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.MeasureTheory.Integral.SetToL1 | {
"line": 1147,
"column": 4
} | {
"line": 1147,
"column": 89
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC C' : ℝ\nc : ℝ≥0∞\nhc_ne_top : c ≠ ∞\nhT : DominatedFinMeasAdditive μ T C\nh... | have h : ∀ s, MeasurableSet s → μ s < ∞ → T s = 0 := fun s hs _ => hT_smul.eq_zero hs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1275,
"column": 2
} | {
"line": 1275,
"column": 40
} | [
{
"pp": "case pos\nα : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nhT : DominatedFinMeasAdditive μ T C\nfs : ℕ → α → E\nf : α → E\nbound ... | rw [tendsto_iff_norm_sub_tendsto_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1297,
"column": 2
} | {
"line": 1311,
"column": 35
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_7\nl : Filter ι\ninst✝ : l.IsCo... | rw [tendsto_iff_seq_tendsto]
intro x xl
have hxl : ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s := by rwa [tendsto_atTop'] at xl
have h :
{ x : ι | (fun n => AEStronglyMeasurable (fs n) μ) x } ∩
{ x : ι | (fun n => ∀ᵐ a ∂μ, ‖fs n a‖ ≤ bound a) x } ∈ l :=
inter_mem hfs_meas h_bound
obtain ⟨k, h⟩ := hxl _ h
r... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.SetToL1 | {
"line": 1297,
"column": 2
} | {
"line": 1311,
"column": 35
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\nm : MeasurableSpace α\nμ : Measure α\nT : Set α → E →L[ℝ] F\nC : ℝ\nhT : DominatedFinMeasAdditive μ T C\nι : Type u_7\nl : Filter ι\ninst✝ : l.IsCo... | rw [tendsto_iff_seq_tendsto]
intro x xl
have hxl : ∀ s ∈ l, ∃ a, ∀ b ≥ a, x b ∈ s := by rwa [tendsto_atTop'] at xl
have h :
{ x : ι | (fun n => AEStronglyMeasurable (fs n) μ) x } ∩
{ x : ι | (fun n => ∀ᵐ a ∂μ, ‖fs n a‖ ≤ bound a) x } ∈ l :=
inter_mem hfs_meas h_bound
obtain ⟨k, h⟩ := hxl _ h
r... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Matrix.Diagonal | {
"line": 62,
"column": 65
} | {
"line": 62,
"column": 90
} | [
{
"pp": "m : Type u_1\ninst✝² : Fintype m\nK : Type u\ninst✝¹ : Semifield K\ninst✝ : DecidableEq m\nw : m → K\nthis : ∀ (i : m), (proj i ∘ₗ toLin' (diagonal w)).ker = comap (toLin' (diagonal w)) (proj i).ker\n⊢ univ ⊆ {i | w i = 0} ∪ {i | w i = 0}ᶜ",
"usedConstants": [
"Eq.mpr",
"congrArg",
... | rw [Set.union_compl_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Marginal | {
"line": 188,
"column": 58
} | {
"line": 193,
"column": 5
} | [
{
"pp": "δ : Type u_1\nX : δ → Type u_3\ninst✝³ : (i : δ) → MeasurableSpace (X i)\nμ : (i : δ) → Measure (X i)\ninst✝² : DecidableEq δ\ninst✝¹ : ∀ (i : δ), SigmaFinite (μ i)\ninst✝ : Fintype δ\nf : ((i : δ) → X i) → ℝ≥0∞\n⊢ ∫⋯∫⁻_Finset.univ, f ∂μ = fun x ↦ ∫⁻ (x : (i : δ) → X i), f x ∂Measure.pi μ",
"usedCo... | by
let e : { j // j ∈ Finset.univ } ≃ δ := Equiv.subtypeUnivEquiv mem_univ
ext1 x
simp_rw [lmarginal, measurePreserving_piCongrLeft μ e |>.lintegral_map_equiv, updateFinset_def]
simp
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 304,
"column": 10
} | {
"line": 304,
"column": 51
} | [
{
"pp": "case convert_2\nX : Type u_1\nE : Type u_3\nmX : MeasurableSpace X\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\nf : X → E\nμ : Measure X\nι : Type u_5\ninst✝¹ : Preorder ι\ninst✝ : atTop.IsCountablyGenerated\ns : ι → Set X\nhsm : ∀ (i : ι), MeasurableSet (s i)\nh_anti : Antitone s\nhne : a... | setIntegral_diff (hsm i) hi₀ (h_anti hi), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Stieltjes | {
"line": 465,
"column": 2
} | {
"line": 465,
"column": 36
} | [
{
"pp": "R : Type u_1\ninst✝¹ : LinearOrder R\ninst✝ : TopologicalSpace R\nf : StieltjesFunction R\nc : R\nt : Set R\nthis : Nonempty R\na b : R\nh : t \\ botSet ⊆ Ioc a b\n⊢ f.length (Ioc a b ∩ Ioi c) + f.length (Ioc a b \\ Ioi c) ≤ ofReal (↑f b - ↑f a)",
"usedConstants": [
"le_total"
]
}
] | rcases le_total a c with hac | hac | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.InnerProductSpace.Defs | {
"line": 358,
"column": 4
} | {
"line": 360,
"column": 14
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nF : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : PreInnerProductSpace.Core 𝕜 F\nx y : F\nt : ℝ\nhzero : ⟪x, y⟫ = 0\n⊢ 0 ≤ normSq x * (t * t) + 2 * ‖⟪x, y⟫‖ * t + normSq y",
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.no... | simp only [← sq, hzero, norm_zero, mul_zero, zero_mul, add_zero]
obtain ⟨hx, hy⟩ : (0 ≤ normSqF x ∧ 0 ≤ normSqF y) := ⟨inner_self_nonneg, inner_self_nonneg⟩
positivity | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Defs | {
"line": 358,
"column": 4
} | {
"line": 360,
"column": 14
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nF : Type u_3\ninst✝² : RCLike 𝕜\ninst✝¹ : AddCommGroup F\ninst✝ : Module 𝕜 F\nc : PreInnerProductSpace.Core 𝕜 F\nx y : F\nt : ℝ\nhzero : ⟪x, y⟫ = 0\n⊢ 0 ≤ normSq x * (t * t) + 2 * ‖⟪x, y⟫‖ * t + normSq y",
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.no... | simp only [← sq, hzero, norm_zero, mul_zero, zero_mul, add_zero]
obtain ⟨hx, hy⟩ : (0 ≤ normSqF x ∧ 0 ≤ normSqF y) := ⟨inner_self_nonneg, inner_self_nonneg⟩
positivity | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 868,
"column": 2
} | {
"line": 869,
"column": 28
} | [
{
"pp": "case pos.h'f\nX : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nf : X → ℝ\ns : Set X\nhs : ∀ x ∈ s, f x ≤ 1\nh's : ∀ x ∈ sᶜ, f x ≤ 0\nH : Integrable f μ\ng : X → ℝ := ⋯\ng_int : Integrable g μ\nthis : ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ ENNReal.ofReal (∫ (x : X), g x ∂μ)\n⊢ ∀ a ∈ sᶜ, ENNReal.ofReal... | · intro x hx
simpa [g] using h's x hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 851,
"column": 42
} | {
"line": 869,
"column": 28
} | [
{
"pp": "X : Type u_1\nmX : MeasurableSpace X\nμ : Measure X\nf : X → ℝ\ns : Set X\nhs : ∀ x ∈ s, f x ≤ 1\nh's : ∀ x ∈ sᶜ, f x ≤ 0\n⊢ ENNReal.ofReal (∫ (x : X), f x ∂μ) ≤ μ s",
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"MeasureTheory.ae",
"le_max_right",
"Eq.mpr",
... | by
by_cases H : Integrable f μ; swap
· simp [integral_undef H]
let g x := max (f x) 0
have g_int : Integrable g μ := H.pos_part
have : ENNReal.ofReal (∫ x, f x ∂μ) ≤ ENNReal.ofReal (∫ x, g x ∂μ) := by
apply ENNReal.ofReal_le_ofReal
exact integral_mono H g_int (fun x ↦ le_max_left _ _)
