module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 303,
"column": 6
} | {
"line": 303,
"column": 24
} | [
{
"pp": "case h\ns : ℂ\nhs : 0 < s.re\nf : ℕ → ℝ → ℂ := ⋯\nf_ible : ∀ (n : ℕ), Integrable (f n) (volume.restrict (Ioi 0))\nx : ℝ\nhx : x ∈ Ioi 0\n⊢ Tendsto (fun n ↦ ↑((1 - x / ↑n) ^ n) * ↑x ^ (s - 1)) atTop (𝓝 (↑(rexp (-x)) * ↑x ^ (s - 1)))",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalN... | simp_rw [mul_comm] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 78
} | [
{
"pp": "case a\nk : Type u_1\nV : Type u_2\ninst✝⁹ : Field k\ninst✝⁸ : LinearOrder k\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : TopologicalSpace V\ninst✝⁴ : TopologicalSpace k\ninst✝³ : OrderTopology k\ninst✝² : IsStrictOrderedRing k\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : ContinuousSMul k V\nv... | simp_rw [vadd_eq_add, add_pure, ← map₂_smul, map_map₂, ← map_prod_eq_map₂] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 269,
"column": 2
} | {
"line": 276,
"column": 35
} | [
{
"pp": "k : Type u_1\nV : Type u_2\ninst✝⁹ : Field k\ninst✝⁸ : LinearOrder k\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : TopologicalSpace V\ninst✝⁴ : TopologicalSpace k\ninst✝³ : OrderTopology k\ninst✝² : IsStrictOrderedRing k\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : ContinuousSMul k V\ns : Set V... | ext v
rw [mem_closure_iff_frequently, ← map_snd_prod (atTop (α := k)) (𝓝 v), frequently_map,
mem_asymptoticCone_iff, asymptoticNhds_eq_smul, ← map₂_smul, ← map_prod_eq_map₂, frequently_map]
apply frequently_congr
filter_upwards [tendsto_fst.eventually (eventually_gt_atTop 0)] with ⟨c, u⟩ hc
refine ⟨fun hu ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.AsymptoticCone | {
"line": 269,
"column": 2
} | {
"line": 276,
"column": 35
} | [
{
"pp": "k : Type u_1\nV : Type u_2\ninst✝⁹ : Field k\ninst✝⁸ : LinearOrder k\ninst✝⁷ : AddCommGroup V\ninst✝⁶ : Module k V\ninst✝⁵ : TopologicalSpace V\ninst✝⁴ : TopologicalSpace k\ninst✝³ : OrderTopology k\ninst✝² : IsStrictOrderedRing k\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : ContinuousSMul k V\ns : Set V... | ext v
rw [mem_closure_iff_frequently, ← map_snd_prod (atTop (α := k)) (𝓝 v), frequently_map,
mem_asymptoticCone_iff, asymptoticNhds_eq_smul, ← map₂_smul, ← map_prod_eq_map₂, frequently_map]
apply frequently_congr
filter_upwards [tendsto_fst.eventually (eventually_gt_atTop 0)] with ⟨c, u⟩ hc
refine ⟨fun hu ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.Beta | {
"line": 567,
"column": 4
} | {
"line": 567,
"column": 31
} | [
{
"pp": "s : ℂ\nh1 : AnalyticOnNhd ℂ (fun z ↦ (Gamma z)⁻¹ * (Gamma (z + 1 / 2))⁻¹) univ\nh2 : AnalyticOnNhd ℂ (fun z ↦ (Gamma (2 * z))⁻¹ * 2 ^ (2 * z - 1) / ↑√π) univ\n⊢ Tendsto ofReal (𝓝[≠] 1) (𝓝[≠] 1)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"... | rw [tendsto_nhdsWithin_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Affine.MazurUlam | {
"line": 70,
"column": 4
} | {
"line": 71,
"column": 48
} | [
{
"pp": "E : Type u_1\nPE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : MetricSpace PE\ninst✝ : NormedAddTorsor E PE\nx y : PE\nz : PE := midpoint ℝ x y\ns : Set (PE ≃ᵢ PE) := {e | e x = x ∧ e y = y}\nthis : Nonempty ↑s\nh_bdd : BddAbove (range fun e ↦ dist (↑e z) z)\nR : PE ≃ᵢ P... | rw [dist_pointReflection_fixed, ← e.dist_eq, e.apply_symm_apply,
dist_pointReflection_self_real, dist_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Affine.MazurUlam | {
"line": 125,
"column": 8
} | {
"line": 125,
"column": 56
} | [
{
"pp": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MetricSpace PE\ninst✝⁴ : NormedAddTorsor E PE\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : MetricSpace PF\ninst✝ : NormedAddTorsor F PF\nf : E ≃ᵢ F\n⊢ (f.tra... | simpa only [sub_eq_add_neg] using sub_self (f 0) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Affine.MazurUlam | {
"line": 125,
"column": 8
} | {
"line": 125,
"column": 56
} | [
{
"pp": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MetricSpace PE\ninst✝⁴ : NormedAddTorsor E PE\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : MetricSpace PF\ninst✝ : NormedAddTorsor F PF\nf : E ≃ᵢ F\n⊢ (f.tra... | simpa only [sub_eq_add_neg] using sub_self (f 0) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Affine.MazurUlam | {
"line": 125,
"column": 8
} | {
"line": 125,
"column": 56
} | [
{
"pp": "E : Type u_1\nPE : Type u_2\nF : Type u_3\nPF : Type u_4\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\ninst✝⁵ : MetricSpace PE\ninst✝⁴ : NormedAddTorsor E PE\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : MetricSpace PF\ninst✝ : NormedAddTorsor F PF\nf : E ≃ᵢ F\n⊢ (f.tra... | simpa only [sub_eq_add_neg] using sub_self (f 0) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Polynomial.Factorization | {
"line": 66,
"column": 2
} | {
"line": 66,
"column": 71
} | [
{
"pp": "f : ℝ[X]\nn : ℕ\nhf : f.IsMonicOfDegree (n + 2)\n⊢ ∃ f₁ f₂, f₁.IsMonicOfDegree 2 ∧ f₂.IsMonicOfDegree n ∧ f = f₁ * f₂",
"usedConstants": [
"instOfNatNat",
"Polynomial.IsMonicOfDegree.eq_isMonicOfDegree_one_or_two_mul",
"instHAdd",
"HAdd.hAdd",
"Nat",
"instAddNat"... | obtain ⟨g₁, g₂, hd₁ | hd₂, h⟩ := hf.eq_isMonicOfDegree_one_or_two_mul | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Normed.Affine.Simplex | {
"line": 172,
"column": 10
} | {
"line": 172,
"column": 13
} | [
{
"pp": "case refine_2.inr.inr.inr.inr.inl\nR : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : Module R V\ninst✝ : NormedAddTorsor V P\nt : Triangle R P\ni₁ : Fin 3\ni j : Fin (2 + 1)\nhij : i ≠ j\nh₁₂ : i₁ ≠ i\nh₁₃ : i₁ ≠ j\nh₂₃ ... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Affine.Simplex | {
"line": 173,
"column": 10
} | {
"line": 173,
"column": 13
} | [
{
"pp": "case refine_2.inr.inr.inr.inr.inr\nR : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring R\ninst✝³ : SeminormedAddCommGroup V\ninst✝² : PseudoMetricSpace P\ninst✝¹ : Module R V\ninst✝ : NormedAddTorsor V P\nt : Triangle R P\ni₁ : Fin 3\ni j : Fin (2 + 1)\nhij : i ≠ j\nh₁₃ : i₁ ≠ i\nh₁₂ : i₁ ≠ j\nh₂₃ ... | h₂, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 243,
