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.Complex.ValueDistribution.LogCounting.Basic | {
"line": 535,
"column": 6
} | {
"line": 535,
"column": 94
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : ProperSpace 𝕜\nf₁ f₂ : 𝕜 → 𝕜\nr : ℝ\nhr : 1 ≤ r\nh₁f₁ : Meromorphic f₁\nh₂f₁ : ∀ (z : 𝕜), meromorphicOrderAt f₁ z ≠ ⊤\nh₁f₂ : Meromorphic f₂\nh₂f₂ : ∀ (z : 𝕜), meromorphicOrderAt f₂ z ≠ ⊤\n⊢ locallyFinsuppWithin.logCounting (divisor (f₁ *... | divisor_mul h₁f₁.meromorphicOn h₁f₂.meromorphicOn (fun z _ ↦ h₂f₁ z) (fun z _ ↦ h₂f₂ z), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.CofilteredSystem | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 38
} | [
{
"pp": "case inl\nJ : Type u\ninst✝³ : Category.{w, u} J\ninst✝² : IsCofilteredOrEmpty J\nF : J ⥤ Type v\ninst✝¹ : ∀ (j : J), Finite (F.obj j)\ninst✝ : ∀ (j : J), Nonempty (F.obj j)\nJ' : Type (max w v u) := AsSmall J\ndown : J' ⥤ J := AsSmall.down\nF' : J' ⥤ Type (max u v w) := down ⋙ F ⋙ uliftFunctor.{max u ... | · fconstructor <;> apply isEmptyElim | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Hall.Basic | {
"line": 100,
"column": 4
} | {
"line": 100,
"column": 31
} | [
{
"pp": "ι : Type u\nα : Type v\nt : ι → Finset α\nι' : Finset ι\ng : ↑(hallMatchingsOn t ι') → ↥ι' → ↥(ι'.biUnion t) := fun f i ↦ ⟨↑f i, ⋯⟩\n⊢ Finite ↑{f | Injective f ∧ ∀ (x : ↥ι'), f x ∈ t ↑x}",
"usedConstants": [
"Finset",
"Pi.finite",
"hallMatchingsOn",
"Classical.propDecidable"... | apply Finite.of_injective g | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Data.PEquiv | {
"line": 116,
"column": 16
} | {
"line": 116,
"column": 83
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\ng : β ≃. γ\na : α\nb : γ\n⊢ (g.symm b).bind ⇑f.symm = some a ↔ (f a).bind ⇑g = some b",
"usedConstants": [
"PEquiv.instFunLikeOption",
"congrArg",
"Option.some",
"Exists",
"Option.bind",
"_private.Mathli... | simp_all [and_comm, eq_some_iff f, eq_some_iff g, bind_eq_some_iff] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.PEquiv | {
"line": 116,
"column": 16
} | {
"line": 116,
"column": 83
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\ng : β ≃. γ\na : α\nb : γ\n⊢ (g.symm b).bind ⇑f.symm = some a ↔ (f a).bind ⇑g = some b",
"usedConstants": [
"PEquiv.instFunLikeOption",
"congrArg",
"Option.some",
"Exists",
"Option.bind",
"_private.Mathli... | simp_all [and_comm, eq_some_iff f, eq_some_iff g, bind_eq_some_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.PEquiv | {
"line": 116,
"column": 16
} | {
"line": 116,
"column": 83
} | [
{
"pp": "α : Type u\nβ : Type v\nγ : Type w\nδ : Type x\nf : α ≃. β\ng : β ≃. γ\na : α\nb : γ\n⊢ (g.symm b).bind ⇑f.symm = some a ↔ (f a).bind ⇑g = some b",
"usedConstants": [
"PEquiv.instFunLikeOption",
"congrArg",
"Option.some",
"Exists",
"Option.bind",
"_private.Mathli... | simp_all [and_comm, eq_some_iff f, eq_some_iff g, bind_eq_some_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Permutation | {
"line": 94,
"column": 21
} | {
"line": 94,
"column": 60
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝² : DecidableEq n\nσ✝ τ✝ : Perm n\ninst✝¹ : Fintype n\ninst✝ : NonAssocSemiring R\nσ τ : Perm n\n⊢ Perm.permMatrix R (σ * τ)⁻¹ = Perm.permMatrix R σ⁻¹ * Perm.permMatrix R τ⁻¹",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | rw [_root_.mul_inv_rev, permMatrix_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Matrix.Permutation | {
"line": 94,
"column": 21
} | {
"line": 94,
"column": 60
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝² : DecidableEq n\nσ✝ τ✝ : Perm n\ninst✝¹ : Fintype n\ninst✝ : NonAssocSemiring R\nσ τ : Perm n\n⊢ Perm.permMatrix R (σ * τ)⁻¹ = Perm.permMatrix R σ⁻¹ * Perm.permMatrix R τ⁻¹",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | rw [_root_.mul_inv_rev, permMatrix_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Permutation | {
"line": 94,
"column": 21
} | {
"line": 94,
"column": 60
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝² : DecidableEq n\nσ✝ τ✝ : Perm n\ninst✝¹ : Fintype n\ninst✝ : NonAssocSemiring R\nσ τ : Perm n\n⊢ Perm.permMatrix R (σ * τ)⁻¹ = Perm.permMatrix R σ⁻¹ * Perm.permMatrix R τ⁻¹",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | rw [_root_.mul_inv_rev, permMatrix_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Stochastic | {
"line": 85,
"column": 57
} | {
"line": 91,
"column": 38
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nM : Matrix n n R\nx : n → R\nhM : M ∈ rowStochastic R n\nhx : ∀ (i : n), 0 ≤ x i\n⊢ ∀ (j : n), 0 ≤ (x ᵥ* M) j",
"usedConstants": [
"Eq.mpr",
"mu... | by
intro j
simp only [Matrix.vecMul, dotProduct]
apply Finset.sum_nonneg
intro k _
apply mul_nonneg (hx k)
exact nonneg_of_mem_rowStochastic hM | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.DoublyStochasticMatrix | {
"line": 167,
"column": 2
} | {
"line": 167,
"column": 24
} | [
{
"pp": "case mpr.inr\nR : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semifield R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nM : Matrix n n R\ns : R\nhs✝ : 0 ≤ s\nhs : 0 < s\n⊢ ((∀ (i j : n), 0 ≤ M i j) ∧ (∀ (i : n), ∑ j, M i j = s) ∧ ∀ (j : n), ∑ i, M i j = s) →\n... | rintro ⟨hM₁, hM₂, hM₃⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.Convex.Caratheodory | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 28
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : DecidableEq E\nt : Finset E\nf : E → 𝕜\nfpos : ∀ y ∈ t, 0 ≤ f y\nfsum : ∑ y ∈ t, f y = 1\ng : E → 𝕜\ngcombo : ∑ e ∈ t, g e • e = 0\ngsum : ∑ e... | let s := {z ∈ t | 0 < g z} | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Convex.Birkhoff | {
"line": 67,
"column": 4
} | {
"line": 67,
"column": 96
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁵ : Fintype n\ninst✝⁴ : DecidableEq n\ninst✝³ : Semifield R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\nM : Matrix n n R\ninst✝ : Nonempty n\ns : R\nhs : 0 < s\nhM : (∀ (i j : n), 0 ≤ M i j) ∧ (∀ (i : n), ∑ j, M i j = s) ∧ ∀ (j : n), ∑ i, M i j = s\nf : n →... | · exact sum_le_sum_of_subset_of_nonneg (by simp) fun _ _ _ => sum_nonneg fun j _ => hM.1 _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Body | {