apply this.tra... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Bochner.Set | {
"line": 1127,
"column": 2
} | {
"line": 1129,
"column": 83
} | [
{
"pp": "Y : Type u_2\nE : Type u_3\nF : Type u_4\nX : Type u_5\nG : Type u_6\n𝕜 : Type u_7\ninst✝¹¹ : TopologicalSpace X\ninst✝¹⁰ : TopologicalSpace Y\ninst✝⁹ : MeasurableSpace Y\ninst✝⁸ : OpensMeasurableSpace Y\nμ : Measure Y\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : Norme... | obtain ⟨v, v_mem, hv⟩ : ∃ v ∈ 𝓝[s] q, ∀ p ∈ v, ∀ x ∈ k, dist (f p x) (f q x) < δ :=
hk.mem_uniformity_of_prod
(hf.mono (Set.prod_mono_right (subset_univ k))) hq (dist_mem_uniformity δpos) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 660,
"column": 58
} | {
"line": 661,
"column": 48
} | [
{
"pp": "F : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nx y : F\nhx : x ≠ 0\nhy : y ≠ 0\nR : ℝ\n⊢ dist ((R / ‖x‖) ^ 2 • x) ((R / ‖y‖) ^ 2 • y) = √(‖(R / ‖x‖) ^ 2 • x - (R / ‖y‖) ^ 2 • y‖ ^ 2)",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mp... | by
rw [dist_eq_norm, sqrt_sq (norm_nonneg _)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Basic | {
"line": 767,
"column": 2
} | {
"line": 770,
"column": 49
} | [
{
"pp": "case mp\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nx y : E\nh₀ : x ≠ 0\nh₀' : ¬↑‖x‖ = 0\nh : ⟪x, y⟫ = ↑‖x‖ * ↑‖y‖\n⊢ (↑‖y‖ / ↑‖x‖) • x = y",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"InnerProductSpace.toNormedS... | · have : x = 0 ∨ y = (⟪x, y⟫ / ⟪x, x⟫ : 𝕜) • x :=
((norm_inner_eq_norm_tfae 𝕜 x y).out 0 1).1 (by simp [h])
rw [this.resolve_left h₀, h]
simp [norm_smul, mul_div_cancel_right₀ _ h₀'] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Group.FundamentalDomain | {
"line": 801,
"column": 47
} | {
"line": 801,
"column": 97
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝⁵ : Group G\ninst✝⁴ : MulAction G α\ninst✝³ : MeasurableSpace α\nν : Measure α\ninst✝² : SMulInvariantMeasure G α ν\ninst✝¹ : Countable G\ninst✝ : MeasurableConstSMul G α\ns : Set α\nfund_dom_s : IsFundamentalDomain G s ν\nt : Set α\nfund_dom_t : IsFundamentalDomain G t... | by rw [fund_dom_s.quotientMeasure_eq _ fund_dom_t] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 294,
"column": 6
} | {
"line": 294,
"column": 19
} | [
{
"pp": "case mp.hre\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\nu v : E\nhv : v ∈ K\nthis : InnerProductSpace ℝ E := ⋯\nK' : Submodule ℝ E := ⋯\nH : ‖u - v‖ = ⨅ w, ‖u - ↑w‖\nA : ∀ w ∈ K, re ⟪u - v, w⟫_𝕜 = 0\nw : E\nhw : w... | simp [A w hw] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 294,
"column": 6
} | {
"line": 294,
"column": 19
} | [
{
"pp": "case mp.hre\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\nu v : E\nhv : v ∈ K\nthis : InnerProductSpace ℝ E := ⋯\nK' : Submodule ℝ E := ⋯\nH : ‖u - v‖ = ⨅ w, ‖u - ↑w‖\nA : ∀ w ∈ K, re ⟪u - v, w⟫_𝕜 = 0\nw : E\nhw : w... | simp [A w hw] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Projection.Minimal | {
"line": 294,
"column": 6
} | {
"line": 294,
"column": 19
} | [
{
"pp": "case mp.hre\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nK : Submodule 𝕜 E\nu v : E\nhv : v ∈ K\nthis : InnerProductSpace ℝ E := ⋯\nK' : Submodule ℝ E := ⋯\nH : ‖u - v‖ = ⨅ w, ‖u - ↑w‖\nA : ∀ w ∈ K, re ⟪u - v, w⟫_𝕜 = 0\nw : E\nhw : w... | simp [A w hw] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.InnerProductSpace.Symmetric | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 15
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : RCLike 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace 𝕜 E\nT : E →ₗ[𝕜] E\nhT : T.IsSymmetric\nh : ∀ (x : E), ⟪T x, x⟫ = 0\nx : E\n⊢ (⟪T (x + T x), x + T x⟫ - ⟪T (x - T x), x - T x⟫ - I * ⟪T (x + I • T x), x + I • T x⟫ +\n I * ⟪T (x - I • T x)... | simp_rw [h _] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.InnerProductSpace.Projection.Basic | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 21
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : InnerProductSpace ℝ F\nK✝ : Submodule 𝕜 E\nK : ClosedSubmodule 𝕜 E\ninst✝ : CompleteSpace E\n⊢ (↑K).HasOrthogonalProjection",
"use... | letI := K.isClosed' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLetI___1 | Lean.Parser.Tactic.tacticLetI__ |
Mathlib.Analysis.Normed.Operator.Banach | {
"line": 218,
"column": 4
} | {
"line": 218,
"column": 42
} | [
{
"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... | rw [tendsto_iff_norm_sub_tendsto_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 204,
"column": 76
} | {
"line": 209,
"column": 40
} | [
{
"pp": "p : ℝ≥0∞\n𝕜 : Type u_1\ninst✝³ : Semiring 𝕜\nη : Type u_5\nιs : η → Type u_6\nMs : η → Type u_7\ninst✝² : (i : η) → AddCommGroup (Ms i)\ninst✝¹ : (i : η) → Module 𝕜 (Ms i)\ninst✝ : DecidableEq η\nv : (j : η) → ιs j → Ms j\nhs : ∀ (i : η), LinearIndependent 𝕜 (v i)\n⊢ LinearIndependent 𝕜 fun ji ↦ s... | by
suffices LinearIndependent 𝕜 ((WithLp.linearEquiv p 𝕜 _).symm.toLinearMap ∘
fun ji : Σ j, ιs j ↦ Pi.single ji.1 (v ji.1 ji.2)) by
simpa
rw [LinearMap.linearIndependent_iff_of_injOn _ (by simp)]
exact Pi.linearIndependent_single v hs | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Lp.ProdLp | {
"line": 979,
"column": 4
} | {
"line": 979,
"column": 38
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝² : SeminormedAddCommGroup α\ninst✝¹ : SeminormedAddCommGroup β\np : ℝ≥0∞\ninst✝ : Fact (1 ≤ p)\nhp : 0 < p.toReal\nx : WithLp p (α × β)\n⊢ (‖toLp p (x.fst, 0)‖ ^ p.toReal + ‖x.snd‖ ^ p.toReal) ^ (1 / p.toReal) =\n (‖idemFst x‖ ^ p.toReal + ‖idemSnd x‖ ^ p.toReal) ^ ... | ← WithLp.norm_toLp_snd p α β x.snd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Haar.OfBasis | {
"line": 192,
"column": 4
} | {
"line": 195,
"column": 42
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝⁵ : Fintype ι\ninst✝⁴ : Fintype ι'\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ E\ninst✝ : NormedSpace ℝ F\nb : Basis ι ℝ E\n⊢ (interior (_root_.parallelepiped ⇑b)).Nonempty",
"usedConstants": [
... | suffices H : Set.Nonempty (interior (b.equivFunL.symm.toHomeomorph '' Icc 0 1)) by
dsimp only [_root_.parallelepiped]
convert! H
exact (b.equivFun_symm_apply _).symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.Normed.Lp.PiLp | {
"line": 1031,
"column": 8
} | {
"line": 1031,
"column": 41
} | [
{