"column": 4
} | {
"line": 243,
"column": 62
} | [
{
"pp": "F : Type u_1\ninst✝⁴ : NormedRing F\ninst✝³ : NormOneClass F\ninst✝² : NormMulClass F\ninst✝¹ : NormedAlgebra ℂ F\ninst✝ : Nontrivial F\n⊢ Function.Bijective ⇑(ofId ℂ F)",
"usedConstants": [
"NormedRing.toRing",
"Complex.commRing",
"AlgHom",
"AlgHom.funLike",
"Complex.... | refine ⟨FaithfulSMul.algebraMap_injective ℂ F, fun x ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 246,
"column": 4
} | {
"line": 246,
"column": 64
} | [
{
"pp": "F : Type u_1\ninst✝⁴ : NormedRing F\ninst✝³ : NormOneClass F\ninst✝² : NormMulClass F\ninst✝¹ : NormedAlgebra ℂ F\ninst✝ : Nontrivial F\nx : F\nz : ℂ\nhz : ‖x - (algebraMap ℂ F) z‖ = 0\n⊢ (ofId ℂ F) z = x",
"usedConstants": [
"norm_eq_zero",
"AddGroup.toSubtractionMonoid",
"Norm.n... | rwa [norm_eq_zero, sub_eq_zero, eq_comm, ← ofId_apply] at hz | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 282,
"column": 12
} | {
"line": 282,
"column": 20
} | [
{
"pp": "case zero\nF : Type u_1\ninst✝³ : NormedRing F\ninst✝² : NormedAlgebra ℝ F\ninst✝¹ : NormOneClass F\ninst✝ : NormMulClass F\nx : F\nM : ℝ\nhM : 0 ≤ M\nh : ∀ (z : ℝ × ℝ), M ≤ ‖φ x z‖\np : ℝ[X]\nhp : p.IsMonicOfDegree (2 * 0)\n⊢ M ^ 0 ≤ ‖(aeval x) p‖",
"usedConstants": [
"Norm.norm",
"Mul... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 282,
"column": 12
} | {
"line": 282,
"column": 20
} | [
{
"pp": "case zero\nF : Type u_1\ninst✝³ : NormedRing F\ninst✝² : NormedAlgebra ℝ F\ninst✝¹ : NormOneClass F\ninst✝ : NormMulClass F\nx : F\nM : ℝ\nhM : 0 ≤ M\nh : ∀ (z : ℝ × ℝ), M ≤ ‖φ x z‖\np : ℝ[X]\nhp : p.IsMonicOfDegree (2 * 0)\n⊢ M ^ 0 ≤ ‖(aeval x) p‖",
"usedConstants": [
"Norm.norm",
"Mul... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 282,
"column": 12
} | {
"line": 282,
"column": 20
} | [
{
"pp": "case zero\nF : Type u_1\ninst✝³ : NormedRing F\ninst✝² : NormedAlgebra ℝ F\ninst✝¹ : NormOneClass F\ninst✝ : NormMulClass F\nx : F\nM : ℝ\nhM : 0 ≤ M\nh : ∀ (z : ℝ × ℝ), M ≤ ‖φ x z‖\np : ℝ[X]\nhp : p.IsMonicOfDegree (2 * 0)\n⊢ M ^ 0 ≤ ‖(aeval x) p‖",
"usedConstants": [
"Norm.norm",
"Mul... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 303,
"column": 2
} | {
"line": 303,
"column": 38
} | [
{
"pp": "F : Type u_1\ninst✝³ : NormedRing F\ninst✝² : NormedAlgebra ℝ F\ninst✝¹ : NormOneClass F\ninst✝ : NormMulClass F\nx : F\nz : ℝ × ℝ\nh : IsMinOn (fun x_1 ↦ ‖φ x x_1‖) Set.univ z\nw✝ : ℝ × ℝ\nM : ℝ := ‖φ x z‖\nH : M ≠ 0\nhM : M = ‖φ x z‖\nhM₀ : 0 < M\nw u : ℝ × ℝ\nhw : ‖φ x w‖ = M\nn : ℕ\nhn : n > 0\nHH ... | rw [HH, le_div_iff₀ (by positivity)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Algebra.GelfandMazur | {
"line": 361,
"column": 10
} | {
"line": 361,
"column": 18
} | [
{
"pp": "F : Type u_1\ninst✝² : NormedRing F\ninst✝¹ : NormedAlgebra ℝ F\ninst✝ : NormOneClass F\nx : F\nc : ℝ\nhc₀ : 0 < c\nhbd : ∀ (r : ℝ), c ≤ ‖x - (algebraMap ℝ F) r‖\nthis : Tendsto (fun y ↦ ‖y.1‖ * c) (cobounded ℝ ×ˢ ⊤) atTop\ny : ℝ × ℝ\nhy : y ∈ {0}ᶜ ×ˢ Set.univ\n⊢ ‖y.1 • x - (y.1 * (y.1⁻¹ * y.2)) • 1‖ =... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Normed.Field.Ultra | {
"line": 124,
"column": 2
} | {
"line": 126,
"column": 18
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedDivisionRing R\nx : R\nm : ℕ\nh : ∀ (x : R) (n : ℕ), ‖n • x‖ ≤ ‖x‖\n⊢ ∑ k ∈ Finset.range (m + 1), ‖x‖ ^ k ≤ ∑ _x ∈ Finset.range (m + 1), max 1 (‖x‖ ^ m)",
"usedConstants": [
"Norm.norm",
"Real",
"NormedDivisionRing.toNorm",
"Monoid.toPow",
"... | rcases max_cases 1 (‖x‖ ^ m) with (⟨hm, hx⟩ | ⟨hm, hx⟩) <;> rw [hm] <;>
-- which we show by comparing the terms in the sum one by one
gcongr with i hi | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.Normed.Algebra.TrivSqZeroExt | {
"line": 128,
"column": 4
} | {
"line": 128,
"column": 34
} | [
{
"pp": "case neg\nR : Type u_3\nM : Type u_4\ninst✝¹⁴ : Ring R\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Module R M\ninst✝¹¹ : Module Rᵐᵒᵖ M\ninst✝¹⁰ : SMulCommClass R Rᵐᵒᵖ M\ninst✝⁹ : TopologicalSpace R\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : IsTopologicalRing R\ninst✝⁶ : IsTopologicalAddGroup M\ninst✝⁵ : Continuou... | simp_rw [← fst_expSeries] at h | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 107,
"column": 6
} | {
"line": 107,
"column": 44
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nx : R\nhx : μ x = 0\nn : ℕ\nhn : 1 ≤ n\n⊢ μ (x ^ n) = 0",
"usedConstants": [
"Real.partialOrder",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"RingSeminorm.ringSeminormClass",
"CommRing.toNonUnitalCommR... | apply le_antisymm _ (apply_nonneg μ _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 76,
"column": 30
} | {
"line": 76,
"column": 43
} | [
{
"pp": "case h\nR : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf : f 0 = 0\nn : ℕ\n⊢ 0 = 0 n",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real",
"Real.instZero",
"congrArg",
"DivisionSemiring.toGroupWithZero",
"Pi.zero_apply",
... | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Unbundled.SeminormFromConst | {
"line": 251,
"column": 4
} | {
"line": 252,
"column": 55
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nc : R\nf : RingSeminorm R\nhf1 : f 1 ≤ 1\nhc : f c ≠ 0\nhpm : IsPowMul ⇑f\nx : R\n⊢ Tendsto (fun n ↦ seminormFromConst_seq c f x (n + 1)) atTop (𝓝 (seminormFromConst' c f x))",
"usedConstants": [
"Real",
"Nat.instIsOrderedAddMonoid",
"covariant_s... | apply (tendsto_seminormFromConst_seq_atTop hf1 hc hpm x).comp
(tendsto_atTop_atTop_of_monotone add_left_mono _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Unbundled.FiniteExtension | {
"line": 155,
"column": 80
} | {
"line": 170,
"column": 38
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁴ : NormedField K\ninst✝³ : Ring L\ninst✝² : Algebra K L\nι : Type u_4\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\nB : Basis ι K L\ni : ι\nhBi : B i = 1\nk : K\ny : L\n⊢ B.norm ((algebraMap K L) k * y) = B.norm ((algebraMap K L) k) * B.norm y",
