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.IntervalIntegral.Basic | {
"line": 1394,
"column": 2
} | {
"line": 1394,
"column": 39
} | [
{
"pp": "f g : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\nhab : a ≤ b\nhf : IntervalIntegrable f μ a b\nhg : IntervalIntegrable g μ a b\nh : ∀ x ∈ Icc a b, f x ≤ g x\nH : ∀ x ∈ Ioc a b, f x ≤ g x := fun x hx ↦ h x (Ioc_subset_Icc_self hx)\n⊢ ∫ (u : ℝ) in a..b, f u ∂μ ≤ ∫ (u : ℝ) in a..b, g u ∂μ",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 605,
"column": 87
} | {
"line": 613,
"column": 100
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : TopologicalSpace X\nμ : Measure ℝ\ninst✝¹ : NoAtoms μ\ninst✝ : IsLocallyFiniteMeasure μ\nf : X → ℝ → E\na₀ : ℝ\nhf : Continuous[instTopologicalSpaceProd, PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (F... | by
gcongr
· exact Eventually.of_forall (fun x ↦ norm_nonneg _)
· exact (hf.uncurry_left _).norm.integrableOn_Icc
· apply uIoc_subset_uIcc.trans (uIcc_subset_Icc ?_ ⟨hs.1.le, hs.2.le⟩ )
simp [δpos.le]
· exact Eventually.of_forall (fun x ↦ norm_nonneg _)
· exact ((hf.uncurry_le... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1426,
"column": 2
} | {
"line": 1426,
"column": 72
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nhfi : Integrable f volume\n⊢ HasSum (fun n ↦ ∫ (x : ℝ) in 0..1, f (x + ↑n)) (∫ (x : ℝ), f x)",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real",
"MeasureTheory.Measure",
"Real.instZero",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 312,
"column": 4
} | {
"line": 312,
"column": 15
} | [
{
"pp": "case h.e'_4.h\nE✝ : Type u_1\ninst✝¹ : NormedAddCommGroup E✝\nf✝ : ℝ → E✝\nT✝ : ℝ\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nT t : ℝ\nh₁f : Periodic f T\nhT✝ : T ≠ 0\nh₂f : IntervalIntegrable f volume t (t + T)\na₁ a₂ : ℝ\nhT : 0 < T\nn₁ : ℕ\nhn₁ : (t - min a₁ a₂) / T ≤ ↑n₁\nn₂ : ℕ\nhn₂ : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 1172,
"column": 2
} | {
"line": 1172,
"column": 34
} | [
{
"pp": "E : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\na b : ℝ\ninst✝ : CompleteSpace E\nf f' : ℝ → E\nhab : a < b\nfa fb : E\nhderiv : ∀ x ∈ Ioo a b, HasDerivAt f (f' x) x\nhint : IntervalIntegrable f' volume a b\nha : Tendsto f (𝓝[>] a) (𝓝 fa)\nhb : Tendsto f (𝓝[<] b) (𝓝 fb)\nF : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 1194,
"column": 2
} | {
"line": 1194,
"column": 36
} | [
{
"pp": "E : Type u_3\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\na b : ℝ\ninst✝ : CompleteSpace E\nf : ℝ → E\nhcont : ContinuousOn f [[a, b]]\nhderiv : ∀ x ∈ uIoo a b, DifferentiableAt ℝ f x\nhint : IntervalIntegrable (deriv f) volume a b\n⊢ ∫ (y : ℝ) in a..b, deriv f y = f b - f a",
"usedCon... | rcases le_total a b with hab | hab | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 100,
"column": 2
} | {
"line": 101,
"column": 9
} | [
{
"pp": "c : ℂ\nR θ : ℝ\n⊢ HasDerivAt (circleMap c R) (circleMap 0 R θ * I) θ",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"Semigroup.toMul",
"Real",
"NormedSpace.toIsBou... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 352,
"column": 6
} | {
"line": 352,
"column": 63
} | [
{
"pp": "case inr.inr\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\nf : ℝ → E\nT : ℝ\ninst✝ : NormedSpace ℝ E\nhf : Periodic f T\nt s : ℝ\nthis :\n ∀ {E : Type u_1} [inst : NormedAddCommGroup E] {f : ℝ → E} {T : ℝ} [inst_1 : NormedSpace ℝ E],\n Periodic f T → ∀ (t s : ℝ), 0 < T → ∫ (x : ℝ) in t..t + T, f x ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 1244,
"column": 4
} | {
"line": 1244,
"column": 82
} | [
{
"pp": "g' g : ℝ → ℝ\na b : ℝ\nhcont : ContinuousOn g (Icc a b)\nhderiv : ∀ x ∈ Ioo a b, HasDerivWithinAt g (g' x) (Ioi x) x\ng'pos : ∀ x ∈ Ioo a b, 0 ≤ g' x\nhab : a < b\nmeas_g' : AEMeasurable g' (volume.restrict (Ioo a b))\nH : ENNReal.ofReal (g b - g a) < ∫⁻ (x : ℝ) in Ioo a b, ↑‖g' x‖₊\nf : SimpleFunc ℝ ℝ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 298,
"column": 2
} | {
"line": 298,
"column": 13