"line": 217,
"column": 8
} | {
"line": 217,
"column": 32
} | [
{
"pp": "case h.refine_1\nV : Type u_1\ninst✝² : SeminormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\ninst✝ : T2Space V\nu : ℕ → ℝ≥0\nK : ConvexBody V\nh_zero : 0 ∈ K\nhu : Tendsto u atTop (𝓝 0)\nx : V\nh : x ∈ ⋂ n, (1 + ↑(u n)) • ↑K\nC : ℝ\nhC_pos : C > 0\nhC_bdd : ∀ x ∈ ↑K, ‖x‖ ≤ C\n⊢ x ∈ ↑K",
"usedConsta... | ← K.isClosed.closure_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Birkhoff | {
"line": 125,
"column": 8
} | {
"line": 125,
"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\nh✝ : Nonempty n\nd : ℕ\nih :\n ∀ m < d,\n ∀ (M : Matrix n n R) (s : R),\n 0 ≤ s →\n (∃ M' ∈ doublyStochastic R n, M = s • M') →\n #{i... | ← hd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Birkhoff | {
"line": 184,
"column": 88
} | {
"line": 202,
"column": 7
} | [
{
"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\n⊢ Set.extremePoints R ↑(doublyStochastic R n) = {x | ∃ σ, Equiv.Perm.permMatrix R σ = x}",
"usedConstants": [
"PEquiv.instFunLikeOption",
"Eq.... | by
refine subset_antisymm ?_ ?_
· rw [doublyStochastic_eq_convexHull_permMatrix]
exact extremePoints_convexHull_subset
rintro _ ⟨σ, rfl⟩
refine ⟨permMatrix_mem_doublyStochastic, fun x₁ hx₁ x₂ hx₂ hσ ↦ ?_⟩
suffices ∀ i j : n, x₁ i j = x₂ i j by
obtain rfl : x₁ = x₂ := by simpa [← Matrix.ext_iff]
si... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Cone.TensorProduct | {
"line": 110,
"column": 6
} | {
"line": 110,
"column": 55
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝⁹ : AddCommGroup E\ninst✝⁸ : Module ℝ E\ninst✝⁷ : AddCommGroup F\ninst✝⁶ : Module ℝ F\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : IsTopologicalAddGroup F\ninst✝³ : T2Space F\ninst✝² : FiniteDimensional ℝ F\ninst✝¹ : ContinuousSMul ℝ F\ninst✝ : LocallyConvexSpace ℝ F\nC₁ : Po... | ← (equivFinsuppOfBasisLeft b).symm_apply_apply z, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Convex.Cone.Dual | {
"line": 104,
"column": 2
} | {
"line": 109,
"column": 98
} | [
{
"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 M\n⊢ dual p ↑(hull R s) = dual p s",
"usedConstants": [
... | refine le_antisymm (dual_anti Submodule.subset_span) (fun x hx y hy => ?_)
induction hy using Submodule.span_induction with
| mem _y h => exact hx h
| zero => simp
| add y z _hy _hz hy hz => rw [map_add, add_apply]; exact add_nonneg hy hz
| smul t y _hy hy => rw [map_smul_of_tower, Nonneg.mk_smul, smul_apply]... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Convex.Cone.Dual | {
"line": 104,
"column": 2
} | {
"line": 109,
"column": 98
} | [
{
"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 M\n⊢ dual p ↑(hull R s) = dual p s",
"usedConstants": [
... | refine le_antisymm (dual_anti Submodule.subset_span) (fun x hx y hy => ?_)
induction hy using Submodule.span_induction with
| mem _y h => exact hx h
| zero => simp
| add y z _hy _hz hy hz => rw [map_add, add_apply]; exact add_nonneg hy hz
| smul t y _hy hy => rw [map_smul_of_tower, Nonneg.mk_smul, smul_apply]... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Exposed | {
"line": 79,
"column": 2
} | {
"line": 81,
"column": 12
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddCommMonoid E\ninst✝¹ : TopologicalSpace E\ninst✝ : Module 𝕜 E\nA B : Set E\nhAB : IsExposed 𝕜 A B\n⊢ B ⊆ A",
"usedConstants": [
"Semiring.toModule",
"ContinuousLinearMap.... | rintro x hx
obtain ⟨_, rfl⟩ := hAB ⟨x, hx⟩
exact hx.1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Exposed | {
"line": 79,
"column": 2
} | {
"line": 81,
"column": 12
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : TopologicalSpace 𝕜\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddCommMonoid E\ninst✝¹ : TopologicalSpace E\ninst✝ : Module 𝕜 E\nA B : Set E\nhAB : IsExposed 𝕜 A B\n⊢ B ⊆ A",
"usedConstants": [
"Semiring.toModule",
"ContinuousLinearMap.... | rintro x hx
obtain ⟨_, rfl⟩ := hAB ⟨x, hx⟩
exact hx.1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Extrema | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 67
} | [
{
"pp": "β : Type u_2\ninst✝⁵ : AddCommGroup β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : Module ℝ β\ninst✝¹ : IsOrderedModule ℝ β\ninst✝ : PosSMulReflectLE ℝ β\nf : ℝ → β\na b : ℝ\na_lt_b : a < b\nh_local_min : IsLocalMinOn f (Icc a b) a\nh_conv : ConvexOn ℝ (Icc a b) f\nc : ℝ\nhc : c ∈ ... | rw [IsLocalMinOn, nhdsWithin_Icc_eq_nhdsGE a_lt_b] at h_local_min | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Continuous | {
"line": 31,
"column": 4
} | {
"line": 31,
"column": 36
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nx₀ : E\nε r M : ℝ\nhf : ConvexOn ℝ (ball x₀ r) f\nhε : 0 < ε\nhM : ∀ (a : E), dist a x₀ < r → |f a| ≤ M\nK : ℝ := 2 * M / ε\nhK : K = 2 * M / ε\nx y : E\nhx : x ∈ ball x₀ (r - ε)\nhy : y ∈ ball x₀ (r - ε)\n⊢ f x - f y ≤ K ... | obtain rfl | hxy := eq_or_ne x y | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Convex.Independent | {
"line": 141,
"column": 8
} | {
"line": 141,
"column": 45
} | [
{
"pp": "case mpr\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nhc : ∀ t ⊆ s, s ∩ (convexHull 𝕜) t ⊆ t\nt : Set { x // x ∈ s }\nx : { x // x ∈ s }\nh : ↑x ∈ (convexHull 𝕜) (Subtype.val '' t)\n⊢ x ∈ t",
"usedConstants"... | ← Subtype.coe_injective.mem_set_image | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Continuous | {
"line": 152,
"column": 16
} | {
"line": 158,
"column": 48
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nC : Set E\nf : E → ℝ\nhC : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] C\nhC' : C.Nonempty\nhf : ConvexOn ℝ C f\ntfae_1_to_2 : LocallyLipschitzOn C f → ContinuousOn f C\ntfae_2_to_3 : ContinuousOn f C → ∃ x₀ ∈ C, Cont... | by
obtain ⟨r, hr⟩ := h hx
obtain ⟨ε, hε, hεD⟩ := Metric.mem_nhds_iff.1 <| Filter.inter_mem (hC.mem_nhds hx) hr
simp only [preimage_setOf_eq, Pi.abs_apply, subset_inter_iff, hC.nhdsWithin_eq hx] at hεD ⊢
obtain ⟨K, hK⟩ := exists_lipschitzOnWith_of_isBounded (hf.subset hεD.1 (convex_ball ..))