"usedConstants": [
... | by
by_cases hk : k = 0
· rw [hk, map_zero, zero_mul, B.norm_zero, zero_mul]
· rw [norm_extends hBi]
obtain ⟨i, _, hi⟩ := exists_mem_eq_sup' univ_nonempty (fun i ↦ ‖B.repr y i‖)
obtain ⟨j, _, hj⟩ := exists_mem_eq_sup' univ_nonempty
(fun i ↦ ‖B.repr ((algebraMap K L) k * y) i‖)
have hij : ‖B.repr ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 298,
"column": 4
} | {
"line": 298,
"column": 52
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nx y : R\nhn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)\nn : ℕ\nhn0 : n = 0\n⊢ ↑(mu μ hn n) / ↑n ≤ 1",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
... | rw [hn0, cast_zero, div_zero]; exact zero_le_one | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 298,
"column": 4
} | {
"line": 298,
"column": 52
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nx y : R\nhn : ∀ (n : ℕ), ∃ m < n + 1, μ ((x + y) ^ n) ^ (1 / ↑n) ≤ (μ (x ^ m) * μ (y ^ (n - m))) ^ (1 / ↑n)\nn : ℕ\nhn0 : n = 0\n⊢ ↑(mu μ hn n) / ↑n ≤ 1",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
... | rw [hn0, cast_zero, div_zero]; exact zero_le_one | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 315,
"column": 2
} | {
"line": 320,
"column": 88
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\ns : ℕ → ℕ\nhs : ∀ (n : ℕ), s n ≤ n\nx : R\nψ : ℕ → ℕ\nhψ : ∀ (n : ℕ), 0 ≤ 1 / ↑(ψ n)\nhx : μ x ≤ 1\n⊢ BddAbove (Set.range fun n ↦ μ (x ^ s (ψ n)) ^ (1 / ↑(ψ n)))",
"usedConstants": [
"Real.instIsOrderedRing",
... | · use 1
simp only [mem_upperBounds, Set.mem_range, forall_exists_index]
rintro _ n rfl
apply le_trans (rpow_le_rpow (apply_nonneg _ _) (map_pow_le_pow' hμ1 _ _) (hψ n))
rw [← rpow_natCast, ← rpow_mul (apply_nonneg _ _), mul_one_div]
exact rpow_le_one (apply_nonneg _ _) hx (div_nonneg (cast_nonneg _)... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Field.Krasner | {
"line": 125,
"column": 4
} | {
"line": 125,
"column": 64
} | [
{
"pp": "K : Type u_1\nL : Type u_2\ninst✝⁵ : NormedField L\ninst✝⁴ : NontriviallyNormedField K\ninst✝³ : CompleteSpace K\ninst✝² : IsUltrametricDist K\ninst✝¹ : NormedAlgebra K L\ninst✝ : Algebra.IsAlgebraic K L\nx y : L\nxsep : IsSeparable K x\nsp : (Polynomial.map (algebraMap K L) (minpoly K x)).Splits\nyint... | let : NormedAlgebra L C := spectralNorm.normedAlgebra' K L C | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Normed.Group.CocompactMap | {
"line": 37,
"column": 2
} | {
"line": 37,
"column": 100
} | [
{
"pp": "E : Type u_2\nF : Type u_3\n𝓕 : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\nf : 𝓕\ninst✝² : ProperSpace F\ninst✝¹ : FunLike 𝓕 E F\ninst✝ : CocompactMapClass 𝓕 E F\nε : ℝ\nh : ∀ s ∈ cocompact F, ⇑f ⁻¹' s ∈ cocompact E\n⊢ ∃ r, ∀ (x : E), r < ‖x‖ → ε < ‖f x‖",
"usedCons... | specialize h (Metric.closedBall 0 ε)ᶜ (mem_cocompact_of_closedBall_compl_subset 0 ⟨ε, rfl.subset⟩) | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.Analysis.Normed.Group.CocompactMap | {
"line": 41,
"column": 49
} | {
"line": 41,
"column": 57
} | [
{
"pp": "E : Type u_2\nF : Type u_3\n𝓕 : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\nf : 𝓕\ninst✝² : ProperSpace F\ninst✝¹ : FunLike 𝓕 E F\ninst✝ : CocompactMapClass 𝓕 E F\nε : ℝ\nh : ⇑f ⁻¹' (closedBall 0 ε)ᶜ ∈ cocompact E\nr : ℝ\nhr : (closedBall 0 r)ᶜ ⊆ ⇑f ⁻¹' (closedBall 0 ε)ᶜ... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Normed.Group.CocompactMap | {
"line": 41,
"column": 49
} | {
"line": 41,
"column": 57
} | [
{
"pp": "E : Type u_2\nF : Type u_3\n𝓕 : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\nf : 𝓕\ninst✝² : ProperSpace F\ninst✝¹ : FunLike 𝓕 E F\ninst✝ : CocompactMapClass 𝓕 E F\nε : ℝ\nh : ⇑f ⁻¹' (closedBall 0 ε)ᶜ ∈ cocompact E\nr : ℝ\nhr : (closedBall 0 r)ᶜ ⊆ ⇑f ⁻¹' (closedBall 0 ε)ᶜ... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.CocompactMap | {
"line": 41,
"column": 49
} | {
"line": 41,
"column": 57
} | [
{
"pp": "E : Type u_2\nF : Type u_3\n𝓕 : Type u_4\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedAddCommGroup F\nf : 𝓕\ninst✝² : ProperSpace F\ninst✝¹ : FunLike 𝓕 E F\ninst✝ : CocompactMapClass 𝓕 E F\nε : ℝ\nh : ⇑f ⁻¹' (closedBall 0 ε)ᶜ ∈ cocompact E\nr : ℝ\nhr : (closedBall 0 r)ᶜ ⊆ ⇑f ⁻¹' (closedBall 0 ε)ᶜ... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Unbundled.SmoothingSeminorm | {
"line": 595,
"column": 6
} | {
"line": 595,
"column": 44
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nμ : RingSeminorm R\nhμ1 : μ 1 ≤ 1\nx : R\nhx : ∀ (y : R), μ (x * y) = μ x * μ y\nn : ℕ\nhn : 1 ≤ n\nhx0 : μ x = 0\n⊢ μ (x ^ n) = 0",
"usedConstants": [
"Real.partialOrder",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"RingSemino... | apply le_antisymm _ (apply_nonneg μ _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 19
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : SeminormedRing R\np : R[X]\nn : ℕ\nh : ¬n < p.natDegree\n⊢ 0 ≤ 0",
"usedConstants": [
"le_refl",
"Real",
"Real.instZero",
"Zero.toOfNat0",
"OfNat.ofNat",
"Real.instPreorder"
]
}
] | · exact le_refl _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 157,
"column": 4
} | {
"line": 157,
"column": 21
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : SeminormedRing R\ninst✝ : Nontrivial R\nr : R\nn : ℕ\nh✝ : n = 0\n⊢ ‖r‖ ≤ ‖r‖",
"usedConstants": [
"Norm.norm",
"SeminormedRing.toNorm",
"le_refl",
"Real",
"Real.instPreorder"
]
},
{
"pp": "case neg\nR : Type u_1\ninst✝¹ : S... | · exact le_refl _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 305,
"column": 6
} | {
"line": 306,
"column": 71
} | [
{
"pp": "case pos\nK : Type u_2\ninst✝³ : NormedField K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : DecidableEq L\nf : AlgebraNorm K L\nhf_pm : IsPowMul ⇑f\nhf_na : IsNonarchimedean ⇑f\nhf1 : f 1 = 1\np : K[X]\ns : Multiset L\nhp : (mapAlg K L) p = (Multiset.map (fun a ↦ X - C a) s).prod\nh_l... | exact norm_root_le_spectralValue hf_pm hf_na
