module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 804,
"column": 4
} | {
"line": 804,
"column": 71
} | [
{
"pp": "case mp\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : F\nh : ⟪x, y⟫_ℝ / (‖x‖ * ‖y‖) = 1\nhx₀ : x ≠ 0\nhy₀ : y ≠ 0\n⊢ y = (‖y‖ / ‖x‖) • x",
"usedConstants": [
"Norm.norm",
"InnerProductSpace.toNormedSpace",
"Real",
"instHSMul",
"inst... | exact ((inner_eq_norm_mul_iff_div hx₀).1 (eq_of_div_eq_one h)).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 872,
"column": 2
} | {
"line": 872,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\nhle : ‖x‖ ≤ ‖y‖\nh : re ⟪x, y⟫ = ‖y‖ ^ 2\nH₁ : ‖x‖ ^ 2 ≤ ‖y‖ ^ 2\nH₂ : re ⟪y, x⟫ = ‖y‖ ^ 2\n⊢ re ⟪x - y, x - y⟫ ≤ 0",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
... | simp only [inner_sub_left, inner_sub_right] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Subspace | {
"line": 119,
"column": 6
} | {
"line": 119,
"column": 48
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : SeminormedAddCommGroup E\ninst✝⁴ : InnerProductSpace 𝕜 E\nι : Type u_4\nG : ι → Type u_5\ninst✝³ : (i : ι) → NormedAddCommGroup (G i)\ninst✝² : (i : ι) → InnerProductSpace 𝕜 (G i)\nV : (i : ι) → G i →ₗᵢ[𝕜] E\nhV : OrthogonalFamily �... | · simp only [LinearIsometry.inner_map_map] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Metrizable.CompletelyMetrizable | {
"line": 269,
"column": 8
} | {
"line": 269,
"column": 26
} | [
{
"pp": "case inr\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : DiscreteTopology X\nx y : X\nh : x ≠ y\n⊢ (if x = y then 0 else 1) = if y = x then 0 else 1",
"usedConstants": [
"Real",
"eq_false",
"Real.instZero",
"congrArg",
"Classical.propDecidable",
... | · simp [h, h.symm] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 30
} | [
{
"pp": "ι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (E i)\ni : ι\nx y : E i\n⊢ dist ⟨i, x⟩ ⟨i, y⟩ = dist x y",
"usedConstants": [
"dite_cond_eq_true",
"Real",
"congrArg",
"Classical.propDecidable",
"cast",
"Metric.Sigma.dist._proof_3",
"Nonempty.s... | simp [Dist.dist, Sigma.dist] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 30
} | [
{
"pp": "ι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (E i)\ni : ι\nx y : E i\n⊢ dist ⟨i, x⟩ ⟨i, y⟩ = dist x y",
"usedConstants": [
"dite_cond_eq_true",
"Real",
"congrArg",
"Classical.propDecidable",
"cast",
"Metric.Sigma.dist._proof_3",
"Nonempty.s... | simp [Dist.dist, Sigma.dist] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 30
} | [
{
"pp": "ι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (E i)\ni : ι\nx y : E i\n⊢ dist ⟨i, x⟩ ⟨i, y⟩ = dist x y",
"usedConstants": [
"dite_cond_eq_true",
"Real",
"congrArg",
"Classical.propDecidable",
"cast",
"Metric.Sigma.dist._proof_3",
"Nonempty.s... | simp [Dist.dist, Sigma.dist] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 407,
"column": 4
} | {
"line": 407,
"column": 19
} | [
{
"pp": "case mpr\nι : Type u_1\nE : ι → Type u_2\ninst✝ : (i : ι) → MetricSpace (E i)\ns : Set ((i : ι) × E i)\nH : ∀ x ∈ s, ∃ ε > 0, ∀ (y : (i : ι) × E i), dist x y < ε → y ∈ s\ni : ι\nx : E i\nhx : x ∈ Sigma.mk i ⁻¹' s\nε : ℝ\nεpos : ε > 0\nhε : ∀ (y : (j : ι) × E j), dist ⟨i, x⟩ y < ε → y ∈ s\ny : E i\nhy :... | apply hε ⟨i, y⟩ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 565,
"column": 6
} | {
"line": 565,
"column": 58
} | [
{
"pp": "case neg\nX : ℕ → Type u\ninst✝ : (n : ℕ) → MetricSpace (X n)\nf : (n : ℕ) → X n → X (n + 1)\nI : ∀ (n : ℕ), Isometry (f n)\nx y : (n : ℕ) × X n\nm : ℕ\nhx : x.fst ≤ m + 1\nhy : y.fst ≤ m + 1\nh : ¬max x.fst y.fst = m + 1\nthis✝ : max x.fst y.fst ≤ m.succ\nthis : max x.fst y.fst ≤ m\n⊢ inductiveLimitDi... | have xm : x.1 ≤ m := le_trans (le_max_left _ _) this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.MetricSpace.Gluing | {
"line": 629,
"column": 2
} | {
"line": 630,
"column": 42
} | [
{
"pp": "case h\nX : ℕ → Type u\ninst✝ : (n : ℕ) → MetricSpace (X n)\nf : (n : ℕ) → X n → X (n + 1)\nI : ∀ (n : ℕ), Isometry (f n)\nn : ℕ\nh : PseudoMetricSpace ((n : ℕ) × X n) := inductivePremetric I\nx✝ : TopologicalSpace ((n : ℕ) × X n) := PseudoMetricSpace.toUniformSpace.toTopologicalSpace\nx : X n\n⊢ induc... | rw [inductiveLimitDist_eq_dist I ⟨n.succ, f n x⟩ ⟨n, x⟩ n.succ, leRecOn_self,
leRecOn_succ, leRecOn_self, dist_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.LinearMap | {
"line": 98,
"column": 21
} | {
"line": 98,
"column": 45
} | [
{
"pp": "V : Type u_4\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℂ V\nS T : V →ₗ[ℂ] V\n⊢ (∀ (x : V), ⟪S x, x⟫_ℂ = ⟪T x, x⟫_ℂ) ↔ S - T = 0",
"usedConstants": [
"Module.End.instRing",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Inn... | ← inner_map_self_eq_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Module.ClosedSubmodule | {
"line": 363,
"column": 56
} | {
"line": 372,
"column": 57
} | [
{
"pp": "R : Type u_2\nM : Type u_3\nN : Type u_4\ninst✝¹⁰ : Semiring R\ninst✝⁹ : AddCommMonoid M\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : Module R M\ninst✝⁶ : AddCommMonoid N\ninst✝⁵ : TopologicalSpace N\ninst✝⁴ : Module R N\ninst✝³ : ContinuousAdd N\ninst✝² : ContinuousConstSMul R N\ninst✝¹ : ContinuousAdd M\ni... | by
ext x
simp only [mapEquiv_apply, toSubmodule_sup, Submodule.carrier_eq_coe, Submodule.map_coe,
LinearEquiv.coe_coe, ContinuousLinearEquiv.coe_toLinearEquiv, coe_toSubmodule,
Submodule.coe_closure, Set.mem_image]
have : f = f.toLinearEquiv.toLinearMap := by
exact LinearMap.ext (congrFun rfl)
rw [←... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 72,
"column": 47
} | {
"line": 72,
"column": 62
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx y : E\n⊢ (starRingEnd 𝕜) ⟪x, T y⟫ = ⟪T y, x⟫",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Inner.inner",