(half_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Piecewise | {
"line": 100,
"column": 2
} | {
"line": 103,
"column": 88
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_3\ninst✝¹¹ : Semiring 𝕜\ninst✝¹⁰ : PartialOrder 𝕜\ninst✝⁹ : AddCommMonoid E\ninst✝⁸ : LinearOrder E\ninst✝⁷ : IsOrderedAddMonoid E\ninst✝⁶ : Module 𝕜 E\ninst✝⁵ : PosSMulMono 𝕜 E\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\... | have h_piecewise_Ici_eq_piecewise_Iic :
(Set.Ici e).piecewise f g = (Set.Iic e).piecewise g f := by
ext x; by_cases hx : x = e
<;> simp [Set.piecewise, @le_iff_lt_or_eq _ _ x e, ← @ite_not _ (e ≤ _), hx, h_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 61
} | [
{
"pp": "X : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\nx y : X\ns t s' t' : ℝ\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\nhs' : 0 ≤ s'\nht' : 0 ≤ t'\nh' : s' + t' = 1\nhss' : s' ≤ s\nf : StdSimplex ℝ (Fin 3) := { weights := Finsupp.equivFunOnFinite.symm ![s', s - s', t], no... | convert dist_iConvexComb_le f ![x, x, y] ![x, y, y] using 1 | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
Mathlib.Analysis.Convex.Radon | {
"line": 276,
"column": 6
} | {
"line": 276,
"column": 22
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module 𝕜 E\ninst✝² : FiniteDimensional 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : T2Space E\nF : Set (Set E)\nh_card : ↑(finrank 𝕜 E) + 1 ≤ F.encard\nh_convex : ... | ← coe_toFinset J | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Basic | {
"line": 357,
"column": 77
} | {
"line": 358,
"column": 33
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\n⊢ G ≠ ⊤ ↔ ∃ a b, a ≠ b ∧ ¬G.Adj a b",
"usedConstants": [
"congrArg",
"SimpleGraph.Adj",
"Exists",
"Ne",
"SimpleGraph",
"iff_self",
"funext",
"And",
"Iff",
"BooleanAlgebra.toTop",
"True",
"of_eq_tr... | by
simp [eq_top_iff_forall_ne_adj] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Strict.Extreme | {
"line": 83,
"column": 77
} | {
"line": 83,
"column": 98
} | [
{
"pp": "A : Type u_1\ninst✝³ : NormedAddCommGroup A\ninst✝² : NormedSpace ℝ A\ninst✝¹ : Nontrivial A\nx : A\nr : ℝ\ninst✝ : StrictConvexSpace ℝ A\n⊢ closedBall x r \\ interior (closedBall x r) = sphere x r",
"usedConstants": [
"Eq.mpr",
"congrArg",
"PseudoMetricSpace.toUniformSpace",
... | interior_closedBall', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.StoneSeparation | {
"line": 36,
"column": 6
} | {
"line": 36,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np q u v x y z : E\nhz : z ∈ segment 𝕜 x y\nhu : u ∈ segment 𝕜 x p\nhv : v ∈ segment 𝕜 y q\n⊢ ¬Disjoint (segment 𝕜 u v) ((convexHull 𝕜) {p, q, z})"... | not_disjoint_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.StoneSeparation | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 40
} | [
{
"pp": "case inr.inr.refine_3\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np q x y : E\naz bz : 𝕜\nhaz : 0 ≤ az\nhbz : 0 ≤ bz\nhabz : az + bz = 1\nhaz' : 0 < az\nav bv : 𝕜\nhav : 0 ≤ av\nhbv : 0 ≤ bv\n... | rw [← add_div, div_self]; positivity | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.StoneSeparation | {
"line": 53,
"column": 4
} | {
"line": 53,
"column": 40
} | [
{
"pp": "case inr.inr.refine_3\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np q x y : E\naz bz : 𝕜\nhaz : 0 ≤ az\nhbz : 0 ≤ bz\nhabz : az + bz = 1\nhaz' : 0 < az\nav bv : 𝕜\nhav : 0 ≤ av\nhbv : 0 ≤ bv\n... | rw [← add_div, div_self]; positivity | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.StrictConvexBetween | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 70
} | [
{
"pp": "case hyz\nE : Type u_3\nPE : Type u_5\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : StrictConvexSpace ℝ E\ninst✝¹ : MetricSpace PE\ninst✝ : NormedAddTorsor E PE\nx y z : PE\nhx : dist x y = dist x z / 2\nhy : dist y z = dist x z / 2\n⊢ dist y z = (1 - ⅟2) * dist x z",
"usedCons... | rwa [invOf_eq_inv, ← one_div, sub_half, one_div, ← div_eq_inv_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.Convex.StrictConvexBetween | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 70
} | [
{
"pp": "case hyz\nE : Type u_3\nPE : Type u_5\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : StrictConvexSpace ℝ E\ninst✝¹ : MetricSpace PE\ninst✝ : NormedAddTorsor E PE\nx y z : PE\nhx : dist x y = dist x z / 2\nhy : dist y z = dist x z / 2\n⊢ dist y z = (1 - ⅟2) * dist x z",
"usedCons... | rwa [invOf_eq_inv, ← one_div, sub_half, one_div, ← div_eq_inv_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.StrictConvexBetween | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 70
} | [
{
"pp": "case hyz\nE : Type u_3\nPE : Type u_5\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : StrictConvexSpace ℝ E\ninst✝¹ : MetricSpace PE\ninst✝ : NormedAddTorsor E PE\nx y z : PE\nhx : dist x y = dist x z / 2\nhy : dist y z = dist x z / 2\n⊢ dist y z = (1 - ⅟2) * dist x z",
"usedCons... | rwa [invOf_eq_inv, ← one_div, sub_half, one_div, ← div_eq_inv_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Side | {
"line": 426,
"column": 4
} | {
"line": 426,
"column": 21
} | [
{
"pp": "case mp.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\ns : AffineSubspace R P\nx y p₂ : P\nh : p₂ ∈ s\nhy : y ∈ s\n⊢ y ∈ s ∨ ∃ p₁ ∈ s, SameRay R (x -ᵥ p₁) (p₂ -... | · exact Or.inl hy | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Side | {
"line": 430,
"column": 4
} | {
"line": 430,
"column": 21
} | [
{
"pp": "case mpr.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\ns : AffineSubspace R P\nx y p₂ : P\nh : p₂ ∈ s\nhy : y ∈ s\n⊢ y ∈ s ∨ ∃ p₂_1 ∈ s, SameRay R (y -ᵥ p₂) (p... | · exact Or.inl hy | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Side | {
"line": 664,
"column": 6
} | {
"line": 664,
"column": 37
} | [
{
"pp": "case h.mp.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\ns : AffineSubspace R P\nx p : P\nhx : x ∉ s\nhp : p ∈ s\ny p₂ : P\nhp₂ : p₂ ∈ s\nh : y = p₂\n⊢ ∃ a,... | refine ⟨0, le_rfl, p₂, hp₂, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Convex.Side | {
"line": 699,
"column": 6
} | {
"line": 699,
"column": 37
} | [
{