(monic_of_monic_mapAlg (hp ▸ monic_multisetProd_X_sub_C s)) hx0 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 388,
"column": 2
} | {
"line": 390,
"column": 30
} | [
{
"pp": "K : Type u_2\ninst✝⁶ : NormedField K\nL : Type u_3\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\nE : Type u_4\ninst✝³ : Field E\ninst✝² : Algebra K E\ninst✝¹ : Algebra E L\ninst✝ : IsScalarTower K E L\nx : E\n⊢ spectralNorm K E x = spectralNorm K L ((algebraMap E L) x)",
"usedConstants": [
"Normed... | have hx : minpoly K (algebraMap E L x) = minpoly K x :=
minpoly.algebraMap_eq (algebraMap E L).injective x
simp only [spectralNorm, hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 388,
"column": 2
} | {
"line": 390,
"column": 30
} | [
{
"pp": "K : Type u_2\ninst✝⁶ : NormedField K\nL : Type u_3\ninst✝⁵ : Field L\ninst✝⁴ : Algebra K L\nE : Type u_4\ninst✝³ : Field E\ninst✝² : Algebra K E\ninst✝¹ : Algebra E L\ninst✝ : IsScalarTower K E L\nx : E\n⊢ spectralNorm K E x = spectralNorm K L ((algebraMap E L) x)",
"usedConstants": [
"Normed... | have hx : minpoly K (algebraMap E L x) = minpoly K x :=
minpoly.algebraMap_eq (algebraMap E L).injective x
simp only [spectralNorm, hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 498,
"column": 2
} | {
"line": 499,
"column": 78
} | [
{
"pp": "case h\nK : Type u_2\ninst✝² : NormedField K\nL : Type u_3\ninst✝¹ : Field L\ninst✝ : Algebra K L\nh_fin : FiniteDimensional K L\nhn : Normal K L\nhu : IsUltrametricDist K\nx : L\nhna : IsNonarchimedean fun x ↦ ‖x‖\nf : AlgebraNorm K L := Classical.choose ⋯\nhf : f = Classical.choose ⋯\n⊢ spectralNorm ... | have hf_pow : IsPowMul f := (Classical.choose_spec
(exists_nonarchimedean_pow_mul_seminorm_of_finiteDimensional h_fin hna)).1 | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Unbundled.SpectralNorm | {
"line": 606,
"column": 4
} | {
"line": 606,
"column": 51
} | [
{
"pp": "K : Type u_2\ninst✝³ : NormedField K\nL : Type u_3\ninst✝² : Field L\ninst✝¹ : Algebra K L\ninst✝ : IsUltrametricDist K\nk : K\ny : L\nhy : IsAlgebraic K y\nE : IntermediateField K L := K⟮y⟯\nh_finiteDimensional_E : FiniteDimensional K ↥E\ng : ↥K⟮y⟯ := AdjoinSimple.gen K y\nhgy : k • y = (algebraMap (↥... | ← spectralNorm.eq_of_normalClosure (k • g) rfl, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.ContinuousInverse | {
"line": 134,
"column": 36
} | {
"line": 138,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝¹² : Semiring R\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nF' : Type u_5\ninst✝¹¹ : TopologicalSpace E\ninst✝¹⁰ : AddCommMonoid E\ninst✝⁹ : Module R E\ninst✝⁸ : TopologicalSpace E'\ninst✝⁷ : AddCommMonoid E'\ninst✝⁶ : Module R E'\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : AddCommMonoid ... | by
obtain ⟨finv, hfinv⟩ := hf
obtain ⟨ginv, hginv⟩ := hg
use finv.prodMap ginv
simp [hfinv, hginv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.ContinuousInverse | {
"line": 300,
"column": 37
} | {
"line": 304,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝¹² : Semiring R\nE : Type u_2\nE' : Type u_3\nF : Type u_4\nF' : Type u_5\ninst✝¹¹ : TopologicalSpace E\ninst✝¹⁰ : AddCommMonoid E\ninst✝⁹ : Module R E\ninst✝⁸ : TopologicalSpace E'\ninst✝⁷ : AddCommMonoid E'\ninst✝⁶ : Module R E'\ninst✝⁵ : TopologicalSpace F\ninst✝⁴ : AddCommMonoid ... | by
obtain ⟨finv, hfinv⟩ := hf
obtain ⟨ginv, hginv⟩ := hg
use finv.prodMap ginv
simp [hfinv, hginv] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.Bases | {
"line": 383,
"column": 6
} | {
"line": 384,
"column": 50
} | [
{
"pp": "case neg.inr\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nX : Type u_2\ninst✝¹ : NormedAddCommGroup X\ninst✝ : NormedSpace 𝕜 X\nP : ℕ → X →L[𝕜] X\nhcomp : ∀ (n m : ℕ) (x : X), (P n) ((P m) x) = (P (min n m)) x\ni j : ℕ\nx : X\nh : ¬i = j\nh' : i > j\n⊢ (P (min i j + 1)) x - (P (min i (j + 1))... | rw [min_eq_right_of_lt h', min_eq_right (Nat.succ_le_of_lt h'),
min_eq_right_of_lt (Nat.lt_succ_of_lt h')] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Module.PiTensorProduct.InjectiveSeminorm | {
"line": 142,
"column": 63
} | {
"line": 192,
"column": 41
} | [
{
"pp": "ι : Type uι\ninst✝⁵ : Fintype ι\n𝕜 : Type u𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : ι → Type uE\ninst✝³ : (i : ι) → SeminormedAddCommGroup (E i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (E i)\nF : Type uF\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : ContinuousMultilinearMap 𝕜 E F\n... | by
/- If `F` were in `Type (max uι u𝕜 uE)` (which is the type of `⨂[𝕜] i, E i`), then the
property that we want to prove would hold by definition of `injectiveSeminorm`. This is
not necessarily true, but we will show that there exists a normed vector space `G` in
`Type (max uι u𝕜 uE)` and an injectiv... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.ODE.Gronwall | {
"line": 57,
"column": 4
} | {
"line": 57,
"column": 52
} | [
{
"pp": "case pos\nδ ε x : ℝ\n⊢ HasDerivAt (gronwallBound δ 0 ε) (0 * gronwallBound δ 0 ε x + ε) x",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"HMul.hMul",
"IsTop... | simp only [gronwallBound_K0, zero_mul, zero_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.PSeriesComplex | {
"line": 27,
"column": 6
} | {
"line": 27,
"column": 39
} | [
{
"pp": "p : ℂ\n⊢ (Summable fun n ↦ 1 / ↑n ^ p) ↔ 1 < p.re",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instPow",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"congrArg",
"Real.instDivInvMonoid",
"Co... | ← Real.summable_one_div_nat_rpow, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 27
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ntmin tmax : ℝ\nt₀ : ↑(Icc tmin tmax)\nx₀ : E\na r L : ℝ≥0\ninst✝ : CompleteSpace E\n⊢ IsComplete (range fun α ↦ toContinuousMap α)",
"usedConstants": [
"ODE.FunSpace.toContinuousMap",
"Real",
"ODE.FunSpace",
"FrechetUrysohnSpace.t... | apply IsClosed.isComplete | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Polynomial.Basic | {
"line": 318,
"column": 2
} | {
"line": 318,
"column": 45