... | inner_conj_symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 93,
"column": 80
} | {
"line": 94,
"column": 80
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nι : Type u_3\nT : ι → E →ₗ[𝕜] E\ns : Finset ι\nhT : ∀ i ∈ s, (T i).IsSymmetric\nx✝¹ x✝ : E\n⊢ ⟪(∑ i ∈ s, T i) x✝¹, x✝⟫ = ⟪x✝¹, (∑ i ∈ s, T i) x✝⟫",
"usedConstants": [
"Eq.mpr",... | by
simpa [sum_inner, inner_sum] using Finset.sum_congr rfl fun _ hi ↦ hT _ hi _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 211,
"column": 16
} | {
"line": 211,
"column": 25
} | [
{
"pp": "case inl\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx y : E\nh : I = 0\n⊢ ↑(re (↑(re ⟪T y, x⟫) + ↑(im ⟪T y, x⟫) * 0)) = ↑(re ⟪T y, x⟫) + ↑(im ⟪T y, x⟫) * 0",
"usedConstants": [
"Eq.mp... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 214,
"column": 66
} | {
"line": 214,
"column": 74
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nx y : E\nh : I * I = -1\n⊢ ⟪T x, y⟫ =\n (⟪T x, x⟫ + ⟪T x, y⟫ + (⟪T y, x⟫ + ⟪T y, y⟫) - (⟪T x, x⟫ - ⟪T x, y⟫ - (⟪T y, x⟫ - ⟪T y, y⟫)) -\n ... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 91
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\np q : E →ₗ[𝕜] E\nhp : p.IsSymmetricProjection\nhq : q.IsSymmetricProjection\nhqp : q ∘ₗ p = p\nx y : E\n⊢ ⟪y, (p * q) x⟫ = ⟪y, p x⟫",
"usedConstants": [
"Module.End.instRing",
... | simp_rw [Module.End.mul_apply, ← hp.isSymmetric _, ← hq.isSymmetric _, ← comp_apply, hqp] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.InnerProductSpace.Projection.Reflection | {
"line": 106,
"column": 6
} | {
"line": 106,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nu v : E\n⊢ (𝕜 ∙ u).reflection v = 2 • (⟪u, v⟫ / ↑‖u‖ ^ 2) • u - v",
"usedConstants": [
"LinearIsometryEquiv.instEquivLike",
"Norm.norm",
"Eq.mpr",
"InnerProductSp... | reflection_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Projection.Reflection | {
"line": 110,
"column": 36
} | {
"line": 110,
"column": 53
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nx : E\n⊢ K.reflection x = x ↔ K.starProjection x = x",
"usedConstants": [
"LinearIsometryEquiv.instEquivLike",
"Eq.mpr"... | reflection_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 366,
"column": 20
} | {
"line": 366,
"column": 64
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nv : E\n⊢ ‖K.orthogonalProjection‖ * ‖v‖ ≤ 1 * ‖v‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Real.instIsOrder... | gcongr; exact orthogonalProjection_norm_le K | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 366,
"column": 20
} | {
"line": 366,
"column": 64
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : K.HasOrthogonalProjection\nv : E\n⊢ ‖K.orthogonalProjection‖ * ‖v‖ ≤ 1 * ‖v‖",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Real.instIsOrder... | gcongr; exact orthogonalProjection_norm_le K | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 401,
"column": 4
} | {
"line": 402,
"column": 14
} | [
{
"pp": "case hvm\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv w : E\n⊢ ⟪v, w⟫ • v ∈ 𝕜 ∙ v",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Submodule",
"instHSMul",
"Inner.inner",
"co... | rw [Submodule.mem_span_singleton]
use ⟪v, w⟫ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 401,
"column": 4
} | {
"line": 402,
"column": 14
} | [
{
"pp": "case hvm\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nv w : E\n⊢ ⟪v, w⟫ • v ∈ 𝕜 ∙ v",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Submodule",
"instHSMul",
"Inner.inner",
"co... | rw [Submodule.mem_span_singleton]
use ⟪v, w⟫ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 458,
"column": 4
} | {
"line": 458,
"column": 65
} | [
{
"pp": "case h.e'_2.h.e'_4\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nU V : Submodule 𝕜 E\ninst✝ : U.HasOrthogonalProjection\nh : U.orthogonalProjection.comp V.subtypeL = 0\nu : E\nhu : u ∈ U\nv : E\nhv : v ∈ V\nthis : U.orthogonalProjecti... | rw [starProjection_apply, this, Submodule.coe_zero, sub_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 81
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\ninst✝ : FiniteDimensional 𝕜 ↥K\nhK : FiniteDimensional 𝕜 ↥Kᗮ\n⊢ LinearMap.det ↑K.reflection.toLinearEquiv = (-1) ^ finrank 𝕜 ↥Kᗮ",
"usedConstants": [
... | let e := K.prodEquivOfIsCompl _ K.isCompl_orthogonal_of_hasOrthogonalProjection | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 188,
"column": 2
} | {
"line": 192,
"column": 44
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜' F\nf : E →SL[σ] F\nσ' : 𝕜' →+* 𝕜\nin... | have ule : ∀ n, ‖u n‖ ≤ (1 / 2) ^ n * (C * ‖y‖) := fun n ↦ by
apply le_trans (hg _).2
calc
C * ‖h^[n] y‖ ≤ C * ((1 / 2) ^ n * ‖y‖) := mul_le_mul_of_nonneg_left (hnle n) C0
_ = (1 / 2) ^ n * (C * ‖y‖) := by ring | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 69
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹⁰ : NontriviallyNormedField 𝕜\ninst✝⁹ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE : Type u_3\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜' F\nf : E →SL[σ] F\nσ' : 𝕜' →+* 𝕜\nin... | have : f (x + w) = z := by rw [f.map_add, wim, fxy, add_sub_cancel] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.InnerProductSpace.Projection.FiniteDimensional | {
"line": 269,
"column": 4
} | {
"line": 271,
"column": 43
} | [
{
"pp": "case mem\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : RCLike 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\nι : Type u_4\ninst✝¹ : Fintype ι\nV : ι → Submodule 𝕜 E\ninst✝ : ∀ (i : ι), CompleteSpace ↥(V i)\nhV : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ\nx✝ : E\ni : ι\... | refine
(Finset.sum_eq_single_of_mem i (Finset.mem_univ _) fun j _ hij => ?_).trans