"pp": "case h.mp.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\ns : AffineSubspace R P\nx p : P\nhx : x ∉ s\nhp : p ∈ s\ny p₂ : P\nhp₂ : p₂ ∈ s\nh : p₂ = y\n⊢ ∃ a ... | refine ⟨0, le_rfl, p₂, hp₂, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Convex.Side | {
"line": 844,
"column": 4
} | {
"line": 844,
"column": 18
} | [
{
"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 : ... | simpa using hj | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Convex.StrictCombination | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 62
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nι : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : TopologicalSpace V\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\ns : Set V\nt : Finset ι\nw : ι → R\nz : ι → V\nhs : StrictConvex R s\nh0 : ∀ i ∈ t, 0 ≤ w i\nh1 : ∑ i ∈ t, w i = ... | exact hs.centerMass_mem_interior h0 i j hi hj hij hi0 hj0 hz | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.StrictCombination | {
"line": 145,
"column": 4
} | {
"line": 147,
"column": 66
} | [
{
"pp": "case h.e_6.h.h.inl\nV : 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... | · simp only [Pi.single_eq_same]
rw [← hw, eq_comm]
exact sum_eq_single i (fun k _ hk ↦ hij k hk.symm) (by simp) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.Layercake | {
"line": 179,
"column": 6
} | {
"line": 179,
"column": 10
} | [
{
"pp": "case hf\nα : Type u_1\ninst✝¹ : MeasurableSpace α\nf : α → ℝ\ng : ℝ → ℝ\nμ : Measure α\ninst✝ : SFinite μ\nf_nn : 0 ≤ f\nf_mble : Measurable f\ng_intble : ∀ t > 0, IntervalIntegrable g volume 0 t\ng_mble : Measurable g\ng_nn : ∀ t > 0, 0 ≤ g t\ng_intble' : ∀ (t : ℝ), 0 ≤ t → IntervalIntegrable g volume... | aux₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.L2Space | {
"line": 110,
"column": 2
} | {
"line": 112,
"column": 71
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup F\nf : ↥(Lp F 2 μ)\n⊢ eLpNorm (fun x ↦ ‖↑↑f x‖ ^ 2) 1 μ < ∞",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Real.instPo... | have h_two : ENNReal.ofReal (2 : ℝ) = 2 := by simp
rw [eLpNorm_norm_rpow f zero_lt_two, one_mul, h_two]
exact ENNReal.rpow_lt_top_of_nonneg zero_le_two (Lp.eLpNorm_ne_top f) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.L2Space | {
"line": 110,
"column": 2
} | {
"line": 112,
"column": 71
} | [
{
"pp": "α : Type u_1\nF : Type u_3\ninst✝¹ : MeasurableSpace α\nμ : Measure α\ninst✝ : NormedAddCommGroup F\nf : ↥(Lp F 2 μ)\n⊢ eLpNorm (fun x ↦ ‖↑↑f x‖ ^ 2) 1 μ < ∞",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Real.instPo... | have h_two : ENNReal.ofReal (2 : ℝ) = 2 := by simp
rw [eLpNorm_norm_rpow f zero_lt_two, one_mul, h_two]
exact ENNReal.rpow_lt_top_of_nonneg zero_le_two (Lp.eLpNorm_ne_top f) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.SmoothApprox | {
"line": 94,
"column": 77
} | {
"line": 94,
"column": 81
} | [
{
"pp": "E : Type u_3\nF : Type u_4\ninst✝⁷ : MeasurableSpace E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : BorelSpace E\ninst✝¹ : NormedSpace ℝ F\nμ : Measure E\ninst✝ : IsFiniteMeasureOnCompacts μ\np : ℝ≥0∞\nhp : p ≠ ∞\nhp₂ ... | hg₂, | Mathlib.Tactic.evalGRewriteSeq | null |
Mathlib.Analysis.Distribution.TestFunction | {
"line": 378,
"column": 4
} | {
"line": 382,
"column": 21
} | [
{
"pp": "𝕜 : Type u_1\n𝕂 : Type u_2\ninst✝¹⁷ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝¹⁶ : NormedAddCommGroup E\ninst✝¹⁵ : NormedSpace ℝ E\nΩ Ω₁ Ω₂ : Opens E\nF : Type u_4\ninst✝¹⁴ : NormedAddCommGroup F\ninst✝¹³ : NormedSpace ℝ F\ninst✝¹² : NormedSpace 𝕜 F\nF' : Type u_5\ninst✝¹¹ : NormedAddCommGrou... | set K : Compacts E := ⟨tsupport f, f.hasCompactSupport⟩
have K_sub_Ω : (K : Set E) ⊆ Ω := f.tsupport_subset
let f_K : 𝓓^{n}_{K}(E, F) := .of_support_subset f.contDiff subset_closure
change toFun (ofSupportedIn K_sub_Ω (c • f_K)) = c • toFun (ofSupportedIn K_sub_Ω f_K)
simp [toFun_eq_T] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Distribution.TestFunction | {
"line": 378,
"column": 4
} | {
"line": 382,
"column": 21
} | [
{
"pp": "𝕜 : Type u_1\n𝕂 : Type u_2\ninst✝¹⁷ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝¹⁶ : NormedAddCommGroup E\ninst✝¹⁵ : NormedSpace ℝ E\nΩ Ω₁ Ω₂ : Opens E\nF : Type u_4\ninst✝¹⁴ : NormedAddCommGroup F\ninst✝¹³ : NormedSpace ℝ F\ninst✝¹² : NormedSpace 𝕜 F\nF' : Type u_5\ninst✝¹¹ : NormedAddCommGrou... | set K : Compacts E := ⟨tsupport f, f.hasCompactSupport⟩
have K_sub_Ω : (K : Set E) ⊆ Ω := f.tsupport_subset
let f_K : 𝓓^{n}_{K}(E, F) := .of_support_subset f.contDiff subset_closure
change toFun (ofSupportedIn K_sub_Ω (c • f_K)) = c • toFun (ofSupportedIn K_sub_Ω f_K)
simp [toFun_eq_T] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Distribution.SchwartzSpace.Deriv | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 49
} | [
{
"pp": "E : Type u_5\nF : Type u_8\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\nf : 𝓢(E, F)\nx : E\n⊢ (Δ f) x = Δ (⇑f) x",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
... | rw [laplacian_eq_sum (stdOrthonormalBasis ℝ E)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.l2Space | {
"line": 514,
"column": 6
} | {
"line": 514,
"column": 57
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\ninst✝⁴ : RCLike 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : CompleteSpace E\nv : ι → E\nhv : Orthonormal 𝕜 v\ninst✝ : DecidableEq ι\nh : ⊤ ≤ (span 𝕜 (Set.range v)).topologicalClosure\ni : ι\n⊢ ⋯.linearIsometryEquiv.symm (lp.s... | IsHilbertSum.linearIsometryEquiv_symm_apply_single, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Fourier.AddCircle | {
"line": 582,
"column": 8
} | {
"line": 582,
"column": 50
} | [
{
"pp": "case e_a\na b : ℝ\nhab : a < b\nf f' : ℝ → ℂ\nn : ℤ\nhn : n ≠ 0\nhf : ContinuousOn f [[a, b]]\nhff' : ∀ x ∈ Ioo (min a b) (max a b), HasDerivWithinAt f (f' x) (Ioi x) x\nhf' : IntervalIntegrable f' volume a b\nhT : Fact (0 < b - a)\nthis : ∀ (u v w : ℂ), u * (↑(b - a) / v * w) = ↑(b - a) / v * (u * w)\... | div_eq_iff (ofReal_ne_zero.mpr hT.out.ne') | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 161,
"column": 39
} | {
"line": 161,
"column": 55
} | [
{