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nc : ℂ\nR : ℝ\nf : ℂ → E\n⊢ IntervalIntegrable (f ∘ circleMap c R) volume (0 + π) (π + 2 * π) ↔\n IntervalIntegrable (fun θ ↦ f (circleMap c R θ)) volume 0 (2 * π)",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.pi",
"HMul.hMul",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 1268,
"column": 2
} | {
"line": 1268,
"column": 36
} | [
{
"pp": "g' g : ℝ → ℝ\na b : ℝ\nhcont : ContinuousOn g [[a, b]]\nhderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt g (g' x) x\nhpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x\n⊢ IntervalIntegrable g' volume a b",
"usedConstants": [
"Real",
"le_total",
"Real.linearOrder"
]
}
] | rcases le_total a b with hab | hab | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 1272,
"column": 2
} | {
"line": 1274,
"column": 56
} | [
{
"pp": "case inr\ng' g : ℝ → ℝ\na b : ℝ\nhcont : ContinuousOn g [[a, b]]\nhderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt g (g' x) x\nhpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x\nhab : b ≤ a\n⊢ IntervalIntegrable g' volume a b",
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"Eq.... | · simp only [uIcc_of_ge, min_eq_right, max_eq_left, hab, IntervalIntegrable, Ioc_eq_empty_of_le,
integrableOn_empty, true_and] at hcont hderiv hpos ⊢
exact integrableOn_deriv_of_nonneg hcont hderiv hpos | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 1267,
"column": 91
} | {
"line": 1274,
"column": 56
} | [
{
"pp": "g' g : ℝ → ℝ\na b : ℝ\nhcont : ContinuousOn g [[a, b]]\nhderiv : ∀ x ∈ Ioo (min a b) (max a b), HasDerivAt g (g' x) x\nhpos : ∀ x ∈ Ioo (min a b) (max a b), 0 ≤ g' x\n⊢ IntervalIntegrable g' volume a b",
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"Eq.mpr",
"Set.Ioc",
... | by
rcases le_total a b with hab | hab
· simp only [uIcc_of_le, min_eq_left, max_eq_right, IntervalIntegrable, hab,
Ioc_eq_empty_of_le, integrableOn_empty, and_true] at hcont hderiv hpos ⊢
exact integrableOn_deriv_of_nonneg hcont hderiv hpos
· simp only [uIcc_of_ge, min_eq_right, max_eq_left, hab, Interv... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 328,
"column": 4
} | {
"line": 329,
"column": 11
} | [
{
"pp": "case neg.refine_1\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nh₀ : ¬R = 0\nh : IntegrableOn (fun θ ↦ (circleMap 0 R θ * I) • f (circleMap c R θ)) (Ι 0 (2 * π)) volume\nH : ∀ {θ : ℝ}, circleMap 0 R θ * I ≠ 0\n⊢ AEStronglyMeasurable (fun θ ↦ f (circleMa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 353,
"column": 4
} | {
"line": 358,
"column": 37
} | [
{
"pp": "case mp\nc : ℂ\nR : ℝ\nn : ℤ\nhR : R ≠ 0\nhn : n < 0\nθ : ℝ\nhθ : θ ∈ [[0, 2 * π]]\nf : ℝ → ℂ := fun θ' ↦ circleMap c R θ' - circleMap c R θ\n⊢ (fun x ↦ (x - θ)⁻¹) =O[𝓝[≠] θ] fun θ_1 ↦ (circleMap 0 R θ_1 * I) • (circleMap c R θ_1 - circleMap c R θ) ^ n",
"usedConstants": [
"instInnerProductS... | have : ∀ᶠ θ' in 𝓝[≠] θ, f θ' ∈ ball (0 : ℂ) 1 \ {0} := by
suffices ∀ᶠ z in 𝓝[≠] circleMap c R θ, z - circleMap c R θ ∈ ball (0 : ℂ) 1 \ {0} from
((differentiable_circleMap c R θ).hasDerivAt.tendsto_nhdsNE
(deriv_circleMap_ne_zero hR)).eventually this
filter_upwards [self_mem_nhdsWithin, ... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 366,
"column": 6
} | {
"line": 367,
"column": 35
} | [
{
"pp": "c : ℂ\nR : ℝ\nn : ℤ\nhR : R ≠ 0\nhn : n < 0\nθ : ℝ\nhθ : θ ∈ [[0, 2 * π]]\nf : ℝ → ℂ := fun θ' ↦ circleMap c R θ' - circleMap c R θ\nthis✝ : ∀ᶠ (θ' : ℝ) in 𝓝[≠] θ, f θ' ∈ ball 0 1 \\ {0}\nθ' : ℝ\nhθ' : f θ' ∈ ball 0 1 \\ {0}\nx : ℝ := ‖f θ'‖\nthis : x⁻¹ ≤ x ^ n\n⊢ ‖circleMap c R θ' - circleMap c R θ‖⁻... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 368,
"column": 35
} | {
"line": 368,
"column": 60
} | [
{
"pp": "c : ℂ\nR : ℝ\nn : ℤ\nhR : R ≠ 0\nhn : n < 0\nθ : ℝ\nhθ : θ ∈ [[0, 2 * π]]\nf : ℝ → ℂ := fun θ' ↦ circleMap c R θ' - circleMap c R θ\nthis : ∀ᶠ (θ' : ℝ) in 𝓝[≠] θ, f θ' ∈ ball 0 1 \\ {0}\nθ' : ℝ\nhθ' : f θ' ∈ ball 0 1 \\ {0}\nx : ℝ := ‖f θ'‖\n⊢ x ∈ Ioo 0 1",
"usedConstants": [
"AddGroup.toSub... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Prod | {
"line": 31,
"column": 4
