
} | {
"line": 355,
"column": 74
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nF₁ : Type u_5\nF₂ : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ F\nx y : E\nf : 𝓢'(E, F)\n⊢ ∂_{x + y} f = ∂_{x} f + ∂_{y} f",
"usedConstants": [
"neg_add_rev",... | ext u
simp [lineDerivOp_left_add, UniformConvergenceCLM.add_apply, add_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Distribution.TemperedDistribution | {
"line": 354,
"column": 4
} | {
"line": 355,
"column": 74
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nF₁ : Type u_5\nF₂ : Type u_6\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℂ F\nx y : E\nf : 𝓢'(E, F)\n⊢ ∂_{x + y} f = ∂_{x} f + ∂_{y} f",
"usedConstants": [
"neg_add_rev",... | ext u
simp [lineDerivOp_left_add, UniformConvergenceCLM.add_apply, add_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.FiniteAbelian.PontryaginDuality | {
"line": 156,
"column": 35
} | {
"line": 156,
"column": 90
} | [
{
"pp": "α : Type u_1\ninst✝¹ : AddCommGroup α\ninst✝ : Finite α\na : α\nha : a ∈ doubleDualEmb.ker\nψ : AddChar α ℂ\n⊢ ψ a = 1",
"usedConstants": [
"DFunLike.congr_fun",
"Equiv.instEquivLike",
"Complex.commRing",
"Additive",
"AddMonoid.toAddZeroClass",
"AddCommGroup.toAd... | by simpa using DFunLike.congr_fun ha (Additive.ofMul ψ) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Fourier.PoissonSummation | {
"line": 113,
"column": 4
} | {
"line": 115,
"column": 87
} | [
{
"pp": "case h.e'_2\nf : C(ℝ, ℂ)\nh_norm : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖\nh_sum : Summable fun n ↦ 𝓕 ⇑f ↑n\nx : ℝ\nF : C(UnitAddCircle, ℂ) := { toFun := ⋯.lift, continuous_toFun := ⋯ }\nthis : Summable (fourierCoeff ⇑F)\n⊢ ∑' (n : ℤ), f... | simpa only [F, coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add]
using (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum x).tsum_eq | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Fourier.PoissonSummation | {
"line": 113,
"column": 4
} | {
"line": 115,
"column": 87
} | [
{
"pp": "case h.e'_2\nf : C(ℝ, ℂ)\nh_norm : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖\nh_sum : Summable fun n ↦ 𝓕 ⇑f ↑n\nx : ℝ\nF : C(UnitAddCircle, ℂ) := { toFun := ⋯.lift, continuous_toFun := ⋯ }\nthis : Summable (fourierCoeff ⇑F)\n⊢ ∑' (n : ℤ), f... | simpa only [F, coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add]
using (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum x).tsum_eq | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Fourier.PoissonSummation | {
"line": 113,
"column": 4
} | {
"line": 115,
"column": 87
} | [
{
"pp": "case h.e'_2\nf : C(ℝ, ℂ)\nh_norm : ∀ (K : Compacts ℝ), Summable fun n ↦ ‖ContinuousMap.restrict (↑K) (f.comp (ContinuousMap.addRight ↑n))‖\nh_sum : Summable fun n ↦ 𝓕 ⇑f ↑n\nx : ℝ\nF : C(UnitAddCircle, ℂ) := { toFun := ⋯.lift, continuous_toFun := ⋯ }\nthis : Summable (fourierCoeff ⇑F)\n⊢ ∑' (n : ℤ), f... | simpa only [F, coe_mk, ← QuotientAddGroup.mk_zero, Periodic.lift_coe, zsmul_one, comp_apply,
coe_addRight, zero_add]
using (hasSum_apply (summable_of_locally_summable_norm h_norm).hasSum x).tsum_eq | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 264,
"column": 2
} | {
"line": 268,
"column": 76
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ℕ → R\nhf : ∀ (n : ℕ), 0 < n → f n ≠ 0\nhg : ∀ (n : ℕ), 0 < n → g n ≠ 0\nn : ℕ\nhn : n > 0\n⊢ ∏ i ∈ n.divisors, f i = g n ↔\n (∏ i ∈ n.divisors, if h : 0 < i then Units.mk0 (f i) ⋯ else 1) = if h : 0 < n then Units.mk0 (g n) ⋯ else 1",
... | · dsimp
rw [dif_pos hn, ← Units.val_inj, ← Units.coeHom_apply, map_prod, Units.val_mk0,
prod_congr rfl _]
intro x hx
rw [dif_pos (pos_of_mem_divisors hx), Units.coeHom_apply, Units.val_mk0] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 350,
"column": 2
} | {
"line": 354,
"column": 76
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝ : CommGroupWithZero R\ns : Set ℕ\nhs : ∀ (m n : ℕ), m ∣ n → n ∈ s → m ∈ s\nf g : ℕ → R\nhf : ∀ n > 0, f n ≠ 0\nhg : ∀ n > 0, g n ≠ 0\nn : ℕ\nhn : n > 0\n⊢ n ∈ s → ∏ i ∈ n.divisors, f i = g n ↔\n n ∈ s →\n ∏ i ∈ n.divisors, (fun n ↦ if h : 0 < n then Units.mk0 ... | · dsimp
rw [dif_pos hn, ← Units.val_inj, ← Units.coeHom_apply, map_prod, Units.val_mk0,
prod_congr rfl _]
intro x hx