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝³ : NormedField 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\nP Q : 𝕜[X]\ninst✝ : OrderTopology 𝕜\nh : P.degree ≤ Q.degree\nhp : P = 0\n⊢ (fun x ↦ eval x P) =O[atTop] fun x ↦ eval x Q",
"usedConstants": [
"Eq.mpr",
"Polynomial.eval",
... | · simpa [hp] using isBigO_zero Q.eval atTop | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.ODE.PicardLindelof | {
"line": 497,
"column": 2
} | {
"line": 499,
"column": 39
} | [
{
"pp": "case a\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf : ℝ → E → E\nα : ℝ → E\nu : Set E\nt₀ tmin tmax : ℝ\nht₀ : t₀ ∈ Icc tmin tmax\nhf : ContinuousOn (uncurry f) (Icc tmin tmax ×ˢ u)\nhα : ContinuousOn α (Icc tmin tmax)\nhmem : ∀ t ∈ Icc tmin tmax, α... | apply intervalIntegral.integral_hasDerivWithinAt_right _ -- need `CompleteSpace E` and `Icc`
(continuousOn_comp hf hα hmem |>.stronglyMeasurableAtFilter_nhdsWithin measurableSet_Icc t)
(continuousOn_comp hf hα hmem _ ht) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.MvPowerSeries.GaussNorm | {
"line": 87,
"column": 2
} | {
"line": 121,
"column": 40
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝ : Semiring R\nv : R → ℝ\nc : σ → ℝ\nf g : MvPowerSeries σ R\nhc : 0 ≤ c\nvNonneg : ∀ (a : R), v a ≥ 0\nhv : ∀ (x y : R), v (x + y) ≤ max (v x) (v y)\nhbfd : HasGaussNorm v c f\nhbgd : HasGaussNorm v c g\n⊢ gaussNorm v c (f + g) ≤ max (gaussNorm v c f) (gaussNorm v c g)... | have H (t : σ →₀ ℕ) : 0 ≤ ∏ i ∈ t.support, c i ^ t i :=
Finset.prod_nonneg (fun i hi ↦ pow_nonneg (hc i) (t i))
have Final (t : σ →₀ ℕ) : v ((coeff t) (f + g)) * ∏ i ∈ t.support, c ↑i ^ t ↑i ≤
max (v ((coeff t) f) * ∏ i ∈ t.support, c ↑i ^ t ↑i)
(v ((coeff t) g) * ∏ i ∈ t.support, c ↑i ^ t ↑i) := by
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.GaussNorm | {
"line": 87,
"column": 2
} | {
"line": 121,
"column": 40
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝ : Semiring R\nv : R → ℝ\nc : σ → ℝ\nf g : MvPowerSeries σ R\nhc : 0 ≤ c\nvNonneg : ∀ (a : R), v a ≥ 0\nhv : ∀ (x y : R), v (x + y) ≤ max (v x) (v y)\nhbfd : HasGaussNorm v c f\nhbgd : HasGaussNorm v c g\n⊢ gaussNorm v c (f + g) ≤ max (gaussNorm v c f) (gaussNorm v c g)... | have H (t : σ →₀ ℕ) : 0 ≤ ∏ i ∈ t.support, c i ^ t i :=
Finset.prod_nonneg (fun i hi ↦ pow_nonneg (hc i) (t i))
have Final (t : σ →₀ ℕ) : v ((coeff t) (f + g)) * ∏ i ∈ t.support, c ↑i ^ t ↑i ≤
max (v ((coeff t) f) * ∏ i ∈ t.support, c ↑i ^ t ↑i)
(v ((coeff t) g) * ∏ i ∈ t.support, c ↑i ^ t ↑i) := by
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Polynomial.Order | {
"line": 91,
"column": 6
} | {
"line": 91,
"column": 74
} | [
{
"pp": "P : ℝ[X]\nx : ℝ\nhroots : ∀ (y : ℝ), P.IsRoot y → x < y\nhlc : 0 ≤ P.leadingCoeff\nhroots' : ∀ (y : ℝ), (P.comp (-X)).IsRoot y → y < -x\n⊢ 0 ≤ ↑↑(↑(P.comp (-X)).natDegree).negOnePow * (P.leadingCoeff * ↑↑(↑P.natDegree).negOnePow)",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Int.... | show (P.comp (-X)).natDegree = P.natDegree by simp [natDegree_comp], | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Polynomial.GaussNorm | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 12
} | [
{
"pp": "case neg\nR : Type u_1\nF : Type u_2\ninst✝² : Semiring R\ninst✝¹ : FunLike F R ℝ\nv : F\nc : ℝ\np : R[X]\ninst✝ : ZeroHomClass F R ℝ\nh_supp : ¬p.support.Nonempty\n⊢ ∃ i, gaussNorm v c p = v (p.coeff i) * c ^ i",
"usedConstants": [
"Real",
"HMul.hMul",
"Classical.not_not._simp_1"... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.RingTheory.Polynomial.GaussNorm | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 12
} | [
{
"pp": "case neg\nR : Type u_1\nF : Type u_2\ninst✝² : Semiring R\ninst✝¹ : FunLike F R ℝ\nv : F\nc : ℝ\np : R[X]\ninst✝ : ZeroHomClass F R ℝ\nh_supp : ¬p.support.Nonempty\n⊢ ∃ i, gaussNorm v c p = v (p.coeff i) * c ^ i",
"usedConstants": [
"Real",
"HMul.hMul",
"Classical.not_not._simp_1"... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.GaussNorm | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 12
} | [
{
"pp": "case neg\nR : Type u_1\nF : Type u_2\ninst✝² : Semiring R\ninst✝¹ : FunLike F R ℝ\nv : F\nc : ℝ\np : R[X]\ninst✝ : ZeroHomClass F R ℝ\nh_supp : ¬p.support.Nonempty\n⊢ ∃ i, gaussNorm v c p = v (p.coeff i) * c ^ i",
"usedConstants": [
"Real",
"HMul.hMul",
"Classical.not_not._simp_1"... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.FilterProduct | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 26
} | [
{
"pp": "α : Type u\nβ : Type v\nφ : Ultrafilter α\ninst✝ : LinearOrder β\nx y : β\n⊢ ↑(max x y) = max ↑x ↑y",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"Filter.Germ.const",
"SemilatticeSup.toMax",
"DistribLattice.toLattice",
"id",
"... | rw [max_def, map₂_const] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Filter.FilterProduct | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 26
} | [
{
"pp": "α : Type u\nβ : Type v\nφ : Ultrafilter α\ninst✝ : LinearOrder β\nx y : β\n⊢ ↑(max x y) = max ↑x ↑y",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"Filter.Germ.const",
"SemilatticeSup.toMax",
"DistribLattice.toLattice",
"id",
"... | rw [max_def, map₂_const] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.FilterProduct | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 26
} | [
{
"pp": "α : Type u\nβ : Type v\nφ : Ultrafilter α\ninst✝ : LinearOrder β\nx y : β\n⊢ ↑(max x y) = max ↑x ↑y",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"Filter.Germ.const",
"SemilatticeSup.toMax",
"DistribLattice.toLattice",
"id",
"... | rw [max_def, map₂_const] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.GaussNorm | {
"line": 245,
"column": 39
} | {
"line": 245,
"column": 47
} | [
{
"pp": "c✝ : ℝ\nR✝ : Type u_3\ninst✝¹ : Ring R✝\nv✝ : AbsoluteValue R✝ ℝ\nc : ℝ\nR : Type u_3\ninst✝ : Ring R\nv : AbsoluteValue R ℝ\nhna : IsNonarchimedean ⇑v\nhc : 0 < c\np q : R[X]\nhc0 : 0 ≤ c\ni : ℕ\nhi_p : gaussNorm v c p = v (p.coeff i) * c ^ i\nhlt_p : ∀ j < i, v (p.coeff j) * c ^ j < gaussNorm v c p\n... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 221,