(starProjection_eq_self_iff.mpr hx) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 418,
"column": 4
} | {
"line": 418,
"column": 15
} | [
{
"pp": "𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝¹⁷ : NontriviallyNormedField 𝕜\ninst✝¹⁶ : NontriviallyNormedField 𝕜'\nσ : 𝕜 →+* 𝕜'\nE✝ : Type u_3\ninst✝¹⁵ : NormedAddCommGroup E✝\ninst✝¹⁴ : NormedSpace 𝕜 E✝\nF✝ : Type u_4\ninst✝¹³ : NormedAddCommGroup F✝\ninst✝¹² : NormedSpace 𝕜' F✝\nf✝¹ : E✝ →SL[σ] F✝\nσ' :... | convert aux | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 549,
"column": 2
} | {
"line": 549,
"column": 59
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : CompleteSpace E\nF : Type u_5\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\ng : E →ₗ[𝕜] F\nhg : IsClosed ↑g.graph\nthis✝ : Complete... | exact (continuous_subtype_val.comp ψ.symm.continuous).snd | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 909,
"column": 86
} | {
"line": 909,
"column": 94
} | [
{
"pp": "case inr\np : ℝ≥0∞\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\nhp✝ : Fact (1 ≤ p)\ninst✝⁶ : SeminormedAddCommGroup α\ninst✝⁵ : SeminormedAddCommGroup β\ninst✝⁴ : SeminormedRing 𝕜\ninst✝³ : Module 𝕜 α\ninst✝² : Module 𝕜 β\ninst✝¹ : IsBoundedSMul 𝕜 α\ninst✝ : IsBoundedSMul 𝕜 β\nc : 𝕜\nf : WithLp p ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 936,
"column": 86
} | {
"line": 936,
"column": 94
} | [
{
"pp": "case inr\np : ℝ≥0∞\n𝕜 : Type u_1\nα : Type u_2\nβ : Type u_3\nhp✝ : Fact (1 ≤ p)\ninst✝⁶ : SeminormedAddCommGroup α\ninst✝⁵ : SeminormedAddCommGroup β\ninst✝⁴ : SeminormedRing 𝕜\ninst✝³ : Module 𝕜 α\ninst✝² : Module 𝕜 β\ninst✝¹ : NormSMulClass 𝕜 α\ninst✝ : NormSMulClass 𝕜 β\nc : 𝕜\nf : WithLp p ... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 50
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddCommGroup (β i)\np : ℝ≥0∞\ni : ι\na b : β i\n⊢ single p i (a - b) = single p i a - single p i b",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"WithLp",
"Eq.mpr",
"PiLp.single",
"congrAr... | simp_rw [← toLp_single, Pi.single_sub, toLp_sub] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 50
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddCommGroup (β i)\np : ℝ≥0∞\ni : ι\na b : β i\n⊢ single p i (a - b) = single p i a - single p i b",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"WithLp",
"Eq.mpr",
"PiLp.single",
"congrAr... | simp_rw [← toLp_single, Pi.single_sub, toLp_sub] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 50
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_4\ninst✝¹ : DecidableEq ι\ninst✝ : (i : ι) → AddCommGroup (β i)\np : ℝ≥0∞\ni : ι\na b : β i\n⊢ single p i (a - b) = single p i a - single p i b",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"WithLp",
"Eq.mpr",
"PiLp.single",
"congrAr... | simp_rw [← toLp_single, Pi.single_sub, toLp_sub] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 643,
"column": 18
} | {
"line": 643,
"column": 88
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_4\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nx y : WithLp ∞ ((i : ι) → β i)\n⊢ edist x.ofLp y.ofLp ≤ edist x y",
"usedConstants": [
"WithLp",
"ENNReal.ofNNReal",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
... | simpa only [ENNReal.coe_one, one_mul] using lipschitzWith_ofLp ∞ β x y | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 643,
"column": 18
} | {
"line": 643,
"column": 88
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_4\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nx y : WithLp ∞ ((i : ι) → β i)\n⊢ edist x.ofLp y.ofLp ≤ edist x y",
"usedConstants": [
"WithLp",
"ENNReal.ofNNReal",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
... | simpa only [ENNReal.coe_one, one_mul] using lipschitzWith_ofLp ∞ β x y | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 643,
"column": 18
} | {
"line": 643,
"column": 88
} | [
{
"pp": "ι : Type u_2\nβ : ι → Type u_4\ninst✝¹ : Fintype ι\ninst✝ : (i : ι) → PseudoEMetricSpace (β i)\nx y : WithLp ∞ ((i : ι) → β i)\n⊢ edist x.ofLp y.ofLp ≤ edist x y",
"usedConstants": [
"WithLp",
"ENNReal.ofNNReal",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
... | simpa only [ENNReal.coe_one, one_mul] using lipschitzWith_ofLp ∞ β x y | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Haar.OfBasis | {
"line": 166,
"column": 8
} | {
"line": 167,
"column": 70
} | [
{
"pp": "case h.mpr.refine_1.inl\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\na x : ι → ℝ\ni : ι\nh : min 0 (a i) ≤ x i ∧ x i ≤ max 0 (a i)\nhai : a i ≤ 0\n⊢ 0 ≤ (fun i ↦ x i / a i) i ∧ (fun i ↦ x i / a i) i ≤ 1",
"usedConstants": [
"Iff.mpr",
"Real.partialOrder",
"Real.instLE... | rw [sup_eq_left.mpr hai, inf_eq_right.mpr hai] at h
exact ⟨div_nonneg_of_nonpos h.2 hai, div_le_one_of_ge h.1 hai⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Haar.OfBasis | {
"line": 166,
"column": 8
} | {
"line": 167,
"column": 70
} | [
{
"pp": "case h.mpr.refine_1.inl\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\na x : ι → ℝ\ni : ι\nh : min 0 (a i) ≤ x i ∧ x i ≤ max 0 (a i)\nhai : a i ≤ 0\n⊢ 0 ≤ (fun i ↦ x i / a i) i ∧ (fun i ↦ x i / a i) i ≤ 1",
"usedConstants": [
"Iff.mpr",
"Real.partialOrder",
"Real.instLE... | rw [sup_eq_left.mpr hai, inf_eq_right.mpr hai] at h
exact ⟨div_nonneg_of_nonpos h.2 hai, div_le_one_of_ge h.1 hai⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 793,
"column": 34
} | {
"line": 793,
"column": 44
} | [
{
"pp": "case inr\np : ℝ≥0∞\n𝕜 : Type u_1\nι : Type u_2\nα : ι → Type u_3\nβ : ι → Type u_4\nhp✝ : Fact (1 ≤ p)\ninst✝⁴ : Fintype ι\ninst✝³ : SeminormedRing 𝕜\ninst✝² : (i : ι) → SeminormedAddCommGroup (β i)\ninst✝¹ : (i : ι) → Module 𝕜 (β i)\ninst✝ : ∀ (i : ι), IsBoundedSMul 𝕜 (β i)\nc : 𝕜\nf : PiLp p β\n... | smul_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 77,