} | {
"line": 31,
"column": 15
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nμ : Measure α\np : ℝ≥0∞\nf : α → ε\nhf : MemLp f p μ\nν : Measure β\ninst✝ : IsFiniteMeasure ν\nhf' : MemLp f p (ν Set.univ • μ)\n⊢ MemLp f p (Measure.map P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Prod | {
"line": 32,
"column": 4
} | {
"line": 32,
"column": 15
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\nμ : Measure α\np : ℝ≥0∞\nf : α → ε\nhf : MemLp f p μ\nν : Measure β\ninst✝ : IsFiniteMeasure ν\nhf' : MemLp f p (ν Set.univ • μ)\n⊢ AEStronglyMeasurable f (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Prod | {
"line": 40,
"column": 4
} | {
"line": 40,
"column": 15
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝³ : TopologicalSpace ε\ninst✝² : ContinuousENorm ε\nν : Measure β\np : ℝ≥0∞\nf : β → ε\nhf : MemLp f p ν\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : SFinite ν\nhf' : MemLp f p (μ Set.univ • ν)\n⊢ MemL... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Prod | {
"line": 41,
"column": 4
} | {
"line": 41,
"column": 15
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nε : Type u_3\nmα : MeasurableSpace α\nmβ : MeasurableSpace β\ninst✝³ : TopologicalSpace ε\ninst✝² : ContinuousENorm ε\nν : Measure β\np : ℝ≥0∞\nf : β → ε\nhf : MemLp f p ν\nμ : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : SFinite ν\nhf' : MemLp f p (μ Set.univ • ν)\n⊢ AESt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 562,
"column": 4
} | {
"line": 562,
"column": 58
} | [
{
"pp": "case inr.hf\nn : ℤ\nc w : ℂ\nR : ℝ\nhn : n < 0\nhw : w ∈ sphere c |R|\nh0 : R ≠ 0\n⊢ ¬CircleIntegrable (fun z ↦ (z - w) ^ n) c R",
"usedConstants": [
"circleIntegrable_sub_zpow_iff._simp_1",
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"False",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 658,
"column": 4
} | {
"line": 658,
"column": 74
} | [
{
"pp": "case refine_4\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nw : ℂ\nhf : CircleIntegrable f c R\nhw : ‖w‖ < R\nhR : 0 < R\nhwR : ‖w‖ / R ∈ Ico 0 1\n⊢ IntervalIntegrable (fun t ↦ ∑' (n : ℕ), (fun n θ ↦ ‖f (circleMap c R θ)‖ * (‖w‖ / R) ^ n) n t) volume 0 ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 663,
"column": 49
} | {
"line": 663,
"column": 76
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nw : ℂ\nhf : CircleIntegrable f c R\nhw : ‖w‖ < R\nhR : 0 < R\nhwR : ‖w‖ / R ∈ Ico 0 1\nθ : ℝ\nx✝ : θ ∈ Ι 0 (2 * π)\n⊢ ‖w / (circleMap c R θ - c)‖ < 1",
"usedConstants": [
"Norm.norm",
"Eq.mpr"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 694,
"column": 66
} | {
"line": 694,
"column": 77
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ≥0\nhf : CircleIntegrable f c ↑R\nhR : 0 < R\ny✝ : ℂ\nhy : y✝ ∈ eball 0 ↑R\n⊢ ‖y✝‖ < ↑R",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 706,
"column": 4
} | {
"line": 706,
"column": 59
} | [
{
"pp": "c w : ℂ\nR : ℝ\nhw : w ∈ ball c R\nhR : 0 < R\nH : HasSum (fun n ↦ ∮ (z : ℂ) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)⁻¹) (2 * ↑π * I)\nA : CircleIntegrable (fun x ↦ 1) c R\n⊢ HasSum (fun n ↦ ∮ (z : ℂ) in C(c, R), ((w - c) / (z - c)) ^ n * (z - c)⁻¹) (∮ (z : ℂ) in C(c, R), (z - w)⁻¹)",
"usedCon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 671,
"column": 4
} | {
"line": 671,
"column": 60
} | [
{
"pp": "case bound_integrable\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\na₀ : ℝ\nhf : IntegrableOn f (Ioi a₀) μ\n⊢ Integrable ((Ioi a₀).indicator (norm ∘ f)) μ",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 693,
"column": 4
} | {
"line": 693,
"column": 20
} | [
{
"pp": "case right\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\ninst✝ : NoAtoms μ\na₀ : ℝ\nhf : IntegrableOn f (Ioi a₀) μ\na : ℝ\nha : a₀ ≤ a\n⊢ ContinuousWithinAt (fun b ↦ ∫ (x : ℝ) in Ioi b, f x ∂μ) (Ici a₀ ∩ Ici a) a",
"usedConstants": [
"Eq.mpr... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 706,
"column": 4
} | {
"line": 706,
"column": 60
} | [
{