"pp": "case coe\nι : Type u_2\nβ : ι → Type u_4\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → SeminormedAddCommGroup (β i)\ninst✝ : DecidableEq ι\ni : ι\nb : β i\nthis : Nonempty ι\np : ℝ≥0\nhp : Fact (1 ≤ ↑p)\nhp0 : ↑p ≠ 0\n⊢ ‖single (↑p) i b‖₊ = ‖b‖₊",
"usedConstants": [
"Eq.mpr",
"PiLp.single",
... | nnnorm_eq_sum ENNReal.coe_ne_top, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 303,
"column": 4
} | {
"line": 303,
"column": 60
} | [
{
"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\nf : E →ₗ[ℝ] E\ns : Set E\nhf : LinearMap.det f ≠ 0\n⊢ μ (⇑f '' s) = ENNReal.ofReal |LinearMap.de... | let g := (f.equivOfDetNeZero hf).toContinuousLinearEquiv | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 303,
"column": 4
} | {
"line": 306,
"column": 9
} | [
{
"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\nf : E →ₗ[ℝ] E\ns : Set E\nhf : LinearMap.det f ≠ 0\n⊢ μ (⇑f '' s) = ENNReal.ofReal |LinearMap.de... | let g := (f.equivOfDetNeZero hf).toContinuousLinearEquiv
change μ (g '' s) = _
rw [ContinuousLinearEquiv.image_eq_preimage_symm g s, addHaar_preimage_continuousLinearEquiv]
congr | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 303,
"column": 4
} | {
"line": 306,
"column": 9
} | [
{
"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\nf : E →ₗ[ℝ] E\ns : Set E\nhf : LinearMap.det f ≠ 0\n⊢ μ (⇑f '' s) = ENNReal.ofReal |LinearMap.de... | let g := (f.equivOfDetNeZero hf).toContinuousLinearEquiv
change μ (g '' s) = _
rw [ContinuousLinearEquiv.image_eq_preimage_symm g s, addHaar_preimage_continuousLinearEquiv]
congr | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 628,
"column": 2
} | {
"line": 628,
"column": 29
} | [
{
"pp": "μ : Measure ℝ\ninst✝ : NoAtoms μ\ns : Set ℝ\np : ℝ → Prop\nh : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ.restrict (s ∩ Ioo a b), p x\nT : ↑s × ↑s → Set ℝ := fun p ↦ Ioo ↑p.1 ↑p.2\n⊢ ∀ᵐ (x : ℝ) ∂μ.restrict s, p x",
"usedConstants": [
"Real",
"Set.Elem",
"Prod",
"Set.... | let u := ⋃ i : ↥s × ↥s, T i | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 647,
"column": 4
} | {
"line": 651,
"column": 27
} | [
{
"pp": "case right\nμ : Measure ℝ\ninst✝ : NoAtoms μ\ns : Set ℝ\np : ℝ → Prop\nh : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ.restrict (s ∩ Ioo a b), p x\nT : ↑s × ↑s → Set ℝ := ⋯\nu : Set ℝ := ⋯\nhfinite : (s \\ u).Finite\nA : Set (↑s × ↑s)\nA_count : A.Countable\nhA : ⋃ i ∈ A, T i = ⋃ i, T i\nthis : ... | rintro ⟨⟨a, as⟩, ⟨b, bs⟩⟩ -
dsimp [T]
rcases le_or_gt b a with (hba | hab)
· simp only [Ioo_eq_empty_of_le hba, inter_empty, restrict_empty, ae_zero, eventually_bot]
· exact h a b as bs hab | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 647,
"column": 4
} | {
"line": 651,
"column": 27
} | [
{
"pp": "case right\nμ : Measure ℝ\ninst✝ : NoAtoms μ\ns : Set ℝ\np : ℝ → Prop\nh : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ.restrict (s ∩ Ioo a b), p x\nT : ↑s × ↑s → Set ℝ := ⋯\nu : Set ℝ := ⋯\nhfinite : (s \\ u).Finite\nA : Set (↑s × ↑s)\nA_count : A.Countable\nhA : ⋃ i ∈ A, T i = ⋃ i, T i\nthis : ... | rintro ⟨⟨a, as⟩, ⟨b, bs⟩⟩ -
dsimp [T]
rcases le_or_gt b a with (hba | hab)
· simp only [Ioo_eq_empty_of_le hba, inter_empty, restrict_empty, ae_zero, eventually_bot]
· exact h a b as bs hab | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 662,
"column": 2
} | {
"line": 662,
"column": 29
} | [
{
"pp": "μ : Measure ℝ\ninst✝ : NoAtoms μ\ns : Set ℝ\np : ℝ → Prop\nh : ∀ (a b : ℝ), a ∈ s → b ∈ s → a < b → ∀ᵐ (x : ℝ) ∂μ, x ∈ s ∩ Ioo a b → p x\nT : ↑s × ↑s → Set ℝ := fun p ↦ Ioo ↑p.1 ↑p.2\n⊢ ∀ᵐ (x : ℝ) ∂μ, x ∈ s → p x",
"usedConstants": [
"Real",
"Set.Elem",
"Prod",
"Set.iUnion",... | let u := ⋃ i : ↥s × ↥s, T i | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.BoxIntegral.Box.SubboxInduction | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 21
} | [
{
"pp": "case h\nι : Type u_1\nI : Box ι\ns t : Set ι\nh : s ≠ t\ny : ι → ℝ\nhs : y ∈ I ∧ ∀ (i : ι), (I.lower i + I.upper i) / 2 < y i ↔ i ∈ s\nht : y ∈ I ∧ ∀ (i : ι), (I.lower i + I.upper i) / 2 < y i ↔ i ∈ t\ni : ι\n⊢ i ∈ s ↔ i ∈ t",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
... | rw [← hs.2, ← ht.2] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 463,
"column": 10
} | {
"line": 463,
"column": 24
} | [
{
"pp": "K : Type u_1\ninst✝⁹ : NormedField K\ninst✝⁸ : LinearOrder K\ninst✝⁷ : IsStrictOrderedRing K\ninst✝⁶ : HasSolidNorm K\ninst✝⁵ : FloorRing K\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\ninst✝² : FiniteDimensional K E\ninst✝¹ : ProperSpace E\nL : Submodule ℤ E\ninst✝ : Discrete... | ← hs.span_top, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.Partition.Split | {
"line": 198,
"column": 65
} | {
"line": 201,
"column": 22
} | [
{
"pp": "ι : Type u_1\nI J : Box ι\ni : ι\nx : ℝ\ny : ι → ℝ\nh₁ : J ∈ split I i x\nh₂ : y ∈ J\nh₃ : y i ≤ x\n⊢ ↑J = ↑I ∩ {y | y i ≤ x}",
"usedConstants": [
"Real.instLE",
"Real",
"BoxIntegral.Prepartition",
"congrArg",
"BoxIntegral.Box.toSet",
"BoxIntegral.Prepartition.sp... | by
refine (mem_split_iff'.1 h₁).resolve_right fun H => ?_
rw [← Box.mem_coe, H] at h₂
exact h₃.not_gt h₂.2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 664,
"column": 41
} | {
"line": 675,
"column": 32
} | [
{
"pp": "E : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nhs : DiscreteTopology ↥(span ℤ s)\n⊢ Set.finrank ℝ s = Set.finrank ℤ s",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Submodule",
"AddCommGroup.intIsScalarTower",
... | by
let F := span ℝ s
let L : Submodule ℤ (span ℝ s) := comap (F.restrictScalars ℤ).subtype (span ℤ s)
let f := Submodule.comapSubtypeEquivOfLe (span_le_restrictScalars ℤ ℝ s)
have : DiscreteTopology L := by
let e : span ℤ s ≃L[ℤ] L :=
⟨f.symm, continuous_of_discreteTopology, Isometry.continuous fun _ ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 721,
"column": 84
} | {
"line": 721,
"column": 93
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : NormedField K\nE : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace K E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace K F\nL : Submodule ℤ E\nhL : span K ↑L = ⊤\ne : F →ₗ[K] E\nhe : ↑L ⊆ ↑e.range\n⊢ comap e ⊤ = ⊤",
"usedConstants": [
"No... | comap_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 251,
"column": 4
} | {
"line": 251,
"column": 80