"column": 38
} | {
"line": 79,
"column": 34
} | [
{
"pp": "E : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁴ : NormedField K\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\nb : Basis ι K E\nF : Type u_4\ninst✝¹ : AddCommGroup F\ninst✝ : Module K F\nf : E ≃ₗ[K] F\n⊢ Submodule.map (↑(LinearEquiv.restrictScalars ℤ f)) (span ℤ (Set.range ⇑b)) = span ℤ (Se... | by
simp_rw [Submodule.map_span, LinearEquiv.coe_coe, LinearEquiv.restrictScalars_apply,
Basis.coe_map, Set.range_comp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 101,
"column": 6
} | {
"line": 101,
"column": 28
} | [
{
"pp": "case h\nE : Type u_1\nι : Type u_2\nK : Type u_3\ninst✝⁵ : NormedField K\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\nb : Basis ι K E\ninst✝² : LinearOrder K\nF : Type u_4\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace K F\nf : E ≃ₗ[K] F\nx : F\n⊢ x ∈ ⇑f '' fundamentalDomain b ↔ x ∈ f... | mem_fundamentalDomain, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 280,
"column": 67
} | {
"line": 280,
"column": 81
} | [
{
"pp": "ι : Type u_1\nI J : Box ι\nπ : Prepartition I\nπi : (J : Box ι) → Prepartition J\n⊢ J ∈ π.biUnion πi ↔ ∃ J' ∈ π, J ∈ πi J'",
"usedConstants": [
"BoxIntegral.Prepartition.biUnion._proof_4",
"BoxIntegral.Prepartition",
"congrArg",
"Finset",
"Finset.mem_biUnion._simp_1",
... | simp [biUnion] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 280,
"column": 67
} | {
"line": 280,
"column": 81
} | [
{
"pp": "ι : Type u_1\nI J : Box ι\nπ : Prepartition I\nπi : (J : Box ι) → Prepartition J\n⊢ J ∈ π.biUnion πi ↔ ∃ J' ∈ π, J ∈ πi J'",
"usedConstants": [
"BoxIntegral.Prepartition.biUnion._proof_4",
"BoxIntegral.Prepartition",
"congrArg",
"Finset",
"Finset.mem_biUnion._simp_1",
... | simp [biUnion] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 280,
"column": 67
} | {
"line": 280,
"column": 81
} | [
{
"pp": "ι : Type u_1\nI J : Box ι\nπ : Prepartition I\nπi : (J : Box ι) → Prepartition J\n⊢ J ∈ π.biUnion πi ↔ ∃ J' ∈ π, J ∈ πi J'",
"usedConstants": [
"BoxIntegral.Prepartition.biUnion._proof_4",
"BoxIntegral.Prepartition",
"congrArg",
"Finset",
"Finset.mem_biUnion._simp_1",
... | simp [biUnion] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 344,
"column": 6
} | {
"line": 344,
"column": 62
} | [
{
"pp": "case intro\nE : Type u_1\nι : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nb : Basis ι ℝ E\ninst✝² : MeasurableSpace E\ninst✝¹ : OpensMeasurableSpace E\ninst✝ : Finite ι\nval✝ : Fintype ι\nthis : FiniteDimensional ℝ E\nD : Set (ι → ℝ) := Set.univ.pi fun x ↦ Set.Ico 0 1\n⊢ Measurab... | (_ : fundamentalDomain b = b.equivFun.toLinearMap ⁻¹' D) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 416,
"column": 4
} | {
"line": 416,
"column": 26
} | [
{
"pp": "E : Type u_1\nι : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nb : Basis ι ℝ E\ninst✝³ : Fintype ι\ninst✝² : MeasurableSpace E\nμ : Measure E\ninst✝¹ : BorelSpace E\ninst✝ : μ.IsAddHaarMeasure\nthis : FiniteDimensional ℝ E\nx : E\nhx : (∀ (i : ι), 0 ≤ (b.repr x) i ∧ (b.repr x) i ≤... | mem_fundamentalDomain, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 710,
"column": 4
} | {
"line": 710,
"column": 25
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) (𝓝[>] 0) (�... | filter_upwards with r | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.BoxIntegral.Partition.Split | {
"line": 185,
"column": 28
} | {
"line": 185,
"column": 76
} | [
{
"pp": "ι : Type u_1\nM : Type u_3\ninst✝ : AddCommMonoid M\nI : Box ι\ni : ι\nx : ℝ\nf : Box ι → M\n⊢ ∑ J ∈ {I.splitLower i x, I.splitUpper i x}, Option.elim' 0 f J =\n Option.elim' 0 f (I.splitLower i x) + Option.elim' 0 f (I.splitUpper i x)",
"usedConstants": [
"Eq.mpr",
"Option.elim'",
... | Finset.sum_pair (I.splitLower_ne_splitUpper i x) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 770,
"column": 4
} | {
"line": 770,
"column": 28
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) ... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 767,
"column": 2
} | {
"line": 775,
"column": 73
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nhs : MeasurableSet s\nx : E\nh : Tendsto (fun r ↦ μ (s ∩ closedBall x r) / μ (closedBall x r)) ... | have I : ∀ u v, μ u ≠ 0 → μ u ≠ ∞ → MeasurableSet v →
μ u / μ u - μ (vᶜ ∩ u) / μ u = μ (v ∩ u) / μ u := by
intro u v uzero utop vmeas
simp_rw [div_eq_mul_inv]
rw [← ENNReal.sub_mul]; swap
· simp only [uzero, ENNReal.inv_eq_top, imp_true_iff, Ne, not_false_iff]
congr 1
rw [inter_comm _ u, int... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.BoxIntegral.Partition.Measure | {
"line": 86,
"column": 2
} | {
"line": 86,
"column": 38
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Finite ι\nI : Box ι\nπ : Prepartition I\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\n⊢ ∀ b ∈ π.boxes, MeasurableSet ↑b",
"usedConstants": [
"BoxIntegral.Box.measurableSet_coe",
"Finset",
"Membership.mem",
"BoxIntegral.Prepartition.boxes",
... | exact fun J _ => J.measurableSet_coe | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Oscillation | {
"line": 97,
"column": 2
} | {
"line": 97,
"column": 71
} | [
{
"pp": "E : Type u\nF : Type v\ninst✝¹ : PseudoEMetricSpace F\ninst✝ : TopologicalSpace E\nf : E → F\nx : E\n⊢ oscillationWithin f univ x = 0 ↔ ContinuousWithinAt f univ x",
"usedConstants": [
"Set.univ",
"Set.mem_univ",
"OscillationWithin.eq_zero_iff_continuousWithinAt"
]
}
] | exact OscillationWithin.eq_zero_iff_continuousWithinAt f (mem_univ x) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.BoxIntegral.Partition.Additive | {