"pp": "case bound_integrable\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\na₀ : ℝ\nhf : IntegrableOn f (Iio a₀) μ\n⊢ Integrable ((Iio a₀).indicator (norm ∘ f)) μ",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 718,
"column": 4
} | {
"line": 718,
"column": 20
} | [
{
"pp": "case left\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nμ : Measure ℝ\nf : ℝ → E\ninst✝ : NoAtoms μ\na₀ : ℝ\nhf : IntegrableOn f (Iio a₀) μ\na : ℝ\nha : a ≤ a₀\n⊢ ContinuousWithinAt (fun b ↦ ∫ (x : ℝ) in Iio b, f x ∂μ) (Iic a₀ ∩ Iic a) a",
"usedConstants": [
"Eq.mpr"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.BoxIntegral.DivergenceTheorem | {
"line": 190,
"column": 8
} | {
"line": 193,
"column": 35
} | [
{
"pp": "case h\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nn : ℕ\ninst✝ : CompleteSpace E\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) → ℝ) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHs : ∀ x ∈ s, ContinuousWithinAt f (Box.Icc I) ... | calc
dist (f y₁) (f y₂) ≤ dist (f y₁) (f x) + dist (f y₂) (f x) := dist_triangle_right _ _ _
_ ≤ ε / 2 / 2 + ε / 2 / 2 := add_le_add (hδ₁ _ <| this hy₁) (hδ₁ _ <| this hy₂)
_ = ε / 2 := add_halves _ | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.BoxIntegral.DivergenceTheorem | {
"line": 197,
"column": 8
} | {
"line": 197,
"column": 19
} | [
{
"pp": "case refine_2\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\nn : ℕ\ninst✝ : CompleteSpace E\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) → ℝ) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHs : ∀ x ∈ s, ContinuousWithinAt f (Box.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 209,
"column": 2
} | {
"line": 211,
"column": 10
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\nX : Type u_4\ninst✝¹ : TopologicalSpace X\nf : β → X\ninst✝ : SFinite ν\nhf : AEStronglyMeasurable (fun x ↦ f x.2) (μ.prod ν)\nhμ : μ ≠ 0\n⊢ AEStronglyMeasurable f ν",
"usedConstants": ... | have := NeZero.mk hμ
obtain ⟨y, hy⟩ := hf.prodMk_left.exists
exact hy | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 209,
"column": 2
} | {
"line": 211,
"column": 10
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\nX : Type u_4\ninst✝¹ : TopologicalSpace X\nf : β → X\ninst✝ : SFinite ν\nhf : AEStronglyMeasurable (fun x ↦ f x.2) (μ.prod ν)\nhμ : μ ≠ 0\n⊢ AEStronglyMeasurable f ν",
"usedConstants": ... | have := NeZero.mk hμ
obtain ⟨y, hy⟩ := hf.prodMk_left.exists
exact hy | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 223,
"column": 2
} | {
"line": 223,
"column": 24
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nf' : ℂ → ℂ →L[ℝ] E\nz w : ℂ\ns : Set ℂ\nhs : s.Countable\nHc : ContinuousOn f ([[z.re, w.re]] ×ℂ [[w.im, z.im]])\nHd : ∀ x ∈ Ioo (min z.re w.re) (max z.re w.re) ×ℂ Ioo (min w.im z.im) (max w.im z.im) \\ s, HasFDerivAt f (f' ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 260,
"column": 6
} | {
"line": 260,
"column": 91
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nz w : ℂ\nHd : DifferentiableOn ℝ f ([[z.re, w.re]] ×ℂ [[z.im, w.im]])\nHi : IntegrableOn (fun z ↦ I • (fderiv ℝ f z) 1 - (fderiv ℝ f z) I) ([[z.re, w.re]] ×ℂ [[z.im, w.im]]) volume\nx : ℂ\nhx : x ∈ Ioo (min z.re w.re) (max z... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 339,
"column": 21
} | {
"line": 339,
"column": 63
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nc : ℂ\nf : ℂ → E\ns : Set ℂ\na : ℝ\nh0 : 0 < rexp a\nb : ℝ\nhle : a ≤ b\nhd : ∀ z ∈ (ball c (rexp b) \\ closedBall c (rexp a)) \\ s, DifferentiableAt ℂ f z\nA : Set ℂ := closedBall c (rexp b) \\ ball c (rexp a)\nhc : ContinuousOn f A\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 344,
"column": 4
} | {
"line": 344,
"column": 53
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nc : ℂ\nf : ℂ → E\ns : Set ℂ\na : ℝ\nh0 : 0 < rexp a\nb : ℝ\nhle : a ≤ b\nhd : ∀ z ∈ (ball c (rexp b) \\ closedBall c (rexp a)) \\ s, DifferentiableAt ℂ f z\nA : Set ℂ := closedBall c (rexp b) \\ ball c (rexp a)\nR : Set ℂ := [[a, b]] ×... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 68