"column": 6
} | {
"line": 221,
"column": 82
} | [
{
"pp": "case neg\np : ℂ[X]\nhp : ¬p = 0\nthis : ∀ x ∈ Multiset.map (fun x ↦ max 1 ‖x‖) p.roots, x ≠ 0\n⊢ (C p.leadingCoeff * (Multiset.map (fun x ↦ X - C x) p.roots).prod).logMahlerMeasure =\n log ‖p.leadingCoeff‖ + (Multiset.map (fun a ↦ log⁺ ‖a‖) p.roots).sum",
"usedConstants": [
"Multiset.sum",... | logMahlerMeasure_mul_eq_add_logMahlerMeasure (by simp [hp, X_sub_C_ne_zero]) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 361,
"column": 4
} | {
"line": 363,
"column": 50
} | [
{
"pp": "case a.refine_2\np : ℂ[X]\n⊢ p.support.card • p.supNorm ^ 2 ≤ (↑p.natDegree + 1) * p.supNorm ^ 2",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real"... | · simp only [nsmul_eq_mul]
gcongr
exact mod_cast p.card_supp_le_succ_natDegree | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 513,
"column": 6
} | {
"line": 513,
"column": 14
} | [
{
"pp": "A : Type u_2\ninst✝ : NormedRing A\np : A[X]\nv : A →+* ℂ\nhv : Isometry ⇑v\n⊢ ((map v p).sum fun x a ↦ ‖a‖) = p.sum fun x a ↦ ‖a‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"NormedRing.toRing",
"congrArg",
"Polynomial.sum",
"Complex.instNorm",
... | sum_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Polynomial.MahlerMeasure | {
"line": 513,
"column": 15
} | {
"line": 513,
"column": 23
} | [
{
"pp": "A : Type u_2\ninst✝ : NormedRing A\np : A[X]\nv : A →+* ℂ\nhv : Isometry ⇑v\n⊢ ∑ n ∈ (map v p).support, ‖(map v p).coeff n‖ = p.sum fun x a ↦ ‖a‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"NormedRing.toRing",
"congrArg",
"Polynomial.sum",
"Complex... | sum_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Real.Pi.Irrational | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 40
} | [
{
"pp": "θ : ℝ\nn : ℕ\nf : ℝ → ℝ := fun x ↦ 1 - x ^ 2\nu₁ : ℝ → ℝ := fun x ↦ f x ^ (n + 1)\nu₁' : ℝ → ℝ := fun x ↦ -(2 * (↑n + 1) * x * f x ^ n)\nv₁ : ℝ → ℝ := fun x ↦ sin (x * θ)\nv₁' : ℝ → ℝ := fun x ↦ cos (x * θ) * θ\nu₂ : ℝ → ℝ := fun x ↦ x * f x ^ n\nu₂' : ℝ → ℝ := fun x ↦ f x ^ n - 2 * ↑n * x ^ 2 * f x ^ ... | have hfd : Continuous f := by fun_prop | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.Complex.Arctan | {
"line": 66,
"column": 9
} | {
"line": 66,
"column": 35
} | [
{
"pp": "case inl.h\nz : ℂ\nk : ℤ\nnr : z.re = ↑(2 * k + 1) * π / 2\nh₂ : 2 * k + 1 ≤ 1\nh₁ : -1 < 2 * k + 1\n⊢ z.re = π / 2",
"usedConstants": [
"Int.cast",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"congrArg",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHA... | show 2 * k + 1 = 1 by lia, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arctan | {
"line": 66,
"column": 4
} | {
"line": 66,
"column": 64
} | [
{
"pp": "case inl.h\nz : ℂ\nk : ℤ\nnr : z.re = ↑(2 * k + 1) * π / 2\nh₂ : 2 * k + 1 ≤ 1\nh₁ : -1 < 2 * k + 1\n⊢ z.re = π / 2",
"usedConstants": [
"Int.cast",
"MulOne.toOne",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"congrArg",
"Real.instDivInvMonoid",
... | rwa [show 2 * k + 1 = 1 by lia, Int.cast_one, one_mul] at nr | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.Real.Pi.Wallis | {
"line": 55,
"column": 2
} | {
"line": 59,
"column": 78
} | [
{
"pp": "k : ℕ\n⊢ 0 < W k",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"mul_nonneg",
"MulOne.toOne",
"Real.partialOrder",
"Real",
"Nat.recAux",
"Preorder.toLT",
"instHDiv",
"IsOrderedRing.toPosMulMono",
"HMul.hMul",
"MulZero... | induction k with
| zero => unfold W; simp
| succ k hk =>
rw [W_succ]
refine mul_pos hk (mul_pos (div_pos ?_ ?_) (div_pos ?_ ?_)) <;> positivity | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.Real.Pi.Wallis | {
"line": 55,
"column": 2
} | {
"line": 59,
"column": 78
} | [
{
"pp": "k : ℕ\n⊢ 0 < W k",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"mul_nonneg",
"MulOne.toOne",
"Real.partialOrder",
"Real",
"Nat.recAux",
"Preorder.toLT",
"instHDiv",
"IsOrderedRing.toPosMulMono",
"HMul.hMul",
"MulZero... | induction k with
| zero => unfold W; simp
| succ k hk =>
rw [W_succ]
refine mul_pos hk (mul_pos (div_pos ?_ ?_) (div_pos ?_ ?_)) <;> positivity | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Real.Pi.Wallis | {
"line": 55,
"column": 2
} | {
"line": 59,
"column": 78
} | [
{
"pp": "k : ℕ\n⊢ 0 < W k",
"usedConstants": [
"Real.instIsOrderedRing",
"Eq.mpr",
"mul_nonneg",
"MulOne.toOne",
"Real.partialOrder",
"Real",
"Nat.recAux",
"Preorder.toLT",
"instHDiv",
"IsOrderedRing.toPosMulMono",
"HMul.hMul",
"MulZero... | induction k with
| zero => unfold W; simp
| succ k hk =>
rw [W_succ]
refine mul_pos hk (mul_pos (div_pos ?_ ?_) (div_pos ?_ ?_)) <;> positivity | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Arcosh | {
"line": 86,
"column": 14
} | {
"line": 86,
"column": 22
} | [
{
"pp": "x : ℝ\nhx : 1 ≤ x\n⊢ sinh (log (x + √(x ^ 2 - 1))) = √(x ^ 2 - 1)",
"usedConstants": [
"Real.sinh_eq",
"Eq.mpr",
"Real",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
"Real.instSub",
"Nat.instAtLeastTwoHAddOfNat",
"HSub.hSub",
"Real.sin... | sinh_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ArithmeticGeometricMean | {
"line": 45,
"column": 48
} | {
"line": 45,
"column": 77
} | [
{
"pp": "x y : ℝ≥0\nh : x ≠ y\nhl : y < x\nkey : 2 * x * y < x ^ 2 + y ^ 2\n⊢ x * y < (x + y) ^ 2 / 2 ^ 2",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"NonAssocSemiring.toAddCommMonoidWithOne",
... | lt_div_iff₀' (by positivity), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 20
} | [
{
"pp": "q : ℕ\np : ℝ\nhp₀ : p ≠ 0\nhp₁ : p ≠ 1\n⊢ deriv (fun p ↦ p * log ↑(↑q - 1) + binEntropy p) p = log (↑q - 1) + log (1 - p) - log p",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real",
"Semiring.toModule",
"HMul.hMul",
"Real.denselyNormedField",
"congrArg",
... | rw [deriv_fun_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 352,
"column": 4
} | {
"line": 354,
"column": 97
} | [
{
"pp": "case neg\nq : ℕ\np : ℝ\nis_x_where_nondiff : ¬(p ≠ 0 ∧ p ≠ 1)\nthis : p = 0 ∨ p = 1\n⊢ 0 = -1 / (p * (1 - p))",