"line": 122,
"column": 2
} | {
"line": 137,
"column": 65
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nn : ℕ\nN : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : AddCommMonoid N\nI₀✝ : WithTop (Box ι)\nI : Box ι\ni : ι\ninst✝ : Finite ι\nf : Box ι → M\nI₀ : WithTop (Box ι)\nhf :\n ∀ (I : Box ι),\n ↑I ≤ I₀ →\n ∀ {i : ι} {x : ℝ},\n x ∈ Set.Ioo (I.lower i) (I.upper ... | refine ⟨f, ?_⟩
replace hf (I : Box ι) (hI : ↑I ≤ I₀) (s) : ∑ J ∈ (splitMany I s).boxes, f J = f I := by
induction s using Finset.induction_on with
| empty => simp
| insert a s _ ihs =>
rw [splitMany_insert, inf_split, ← ihs, biUnion_boxes, sum_biUnion_boxes]
refine Finset.sum_congr rfl fun J' ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.Partition.Additive | {
"line": 122,
"column": 2
} | {
"line": 137,
"column": 65
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nn : ℕ\nN : Type u_3\ninst✝² : AddCommMonoid M\ninst✝¹ : AddCommMonoid N\nI₀✝ : WithTop (Box ι)\nI : Box ι\ni : ι\ninst✝ : Finite ι\nf : Box ι → M\nI₀ : WithTop (Box ι)\nhf :\n ∀ (I : Box ι),\n ↑I ≤ I₀ →\n ∀ {i : ι} {x : ℝ},\n x ∈ Set.Ioo (I.lower i) (I.upper ... | refine ⟨f, ?_⟩
replace hf (I : Box ι) (hI : ↑I ≤ I₀) (s) : ∑ J ∈ (splitMany I s).boxes, f J = f I := by
induction s using Finset.induction_on with
| empty => simp
| insert a s _ ihs =>
rw [splitMany_insert, inf_split, ← ihs, biUnion_boxes, sum_biUnion_boxes]
refine Finset.sum_congr rfl fun J' ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Oscillation | {
"line": 119,
"column": 8
} | {
"line": 119,
"column": 49
} | [
{
"pp": "E : Type u\nF : Type v\ninst✝¹ : PseudoEMetricSpace F\ninst✝ : PseudoEMetricSpace E\nK : Set E\nf : E → F\nD : Set E\nε : ℝ≥0∞\ncomp : IsCompact K\nhK : ∀ x ∈ K, oscillationWithin f D x < ε\nS : ℝ → Set E := fun r ↦ {x | ∃ a > r, ediam (f '' (eball x (ENNReal.ofReal a) ∩ D)) ≤ ε}\nr : ℝ\nx✝¹ : r > 0\nx... | ← ofReal_add (by linarith) (by linarith), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 1138,
"column": 2
} | {
"line": 1139,
"column": 68
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_3\ninst✝⁵ : RCLike 𝕜\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : Fintype ι\ninst✝¹ : FiniteDimensional 𝕜 E\nn : ℕ\nhn : finrank 𝕜 E = n\ninst✝ : DecidableEq ι\nV : ι → Submodule 𝕜 E\nhV : IsInternal V\nhV' : OrthogonalFamily 𝕜 (... | apply Finset.card_eq_of_equiv_fin
simpa using hV.subordinateOrthonormalBasisIndexFiberEquiv hn hV' i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 1138,
"column": 2
} | {
"line": 1139,
"column": 68
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_3\ninst✝⁵ : RCLike 𝕜\nE : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : Fintype ι\ninst✝¹ : FiniteDimensional 𝕜 E\nn : ℕ\nhn : finrank 𝕜 E = n\ninst✝ : DecidableEq ι\nV : ι → Submodule 𝕜 E\nhV : IsInternal V\nhV' : OrthogonalFamily 𝕜 (... | apply Finset.card_eq_of_equiv_fin
simpa using hV.subordinateOrthonormalBasisIndexFiberEquiv hn hV' i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.Integrability | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 44
} | [
{
"pp": "case h\nι : Type u\nE : Type v\ninst✝³ : Fintype ι\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nl : IntegrationParams\nI : Box ι\nf : (ι → ℝ) → E\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\nhl : l.bRiemann = false\nε : ℝ≥0\nε0 : 0 < ε\nδ : ℕ → ℝ≥0\nδ0 : ∀ (i : ℕ), 0 < δ i\nc✝ :... | refine (norm_sum_le_of_le _ this).trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 1207,
"column": 8
} | {
"line": 1207,
"column": 90
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝¹⁰ : RCLike 𝕜\nE✝ : Type u_4\ninst✝⁹ : NormedAddCommGroup E✝\ninst✝⁸ : InnerProductSpace 𝕜 E✝\nF : Type u_5\ninst✝⁷ : NormedAddCommGroup F\ninst✝⁶ : InnerProductSpace ℝ F\nF' : Type u_6\ninst✝⁵ : NormedAddCommGroup F'\ninst✝⁴ : InnerProductSpace ℝ F'\n... | norm_add_sq_eq_norm_sq_add_norm_sq_of_inner_eq_zero (L (p1 x)) (L3 (p2 x)) Mx_orth | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Prime.Int | {
"line": 27,
"column": 6
} | {
"line": 28,
"column": 96
} | [
{
"pp": "p : ℕ\nhp : Prime p\na b : ℤ\nh : ↑p ∣ a * b\n⊢ ↑p ∣ a ∨ ↑p ∣ b",
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"HMul.hMul",
"Int.dvd_natAbs",
"Int.natAbs_mul",
"MulZeroClass.toMul",
"congrArg",
"semigroupDvd",
"CommSemiring.toCommMonoidWithZero",
... | rw [← Int.dvd_natAbs, Int.natCast_dvd_natCast, Int.natAbs_mul, hp.dvd_mul] at h
rwa [← Int.dvd_natAbs, Int.natCast_dvd_natCast, ← Int.dvd_natAbs, Int.natCast_dvd_natCast] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Prime.Int | {
"line": 27,
"column": 6
} | {
"line": 28,
"column": 96
} | [
{
"pp": "p : ℕ\nhp : Prime p\na b : ℤ\nh : ↑p ∣ a * b\n⊢ ↑p ∣ a ∨ ↑p ∣ b",
"usedConstants": [
"Eq.mpr",
"Dvd.dvd",
"HMul.hMul",
"Int.dvd_natAbs",
"Int.natAbs_mul",
"MulZeroClass.toMul",
"congrArg",
"semigroupDvd",
"CommSemiring.toCommMonoidWithZero",
... | rw [← Int.dvd_natAbs, Int.natCast_dvd_natCast, Int.natAbs_mul, hp.dvd_mul] at h
rwa [← Int.dvd_natAbs, Int.natCast_dvd_natCast, ← Int.dvd_natAbs, Int.natCast_dvd_natCast] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.UnitPartition | {
"line": 395,
"column": 17
} | {
"line": 395,
"column": 26
} | [
{
"pp": "ι : Type u_1\nn : ℕ\ninst✝¹ : NeZero n\ns : Set (ι → ℝ)\nF : (ι → ℝ) → ℝ\ninst✝ : Fintype ι\nB : Box ι\nhB : hasIntegralVertices B\nhs₀ : s ≤ ↑B\nthis : Fintype ↑(s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))\n⊢ ∑ i ∈ (s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι)))).toFinset, F i / ↑n ^... | mul_zero, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 95,
"column": 2
} | {
"line": 101,
"column": 25