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nc : ℂ\nf : ℂ → E\ns : Set ℂ\na : ℝ\nh0 : 0 < rexp a\nb : ℝ\nhle : a ≤ b\nA : Set ℂ := closedBall c (rexp b) \\ ball c (rexp a)\nR : Set ℂ := [[a, b]] ×ℂ [[0, 2 * π]]\ng : ℂ → ℂ := fun x ↦ c + cexp x\nhdg : Differentiable ℂ g\nhs : (g ⁻... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 344,
"column": 39
} | {
"line": 344,
"column": 50
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nE : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝³ : NormedAddCommGroup E\nR : Type u_4\ninst✝² : NormedRing R\ninst✝¹ : Module R E\ninst✝ : IsBoundedSMul R E\nf : α → R\ng : β → E\nhf : Integrable f μ\nhg : Integrable ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Liouville | {
"line": 49,
"column": 4
} | {
"line": 49,
"column": 91
} | [
{
"pp": "F : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℂ F\ninst✝ : CompleteSpace F\nc : ℂ\nR C : ℝ\nf : ℂ → F\nn : ℕ\nhR : 0 < R\nhf : DiffContOnCl ℂ f (ball c R)\nhC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C\nz : ℂ\nhz : z ∈ sphere c R\n⊢ ‖(z - c)⁻¹ ^ (n + 1) • f z‖ ≤ C / (R ^ n * R)",
"usedConstant... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Liouville | {
"line": 69,
"column": 2
} | {
"line": 69,
"column": 13
} | [
{
"pp": "F : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℂ F\ninst✝ : CompleteSpace F\nc : ℂ\nR C : ℝ\nf : ℂ → F\nhR : 0 < R\nhf : DiffContOnCl ℂ f (ball c R)\nhC : ∀ z ∈ sphere c R, ‖f z‖ ≤ C\n⊢ ‖deriv f c‖ ≤ C / R",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 559,
"column": 4
} | {
"line": 559,
"column": 48
} | [
{
"pp": "case refine_1\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\nhf : DiffContOnCl ℂ f (ball c R)\nhw : w ∈ ball c R\nhR : 0 < R\n⊢ ContinuousOn f (closedBall c R)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 560,
"column": 4
} | {
"line": 560,
"column": 33
} | [
{
"pp": "case refine_2\nE : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nc w : ℂ\nf : ℂ → E\nhf : DiffContOnCl ℂ f (ball c R)\nhw : w ∈ ball c R\nhR : 0 < R\n⊢ ∀ x ∈ ball c R \\ ∅, DifferentiableAt ℂ f x",
"usedConstants": [
"Eq.mpr",
"NormedCo... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 577,
"column": 2
} | {
"line": 577,
"column": 48
} | [
{
"pp": "R : ℝ\nc w : ℂ\ns : Set ℂ\nhs : s.Countable\nhw : w ∈ ball c R\nf : ℂ → ℂ\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ z ∈ ball c R \\ s, DifferentiableAt ℂ f z\n⊢ ∮ (z : ℂ) in C(c, R), f z / (z - w) = 2 * ↑π * I * f w",
"usedConstants": [
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Liouville | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 49
} | [
{
"pp": "E : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℂ F\ninst✝ : Nontrivial E\nf : E → F\nhf : Differentiable ℂ f\nc : F\nhb : Tendsto f (cocompact E) (𝓝 c)\ns : Set E\nhs : s ∈ cocompact E\nhs_bdd : Bornology.IsBounded (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 598,
"column": 6
} | {
"line": 599,
"column": 46
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ≥0\nc : ℂ\nf : ℂ → E\ns : Set ℂ\nhs : s.Countable\nhc : ContinuousOn f (closedBall c ↑R)\nhd : ∀ z ∈ ball c ↑R \\ s, DifferentiableAt ℂ f z\nhR : 0 < R\nw : ℂ\nhw : w ∈ eball 0 ↑R\n⊢ c + w ∈ ball c ↑R",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Liouville | {
"line": 146,
"column": 35
} | {
"line": 146,
"column": 52
} | [
{
"pp": "E : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℂ F\ninst✝ : Nontrivial E\nf : E → F\nhf : Differentiable ℂ f\nc : F\nhb : Tendsto f (cocompact E) (𝓝 c)\nh_bdd : Bornology.IsBounded (range f)\nc' : F\nhc' : f = const ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 735,
"column": 2
} | {
"line": 735,
"column": 13
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nf : ℂ → E\nc : ℂ\ns : Set ℂ\nh0 : 0 < R\nhs : s.Countable\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ z ∈ ball c R \\ s, DifferentiableAt ℂ f z\n⊢ ∮ (z : ℂ) in C(c, R), (1 / (z - c) ^ 2) • f z = (2 * ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 752,
"column": 2
} | {
"line": 752,
"column": 13