"pp": "E : Type u_5\nF : Type u_6\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nf : 𝓢(E, F)\ninst✝ : ProperSpace E\ns : ℝ\nk : ℕ := ⌈-s⌉₊\nhk : -↑k ≤ s\nx : ℝ\nhx : 1 ≤ x\n⊢ x ^ (-↑↑k) ≤ x ^ s",
"usedConstants": [
"AddGroup.toSubt... | Int.cast_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 567,
"column": 4
} | {
"line": 573,
"column": 21
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nh₁ : HasCompactSupport f\nh₂ : ContDiff ℝ ∞ f\nk n... | set g := fun x ↦ ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖
have hg₁ : Continuous g := by
apply Continuous.mul (by fun_prop)
exact (h₂.of_le (mod_cast le_top)).continuous_iteratedFDeriv'.norm
have hg₂ : HasCompactSupport g := (h₁.iteratedFDeriv _).norm.mul_left
obtain ⟨x₀, hx₀⟩ := hg₁.exists_forall_ge_o... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 567,
"column": 4
} | {
"line": 573,
"column": 21
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\nh₁ : HasCompactSupport f\nh₂ : ContDiff ℝ ∞ f\nk n... | set g := fun x ↦ ‖x‖ ^ k * ‖iteratedFDeriv ℝ n f x‖
have hg₁ : Continuous g := by
apply Continuous.mul (by fun_prop)
exact (h₂.of_le (mod_cast le_top)).continuous_iteratedFDeriv'.norm
have hg₂ : HasCompactSupport g := (h₁.iteratedFDeriv _).norm.mul_left
obtain ⟨x₀, hx₀⟩ := hg₁.exists_forall_ge_o... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 306,
"column": 14
} | {
"line": 307,
"column": 91
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : BorelSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ng : α → E\nx₀ : α\ns : Set α\ninst✝² : CompleteSpace E\ninst✝¹ : MetrizableSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nhs : IsCo... | refine integrableOn_const (C := t' ^ n) ?_
exact (lt_of_le_of_lt (measure_mono inter_subset_right) hs.measure_lt_top).ne | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.PeakFunction | {
"line": 306,
"column": 14
} | {
"line": 307,
"column": 91
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nhm : MeasurableSpace α\nμ : Measure α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : BorelSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ng : α → E\nx₀ : α\ns : Set α\ninst✝² : CompleteSpace E\ninst✝¹ : MetrizableSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nhs : IsCo... | refine integrableOn_const (C := t' ^ n) ?_
exact (lt_of_le_of_lt (measure_mono inter_subset_right) hs.measure_lt_top).ne | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 780,
"column": 51
} | {
"line": 783,
"column": 6
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_5\nF : Type u_6\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAlgebra ℝ 𝕜\ninst✝ : NormedSpace 𝕜 F\ng : E → 𝕜\nhg : Function.HasTemperateGrowth g\nc : 𝕜... | by
have : (fun (_ : E) ↦ c).HasTemperateGrowth := by fun_prop
convert! (smulLeftCLM_compL_smulLeftCLM this hg).symm using 1
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 897,
"column": 15
} | {
"line": 897,
"column": 25
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace ℝ E\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace ℝ F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAlgebra... | le_opNorm, | Mathlib.Tactic.evalGRewriteSeq | null |
Mathlib.Analysis.Distribution.SchwartzSpace.Basic | {
"line": 935,
"column": 4
} | {
"line": 935,
"column": 31
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\n𝕜' : Type u_3\nD : Type u_4\nE : Type u_5\nF : Type u_6\nG : Type u_7\nH : Type u_8\nV : Type u_9\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace ℝ F\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup D\ninst✝¹ : ... | rw [norm_zero] at hg_upper' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 68
} | [
{
"pp": "s : ℂ\nt : ℝ\nht : 0 < t\n⊢ IntegrableOn (fun x ↦ ‖↑x ^ s‖) (Ioi t) volume ↔ s.re < -1",
"usedConstants": [
"Norm.norm",
"NormedCommRing.toSeminormedCommRing",
"Real",
"Set.Ioi",
"integrableOn_Ioi_norm_cpow_of_lt",
"MeasureTheory.MeasureSpace.toMeasurableSpace",
... | refine ⟨fun h ↦ ?_, fun h ↦ integrableOn_Ioi_norm_cpow_of_lt h ht⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 216,
"column": 2
} | {
"line": 222,
"column": 60
} | [
{
"pp": "s : ℂ\nt : ℝ\nht : 0 < t\nhs : s.re ≤ 0\n⊢ IntegrableOn (deriv fun x ↦ ‖↑x ^ s‖) (Ioi t) volume",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"MeasureTheory.integrableOn_congr_fun",
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
... | rw [integrableOn_congr_fun (fun x hx ↦ by
rw [deriv_norm_ofReal_cpow _ (ht.trans hx)]) measurableSet_Ioi]
obtain hs | hs := eq_or_lt_of_le hs
· simp_rw [hs, zero_mul]
exact integrableOn_zero
· replace hs : s.re - 1 < -1 := by rwa [sub_lt_iff_lt_add, neg_add_cancel]
exact (integrableOn_Ioi_rpow_of_lt h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ImproperIntegrals | {
"line": 216,
"column": 2
} | {
"line": 222,
"column": 60
} | [
{
"pp": "s : ℂ\nt : ℝ\nht : 0 < t\nhs : s.re ≤ 0\n⊢ IntegrableOn (deriv fun x ↦ ‖↑x ^ s‖) (Ioi t) volume",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"MeasureTheory.integrableOn_congr_fun",
"NormedCommRing.toNormedRing",
"Norm.norm",
"Eq.mpr",
... | rw [integrableOn_congr_fun (fun x hx ↦ by
rw [deriv_norm_ofReal_cpow _ (ht.trans hx)]) measurableSet_Ioi]
obtain hs | hs := eq_or_lt_of_le hs
· simp_rw [hs, zero_mul]
exact integrableOn_zero
· replace hs : s.re - 1 < -1 := by rwa [sub_lt_iff_lt_add, neg_add_cancel]
exact (integrableOn_Ioi_rpow_of_lt h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 122,
"column": 75
} | {
"line": 130,
"column": 6
} | [
{
"pp": "s : ℝ\n⊢ (↑s).GammaIntegral = ↑(∫ (x : ℝ) in Ioi 0, rexp (-x) * x ^ (s - 1))",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real.instPow",
"instClosedIicTopology",
... | by
have : ∀ r : ℝ, Complex.ofReal r = @RCLike.ofReal ℂ _ r := fun r => rfl
rw [GammaIntegral]
conv_rhs => rw [this, ← _root_.integral_ofReal]
refine setIntegral_congr_fun measurableSet_Ioi ?_
intro x hx; dsimp only
conv_rhs => rw [← this]
rw [ofReal_mul, ofReal_cpow (mem_Ioi.mp hx).le]
simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 277,
"column": 2
} | {
"line": 283,
"column": 13
} | [
{
"pp": "case succ\ns : ℂ\nn : ℕ\nh1 : -s.re < ↑(n + 1)\n⊢ GammaAux (n + 1) s = GammaAux (n + 1 + 1) s",