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nI : Box ι\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nπ : TaggedPrepartition I\nπi : (J : Box ι) → Prepartition J\nhπi : ∀ J ∈ π, (πi J).IsPartition\nJ : ... | calc
(∑ J' ∈ (πi J).boxes, vol J' (f (π.tag <| π.toPrepartition.biUnionIndex πi J'))) =
∑ J' ∈ (πi J).boxes, vol J' (f (π.tag J)) :=
sum_congr rfl fun J' hJ' => by rw [Prepartition.biUnionIndex_of_mem _ hJ hJ']
_ = vol J (f (π.tag J)) :=
(vol.map ⟨⟨fun g : E →L[ℝ] F => g (f (π.tag J)), rfl⟩,... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 331,
"column": 2
} | {
"line": 332,
"column": 64
} | [
{
"pp": "case neg\nι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nc : ℝ\nhc : c ≠ 0\nhf : ¬Integrable I l f vol\n... | · have : ¬Integrable I l (fun x => c • f x) vol := mt (fun h => h.of_smul hc) hf
rw [integral, integral, dif_neg hf, dif_neg this, smul_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 410,
"column": 2
} | {
"line": 410,
"column": 19
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nI : Box ι\ninst✝ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nh : Integrable I l f vol\nε : ℝ\nc : ℝ≥0\n⊢ l.RCond (h.co... | rw [convergenceR] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 87,
"column": 6
} | {
"line": 87,
"column": 23
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\na b : ι\nh₀ : a ≠ b\nthis : ∀ (a b : ι), a < b → ⟪gramSchmidt 𝕜 f a, gr... | exact this _ _ hb | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 303,
"column": 2
} | {
"line": 307,
"column": 82
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\nh₀ : LinearIndependent 𝕜 f\n⊢ LinearIndependent 𝕜 (gramSchmidtNormed 𝕜 f)",
... | unfold gramSchmidtNormed
have (i : ι) : IsUnit (‖gramSchmidt 𝕜 f i‖⁻¹ : 𝕜) :=
isUnit_iff_ne_zero.mpr (by simp [gramSchmidt_ne_zero i h₀])
let w : ι → 𝕜ˣ := fun i ↦ (this i).unit
apply (gramSchmidt_linearIndependent h₀).units_smul (w := fun i ↦ (this i).unit) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.GramSchmidtOrtho | {
"line": 303,
"column": 2
} | {
"line": 307,
"column": 82
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : InnerProductSpace 𝕜 E\nι : Type u_3\ninst✝² : LinearOrder ι\ninst✝¹ : LocallyFiniteOrderBot ι\ninst✝ : WellFoundedLT ι\nf : ι → E\nh₀ : LinearIndependent 𝕜 f\n⊢ LinearIndependent 𝕜 (gramSchmidtNormed 𝕜 f)",
... | unfold gramSchmidtNormed
have (i : ι) : IsUnit (‖gramSchmidt 𝕜 f i‖⁻¹ : 𝕜) :=
isUnit_iff_ne_zero.mpr (by simp [gramSchmidt_ne_zero i h₀])
let w : ι → 𝕜ˣ := fun i ↦ (this i).unit
apply (gramSchmidt_linearIndependent h₀).units_smul (w := fun i ↦ (this i).unit) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 238,
"column": 4
} | {
"line": 239,
"column": 36
} | [
{
"pp": "case succ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn✝ : ℕ\n_i : Fact (finrank ℝ E = n✝ + 1)\no : Orientation ℝ E (Fin (n✝ + 1))\nb : OrthonormalBasis (Fin (n✝ + 1)) ℝ E\nv : Fin (n✝ + 1) → E\n⊢ |o.volumeForm v| = |b.toBasis.det v|",
"usedConstants": [
"Alte... | rw [o.volumeForm_robust (b.adjustToOrientation o) (b.orientation_adjustToOrientation o),
b.abs_det_adjustToOrientation] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 238,
"column": 4
} | {
"line": 239,
"column": 36
} | [
{
"pp": "case succ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn✝ : ℕ\n_i : Fact (finrank ℝ E = n✝ + 1)\no : Orientation ℝ E (Fin (n✝ + 1))\nb : OrthonormalBasis (Fin (n✝ + 1)) ℝ E\nv : Fin (n✝ + 1) → E\n⊢ |o.volumeForm v| = |b.toBasis.det v|",
"usedConstants": [
"Alte... | rw [o.volumeForm_robust (b.adjustToOrientation o) (b.orientation_adjustToOrientation o),
b.abs_det_adjustToOrientation] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Orientation | {
"line": 238,
"column": 4
} | {
"line": 239,
"column": 36
} | [
{
"pp": "case succ\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℝ E\nn✝ : ℕ\n_i : Fact (finrank ℝ E = n✝ + 1)\no : Orientation ℝ E (Fin (n✝ + 1))\nb : OrthonormalBasis (Fin (n✝ + 1)) ℝ E\nv : Fin (n✝ + 1) → E\n⊢ |o.volumeForm v| = |b.toBasis.det v|",
"usedConstants": [
"Alte... | rw [o.volumeForm_robust (b.adjustToOrientation o) (b.orientation_adjustToOrientation o),
b.abs_det_adjustToOrientation] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.PSeries | {
"line": 289,
"column": 6
} | {
"line": 290,
"column": 12
} | [
{
"pp": "case inl.h_mono\np : ℝ\nhp : 0 ≤ p\n⊢ ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → (↑n ^ p)⁻¹ ≤ (↑m ^ p)⁻¹",
"usedConstants": [
"Iff.mpr",
"Real.instPow",
"Real.partialOrder",
"Real.rpow_pos_of_pos",
"Real",
"Preorder.toLT",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",... | intro m n hm hmn
gcongr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.PSeries | {
"line": 289,
"column": 6
} | {
"line": 290,
"column": 12
} | [
{
"pp": "case inl.h_mono\np : ℝ\nhp : 0 ≤ p\n⊢ ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → (↑n ^ p)⁻¹ ≤ (↑m ^ p)⁻¹",
"usedConstants": [
"Iff.mpr",
"Real.instPow",
"Real.partialOrder",
"Real.rpow_pos_of_pos",
"Real",
"Preorder.toLT",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",... | intro m n hm hmn
gcongr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.Cardinal | {
"line": 38,
"column": 49
} | {
"line": 38,
"column": 82
} | [
{
"pp": "R : Type u\nM' : Type v\ninst✝² : Semiring R\ninst✝¹ : Infinite M'\ninst✝ : Nontrivial R\n⊢ #R[M'] = max (lift.{v, u} #R) (lift.{u, v} #M')",
"usedConstants": [
"Lattice.toSemilatticeSup",
"Cardinal.mk_finsupp_lift_of_infinite",
"Cardinal",
"congrArg",
"Cardinal.lift",... | by simp [MonoidAlgebra, max_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MonoidAlgebra.Cardinal | {
"line": 48,
"column": 49
} | {
"line": 48,
"column": 82
} | [
{