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\n𝕜 : Type u_3\ninst✝⁸ : RCLike 𝕜\nE : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : InnerProductSpace 𝕜 E\nF : Type u_5\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : InnerProductSpace ℝ F\nF' : Type u_6\ninst✝³ : NormedAddCommGroup F'\ninst✝² : InnerProductSpace ℝ F'\ninst... | convert! this (toLp 2 (e₁ (e₂.symm v₀))) (toLp 2 (e₁ (e₂.symm w₀))) <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 262,
"column": 69
} | {
"line": 268,
"column": 31
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_3\ninst✝⁴ : RCLike 𝕜\nE : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : Fintype ι\ninst✝ : DecidableEq ι\nV : ι → Submodule 𝕜 E\nhV : IsInternal V\nhV' : OrthogonalFamily 𝕜 (fun i ↦ ↥(V i)) fun i ↦ (V i).subtypeₗᵢ\nw : PiLp 2 fun i ↦ ↥(V... | by
classical
let e₁ := DirectSum.linearEquivFunOnFintype 𝕜 ι fun i => V i
let e₂ := LinearEquiv.ofBijective (DirectSum.coeLinearMap V) hV
suffices ∀ v : ⨁ i, V i, e₂ v = ∑ i, e₁ v i by exact this (e₁.symm w)
simp [e₁, e₂, DirectSum.coeLinearMap, DirectSum.toModule, DFinsupp.lsum,
DFinsupp.sumAd... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 507,
"column": 2
} | {
"line": 507,
"column": 22
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_3\ninst✝³ : RCLike 𝕜\nE : Type u_4\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : Fintype ι\nb : OrthonormalBasis ι 𝕜 E\nx y : E\nthis : ((innerSL 𝕜) x) (∑ i, (b.repr y).ofLp i • b i) = ((innerSL 𝕜) x) y\n⊢ ∑ i, ⟪x, b i⟫ * ⟪b i, y⟫ = ⟪x, y⟫",
... | rw [map_sum] at this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 1044,
"column": 4
} | {
"line": 1044,
"column": 13
} | [
{
"pp": "𝕜 : Type u_3\ninst✝⁴ : RCLike 𝕜\nE : Type u_4\ninst✝³ : NormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : FiniteDimensional 𝕜 E\nι : Type u_7\ninst✝ : Fintype ι\ncard_ι : finrank 𝕜 E = Fintype.card ι\nv : ι → E\ns : Set ι\nhv : Orthonormal 𝕜 (s.restrict v)\nhsv : Injective (s.restric... | exact hsv | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.PiL2 | {
"line": 1179,
"column": 2
} | {
"line": 1179,
"column": 46
} | [
{
"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... | let p1 := S.orthogonalProjection.toLinearMap | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 119,
"column": 72
} | {
"line": 122,
"column": 54
} | [
{
"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\nπ₁ π₂ : TaggedPrepartition I\nh₁ : π₁.IsPartition\nh₂ : π₂.IsPartition\n⊢ integralSum f vol π₁ - int... | by
rw [← integralSum_inf_partition f vol π₁ h₂, ← integralSum_inf_partition f vol π₂ h₁,
integralSum, integralSum, Finset.sum_sub_distrib]
simp only [infPrepartition_toPrepartition, inf_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 65
} | [
{
"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\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\n⊢ HasIntegral I l (fun x ↦ 0) vol 0",
"usedConstants": [
"Eq.mpr... | simpa only [← (vol I).map_zero] using hasIntegral_const (0 : E) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 65
} | [
{
"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\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\n⊢ HasIntegral I l (fun x ↦ 0) vol 0",
"usedConstants": [
"Eq.mpr... | simpa only [← (vol I).map_zero] using hasIntegral_const (0 : E) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 65
} | [
{
"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\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\n⊢ HasIntegral I l (fun x ↦ 0) vol 0",
"usedConstants": [
"Eq.mpr... | simpa only [← (vol I).map_zero] using hasIntegral_const (0 : E) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 543,
"column": 52
} | {
"line": 543,
"column": 73
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\nI : Box ι\nπ : TaggedPrepartition I\ninst✝¹ : Fintype ι\nl : IntegrationParams\nf : (ι → ℝ) → E\nvol : ι →ᵇᵃ[⊤] E →L[ℝ] F\nc : ℝ≥0\nε : ℝ\ninst✝ : Comple... | exact min_le_left _ _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.PSeries | {
"line": 252,
"column": 4
} | {
"line": 252,
"column": 70
} | [
{
"pp": "case h.e'_1.a\nf : ℕ → ℝ\nn : ℕ\nhn : ∀ b ≥ n, 0 b ≤ f b\nm : ℕ\nhm : ∀ b ≥ m, f (b + 1) ≤ f b\n⊢ (fun k ↦ 2 ^ k * f (2 ^ k)) =ᶠ[Filter.atTop] fun k ↦ 2 ^ k * f (max (2 ^ k) (n + m))",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Preorder.toLT",
"instReflLe",
... | have h_pow := tendsto_pow_atTop_atTop_of_one_lt (r := 2) (by simp) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.MonoidAlgebra.Ideal | {
"line": 35,
"column": 31
} | {
"line": 42,
"column": 46
} | [
{
"pp": "k : Type u_1\nG : Type u_3\ninst✝¹ : Monoid G\ninst✝ : Semiring k\ns : Set G\nx✝ x y : k[G]\nhy : y ∈ {p | ∀ m ∈ p.support, ∃ m' ∈ s, ∃ d, m = d * m'}\nm : G\nhm : m ∈ (x • y).support\n⊢ ∃ m' ∈ s, ∃ d, m = d * m'",
"usedConstants": [
"MonoidAlgebra.semiring",
"MonoidAlgebra.addCommMonoi... | by
simp only [smul_eq_mul, mul_def] at hm
obtain ⟨xm, -, hm⟩ := Finset.mem_biUnion.mp (Finsupp.support_sum hm)
obtain ⟨ym, hym, hm⟩ := Finset.mem_biUnion.mp (Finsupp.support_sum hm)
obtain rfl := Finset.mem_singleton.mp (Finsupp.support_single_subset hm)
refine (hy _ hym).imp fun... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.ZLattice.Summable | {
"line": 257,
"column": 2
} | {
"line": 259,
"column": 88
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : FiniteDimensional ℝ E\nL : Submodule ℤ E\ninst✝ : DiscreteTopology ↥L\nr : ℝ\nhr : r < -↑(finrank ℤ ↥L)\nx : E\nh✝ : Nontrivial ↥L\nH : IsClosed ↑L\nt : ↥L\nht : t ∈ {x_1 | (fun i ↦ ‖‖↑i - x‖ ^ r‖ ≤ (1 / 2) ^ r * ... | have : ‖t - x‖ < 2⁻¹ * ‖t‖ := by
rw [← Real.rpow_lt_rpow_iff_of_neg (by positivity) (by positivity) (hr.trans (by simpa))]
simpa [Real.mul_rpow, abs_eq_self.mpr (show 0 ≤ ‖t - x‖ ^ r by positivity)] using ht | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 79,
"column": 60
} | {
"line": 79,
"column": 78
} | [
{
"pp": "α : Type u_1\nf : α → ℕ\n⊢ multinomial ∅ f = 1",
"usedConstants": [
"Preorder.toLT",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"Nat.multinomial",
"congrArg",
"Finset",
"AddMonoid.toAddZeroClass",
"PartialOrder.toPreorder",
"SemilatticeInf.to... | simp [multinomial] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 79,
"column": 60
} | {
"line": 79,