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"Real",
"instHDiv",
"HMul.hMul",
"sub_self",
"Real.instZero",
"congrArg",
"sub_zero",
... | · simp_all only [ne_eq, not_and, Decidable.not_not]
cases this <;>
simp_all only [mul_zero, one_ne_zero, zero_ne_one, sub_zero, mul_one, div_zero, sub_self] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 405,
"column": 6
} | {
"line": 405,
"column": 36
} | [
{
"pp": "q : ℕ\nqLe2 : 2 ≤ q\np1 : ℝ\nhp1 : p1 ∈ Icc (1 - 1 / ↑q) 1\np2 : ℝ\nhp2 : p2 ∈ Icc (1 - 1 / ↑q) 1\np1le2 : p1 < p2\np : ℝ\nthis : 2 ≤ ↑q\nqinv_lt_1 : (↑q)⁻¹ < 1\nzero_lt_1_sub_p : 0 < 1 - p\nhp : 1 - (↑q)⁻¹ < p ∧ p < 1\n⊢ log (↑q - 1) + log (1 - p) - log p < 0",
"usedConstants": [
"IsRightCan... | simp only [sub_neg, gt_iff_lt] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.BinaryEntropy | {
"line": 440,
"column": 4
} | {
"line": 440,
"column": 45
} | [
{
"pp": "case ha\nq : ℕ\np : ℝ\nhp : 0 < p ∧ p < 1\n⊢ -1 < 0",
"usedConstants": [
"Real.instIsOrderedRing",
"NegZeroClass.toNeg",
"Real.partialOrder",
"Real",
"AddGroupWithOne.toAddMonoidWithOne",
"Int.ofNat",
"Real.instRing",
"SubtractionMonoid.toSubNegZeroMo... | · norm_num [show 0 < log 2 by positivity] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Log.RpowTendsto | {
"line": 82,
"column": 50
} | {
"line": 85,
"column": 71
} | [
{
"pp": "s : Set ℝ\nhs : s ⊆ Ioi 0\nhs' : IsCompact s\nε : ℝ\nhε : 0 < ε\npbound : ℝ := ε / (sSup ((fun x ↦ ‖log x‖ ^ 2) '' s) + 1)\nhxs : ∀ x ∈ s, x ≠ 0\nsSup_nonneg : 0 ≤ sSup ((fun x ↦ ‖log x‖ ^ 2) '' s)\nsSup_nonneg' : 0 ≤ sSup ((fun x ↦ ‖log x‖) '' s)\npbound_pos : 0 < pbound\nh₁ : ∀ᶠ (p : ℝ) in 𝓝[>] 0, 0... | by
gcongr
refine le_csSup ?_ (by grind)
grind [IsCompact.bddAbove, ← IsCompact.image_of_continuousOn] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Order | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 12
} | [
{
"pp": "case neg\nA : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : ¬IsStrictlyPositive a\n⊢ Tendsto (fun i ↦ if a ∈ {b | IsStrictlyPositive b} then cfc (fun x ↦ i⁻¹ * (x ^ i - 1)) a else 0) (𝓝[>] 0)\n (𝓝 (if a ∈ {b | IsStrictlyPositive b} then log a el... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Order | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 12
} | [
{
"pp": "case neg\nA : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : ¬IsStrictlyPositive a\n⊢ Tendsto (fun i ↦ if a ∈ {b | IsStrictlyPositive b} then cfc (fun x ↦ i⁻¹ * (x ^ i - 1)) a else 0) (𝓝[>] 0)\n (𝓝 (if a ∈ {b | IsStrictlyPositive b} then log a el... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.ExpLog.Order | {
"line": 61,
"column": 4
} | {
"line": 61,
"column": 12
} | [
{
"pp": "case neg\nA : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : ¬IsStrictlyPositive a\n⊢ Tendsto (fun i ↦ if a ∈ {b | IsStrictlyPositive b} then cfc (fun x ↦ i⁻¹ * (x ^ i - 1)) a else 0) (𝓝[>] 0)\n (𝓝 (if a ∈ {b | IsStrictlyPositive b} then log a el... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.Deligne | {
"line": 89,
"column": 10
} | {
"line": 89,
"column": 44
} | [
{
"pp": "| Differentiable ℂ fun s ↦ s.Gammaℝ⁻¹",
"usedConstants": [
"Differentiable",
"NormedCommRing.toSeminormedCommRing",
"Semiring.toModule",
"Complex.Gammaℝ",
"Complex.instNormedField",
"PseudoMetricSpace.toUniformSpace",
"NormedField.toField",
"Eq.rec",
... | enter [2, s]; rw [Gammaℝ, mul_inv] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Gamma.Deligne | {
"line": 89,
"column": 10
} | {
"line": 89,
"column": 44
} | [
{
"pp": "| Differentiable ℂ fun s ↦ s.Gammaℝ⁻¹",
"usedConstants": [
"Differentiable",
"NormedCommRing.toSeminormedCommRing",
"Semiring.toModule",
"Complex.Gammaℝ",
"Complex.instNormedField",
"PseudoMetricSpace.toUniformSpace",
"NormedField.toField",
"Eq.rec",
... | enter [2, s]; rw [Gammaℝ, mul_inv] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.Analysis.SpecialFunctions.Gamma.Deligne | {
"line": 169,
"column": 4
} | {
"line": 169,
"column": 12
} | [
{
"pp": "s : ℂ\nn : ℕ\nh : s = -(2 * ↑n)\nhs : s ≠ -↑(2 * n)\n⊢ False",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"HMul.hMul",
"congrArg",
"False.elim",
"Nat.instAtLeastTwoHAddOfNat",
"Complex.instMul",
"Eq.mp",
"not_true_eq_... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.SpecialFunctions.Gamma.Deligne | {
"line": 173,
"column": 4
} | {
"line": 173,
"column": 12
} | [
{
"pp": "s : ℂ\nh1 : s.Gammaℝ ≠ 0\nn : ℕ\nh : s = -(2 * ↑n + 1)\nhs : s ≠ -↑(2 * n + 1)\n⊢ False",
"usedConstants": [
"neg_add_rev",
"NonAssocSemiring.toAddCommMonoidWithOne",
"False",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"congrArg",
"False.elim",
"AddMono... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.NumberTheory.Harmonic.EulerMascheroni | {
"line": 102,
"column": 8
} | {
"line": 102,
"column": 27
} | [
{
"pp": "⊢ eulerMascheroniSeq' 6 = 49 / 20 - log 6",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
"Real.instSub",
"Real.instRatCast",
"Nat.instAtLeastTwoHAddOfNat",
"HSub.hSub",
"Real.eulerMascheroniSeq'.eq_... | eulerMascheroniSeq' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 170,
"column": 2
} | {
"line": 172,
"column": 46
} | [
{
"pp": "n : ℕ\n⊢ HasDerivAt Gamma (↑n ! * (-↑γ + ↑(harmonic n))) (↑n + 1)",
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"NormedCommRing.toSeminormedCommRing",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"HMul.hMul",
"Ad... | exact_mod_cast HasDerivAt.complex_of_real
(by exact_mod_cast differentiableAt_Gamma_nat_add_one n)
(Real.hasDerivAt_Gamma_nat n) Gamma_ofReal | Lean.Parser.Tactic._aux_Init_TacticsExtra___macroRules_Lean_Parser_Tactic_tacticExact_mod_cast__1 | Lean.Parser.Tactic.tacticExact_mod_cast_ |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 170,
"column": 2
} | {
"line": 172,
"column": 46
} | [
{
"pp": "n : ℕ\n⊢ HasDerivAt Gamma (↑n ! * (-↑γ + ↑(harmonic n))) (↑n + 1)",
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"NormedCommRing.toSeminormedCommRing",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"HMul.hMul",