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib... | · dsimp only [GammaAux]
have : GammaAux n (s + 1 + 1) / (s + 1) = GammaAux n (s + 1) := by
have hh1 : -(s + 1).re < n := by
rw [Nat.cast_add, Nat.cast_one] at h1
rw [add_re, one_re]; linarith
rw [GammaAux_recurrence1 (s + 1) n hh1]
rw [this] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 391,
"column": 15
} | {
"line": 391,
"column": 35
} | [
{
"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 * (1 / ↑r) ^ (a - 1) * Gamma a",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"Inne... | Gamma_eq_integral ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 416,
"column": 35
} | {
"line": 417,
"column": 67
} | [
{
"pp": "⊢ Gamma 1 = 1",
"usedConstants": [
"Eq.mpr",
"Real",
"Complex.Gamma_one",
"congrArg",
"Complex.Gamma",
"id",
"Complex.ofReal_one",
"Complex.ofReal",
"Complex.re",
"Real.instOne",
"Real.Gamma",
"One.toOfNat1",
"Eq.refl",
... | by
rw [Gamma, Complex.ofReal_one, Complex.Gamma_one, Complex.one_re] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 444,
"column": 2
} | {
"line": 455,
"column": 37
} | [
{
"pp": "s : ℝ\nhs : 0 < s\n⊢ 0 < Gamma s",
"usedConstants": [
"MeasureTheory.ae",
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real.instPow",
"instClosedIicTopology",
"Real.partialOrder",
"Real.instLE",
"Real.rpow_pos_of_... | rw [Gamma_eq_integral hs]
have : (Function.support fun x : ℝ => exp (-x) * x ^ (s - 1)) ∩ Ioi 0 = Ioi 0 := by
rw [inter_eq_right]
intro x hx
rw [Function.mem_support]
exact mul_ne_zero (exp_pos _).ne' (rpow_pos_of_pos hx _).ne'
rw [setIntegral_pos_iff_support_of_nonneg_ae]
· rw [this, volume_Ioi, ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gamma.Basic | {
"line": 444,
"column": 2
} | {
"line": 455,
"column": 37
} | [
{
"pp": "s : ℝ\nhs : 0 < s\n⊢ 0 < Gamma s",
"usedConstants": [
"MeasureTheory.ae",
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real.instPow",
"instClosedIicTopology",
"Real.partialOrder",
"Real.instLE",
"Real.rpow_pos_of_... | rw [Gamma_eq_integral hs]
have : (Function.support fun x : ℝ => exp (-x) * x ^ (s - 1)) ∩ Ioi 0 = Ioi 0 := by
rw [inter_eq_right]
intro x hx
rw [Function.mem_support]
exact mul_ne_zero (exp_pos _).ne' (rpow_pos_of_pos hx _).ne'
rw [setIntegral_pos_iff_support_of_nonneg_ae]
· rw [this, volume_Ioi, ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 79
} | [
{
"pp": "b : ℂ\nx : ℝ\n⊢ ‖cexp (-b * ↑x ^ 2)‖ = rexp (-b.re * x ^ 2)",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Norm.norm",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"congrArg",
"Com... | rw [norm_exp, ← ofReal_pow, mul_comm (-b) _, re_ofReal_mul, neg_re, mul_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 79
} | [
{
"pp": "b : ℂ\nx : ℝ\n⊢ ‖cexp (-b * ↑x ^ 2)‖ = rexp (-b.re * x ^ 2)",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Norm.norm",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"congrArg",
"Com... | rw [norm_exp, ← ofReal_pow, mul_comm (-b) _, re_ofReal_mul, neg_re, mul_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 79
} | [
{
"pp": "b : ℂ\nx : ℝ\n⊢ ‖cexp (-b * ↑x ^ 2)‖ = rexp (-b.re * x ^ 2)",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Norm.norm",
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"congrArg",
"Com... | rw [norm_exp, ← ofReal_pow, mul_comm (-b) _, re_ofReal_mul, neg_re, mul_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 193,
"column": 54
} | {
"line": 221,
"column": 10
} | [
{
"pp": "b : ℂ\nhb : 0 < b.re\n⊢ (∫ (x : ℝ), cexp (-b * ↑x ^ 2)) ^ 2 = ↑π / b",
"usedConstants": [
"instWeaklyLocallyCompactSpaceOfLocallyCompactSpace",
"instInnerProductSpaceRealComplex",
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Set.instSProd",
"Distrib.leftDistribClass",
... | by
/- We compute `(∫ exp (-b x^2))^2` as an integral over `ℝ^2`, and then make a polar change
of coordinates. We are left with `∫ r * exp (-b r^2)`, which has been computed in
`integral_mul_cexp_neg_mul_sq` using the fact that this function has an obvious primitive. -/
calc
(∫ x : ℝ, cexp (-b * (x : ℂ) ^ 2)... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 226,
"column": 4
} | {
"line": 228,
"column": 65
} | [
{
"pp": "case inl\nb : ℝ\nhb : b ≤ 0\n⊢ ∫ (x : ℝ), rexp (-b * x ^ 2) = √(π / b)",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real.partialOrder",
"Real",
"instHDiv",
"IsOrderedRing.toPosMulMono",
"Real.pi",
... | rw [integral_undef, sqrt_eq_zero_of_nonpos]
· exact div_nonpos_of_nonneg_of_nonpos pi_pos.le hb
· simpa only [not_lt, integrable_exp_neg_mul_sq_iff] using hb | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 226,
"column": 4
} | {
"line": 228,
"column": 65
} | [
{
"pp": "case inl\nb : ℝ\nhb : b ≤ 0\n⊢ ∫ (x : ℝ), rexp (-b * x ^ 2) = √(π / b)",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real.partialOrder",
"Real",
"instHDiv",
"IsOrderedRing.toPosMulMono",
"Real.pi",
... | rw [integral_undef, sqrt_eq_zero_of_nonpos]
· exact div_nonpos_of_nonneg_of_nonpos pi_pos.le hb
· simpa only [not_lt, integrable_exp_neg_mul_sq_iff] using hb | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Gaussian.GaussianIntegral | {
"line": 316,
"column": 2
} | {
"line": 318,
"column": 71
} | [
{
"pp": "case inl\nb : ℝ\nhb : b ≤ 0\n⊢ ∫ (x : ℝ) in Ioi 0, rexp (-b * x ^ 2) = √(π / b) / 2",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Real.partialO... | · rw [integral_undef, sqrt_eq_zero_of_nonpos, zero_div]
· exact div_nonpos_of_nonneg_of_nonpos pi_pos.le hb
· rwa [← IntegrableOn, integrableOn_Ioi_exp_neg_mul_sq_iff, not_lt] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 75,
"column": 4
} | {
"line": 76,
"column": 25
} | [
{
"pp": "case a\nd : Type u_1\ninst✝ : Fintype d\nn : d → ℤ\n⊢ 1 = ‖(mFourier n) fun x ↦ 0‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Norm.norm",
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"Real",
"Finset.univ",
"Complex.commRing",
"Pi.t... | simp only [mFourier, ContinuousMap.coe_mk, fourier_eval_zero, Finset.prod_const_one,
CStarRing.norm_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Fourier.AddCircleMulti | {
"line": 280,
"column": 2
} | {