"column": 78
} | [
{
"pp": "α : Type u_1\nf : α → ℕ\n⊢ multinomial ∅ f = 1",
"usedConstants": [
"Preorder.toLT",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"Nat.multinomial",
"congrArg",
"Finset",
"AddMonoid.toAddZeroClass",
"PartialOrder.toPreorder",
"SemilatticeInf.to... | simp [multinomial] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 79,
"column": 60
} | {
"line": 79,
"column": 78
} | [
{
"pp": "α : Type u_1\nf : α → ℕ\n⊢ multinomial ∅ f = 1",
"usedConstants": [
"Preorder.toLT",
"LinearOrderedCommMonoidWithZero.toIsBotZeroClass",
"Nat.multinomial",
"congrArg",
"Finset",
"AddMonoid.toAddZeroClass",
"PartialOrder.toPreorder",
"SemilatticeInf.to... | simp [multinomial] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Division | {
"line": 329,
"column": 2
} | {
"line": 336,
"column": 40
} | [
{
"pp": "case mpr\nσ : Type u_1\nR : Type u_3\ninst✝¹ : CommRing R\ni : σ\np q : MvPolynomial σ R\ninst✝ : IsCancelMulZero R\n⊢ p ∣ q ∨ X i ∣ p ∧ p.divMonomial (Finsupp.single i 1) ∣ q → p ∣ X i * q",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoi... | · rintro (hp | ⟨hi, hq⟩)
· exact dvd_mul_of_dvd_right hp (X i)
· suffices p = X i * p.divMonomial (Finsupp.single i 1) by
rw [this]
exact mul_dvd_mul_left (X i) hq
conv_lhs => rw [← p.modMonomial_add_divMonomial (Finsupp.single i 1)]
simpa only [← C_mul_X_eq_monomial, C_1, one_mul, a... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.MvPolynomial.Nilpotent | {
"line": 59,
"column": 2
} | {
"line": 72,
"column": 70
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\nP : MvPolynomial σ R\n⊢ IsUnit P ↔ IsUnit (coeff 0 P) ∧ ∀ (i : σ →₀ ℕ), i ≠ 0 → IsNilpotent (coeff i P)",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemirin... | classical
refine ⟨fun H ↦ ⟨H.map constantCoeff, ?_⟩, fun ⟨h₁, h₂⟩ ↦ ?_⟩
· intro n hn
obtain ⟨i, hi⟩ : ∃ i, n i ≠ 0 := by simpa [Finsupp.ext_iff] using hn
let e := (optionEquivLeft _ _).symm.trans (renameEquiv R (Equiv.optionSubtypeNe i))
have H := (Polynomial.coeff_isUnit_isNilpotent_of_isUnit (H.map e.... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Algebra.MvPolynomial.Nilpotent | {
"line": 59,
"column": 2
} | {
"line": 72,
"column": 70
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\nP : MvPolynomial σ R\n⊢ IsUnit P ↔ IsUnit (coeff 0 P) ∧ ∀ (i : σ →₀ ℕ), i ≠ 0 → IsNilpotent (coeff i P)",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemirin... | classical
refine ⟨fun H ↦ ⟨H.map constantCoeff, ?_⟩, fun ⟨h₁, h₂⟩ ↦ ?_⟩
· intro n hn
obtain ⟨i, hi⟩ : ∃ i, n i ≠ 0 := by simpa [Finsupp.ext_iff] using hn
let e := (optionEquivLeft _ _).symm.trans (renameEquiv R (Equiv.optionSubtypeNe i))
have H := (Polynomial.coeff_isUnit_isNilpotent_of_isUnit (H.map e.... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Nilpotent | {
"line": 59,
"column": 2
} | {
"line": 72,
"column": 70
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : CommRing R\nP : MvPolynomial σ R\n⊢ IsUnit P ↔ IsUnit (coeff 0 P) ∧ ∀ (i : σ →₀ ℕ), i ≠ 0 → IsNilpotent (coeff i P)",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Finsupp.instFunLike",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemirin... | classical
refine ⟨fun H ↦ ⟨H.map constantCoeff, ?_⟩, fun ⟨h₁, h₂⟩ ↦ ?_⟩
· intro n hn
obtain ⟨i, hi⟩ : ∃ i, n i ≠ 0 := by simpa [Finsupp.ext_iff] using hn
let e := (optionEquivLeft _ _).symm.trans (renameEquiv R (Equiv.optionSubtypeNe i))
have H := (Polynomial.coeff_isUnit_isNilpotent_of_isUnit (H.map e.... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Choose.Multinomial | {
"line": 392,
"column": 6
} | {
"line": 392,
"column": 37
} | [
{
"pp": "n : ℕ\nα : Type u_1\ninst✝ : DecidableEq α\nm : Fin (n + 1)\ns : Sym α (n - ↑m)\nx : α\nhx : x ∉ s\n⊢ (↑(fill x m s)).countPerms = n.choose ↑m * (↑s).countPerms",
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"Nat.choose",
"HMul.hMul",
"congrArg",
"HSub.hSub",... | Multiset.countPerms_filter_ne x | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Funext | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 46
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nσ : Type u_2\np✝ q : MvPolynomial σ R\ns : σ → Set R\nhs : ∀ (i : σ), (s i).Infinite\nh✝ : ∀ x ∈ Set.univ.pi s, (eval x) p✝ = (eval x) q\nn : ℕ\nf : Fin n → σ\nhf : Function.Injective f\np : MvPolynomial (Fin n) R\nh : ∀ x ∈ Set.univ.pi s... | simp_rw [Function.extend, dif_neg nex, hg] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.MvPolynomial.Funext | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 46
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nσ : Type u_2\np✝ q : MvPolynomial σ R\ns : σ → Set R\nhs : ∀ (i : σ), (s i).Infinite\nh✝ : ∀ x ∈ Set.univ.pi s, (eval x) p✝ = (eval x) q\nn : ℕ\nf : Fin n → σ\nhf : Function.Injective f\np : MvPolynomial (Fin n) R\nh : ∀ x ∈ Set.univ.pi s... | simp_rw [Function.extend, dif_neg nex, hg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Funext | {
"line": 73,
"column": 4
} | {
"line": 73,
"column": 46
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nσ : Type u_2\np✝ q : MvPolynomial σ R\ns : σ → Set R\nhs : ∀ (i : σ), (s i).Infinite\nh✝ : ∀ x ∈ Set.univ.pi s, (eval x) p✝ = (eval x) q\nn : ℕ\nf : Fin n → σ\nhf : Function.Injective f\np : MvPolynomial (Fin n) R\nh : ∀ x ∈ Set.univ.pi s... | simp_rw [Function.extend, dif_neg nex, hg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MvPolynomial.Funext | {
"line": 60,
"column": 2
} | {
"line": 73,
"column": 46
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nσ : Type u_2\np q : MvPolynomial σ R\ns : σ → Set R\nhs : ∀ (i : σ), (s i).Infinite\nh : ∀ x ∈ Set.univ.pi s, (eval x) p = (eval x) q\n⊢ p = q",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Finsupp.instAddZeroClass",
"E... | suffices ∀ p, (∀ x ∈ Set.pi .univ s, eval x p = 0) → p = 0 by
rw [← sub_eq_zero, this (p - q)]
intro x hx
simp_rw [map_sub, h x hx, sub_self]
intro p h
obtain ⟨n, f, hf, p, rfl⟩ := exists_fin_rename p
suffices p = 0 by rw [this, map_zero]
refine funext_fin (s ∘ f) (fun _ ↦ hs _) fun x hx ↦ ?_
choo... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Funext | {
"line": 60,
"column": 2
} | {
"line": 73,
"column": 46