"pp": "R : Type u\nM' : Type v\ninst✝² : Semiring R\ninst✝¹ : Nonempty M'\ninst✝ : Infinite R\n⊢ #R[M'] = max (lift.{v, u} #R) (lift.{u, v} #M')",
"usedConstants": [
"Lattice.toSemilatticeSup",
"Cardinal",
"congrArg",
"Cardinal.lift",
"NonUnitalNonAssocSemiring.toMulZeroClass... | by simp [MonoidAlgebra, max_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MonoidAlgebra.Grading | {
"line": 155,
"column": 26
} | {
"line": 155,
"column": 30
} | [
{
"pp": "case refine_2\nM : Type u_1\nι : Type u_2\nR : Type u_3\ninst✝³ : AddMonoid M\ninst✝² : DecidableEq ι\ninst✝¹ : AddMonoid ι\ninst✝ : CommSemiring R\nf : M →+ ι\ni : ι\nx : R[M]\nm : M\nb : R\ny : M →₀ R\nhmy : m ∉ y.support\nhb : b ≠ 0\nih : ∀ (hx : y ∈ gradeBy R (⇑f) i), (decomposeAux f) ↑⟨y, hx⟩ = (D... | hmby | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Data.Nat.Factorial.DoubleFactorial | {
"line": 64,
"column": 8
} | {
"line": 64,
"column": 16
} | [
{
"pp": "n : ℕ\n⊢ (2 * (n + 1))‼ = 2 ^ (n + 1) * (n + 1)!",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"HMul.hMul",
"congrArg",
"Nat.instMonoid",
"Nat.doubleFactorial",
"id",
"instMulNat",
"instOfNatNat",
"Monoid.toPow",
"instHA... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Comap | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 8
} | [
{
"pp": "case h.e'_2.h.e'_5\nσ : Type u_1\nR : Type u_4\ninst✝ : CommSemiring R\nf : MvPolynomial σ R →ₐ[R] MvPolynomial σ R\nhf : ∀ (φ : MvPolynomial σ R), f φ = φ\nx : σ → R\n⊢ f = AlgHom.id R (MvPolynomial σ R)",
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"C... | ext1 φ | Lean.Elab.Tactic.Ext._aux_Init_Ext___macroRules_Lean_Elab_Tactic_Ext_tacticExt1____1 | Lean.Elab.Tactic.Ext.tacticExt1___ |
Mathlib.Algebra.MvPolynomial.Polynomial | {
"line": 24,
"column": 2
} | {
"line": 29,
"column": 13
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nσ : Type u_3\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nx : S\nf : R →+* Polynomial S\ng : σ → Polynomial S\np : MvPolynomial σ R\n⊢ Polynomial.eval x (eval₂ f g p) = eval₂ ((Polynomial.evalRingHom x).comp f) (fun s ↦ Polynomial.eval x (g s)) p",
"usedConstants": ... | apply induction_on p
· simp
· intro p q hp hq
simp [hp, hq]
· intro p n hp
simp [hp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Polynomial | {
"line": 24,
"column": 2
} | {
"line": 29,
"column": 13
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nσ : Type u_3\ninst✝¹ : CommSemiring R\ninst✝ : CommSemiring S\nx : S\nf : R →+* Polynomial S\ng : σ → Polynomial S\np : MvPolynomial σ R\n⊢ Polynomial.eval x (eval₂ f g p) = eval₂ ((Polynomial.evalRingHom x).comp f) (fun s ↦ Polynomial.eval x (g s)) p",
"usedConstants": ... | apply induction_on p
· simp
· intro p q hp hq
simp [hp, hq]
· intro p n hp
simp [hp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Expand | {
"line": 96,
"column": 4
} | {
"line": 97,
"column": 37
} | [
{
"pp": "case a\nσ : Type u_1\nR : Type u_3\ninst✝ : CommSemiring R\nn : ℕ\nhn : 0 < n\ng g' : MvPolynomial σ R\nH : (expand n) g = (expand n) g'\nd : σ →₀ ℕ\n⊢ coeff d g = coeff d g'",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
... | rw [← coeff_expand_smul _ (n.ne_zero_iff_zero_lt.mpr hn), H, coeff_expand_smul _
(n.ne_zero_iff_zero_lt.mpr hn)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.MvPolynomial.Division | {
"line": 340,
"column": 2
} | {
"line": 379,
"column": 36
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\np q : MvPolynomial σ R\ninst✝ : IsCancelMulZero R\nn : σ →₀ ℕ\n⊢ p ∣ (monomial n) 1 * q ↔ ∃ m r, m ≤ n ∧ r ∣ q ∧ p = (monomial m) 1 * r",
"usedConstants": [
"Finsupp.instCanonicallyOrderedAddOfAddLeftMono",
"Nontrivial",
"Finsupp.me... | rcases subsingleton_or_nontrivial R with hR | hR
· simp only [Subsingleton.elim _ p, dvd_refl, and_self, and_true, exists_const, true_iff]
refine ⟨n, le_refl n⟩
suffices ∀ (d) (n : σ →₀ ℕ) (hd : n.degree = d) (p q : MvPolynomial σ R),
p ∣ monomial n 1 * q ↔ ∃ m r, m ≤ n ∧ r ∣ q ∧ p = monomial m 1 * r from t... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Division | {
"line": 340,
"column": 2
} | {
"line": 379,
"column": 36
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\np q : MvPolynomial σ R\ninst✝ : IsCancelMulZero R\nn : σ →₀ ℕ\n⊢ p ∣ (monomial n) 1 * q ↔ ∃ m r, m ≤ n ∧ r ∣ q ∧ p = (monomial m) 1 * r",
"usedConstants": [
"Finsupp.instCanonicallyOrderedAddOfAddLeftMono",
"Nontrivial",
"Finsupp.me... | rcases subsingleton_or_nontrivial R with hR | hR
· simp only [Subsingleton.elim _ p, dvd_refl, and_self, and_true, exists_const, true_iff]
refine ⟨n, le_refl n⟩
suffices ∀ (d) (n : σ →₀ ℕ) (hd : n.degree = d) (p q : MvPolynomial σ R),
p ∣ monomial n 1 * q ↔ ∃ m r, m ≤ n ∧ r ∣ q ∧ p = monomial m 1 * r from t... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.GameAdd | {
"line": 151,
"column": 55
} | {
"line": 151,
"column": 96
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nrα✝ : α → α → Prop\nrβ : β → β → Prop\na : α\nb : β\nrα : α → α → Prop\na₁ b₁ a₂ b₂ : α\n⊢ (Prod.GameAdd rα rα (a₁, b₁) (a₂, b₂) ∨ Prod.GameAdd rα rα (b₁, a₁) (a₂, b₂)) =\n (Prod.GameAdd rα rα (b₁, a₁) (a₂, b₂) ∨ Prod.GameAdd rα rα (a₁, b₁) (a₂, b₂)) ∧\n (Prod.GameAd... | Prod.gameAdd_swap_swap_mk _ _ a₁ b₂ b₁ a₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.GameAdd | {
"line": 155,
"column": 97
} | {
"line": 157,
"column": 5
} | [
{
"pp": "α : Type u_1\nrα : α → α → Prop\n⊢ ∀ {x y : α × α}, GameAdd rα s(x.1, x.2) s(y.1, y.2) ↔ Prod.GameAdd rα rα x y ∨ Prod.GameAdd rα rα x.swap y",
"usedConstants": [
"Sym2.mk",
"Prod.GameAdd",
"Iff.rfl",
"Prod.mk",
"Prod.fst",
"Iff",
"Sym2.GameAdd",
"Or"... | by
rintro ⟨_, _⟩ ⟨_, _⟩
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.NoZeroDivisors | {
"line": 61,