"line": 280,
"column": 42
} | [
{
"pp": "d : Type u_1\ninst✝ : Fintype d\nf : ↥(Lp ℂ 2 volume)\ni : d → ℤ\n⊢ ↑(mFourierBasis.repr f) i = mFourierCoeff (↑↑f) i",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"LinearIsometryEquiv.instEquivLike",
"NormedCommRing.toNormedRing",
"InnerProductSpace.toNormedSpace... | trans ∫ t, conj (mFourierLp 2 i t) * f t | Batteries.Tactic._aux_Batteries_Tactic_Trans___elabRules_Batteries_Tactic_tacticTrans____1 | Batteries.Tactic.tacticTrans___ |
Mathlib.NumberTheory.ArithmeticFunction.Moebius | {
"line": 173,
"column": 16
} | {
"line": 173,
"column": 36
} | [
{
"pp": "⊢ μ * ↑ζ = 1",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"ArithmeticFunction.instMul",
"congrArg",
"Arithmeti... | moebius_mul_coe_zeta | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.RootsOfUnity.Complex | {
"line": 39,
"column": 53
} | {
"line": 50,
"column": 93
} | [
{
"pp": "i : ℤ\nn : ℕ\nh0 : n ≠ 0\nhi : IsCoprime i ↑n\n⊢ IsPrimitiveRoot (cexp (2 * ↑π * I * (↑i / ↑n))) n",
"usedConstants": [
"Mathlib.Tactic.FieldSimp.zpow'_one",
"Int.cast",
"Mathlib.Tactic.FieldSimp.NF.div_eq_eval₁",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Int.... | by
rw [IsPrimitiveRoot.iff_def]
simp only [← exp_nat_mul, exp_eq_one_iff]
constructor
· use i
simp (discharger := norm_cast) [field]
· simp only [forall_exists_index]
have hn0 : (n : ℂ) ≠ 0 := mod_cast h0
rintro l k hk
field_simp at hk
norm_cast at hk
exact Int.natCast_dvd_natCast.mp <... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.RootsOfUnity.PrimitiveRoots | {
"line": 343,
"column": 2
} | {
"line": 343,
"column": 63
} | [
{
"pp": "case pos\nG : Type u_3\ninst✝ : DivisionCommMonoid G\nk : ℕ\nζ : G\nh : IsPrimitiveRoot ζ k\nl : ℤ\nh0 : 0 ≤ l\n⊢ ζ ^ l = 1 ↔ ↑k ∣ l",
"usedConstants": [
"zpow_natCast",
"Dvd.dvd",
"InvOneClass.toOne",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneCla... | · lift l to ℕ using h0; exact_mod_cast h.pow_eq_one_iff_dvd l | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 210,
"column": 22
} | {
"line": 210,
"column": 63
} | [
{
"pp": "case h\nK : Type u_2\ninst✝¹ : CommRing K\ninst✝ : IsDomain K\nk : ℕ\nihk : ∀ m < k, ∀ {ζ : K}, IsPrimitiveRoot ζ m → cyclotomic' m K ∈ lifts (Int.castRingHom K)\nζ : K\nh : IsPrimitiveRoot ζ k\nhpos : k > 0\nB : K[X] := ∏ i ∈ k.properDivisors, cyclotomic' i K\nBmo : B.Monic\nB₁ : ℤ[X]\nhB₁ : map (Int.... | map_divByMonic (Int.castRingHom K) hB₁mo, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 433,
"column": 2
} | {
"line": 434,
"column": 51
} | [
{
"pp": "case inr\nn : ℕ\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nhpos : n > 0\nh :\n ∀ (n : ℕ),\n 0 < n → ∏ i ∈ n.divisors, (algebraMap R[X] (RatFunc R)) (cyclotomic i R) = (algebraMap R[X] (RatFunc R)) (X ^ n - 1)\n⊢ (algebraMap R[X] (RatFunc R)) (cyclotomic n R) =\n ∏ i ∈ n.divisorsAnti... | rw [(prod_eq_iff_prod_pow_moebius_eq_of_nonzero (fun n hn => _) fun n hn => _).1 h n hpos] <;>
simp_rw [Ne, IsFractionRing.to_map_eq_zero_iff] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.RingTheory.Polynomial.Cyclotomic.Basic | {
"line": 587,
"column": 82
} | {
"line": 595,
"column": 48
} | [
{
"pp": "n : ℕ\nhpos : 0 < n\np : ℕ\ninst✝ : Fact (Nat.Prime p)\na : ℕ\nhroot : (cyclotomic n (ZMod p)).IsRoot ((Nat.castRingHom (ZMod p)) a)\n⊢ orderOf (ZMod.unitOfCoprime a ⋯) ∣ n",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Units.val",
"Eq.mpr",
"Polynomial.eval",
"... | by
apply orderOf_dvd_of_pow_eq_one
suffices hpow : eval (Nat.castRingHom (ZMod p) a) (X ^ n - 1 : (ZMod p)[X]) = 0 by
simp only [eval_X, eval_one, eval_pow, eval_sub, eq_natCast] at hpow
apply Units.val_eq_one.1
simp only [sub_eq_zero.mp hpow, ZMod.coe_unitOfCoprime, Units.val_pow_eq_pow_val]
rw [IsRo... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.FieldTheory.KrullTopology | {
"line": 315,
"column": 8
} | {
"line": 315,
"column": 51
} | [
{
"pp": "case inr\nk : Type u_1\nK : Type u_3\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nL : IntermediateField k K\ninst✝ : IsGalois k K\nthis : ∀ (L : IntermediateField k K), FiniteDimensional k ↥L → Module.finrank k ↥L = L.fixingSubgroup.index\nhnfd : ¬FiniteDimensional k ↥L\n⊢ Module.finrank ... | Module.finrank_of_infinite_dimensional hnfd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Adjoin.PowerBasis | {
"line": 45,
"column": 4
} | {
"line": 45,
"column": 48
} | [
{
"pp": "case hsp\nK : Type u_1\nS : Type u_2\ninst✝² : Field K\ninst✝¹ : CommRing S\ninst✝ : Algebra K S\nx : S\nhx : IsIntegral K x\nhST : Function.Injective ⇑(algebraMap (↥K[x]) S)\nhx' : IsIntegral K ⟨x, ⋯⟩\ny : S\nhy✝ : y ∈ K[x]\nhy : y ∈ (aeval x).range\na✝ : ⟨y, hy✝⟩ ∈ ⊤\nthis : (∃ f, ⟨y, hy✝⟩ = (aeval ⟨... | obtain ⟨f, rfl⟩ := (aeval x).mem_range.mp hy | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.Adjoin.PowerBasis | {
"line": 40,
"column": 4
} | {
"line": 48,
"column": 52
} | [
{
"pp": "case hsp\nK : Type u_1\nS : Type u_2\ninst✝² : Field K\ninst✝¹ : CommRing S\ninst✝ : Algebra K S\nx : S\nhx : IsIntegral K x\nhST : Function.Injective ⇑(algebraMap (↥K[x]) S)\nhx' : IsIntegral K ⟨x, ⋯⟩\n⊢ ⊤ ≤ Submodule.span K (Set.range fun i ↦ ⟨x, ⋯⟩ ^ ↑i)",
"usedConstants": [
"Subalgebra.in... | rintro ⟨y, hy⟩ _
have := hx'.mem_span_pow (y := ⟨y, hy⟩)
rw [← minpoly.algebraMap_eq hST] at this
apply this
rw [adjoin_singleton_eq_range_aeval] at hy
obtain ⟨f, rfl⟩ := (aeval x).mem_range.mp hy
use f
ext
exact aeval_algebraMap_apply S (⟨x, _⟩ : K[x]) _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Adjoin.PowerBasis | {
"line": 40,
"column": 4
} | {
"line": 48,
"column": 52
} | [
{
"pp": "case hsp\nK : Type u_1\nS : Type u_2\ninst✝² : Field K\ninst✝¹ : CommRing S\ninst✝ : Algebra K S\nx : S\nhx : IsIntegral K x\nhST : Function.Injective ⇑(algebraMap (↥K[x]) S)\nhx' : IsIntegral K ⟨x, ⋯⟩\n⊢ ⊤ ≤ Submodule.span K (Set.range fun i ↦ ⟨x, ⋯⟩ ^ ↑i)",
"usedConstants": [
"Subalgebra.in... | rintro ⟨y, hy⟩ _
have := hx'.mem_span_pow (y := ⟨y, hy⟩)
rw [← minpoly.algebraMap_eq hST] at this
apply this
rw [adjoin_singleton_eq_range_aeval] at hy
obtain ⟨f, rfl⟩ := (aeval x).mem_range.mp hy
use f
ext