"Ad... | exact_mod_cast HasDerivAt.complex_of_real
(by exact_mod_cast differentiableAt_Gamma_nat_add_one n)
(Real.hasDerivAt_Gamma_nat n) Gamma_ofReal | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.Harmonic.GammaDeriv | {
"line": 170,
"column": 2
} | {
"line": 172,
"column": 46
} | [
{
"pp": "n : ℕ\n⊢ HasDerivAt Gamma (↑n ! * (-↑γ + ↑(harmonic n))) (↑n + 1)",
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"NormedCommRing.toSeminormedCommRing",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"HMul.hMul",
"Ad... | exact_mod_cast HasDerivAt.complex_of_real
(by exact_mod_cast differentiableAt_Gamma_nat_add_one n)
(Real.hasDerivAt_Gamma_nat n) Gamma_ofReal | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.RingInverseOrder | {
"line": 36,
"column": 2
} | {
"line": 36,
"column": 47
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\n⊢ ConvexOn ℝ {a | IsStrictlyPositive a} inverse",
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"instHSMul",
"NormedRing.toRing",
... | refine ⟨by grind [convex_iff_forall_pos], ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.RingInverseOrder | {
"line": 76,
"column": 10
} | {
"line": 81,
"column": 59
} | [
{
"pp": "case h.refine_3\nA : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx : A\nxpos : IsStrictlyPositive x\ny : A\nypos : IsStrictlyPositive y\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nz : A := (conjSqrt x⁻¹ʳ) y\nzpos : IsStrictlyPositive z\nxinvpos : IsStri... | intro r hr
suffices (a • 1 + b • r) ^ (-1 : ℤ) ≤ a • 1 ^ (-1 : ℤ) + b • r ^ (-1 : ℤ) by
simp_rw [← Real.rpow_intCast] at this
simpa using this
have : ConvexOn ℝ (Set.Ioi 0) (fun z : ℝ => z ^ (-1 : ℤ)) := convexOn_zpow (-1)
grind [ConvexOn, IsStrictlyPositive.spectru... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.RingInverseOrder | {
"line": 76,
"column": 10
} | {
"line": 81,
"column": 59
} | [
{
"pp": "case h.refine_3\nA : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\nx : A\nxpos : IsStrictlyPositive x\ny : A\nypos : IsStrictlyPositive y\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nz : A := (conjSqrt x⁻¹ʳ) y\nzpos : IsStrictlyPositive z\nxinvpos : IsStri... | intro r hr
suffices (a • 1 + b • r) ^ (-1 : ℤ) ≤ a • 1 ^ (-1 : ℤ) + b • r ^ (-1 : ℤ) by
simp_rw [← Real.rpow_intCast] at this
simpa using this
have : ConvexOn ℝ (Set.Ioi 0) (fun z : ℝ => z ^ (-1 : ℤ)) := convexOn_zpow (-1)
grind [ConvexOn, IsStrictlyPositive.spectru... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.IntegralRepresentation | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 51
} | [
{
"pp": "p : ℝ\nhp : p ∈ Ioo 0 1\ns : Set ℝ\nhs : s ⊆ Ici 0\ng : ℝ × ℝ → ℝ := fun q ↦ q.1 ^ (p - 1) * q.2 / (q.1 + q.2)\n⊢ ContinuousOn (Function.uncurry p.rpowIntegrand₀₁) (Ioi 0 ×ˢ s)",
"usedConstants": [
"Set.instSProd",
"Real",
"Set.Ioi",
"Real.instZero",
"SProd.sprod",
... | refine ContinuousOn.congr (f := g) ?_ fun q => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.Int.Log | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 44
} | [
{
"pp": "case inr\nR : Type u_1\ninst✝³ : Semifield R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : FloorSemiring R\nb : ℕ\nr : R\nhb : 1 < b\nhr : 0 < r\nhr1 : r ≤ 1\n⊢ ↑b ^ log b r ≤ r",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"GroupWith... | rw [log_of_right_le_one _ hr1, zpow_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Int.Log | {
"line": 219,
"column": 74
} | {
"line": 219,
"column": 96
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semifield R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : FloorSemiring R\nb : ℕ\nr : R\n⊢ -clog b r⁻¹ = log b r",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"Divisio... | rw [clog_inv, neg_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Int.Log | {
"line": 219,
"column": 74
} | {
"line": 219,
"column": 96
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semifield R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : FloorSemiring R\nb : ℕ\nr : R\n⊢ -clog b r⁻¹ = log b r",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"Divisio... | rw [clog_inv, neg_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Int.Log | {
"line": 219,
"column": 74
} | {
"line": 219,
"column": 96
} | [
{
"pp": "R : Type u_1\ninst✝³ : Semifield R\ninst✝² : LinearOrder R\ninst✝¹ : IsStrictOrderedRing R\ninst✝ : FloorSemiring R\nb : ℕ\nr : R\n⊢ -clog b r⁻¹ = log b r",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"CommRing.toNonUnitalCommRing",
"Divisio... | rw [clog_inv, neg_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 57,
"column": 83
} | {
"line": 57,
"column": 96
} | [
{
"pp": "case h\nx✝ : ℝ\n⊢ 0 = 0 x✝",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Pi.zero_apply",
"id",
"Pi.instZero",
"Zero.toOfNat0",
"OfNat.ofNat",
"Eq"
]
}
] | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 61,
"column": 81
} | {
"line": 61,
"column": 94
} | [
{
"pp": "case h\nx✝ : ℝ\n⊢ 0 = 0 x✝",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Pi.zero_apply",
"id",
"Pi.instZero",
"Zero.toOfNat0",
"OfNat.ofNat",
"Eq"
]
}
] | Pi.zero_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 255,
"column": 6
} | {
"line": 255,
"column": 21
} | [
{
"pp": "b : ℝ\nhb : 1 < b\nx : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x < y\n⊢ logb b y < logb b x",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"Real.lattice",
"abs",
"congrArg",
"id",
"Real.instAddGroup",
"Real.logb",
"LT.lt",
"... | ← logb_abs b y, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Base | {
"line": 257,
"column": 37
} | {
"line": 257,
"column": 51
} | [
{
"pp": "b : ℝ\nhb : 1 < b\nx : ℝ\nhx : x < 0\ny : ℝ\nhy : y < 0\nhxy : x < y\n⊢ -y < -x",
"usedConstants": [
"IsRightCancelAdd.addRightStrictMono_of_addRightMono",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"neg_lt_neg_iff",
"Real.partialOrder",
... | neg_lt_neg_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
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