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\nσ : Type u_2\np q : MvPolynomial σ R\ns : σ → Set R\nhs : ∀ (i : σ), (s i).Infinite\nh : ∀ x ∈ Set.univ.pi s, (eval x) p = (eval x) q\n⊢ p = q",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Finsupp.instAddZeroClass",
"E... | suffices ∀ p, (∀ x ∈ Set.pi .univ s, eval x p = 0) → p = 0 by
rw [← sub_eq_zero, this (p - q)]
intro x hx
simp_rw [map_sub, h x hx, sub_self]
intro p h
obtain ⟨n, f, hf, p, rfl⟩ := exists_fin_rename p
suffices p = 0 by rw [this, map_zero]
refine funext_fin (s ∘ f) (fun _ ↦ hs _) fun x hx ↦ ?_
choo... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.GameAdd | {
"line": 215,
"column": 8
} | {
"line": 215,
"column": 62
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nrα : α → α → Prop\nrβ : β → β → Prop\na✝ : α\nb✝ : β\nC : α → α → Sort u_3\nhr : WellFounded rα\nIH : (a₁ b₁ : α) → ((a₂ b₂ : α) → GameAdd rα s(a₂, b₂) s(a₁, b₁) → C a₂ b₂) → C a₁ b₁\na b : α\n⊢ WellFounded fun x y ↦ Prod.GameAdd rα rα x y ∨ Prod.GameAdd rα rα x.swap y",
... | simpa [← Sym2.gameAdd_iff] using hr.sym2_gameAdd.onFun | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Order.GameAdd | {
"line": 215,
"column": 8
} | {
"line": 215,
"column": 62
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nrα : α → α → Prop\nrβ : β → β → Prop\na✝ : α\nb✝ : β\nC : α → α → Sort u_3\nhr : WellFounded rα\nIH : (a₁ b₁ : α) → ((a₂ b₂ : α) → GameAdd rα s(a₂, b₂) s(a₁, b₁) → C a₂ b₂) → C a₁ b₁\na b : α\n⊢ WellFounded fun x y ↦ Prod.GameAdd rα rα x y ∨ Prod.GameAdd rα rα x.swap y",
... | simpa [← Sym2.gameAdd_iff] using hr.sym2_gameAdd.onFun | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.GameAdd | {
"line": 215,
"column": 8
} | {
"line": 215,
"column": 62
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nrα : α → α → Prop\nrβ : β → β → Prop\na✝ : α\nb✝ : β\nC : α → α → Sort u_3\nhr : WellFounded rα\nIH : (a₁ b₁ : α) → ((a₂ b₂ : α) → GameAdd rα s(a₂, b₂) s(a₁, b₁) → C a₂ b₂) → C a₁ b₁\na b : α\n⊢ WellFounded fun x y ↦ Prod.GameAdd rα rα x y ∨ Prod.GameAdd rα rα x.swap y",
... | simpa [← Sym2.gameAdd_iff] using hr.sym2_gameAdd.onFun | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.DFinsupp.WellFounded | {
"line": 89,
"column": 4
} | {
"line": 90,
"column": 38
} | [
{
"pp": "case neg\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\ninst✝ : (i : ι) → (s : Set ι) → Decidable (i ∈ s)\np : Set ι\nx₁ x₂ x : Π₀ (i : ι), α i\ni : ι\nhr : ∀ (j : ι), r j i → x j = if j ∈ p then x₁ j else x₂ j\nhp : i ∉ p\nhs : s i (x ... | refine ⟨⟨{ j | r j i ∧ j ∈ p }, x₁, piecewise x₂ x { j | r j i }⟩,
.snd ⟨i, fun j hj ↦ ?_, ?_⟩, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.MvPolynomial.NoZeroDivisors | {
"line": 189,
"column": 6
} | {
"line": 189,
"column": 26
} | [
{
"pp": "case h.mpr.inl\nR : Type u_1\nσ : Type u_2\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\np : MvPolynomial σ R\ni : σ\nr : R\nhr : r ≠ 0\nb : R\nhb : b ∣ r\nhp : p = C b\n⊢ ∃ a ≤ Finsupp.single i 1, b ∣ r ∧ p = (monomial a) b",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Finsupp.in... | use 0; simp [hb, hp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.NoZeroDivisors | {
"line": 189,
"column": 6
} | {
"line": 189,
"column": 26
} | [
{
"pp": "case h.mpr.inl\nR : Type u_1\nσ : Type u_2\ninst✝¹ : CommRing R\ninst✝ : NoZeroDivisors R\np : MvPolynomial σ R\ni : σ\nr : R\nhr : r ≠ 0\nb : R\nhb : b ∣ r\nhp : p = C b\n⊢ ∃ a ≤ Finsupp.single i 1, b ∣ r ∧ p = (monomial a) b",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Finsupp.in... | use 0; simp [hb, hp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.NonAssoc.LieAdmissible.Defs | {
"line": 97,
"column": 4
} | {
"line": 97,
"column": 54
} | [
{
"pp": "R : Type u_1\nL : Type u_2\ninst✝¹ : CommRing R\ninst✝ : LeftPreLieRing L\nx y z : L\nassoc_xyz : associator x y z = associator y x z\n⊢ associator x y z + associator z x y + associator y z x = associator y x z + associator z y x + associator x z y",
"usedConstants": [
"LeftPreLieRing.toNonUn... | have assoc_zxy := LeftPreLieRing.assoc_symm' z x y | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 159,
"column": 6
} | {
"line": 159,
"column": 29
} | [
{
"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 f = 0",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"congrArg",
"CommSemiring.toSemiring",
"Fi... | Subsingleton.eq_zero f, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Squarefree | {
"line": 83,
"column": 6
} | {
"line": 83,
"column": 49
} | [
{
"pp": "case pos\nn m : ℕ\nhn : Squarefree n\nhm : Squarefree m\nh : ∀ (p : ℕ), Prime p → (p ∣ n ↔ p ∣ m)\np : ℕ\nhp : Prime p\nh₁ : n.factorization p = 0 ↔ ¬p ∣ m\n⊢ n.factorization p = m.factorization p",
"usedConstants": [
"Finsupp.instFunLike",
"Nat.instMulZeroClass",
"Dvd.dvd",
... | hp.dvd_iff_one_le_factorization hm.ne_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Squarefree | {
"line": 292,
"column": 6
} | {
"line": 295,
"column": 19
} | [
{
"pp": "n : ℕ\nh0 : n ≠ 0\nx : Finset ℕ\nhx : x ⊆ (normalizedFactors n).toFinset\ny : Finset ℕ\nhy : y ⊆ (normalizedFactors n).toFinset\nh : x.val.prod = y.val.prod\n⊢ ∀ x ∈ y.val, Irreducible x",
"usedConstants": [
"UniqueFactorizationMonoid.normalizedFactors",
"Multiset.toFinset",
"Eq.m... | intro z hz
apply irreducible_of_normalized_factor z
· rw [← Multiset.mem_toFinset]
apply hy hz | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Nat.Squarefree | {
"line": 292,
"column": 6
} | {
"line": 295,
"column": 19
} | [
{
"pp": "n : ℕ\nh0 : n ≠ 0\nx : Finset ℕ\nhx : x ⊆ (normalizedFactors n).toFinset\ny : Finset ℕ\nhy : y ⊆ (normalizedFactors n).toFinset\nh : x.val.prod = y.val.prod\n⊢ ∀ x ∈ y.val, Irreducible x",
"usedConstants": [
"UniqueFactorizationMonoid.normalizedFactors",
"Multiset.toFinset",
"Eq.m... | intro z hz
apply irreducible_of_normalized_factor z
· rw [← Multiset.mem_toFinset]
apply hy hz | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.Squarefree | {
"line": 382,
"column": 2
} | {
"line": 384,
"column": 75
} | [
{
"pp": "case h\nm n : ℕ\nhm : Squarefree m\nhn : n ≠ 0\np : ℕ\nthis : m / m.gcd n ≠ 0\n⊢ Prime p ∧ m.gcd n * p ∣ m ↔ (Prime p ∧ p ∣ m) ∧ (Prime p → ¬p ∣ n)",
"usedConstants": [
"Nat.gcd",
"Nat.gcd_dvd_left",
"Nat.Prime",
"Dvd.dvd",
"HMul.hMul",
"Monoid.toMulOneClass",
... | refine ⟨fun hp ↦ ⟨⟨hp.1, dvd_of_mul_left_dvd hp.2⟩, fun _ hpn ↦ hp.1.not_isUnit <| hm _ <|