"column": 9
} | {
"line": 61,
"column": 43
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\nn : σ\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\nι : Type u_3\nf : ι → MvPolynomial σ R\nnontrivial : Nontrivial (MvPolynomial σ R)\na : ι\ns : Finset ι\na_not_mem : a ∉ s\nih : (∀ i ∈ s, f i ≠ 0) → degreeOf n (∏ i ∈ s, f i) = ∑ i ∈ s, degreeOf n (f i)\nha : ¬f a = ... | by rw [prod_ne_zero_iff]; exact hs | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.MvPolynomial.NoZeroDivisors | {
"line": 158,
"column": 2
} | {
"line": 164,
"column": 30
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\np : MvPolynomial σ R\nn : σ →₀ ℕ\n⊢ p ∣ (monomial n) 1 ↔ ∃ m u, m ≤ n ∧ IsUnit u ∧ p = (monomial m) u",
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"Finsupp.instLE",
"MulOne.toOne",
"le_refl",
... | rcases subsingleton_or_nontrivial R with hR | hR
· suffices ∃ m, m ≤ n by simpa [Subsingleton.elim _ p]
use n
rw [dvd_monomial_iff_exists (one_ne_zero' R)]
apply exists_congr
intro m
simp_rw [isUnit_iff_dvd_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.NoZeroDivisors | {
"line": 158,
"column": 2
} | {
"line": 164,
"column": 30
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\np : MvPolynomial σ R\nn : σ →₀ ℕ\n⊢ p ∣ (monomial n) 1 ↔ ∃ m u, m ≤ n ∧ IsUnit u ∧ p = (monomial m) u",
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"Finsupp.instLE",
"MulOne.toOne",
"le_refl",
... | rcases subsingleton_or_nontrivial R with hR | hR
· suffices ∃ m, m ≤ n by simpa [Subsingleton.elim _ p]
use n
rw [dvd_monomial_iff_exists (one_ne_zero' R)]
apply exists_congr
intro m
simp_rw [isUnit_iff_dvd_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 159,
"column": 30
} | {
"line": 159,
"column": 41
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Subsingleton R\nf : MvPolynomial σ R\n⊢ m.degree 0 = 0",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"congrArg",
"CommSemiring.toSemiring",
"AddMonoid.toAddZeroClass",
"Add... | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.Zeta | {
"line": 62,
"column": 6
} | {
"line": 62,
"column": 16
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : Semiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : MulAction R M\nf : ArithmeticFunction M\nx : ℕ\n⊢ (↑ζ • f) x = ∑ i ∈ x.divisors, f i",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"Nat.divisorsAntidia... | smul_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 574,
"column": 23
} | {
"line": 574,
"column": 34
} | [
{
"pp": "case neg\nσ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : IsReduced R\nf : MvPolynomial σ R\nn : ℕ\nhf : f = 0\nhn : ¬n = 0\n⊢ m.degree 0 = 0",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"congrArg",
... | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.Zeta | {
"line": 201,
"column": 3
} | {
"line": 201,
"column": 19
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\n⊢ (f.pmul g) 1 = 1",
"usedConstants": [
"ArithmeticFunction.pmul",
"MulOne.toOne",
"HMul.hMul",
"ArithmeticFunction.instFunLikeNat",
"MulZeroClass.toMul"... | by simp [hf, hg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ArithmeticFunction.Zeta | {
"line": 208,
"column": 3
} | {
"line": 208,
"column": 19
} | [
{
"pp": "R : Type u_1\ninst✝ : CommGroupWithZero R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\n⊢ (f.pdiv g) 1 = 1",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"False",
"instHDiv",
"NeZero.one",
"GroupWithZero.to... | by simp [hf, hg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Factorization.PrimePow | {
"line": 62,
"column": 47
} | {
"line": 62,
"column": 82
} | [
{
"pp": "n : ℕ\nhn : 2 ≤ n\n⊢ ¬n.primeFactors.card = 1 ↔ n.primeFactors.Nontrivial",
"usedConstants": [
"Eq.mpr",
"Finset.one_lt_card_iff_nontrivial",
"congrArg",
"id",
"instOfNatNat",
"Iff",
"Nat",
"LT.lt",
"Finset.Nontrivial",
"propext",
"F... | ← Finset.one_lt_card_iff_nontrivial | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 880,
"column": 34
} | {
"line": 882,
"column": 18
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nf : MvPolynomial σ R\n⊢ m.degree (-f) = m.degree f",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Nat.instMulZeroClass",
"Lattice.toSemilatticeSup",
"congrArg",
"Finset",
"MonomialO... | by
unfold degree
rw [support_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 962,
"column": 17
} | {
"line": 962,
"column": 54
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nf g : MvPolynomial σ R\n⊢ m.leadingCoeff g * coeff (m.degree f) f -\n coeff (m.degree f ⊔ m.degree g) ((monomial (m.degree f ⊔ m.degree g - m.degree g)) (m.leadingCoeff f) * g) =\n 0",
"usedConstants": [
"Finsupp.ins... | ← tsub_add_cancel_of_le le_sup_right, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 175,
"column": 82
} | {
"line": 176,
"column": 38
} | [
{
"pp": "k n : ℕ\n⊢ (σ k) n = ∑ d ∈ n.divisors, (n / d) ^ k",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"instHDiv",
"ArithmeticFunction.instFunLikeNat",
"ArithmeticFunction.sigma_apply",
"congrArg",
"Nat.instMonoid",
"id",
"HDiv.hDiv",
"... | by
rw [sigma_apply, ← sum_div_divisors] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 321,
"column": 2
} | {
"line": 321,
"column": 76
} | [
{
"pp": "n : ℕ\n⊢ Ω n = n.factorization.sum fun x k ↦ k",
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instMulZeroClass",
"ArithmeticFunction.instFunLikeNat",
"congrArg",
"Finset",
"_private.Mathlib.NumberTheory.ArithmeticFunction.Misc.0.ArithmeticFunction.cardFactors_e... | simp [cardFactors_apply, ← List.sum_toFinset_count_eq_length, Finsupp.sum] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
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