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nf : ℂ → E\nc : ℂ\nh0 : 0 < R\nhc : DiffContOnCl ℂ f (ball c R)\n⊢ ∮ (z : ℂ) in C(c, R), (1 / (z - c) ^ 2) • f z = (2 * ↑π * I) • deriv f c",
"usedConstants": [
"Eq.mpr",
"Real",
"D... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 767,
"column": 2
} | {
"line": 767,
"column": 13
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℂ E\ninst✝ : CompleteSpace E\nR : ℝ\nf : ℂ → E\nc : ℂ\nh0 : 0 < R\nhc : DifferentiableOn ℂ f (closedBall c R)\n⊢ ∮ (z : ℂ) in C(c, R), (1 / (z - c) ^ 2) • f z = (2 * ↑π * I) • deriv f c",
"usedConstants": [
"Eq.mpr",
"Real"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 130,
"column": 41
} | {
"line": 141,
"column": 92
} | [
{
"pp": "F : Type u_1\ninst✝² : Field F\np : F[X]\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nh : Fact (map (algebraMap F E) p).Splits\n⊢ Function.Bijective (mapRoots p E)",
"usedConstants": [
"Multiset.toFinset",
"Iff.mpr",
"_private.Mathlib.FieldTheory.PolynomialGaloisGroup.0.P... | by
constructor
· exact fun _ _ h => Subtype.ext (RingHom.injective _ (Subtype.ext_iff.mp h))
· intro y
-- this is just an equality of two different ways to write the roots of `p` as an `E`-polynomial
have key := (IsSplittingField.splits p.SplittingField p).roots_map
(IsScalarTower.toAlgHom F p.Split... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 593,
"column": 2
} | {
"line": 593,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nE : Type u_3\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝³ : NormedAddCommGroup E\ninst✝² : SFinite ν\ninst✝¹ : NormedSpace ℝ E\ninst✝ : SFinite μ\nf : β → E\n⊢ ∫ (z : α × β), f z.2 ∂μ.prod ν = μ.real univ • ∫ (y : β), f y ∂ν",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 158,
"column": 15
} | {
"line": 158,
"column": 26
} | [
{
"pp": "σ : Type u_1\nτ : Type u_2\nR : Type u_3\ninst✝ : CommSemiring R\nφ : MvPolynomial σ R\ne : σ ≃ τ\nh : ((rename ⇑e) φ).IsSymmetric\n⊢ φ.IsSymmetric",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 235,
"column": 2
} | {
"line": 235,
"column": 48
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\np q : F[X]\nhpq : p ∣ q\nhq : q ≠ 0\nthis : Fact (map (algebraMap F q.SplittingField) p).Splits\n⊢ Function.Surjective ⇑(restrictDvd hpq)",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Dvd.dvd",
"MonoidHom.instFunLike",
"CommRing.toNonUnitalC... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 241,
"column": 6
} | {
"line": 241,
"column": 50
} | [
{
"pp": "τ : Type u_2\nσ : Type u_5\nR : Type u_6\ninst✝² : CommSemiring R\ninst✝¹ : Fintype σ\ninst✝ : Fintype τ\nn : ℕ\ne : σ ≃ τ\n⊢ (rename ⇑e) (esymm σ R n) = ∑ x ∈ powersetCard n univ, ∏ i ∈ x, X (e i)",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroClass",
... | simp_rw [esymm, map_sum, map_prod, rename_X] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 241,
"column": 6
} | {
"line": 241,
"column": 50
} | [
{
"pp": "τ : Type u_2\nσ : Type u_5\nR : Type u_6\ninst✝² : CommSemiring R\ninst✝¹ : Fintype σ\ninst✝ : Fintype τ\nn : ℕ\ne : σ ≃ τ\n⊢ (rename ⇑e) (esymm σ R n) = ∑ x ∈ powersetCard n univ, ∏ i ∈ x, X (e i)",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroClass",
... | simp_rw [esymm, map_sum, map_prod, rename_X] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 241,
"column": 6
} | {
"line": 241,
"column": 50
} | [
{
"pp": "τ : Type u_2\nσ : Type u_5\nR : Type u_6\ninst✝² : CommSemiring R\ninst✝¹ : Fintype σ\ninst✝ : Fintype τ\nn : ℕ\ne : σ ≃ τ\n⊢ (rename ⇑e) (esymm σ R n) = ∑ x ∈ powersetCard n univ, ∏ i ∈ x, X (e i)",
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMulZeroClass",
... | simp_rw [esymm, map_sum, map_prod, rename_X] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 276,
"column": 6
} | {
"line": 276,
"column": 20
} | [
{
"pp": "σ : Type u_5\nR : Type u_6\ninst✝³ : CommSemiring R\ninst✝² : Fintype σ\ninst✝¹ : DecidableEq σ\ninst✝ : Nontrivial R\nn : ℕ\n⊢ (esymm σ R n).support = image (fun t ↦ ∑ i ∈ t, Finsupp.single i 1) (powersetCard n univ)",
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"Finset.... | support_esymm' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 292,