(mul_dvd_mul_right (dvd_gcd (dvd_of_mul_left_dvd hp.2) hpn) _).trans hp.2⟩, fun hp ↦
⟨hp.1.1, Coprime.mul_dvd_of_dvd_of_dvd ?_ (gcd_dvd_left _ _) hp.1.2⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.NumberTheory.ArithmeticFunction.Zeta | {
"line": 214,
"column": 17
} | {
"line": 214,
"column": 32
} | [
{
"pp": "case succ\nR : Type u_1\ninst✝ : CommSemiring R\nf : ArithmeticFunction R\nhf : f.IsMultiplicative\nk : ℕ\nhi : (f.ppow k).IsMultiplicative\n⊢ (f.ppow (k + 1)).IsMultiplicative",
"usedConstants": [
"Eq.mpr",
"ArithmeticFunction.pmul",
"congrArg",
"CommSemiring.toSemiring",
... | rw [ppow_succ'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 708,
"column": 86
} | {
"line": 709,
"column": 69
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\np : MvPolynomial σ R\n⊢ m.leadingTerm p = 0 ↔ p = 0",
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Semiring.toModule",
"AddMonoidAlgebra.addAddCommMonoid",
"congrArg",
... | by
simp only [leadingTerm, monomial_eq_zero, leadingCoeff_eq_zero_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 276,
"column": 2
} | {
"line": 276,
"column": 74
} | [
{
"pp": "n : ℕ\n⊢ Ω n = 0 ↔ n = 0 ∨ n = 1",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"ArithmeticFunction.cardFactors_apply",
"ArithmeticFunction.instFunLikeNat",
"congrArg",
"Iff.rfl",
"id",
"ArithmeticFunction.cardFactors",
"instOfNatNat",
... | rw [cardFactors_apply, List.length_eq_zero_iff, primeFactorsList_eq_nil] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 276,
"column": 2
} | {
"line": 276,
"column": 74
} | [
{
"pp": "n : ℕ\n⊢ Ω n = 0 ↔ n = 0 ∨ n = 1",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"ArithmeticFunction.cardFactors_apply",
"ArithmeticFunction.instFunLikeNat",
"congrArg",
"Iff.rfl",
"id",
"ArithmeticFunction.cardFactors",
"instOfNatNat",
... | rw [cardFactors_apply, List.length_eq_zero_iff, primeFactorsList_eq_nil] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 276,
"column": 2
} | {
"line": 276,
"column": 74
} | [
{
"pp": "n : ℕ\n⊢ Ω n = 0 ↔ n = 0 ∨ n = 1",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"ArithmeticFunction.cardFactors_apply",
"ArithmeticFunction.instFunLikeNat",
"congrArg",
"Iff.rfl",
"id",
"ArithmeticFunction.cardFactors",
"instOfNatNat",
... | rw [cardFactors_apply, List.length_eq_zero_iff, primeFactorsList_eq_nil] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 353,
"column": 4
} | {
"line": 353,
"column": 31
} | [
{
"pp": "case mpr\nn : ℕ\nh0 : n ≠ 0\nh : n.primeFactorsList.Nodup\n⊢ n.primeFactorsList.dedup.length = Ω n",
"usedConstants": [
"congrArg",
"List.dedup",
"List",
"Nat",
"True",
"eq_self",
"of_eq_true",
"congrFun'",
"instDecidableEqNat",
"List.Nodu... | simp [h.dedup, cardFactors] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 353,
"column": 4
} | {
"line": 353,
"column": 31
} | [
{
"pp": "case mpr\nn : ℕ\nh0 : n ≠ 0\nh : n.primeFactorsList.Nodup\n⊢ n.primeFactorsList.dedup.length = Ω n",
"usedConstants": [
"congrArg",
"List.dedup",
"List",
"Nat",
"True",
"eq_self",
"of_eq_true",
"congrFun'",
"instDecidableEqNat",
"List.Nodu... | simp [h.dedup, cardFactors] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ArithmeticFunction.Misc | {
"line": 353,
"column": 4
} | {
"line": 353,
"column": 31
} | [
{
"pp": "case mpr\nn : ℕ\nh0 : n ≠ 0\nh : n.primeFactorsList.Nodup\n⊢ n.primeFactorsList.dedup.length = Ω n",
"usedConstants": [
"congrArg",
"List.dedup",
"List",
"Nat",
"True",
"eq_self",
"of_eq_true",
"congrFun'",
"instDecidableEqNat",
"List.Nodu... | simp [h.dedup, cardFactors] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 927,
"column": 2
} | {
"line": 935,
"column": 42
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nf : MvPolynomial σ R\nh : m.degree f ≠ 0\n⊢ m.toSyn (m.degree (f - m.leadingTerm f)) < m.toSyn (m.degree f)",
"usedConstants": [
"Finsupp.instAddZeroClass",
"AddEquivClass.instAddMonoidHomClass",
"Eq.mpr",
... | classical
by_cases hl : f - m.leadingTerm f = 0
· simpa [hl, toSyn_lt_iff_ne_zero]
· apply lt_of_le_of_ne (m.degree_sub_leadingTerm_le f)
by_contra! h'
simp only [EmbeddingLike.apply_eq_iff_eq] at h'
apply m.degree_mem_support at hl
rw [h', mem_support_iff] at hl
simp [leadingTerm, leadingCoef... | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 927,
"column": 2
} | {
"line": 935,
"column": 42
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nf : MvPolynomial σ R\nh : m.degree f ≠ 0\n⊢ m.toSyn (m.degree (f - m.leadingTerm f)) < m.toSyn (m.degree f)",
"usedConstants": [
"Finsupp.instAddZeroClass",
"AddEquivClass.instAddMonoidHomClass",
"Eq.mpr",
... | classical
by_cases hl : f - m.leadingTerm f = 0
· simpa [hl, toSyn_lt_iff_ne_zero]
· apply lt_of_le_of_ne (m.degree_sub_leadingTerm_le f)
by_contra! h'
simp only [EmbeddingLike.apply_eq_iff_eq] at h'
apply m.degree_mem_support at hl
rw [h', mem_support_iff] at hl
simp [leadingTerm, leadingCoef... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 927,
"column": 2
} | {
"line": 935,
"column": 42
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nf : MvPolynomial σ R\nh : m.degree f ≠ 0\n⊢ m.toSyn (m.degree (f - m.leadingTerm f)) < m.toSyn (m.degree f)",
"usedConstants": [
"Finsupp.instAddZeroClass",
"AddEquivClass.instAddMonoidHomClass",
"Eq.mpr",
... | classical
by_cases hl : f - m.leadingTerm f = 0
· simpa [hl, toSyn_lt_iff_ne_zero]
· apply lt_of_le_of_ne (m.degree_sub_leadingTerm_le f)
by_contra! h'
simp only [EmbeddingLike.apply_eq_iff_eq] at h'
apply m.degree_mem_support at hl
rw [h', mem_support_iff] at hl
simp [leadingTerm, leadingCoef... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 611,
"column": 2
} | {
"line": 611,
"column": 28
} | [
{
"pp": "R : Type u_1\ninst✝ : CommMonoidWithZero R\nf : ArithmeticFunction R\nhf : f.IsMultiplicative\nx y : ℕ\nhx : ¬x = 0\nhy : ¬y = 0\nhgcd_ne_zero : x.gcd y ≠ 0\nhlcm_ne_zero : x.lcm y ≠ 0\nhfi_zero : ∀ {i : ℕ}, f (i ^ 0) = 1\n⊢ y.factorization.support ⊆ x.primeFactors ∪ y.primeFactors",
"usedConstants... | · apply subset_union_right | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.CompleteField | {
"line": 174,
"column": 2
} | {
"line": 181,
"column": 33