exact aeval_algebraMap_apply S (⟨x, _⟩ : K[x]) _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 29
} | [
{
"pp": "case h.inl\nR : Type u_1\ninst✝² : CommRing R\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nx : R\nn : ℕ\nih : ∀ m < n, 2 < m → 0 < eval x (cyclotomic m R)\nhn : 2 < n\nhn' : 0 < n\nhn'' : 1 < n\nthis : eval x (cyclotomic n R) * eval x (∏ x ∈ n.properDivisors.erase 1, cyclotomic x R) = ∑ i ∈ ... | apply pos_of_mul_pos_left | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Polynomial.Cyclotomic.Expand | {
"line": 115,
"column": 2
} | {
"line": 120,
"column": 58
} | [
{
"pp": "case inr\np : ℕ\nhp : Nat.Prime p\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nm : ℕ\nhm : m > 0\nk : ℕ\nhmn : m ≤ m + k\nh : Irreducible (cyclotomic (p ^ (m + k)) R)\n⊢ Irreducible (cyclotomic (p ^ m) R)",
"usedConstants": [
"Nat.zero_le",
"Nat.recAux",
"Nat.instIsOrde... | induction k with
| zero => simpa using h
| succ k hk =>
have : m + k ≠ 0 := (add_pos_of_pos_of_nonneg hm k.zero_le).ne'
rw [Nat.add_succ, pow_succ, ← cyclotomic_expand_eq_cyclotomic hp <| dvd_pow_self p this] at h
exact hk (by lia) (of_irreducible_expand hp.ne_zero h) | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.FieldTheory.Galois.Infinite | {
"line": 135,
"column": 4
} | {
"line": 135,
"column": 42
} | [
{
"pp": "case h.h.refine_2\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nH : Subgroup Gal(K/k)\nL : IntermediateField k K\ninst✝ : Normal k ↥L\nx : K\nh : x ∈ ↑(lift (fixedField (Subgroup.map (restrictNormalHom ↥L) H)))\nxL : x ∈ L\n⊢ x ∈ ↑(fixedField H ⊓ L)",
"usedC... | apply (mem_lift (⟨x, xL⟩ : L)).mp at h | Mathlib.Tactic._aux_Mathlib_Tactic_ApplyAt___elabRules_Mathlib_Tactic_tacticApply_At__1 | Mathlib.Tactic.tacticApply_At_ |
Mathlib.FieldTheory.Galois.Infinite | {
"line": 136,
"column": 4
} | {
"line": 137,
"column": 36
} | [
{
"pp": "case h.h.refine_2\nk : Type u_1\nK : Type u_2\ninst✝³ : Field k\ninst✝² : Field K\ninst✝¹ : Algebra k K\nH : Subgroup Gal(K/k)\nL : IntermediateField k K\ninst✝ : Normal k ↥L\nx : K\nxL : x ∈ L\nh : ⟨x, xL⟩ ∈ fixedField (Subgroup.map (restrictNormalHom ↥L) H)\n⊢ x ∈ ↑(fixedField H ⊓ L)",
"usedConst... | simp only [mem_fixedField_iff, Subgroup.mem_map, forall_exists_index, and_imp,
forall_apply_eq_imp_iff₂] at h | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Polynomial.Cyclotomic.Eval | {
"line": 163,
"column": 4
} | {
"line": 164,
"column": 29
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\nn : ℕ\nh : ∀ {p : ℕ}, Nat.Prime p → ∀ (k : ℕ), p ^ k ≠ n\nhn' : n > 0\nhn : 1 < n\np : ℕ\nhp : Nat.Prime p\nthis✝¹ : Fact (Nat.Prime p)\nt : ℤ\nthis✝ :\n ↑n =\n ↑p ^ padicValNat p n * ↑p *\n (t *\n ∏ x ∈ (n.divisors.erase 1 \\ image (fun t ↦ p ^ (t + 1)) (r... | simp only [← _root_.pow_succ, ← Int.natAbs_dvd_natAbs, Int.natAbs_natCast,
Int.natAbs_pow] at this | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Cyclotomic.PrimitiveRoots | {
"line": 181,
"column": 2
} | {
"line": 182,
"column": 86
} | [
{
"pp": "n : ℕ\ninst✝⁵ : NeZero n\nK : Type u\nL : Type v\ninst✝⁴ : Field K\ninst✝³ : CommRing L\ninst✝² : IsDomain L\ninst✝¹ : Algebra K L\ninst✝ : IsCyclotomicExtension {n} K L\nhirr : Irreducible (cyclotomic n K)\nthis : NeZero ↑n\n⊢ Module.finrank K L = φ n",
"usedConstants": [
"Eq.mpr",
"No... | rw [((zeta_spec n K L).powerBasis K).finrank, IsPrimitiveRoot.powerBasis_dim, ←
(zeta_spec n K L).minpoly_eq_cyclotomic_of_irreducible hirr, natDegree_cyclotomic] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 351,
"column": 29
} | {
"line": 351,
"column": 44
} | [
{
"pp": "case h.add\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : IsCyclotomicExtension S A B\ninst✝ : IsDomain B\nf g : B ≃ₐ[A] B\nH : ∀ n ∈ S, n ≠ 0 → ∃ r, IsPrimitiveRoot r n ∧ f r = g r\nx✝ x y : B\nhx : x ∈ adjoin A {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n ... | simp [ihx, ihy] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 351,
"column": 29
} | {
"line": 351,
"column": 44
} | [
{
"pp": "case h.add\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : IsCyclotomicExtension S A B\ninst✝ : IsDomain B\nf g : B ≃ₐ[A] B\nH : ∀ n ∈ S, n ≠ 0 → ∃ r, IsPrimitiveRoot r n ∧ f r = g r\nx✝ x y : B\nhx : x ∈ adjoin A {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n ... | simp [ihx, ihy] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 351,
"column": 29
} | {
"line": 351,
"column": 44
} | [
{
"pp": "case h.add\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : IsCyclotomicExtension S A B\ninst✝ : IsDomain B\nf g : B ≃ₐ[A] B\nH : ∀ n ∈ S, n ≠ 0 → ∃ r, IsPrimitiveRoot r n ∧ f r = g r\nx✝ x y : B\nhx : x ∈ adjoin A {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n ... | simp [ihx, ihy] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 352,
"column": 29
} | {
"line": 352,
"column": 44
} | [
{
"pp": "case h.mul\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : IsCyclotomicExtension S A B\ninst✝ : IsDomain B\nf g : B ≃ₐ[A] B\nH : ∀ n ∈ S, n ≠ 0 → ∃ r, IsPrimitiveRoot r n ∧ f r = g r\nx✝ x y : B\nhx : x ∈ adjoin A {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n ... | simp [ihx, ihy] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 352,
"column": 29
} | {
"line": 352,
"column": 44
} | [
{
"pp": "case h.mul\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : IsCyclotomicExtension S A B\ninst✝ : IsDomain B\nf g : B ≃ₐ[A] B\nH : ∀ n ∈ S, n ≠ 0 → ∃ r, IsPrimitiveRoot r n ∧ f r = g r\nx✝ x y : B\nhx : x ∈ adjoin A {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n ... | simp [ihx, ihy] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Cyclotomic.Basic | {
"line": 352,
"column": 29
} | {
"line": 352,
"column": 44
} | [
{
"pp": "case h.mul\nS : Set ℕ\nA : Type u\nB : Type v\ninst✝⁴ : CommRing A\ninst✝³ : CommRing B\ninst✝² : Algebra A B\ninst✝¹ : IsCyclotomicExtension S A B\ninst✝ : IsDomain B\nf g : B ≃ₐ[A] B\nH : ∀ n ∈ S, n ≠ 0 → ∃ r, IsPrimitiveRoot r n ∧ f r = g r\nx✝ x y : B\nhx : x ∈ adjoin A {b | ∃ n ∈ S, n ≠ 0 ∧ b ^ n ... | simp [ihx, ihy] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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