rw [dif_pos (pos_of_mem_divisors hx), Units.coeHom_apply, Units.val_mk0] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.RootsOfUnity.Complex | {
"line": 46,
"column": 4
} | {
"line": 46,
"column": 41
} | [
{
"pp": "case right\ni : ℤ\nn : ℕ\nh0 : n ≠ 0\nhi : IsCoprime i ↑n\n⊢ ∀ (l : ℕ) (x : ℤ), ↑l * (2 * ↑π * I * (↑i / ↑n)) = ↑x * (2 * ↑π * I) → n ∣ l",
"usedConstants": [
"congrArg",
"AddMonoid.toAddZeroClass",
"AddGroupWithOne.toAddMonoidWithOne",
"Complex.instZero",
"cast",
... | have hn0 : (n : ℂ) ≠ 0 := mod_cast h0 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 364,
"column": 4
} | {
"line": 365,
"column": 40
} | [
{
"pp": "case pos\nG : Type u_3\ninst✝ : DivisionCommMonoid G\nk : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nhi : i.gcd ↑k = 1\nh0 : 0 ≤ i\n⊢ IsPrimitiveRoot (ζ ^ i) k",
"usedConstants": [
"zpow_natCast",
"Int.gcd",
"DivisionCommMonoid.toDivisionMonoid",
"congrArg",
"DivInvMono... | lift i to ℕ using h0
exact_mod_cast h.pow_of_coprime i hi | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 364,
"column": 4
} | {
"line": 365,
"column": 40
} | [
{
"pp": "case pos\nG : Type u_3\ninst✝ : DivisionCommMonoid G\nk : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\ni : ℤ\nhi : i.gcd ↑k = 1\nh0 : 0 ≤ i\n⊢ IsPrimitiveRoot (ζ ^ i) k",
"usedConstants": [
"zpow_natCast",
"Int.gcd",
"DivisionCommMonoid.toDivisionMonoid",
"congrArg",
"DivInvMono... | lift i to ℕ using h0
exact_mod_cast h.pow_of_coprime i hi | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Fourier.RiemannLebesgueLemma | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 20
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℝ → E\n⊢ Tendsto (fun w ↦ ∫ (v : ℝ), 𝐞 (-(v * w)) • f v) (cocompact ℝ) (𝓝 0)",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Real",
"instHSMul",
"No... | simp_rw [mul_comm] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 456,
"column": 8
} | {
"line": 456,
"column": 29
} | [
{
"pp": "M : Type u_1\nN : Type u_2\nG : Type u_3\nR : Type u_4\nS : Type u_5\nF : Type u_6\ninst✝³ : CommMonoid M\ninst✝² : CommMonoid N\ninst✝¹ : DivisionCommMonoid G\nk l : ℕ\ninst✝ : CommRing R\nζ : Rˣ\nh✝ h : IsPrimitiveRoot ζ k\ni : ℤ\nhi : ↑k ∣ i\n⊢ { toFun := fun i ↦ Additive.ofMul ⟨ζ ^ i, ⋯⟩, map_zero'... | obtain ⟨i, rfl⟩ := hi | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 552,
"column": 4
} | {
"line": 552,
"column": 50
} | [
{
"pp": "case refine_1\nR : Type u_4\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\nk : ℕ\ninst✝ : NeZero k\nζ : Rˣ\nh : IsPrimitiveRoot ζ k\nn : ℤ\nhξ : ζ ^ n ∈ rootsOfUnity k R\nhk0 : 0 < ↑k\ni : ℤ := n % ↑k\ni₀ : ℕ\nhi₀ : ↑i₀ = i\n⊢ i₀ < k",
"usedConstants": [
"Eq.mpr",
"Int.ofNat_lt._simp_2",
... | zify; rw [hi₀]; exact Int.emod_lt_of_pos _ hk0 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 552,
"column": 4
} | {
"line": 552,
"column": 50
} | [
{
"pp": "case refine_1\nR : Type u_4\ninst✝² : CommRing R\ninst✝¹ : IsDomain R\nk : ℕ\ninst✝ : NeZero k\nζ : Rˣ\nh : IsPrimitiveRoot ζ k\nn : ℤ\nhξ : ζ ^ n ∈ rootsOfUnity k R\nhk0 : 0 < ↑k\ni : ℤ := n % ↑k\ni₀ : ℕ\nhi₀ : ↑i₀ = i\n⊢ i₀ < k",
"usedConstants": [
"Eq.mpr",
"Int.ofNat_lt._simp_2",
... | zify; rw [hi₀]; exact Int.emod_lt_of_pos _ hk0 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.RootsOfUnity.Minpoly | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 12
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝² : CommRing K\nμ : K\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nh : IsPrimitiveRoot μ 0\nhdiv : ¬p ∣ 0\n⊢ minpoly ℤ μ ∣ (expand ℤ p) (minpoly ℤ (μ ^ p))",
"usedConstants": [
"False",
"Dvd.dvd",
"CommRing.toNonUnitalCommRing",
"Nat.instSemi... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.RootsOfUnity.Minpoly | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 12
} | [
{
"pp": "case inl\nK : Type u_1\ninst✝² : CommRing K\nμ : K\ninst✝¹ : IsDomain K\ninst✝ : CharZero K\np : ℕ\nh : IsPrimitiveRoot μ 0\nhdiv : ¬p ∣ 0\n⊢ minpoly ℤ μ ∣ (expand ℤ p) (minpoly ℤ (μ ^ p))",
"usedConstants": [
"False",
"Dvd.dvd",
"CommRing.toNonUnitalCommRing",
"Nat.instSemi... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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