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝⁶ : Field α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : IsStrictOrderedRing α\ninst✝³ : Field β\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : Archimedean α\nq : ℚ\n⊢ inducedMap α β ↑q = ↑q",
"usedConstants": [
"Eq.mpr",
"Pre... | refine csSup_eq_of_forall_le_of_forall_lt_exists_gt
(cutMap_nonempty β (q : α)) (fun x h => ?_) fun w h => ?_
· rw [cutMap_coe] at h
obtain ⟨r, h, rfl⟩ := h
exact le_of_lt h
· obtain ⟨q', hwq, hq⟩ := exists_rat_btwn h
rw [cutMap_coe]
exact ⟨q', ⟨_, hq, rfl⟩, hwq⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.CompleteField | {
"line": 174,
"column": 2
} | {
"line": 181,
"column": 33
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝⁶ : Field α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : IsStrictOrderedRing α\ninst✝³ : Field β\ninst✝² : ConditionallyCompleteLinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : Archimedean α\nq : ℚ\n⊢ inducedMap α β ↑q = ↑q",
"usedConstants": [
"Eq.mpr",
"Pre... | refine csSup_eq_of_forall_le_of_forall_lt_exists_gt
(cutMap_nonempty β (q : α)) (fun x h => ?_) fun w h => ?_
· rw [cutMap_coe] at h
obtain ⟨r, h, rfl⟩ := h
exact le_of_lt h
· obtain ⟨q', hwq, hq⟩ := exists_rat_btwn h
rw [cutMap_coe]
exact ⟨q', ⟨_, hq, rfl⟩, hwq⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 455,
"column": 6
} | {
"line": 455,
"column": 37
} | [
{
"pp": "M : Type u_1\ninst✝³ : CommGroup M\ninst✝² : LinearOrder M\ninst✝¹ : IsOrderedMonoid M\nι : Type u_2\ninst✝ : LinearOrder ι\ns✝ : Finset ι\na : ι → M\ni : ι\ns : Finset ι\nhi✝ : i ∉ s\nhs : s.Nonempty\nhi : s.min' hs ∉ s\nih✝ : StrictMonoOn (mk ∘ a) ↑s → mk (∏ i ∈ s, a i) = mk (a (s.min' hs))\nhmono : ... | exact hi (Finset.min'_mem _ hs) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Floor.Extended | {
"line": 95,
"column": 35
} | {
"line": 95,
"column": 53
} | [
{
"pp": "case coe\nn : ℕ\nx✝ : ℝ≥0\nhn₀ : ¬n = 0\n⊢ n ≤ ⌈x✝⌉₊ ↔ n - 1 + 1 ≤ ⌈x✝⌉₊",
"usedConstants": [
"Eq.mpr",
"congrArg",
"HSub.hSub",
"SemilatticeInf.toPartialOrder",
"DistribLattice.toLattice",
"id",
"instSubNat",
"NNReal",
"instOfNatNat",
"LE... | Nat.sub_add_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Group.Cone | {
"line": 57,
"column": 54
} | {
"line": 57,
"column": 74
} | [
{
"pp": "case mk.mk.e_toSubmonoid\nG : Type u_1\ninst✝ : CommGroup G\ntoSubmonoid✝¹ : Submonoid G\neq_one_of_mem_of_inv_mem'✝¹ : ∀ {a : G}, a ∈ toSubmonoid✝¹.carrier → a⁻¹ ∈ toSubmonoid✝¹.carrier → a = 1\ntoSubmonoid✝ : Submonoid G\neq_one_of_mem_of_inv_mem'✝ : ∀ {a : G}, a ∈ toSubmonoid✝.carrier → a⁻¹ ∈ toSubm... | exact SetLike.ext' h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Module.Archimedean | {
"line": 38,
"column": 4
} | {
"line": 40,
"column": 51
} | [
{
"pp": "case refine_2\nM : Type u_1\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LinearOrder M\ninst✝⁶ : IsOrderedAddMonoid M\nK : Type u_2\ninst✝⁵ : Ring K\ninst✝⁴ : LinearOrder K\ninst✝³ : IsOrderedRing K\ninst✝² : Archimedean K\ninst✝¹ : Module K M\ninst✝ : PosSMulMono K M\na : M\nk : K\nh : k ≠ 0\nthis : Nontrivial ... | have : n • |a| = (n • (1 : K)) • |a| := by rw [smul_assoc, one_smul]
rw [this]
exact smul_le_smul_of_nonneg_right hn (by simp) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Module.Archimedean | {
"line": 38,
"column": 4
} | {
"line": 40,
"column": 51
} | [
{
"pp": "case refine_2\nM : Type u_1\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LinearOrder M\ninst✝⁶ : IsOrderedAddMonoid M\nK : Type u_2\ninst✝⁵ : Ring K\ninst✝⁴ : LinearOrder K\ninst✝³ : IsOrderedRing K\ninst✝² : Archimedean K\ninst✝¹ : Module K M\ninst✝ : PosSMulMono K M\na : M\nk : K\nh : k ≠ 0\nthis : Nontrivial ... | have : n • |a| = (n • (1 : K)) • |a| := by rw [smul_assoc, one_smul]
rw [this]
exact smul_le_smul_of_nonneg_right hn (by simp) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.Basic | {
"line": 217,
"column": 62
} | {
"line": 218,
"column": 42
} | [
{
"pp": "Γ : Type u_1\nR : Type u_3\ninst✝¹ : PartialOrder Γ\ninst✝ : Zero R\na : Γ\nr : R\n⊢ ((single a) r).support ⊆ {a}",
"usedConstants": [
"Classical.propDecidable",
"Pi.support_single_subset",
"Eq"
]
}
] | by
classical exact Pi.support_single_subset | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 102,
"column": 4
} | {
"line": 102,
"column": 44
} | [
{
"pp": "case mp.inr\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrder Γ\ninst✝¹ : Zero R\ninst✝ : LinearOrder R\nx : Lex R⟦Γ⟧\nh : 0 ≤ (ofLex x).leadingCoeff\nhlt : 0 < (ofLex x).leadingCoeff\n⊢ 0 ≤ x",
"usedConstants": [
"Preorder.toLT",
"Equiv.instEquivLike",
"Lex",
"PartialOrder.... | · exact (leadingCoeff_pos_iff.mp hlt).le | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.Ring.Cone | {
"line": 39,
"column": 54
} | {
"line": 39,
"column": 74
} | [
{
"pp": "case mk.mk.e_toSubsemiring\nR : Type u_1\ninst✝ : Ring R\ntoSubsemiring✝¹ : Subsemiring R\neq_zero_of_mem_of_neg_mem'✝¹ : ∀ {a : R}, a ∈ toSubsemiring✝¹.carrier → -a ∈ toSubsemiring✝¹.carrier → a = 0\ntoSubsemiring✝ : Subsemiring R\neq_zero_of_mem_of_neg_mem'✝ : ∀ {a : R}, a ∈ toSubsemiring✝.carrier → ... | exact SetLike.ext' h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Ring.Ordering.Defs | {
"line": 57,
"column": 54
} | {
"line": 57,
"column": 74
} | [
{
"pp": "case mk.mk.e_toSubsemiring\nR : Type u_1\ninst✝ : CommRing R\ntoSubsemiring✝¹ : Subsemiring R\nmem_of_isSquare'✝¹ : ∀ {x : R}, IsSquare x → x ∈ toSubsemiring✝¹.carrier\nneg_one_notMem'✝¹ : -1 ∉ toSubsemiring✝¹.carrier\ntoSubsemiring✝ : Subsemiring R\nmem_of_isSquare'✝ : ∀ {x : R}, IsSquare x → x ∈ toSu... | exact SetLike.ext' h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Order.Ring.Archimedean | {
"line": 298,
"column": 42
} | {
"line": 303,
"column": 42
} | [
{
"pp": "R : Type u_1\ninst✝² : LinearOrder R\ninst✝¹ : Field R\ninst✝ : IsOrderedRing R\nn : ℕ\nx : ArchimedeanClass R\n⊢ ↑n.succ • x = ↑n • x + x",
"usedConstants": [
"zero_zpow",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Nat.cast_succ",
"instHSMul",
"IsDomain.to_noZ... | by
induction x with | mk x
rw [← mk_zpow, Nat.cast_succ]
obtain rfl | hx := eq_or_ne x 0
· simp [zero_zpow _ n.cast_add_one_ne_zero]
· rw [zpow_add_one₀ hx, mk_mul, mk_zpow] | [anonymous] | Lean.Parser.Term.byTactic |
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