"column": 4
} | {
"line": 292,
"column": 15
} | [
{
"pp": "σ : Type u_5\nR : Type u_6\ninst✝² : CommSemiring R\ninst✝¹ : Fintype σ\ninst✝ : Nontrivial R\nthis✝ : (⇑Finsupp.toMultiset ∘ fun t ↦ ∑ i ∈ t, Finsupp.single i 1) = val\nk : ℕ\nhpos : 0 < k.succ\nhn : k.succ ≤ Fintype.card σ\nthis : ((powersetCard k.succ univ).sup fun x ↦ x).val = (powersetCard k.succ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 350,
"column": 54
} | {
"line": 350,
"column": 65
} | [
{
"pp": "σ : Type u_5\nR : Type u_6\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\n⊢ psum σ R 0 = ↑(Fintype.card σ)",
"usedConstants": [
"Finsupp.instAddZeroClass",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
... | simp [psum] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 350,
"column": 54
} | {
"line": 350,
"column": 65
} | [
{
"pp": "σ : Type u_5\nR : Type u_6\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\n⊢ psum σ R 0 = ↑(Fintype.card σ)",
"usedConstants": [
"Finsupp.instAddZeroClass",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
... | simp [psum] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 350,
"column": 54
} | {
"line": 350,
"column": 65
} | [
{
"pp": "σ : Type u_5\nR : Type u_6\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\n⊢ psum σ R 0 = ↑(Fintype.card σ)",
"usedConstants": [
"Finsupp.instAddZeroClass",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
... | simp [psum] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 353,
"column": 47
} | {
"line": 353,
"column": 58
} | [
{
"pp": "σ : Type u_5\nR : Type u_6\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\n⊢ psum σ R 1 = ∑ i, X i",
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Finset.univ",
"AddMonoidAlgebra.addAddCommMonoid",
"congrArg",
"CommSemiring.toSemiring",
... | simp [psum] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 353,
"column": 47
} | {
"line": 353,
"column": 58
} | [
{
"pp": "σ : Type u_5\nR : Type u_6\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\n⊢ psum σ R 1 = ∑ i, X i",
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Finset.univ",
"AddMonoidAlgebra.addAddCommMonoid",
"congrArg",
"CommSemiring.toSemiring",
... | simp [psum] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 353,
"column": 47
} | {
"line": 353,
"column": 58
} | [
{
"pp": "σ : Type u_5\nR : Type u_6\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\n⊢ psum σ R 1 = ∑ i, X i",
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Finset.univ",
"AddMonoidAlgebra.addAddCommMonoid",
"congrArg",
"CommSemiring.toSemiring",
... | simp [psum] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 313,
"column": 2
} | {
"line": 325,
"column": 27
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\np q : F[X]\nhq : q.natDegree ≠ 0\nP : F[X] → Prop := fun r ↦ (map (algebraMap F (r.comp q).SplittingField) r).Splits\nkey1 : ∀ {r : F[X]}, Irreducible r → P r\n⊢ (map (algebraMap F (p.comp q).SplittingField) p).Splits",
"usedConstants": [
"Eq.mpr",
"Polyno... | have key2 : ∀ {p₁ p₂ : F[X]}, P p₁ → P p₂ → P (p₁ * p₂) := by
intro p₁ p₂ hp₁ hp₂
by_cases h₁ : p₁.comp q = 0
· rcases comp_eq_zero_iff.mp h₁ with h | h
· rw [h, zero_mul]
simp [P]
· exact False.elim (hq (by rw [h.2, natDegree_C]))
by_cases h₂ : p₂.comp q = 0
· rcases comp_eq_zer... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 33
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\np q : F[X]\nhq : q.natDegree ≠ 0\nthis : Fact (map (algebraMap F (p.comp q).SplittingField) p).Splits\n⊢ Function.Surjective ⇑(restrictComp p q hq)",
"usedConstants": [
"MonoidHom.instFunLike",
"MonoidHom",
"Monoid.toMulOneClass",
"Polynomial.G... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 350,
"column": 8
} | {
"line": 351,
"column": 71
} | [
{
"pp": "case refine_2\nE : Type u\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\nn : ℕ\nF : Type u_1\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace ℝ F\ninst✝² : Preorder F\ninst✝¹ : MeasureSpace F\ninst✝ : BorelSpace F\neL : F ≃L[ℝ] Fin (n + 1) → ℝ\nhe_ord : ∀ (x y : F), eL x ≤ eL y ↔ x ≤ y\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 411,
"column": 2
} | {
"line": 411,
"column": 36
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf f' : ℝ → E\na b : ℝ\ns : Set ℝ\nhs : s.Countable\nHc : ContinuousOn f [[a, b]]\nHd : ∀ x ∈ Set.Ioo (min a b) (max a b) \\ s, HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\n⊢ ∫ (x : ℝ) in a..b,... | rcases le_total a b with hab | hab | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Algebra.Polynomial | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 29
} | [
{
"pp": "case refine_1\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝⁵ : Semiring R\ninst✝⁴ : Ring S\ninst✝³ : Field k\ninst✝² : LinearOrder k\ninst✝¹ : IsStrictOrderedRing k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\na✝ : R\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Polynomial | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 29
} | [
{
"pp": "case refine_2\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝⁵ : Semiring R\ninst✝⁴ : Ring S\ninst✝³ : Field k\ninst✝² : LinearOrder k\ninst✝¹ : IsStrictOrderedRing k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\np✝ : R[X... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Polynomial | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 15
} | [
{
"pp": "case refine_3\nR : Type u_1\nS : Type u_2\nk : Type u_3\nα : Type u_4\ninst✝⁵ : Semiring R\ninst✝⁴ : Ring S\ninst✝³ : Field k\ninst✝² : LinearOrder k\ninst✝¹ : IsStrictOrderedRing k\nf : R →+* S\nabv : S → k\ninst✝ : IsAbsoluteValue abv\nl : Filter α\nz : α → S\nhz : Tendsto (abv ∘ z) l atTop\np✝ : R[X... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 411,
"column": 2
} | {
"line": 416,
"column": 81
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf f' : ℝ → E\na b : ℝ\ns : Set ℝ\nhs : s.Countable\nHc : ContinuousOn f [[a, b]]\nHd : ∀ x ∈ Set.Ioo (min a b) (max a b) \\ s, HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\n⊢ ∫ (x : ℝ) in a..b,... | rcases le_total a b with hab | hab
· simp only [uIcc_of_le hab, min_eq_left hab, max_eq_right hab] at *
exact integral_eq_of_hasDerivAt_off_countable_of_le f f' hab hs Hc Hd Hi
· simp only [uIcc_of_ge hab, min_eq_right hab, max_eq_left hab] at *
rw [intervalIntegral.integral_symm, neg_eq_iff_eq_neg, neg_sub... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 411,
"column": 2
} | {
"line": 416,
"column": 81
} | [
{
"pp": "E : Type u\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf f' : ℝ → E\na b : ℝ\ns : Set ℝ\nhs : s.Countable\nHc : ContinuousOn f [[a, b]]\nHd : ∀ x ∈ Set.Ioo (min a b) (max a b) \\ s, HasDerivAt f (f' x) x\nHi : IntervalIntegrable f' volume a b\n⊢ ∫ (x : ℝ) in a..b,... | rcases le_total a b with hab | hab
· simp only [uIcc_of_le hab, min_eq_left hab, max_eq_right hab] at *
exact integral_eq_of_hasDerivAt_off_countable_of_le f f' hab hs Hc Hd Hi
· simp only [uIcc_of_ge hab, min_eq_right hab, max_eq_left hab] at *
rw [intervalIntegral.integral_symm, neg_eq_iff_eq_neg, neg_sub... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Polynomial | {
"line": 149,
"column": 34
} | {
"line": 149,
"column": 45
} | [
{
"pp": "case inl\nR : Type u_2\ninst✝² : NormedRing R\ninst✝¹ : IsAbsoluteValue norm\ninst✝ : ProperSpace R\np : R[X]\nh : p.degree ≤ 0\n⊢ IsClosedMap fun x ↦ eval x (C (p.coeff 0))",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"Polynomial.eval",
"NormedRing.toRing",
"Polynom... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.Algebra.Polynomial | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 26
} | [
{
"pp": "R : Type u_2\ninst✝² : NormedRing R\ninst✝¹ : IsAbsoluteValue norm\ninst✝ : ProperSpace R\nn : ℕ\n⊢ IsClosedMap fun x ↦ x ^ n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Polynomial.Basic | {
"line": 44,
"column": 8
} | {
"line": 45,
"column": 15
} | [
{
"pp": "f : ℂ[X]\nhf : 0 < f.degree\nhf' : ∀ (z : ℂ), ¬f.IsRoot z\nz : ℂ\n⊢ Filter.Tendsto (fun x ↦ eval x f) (cobounded ℂ) (cobounded ℂ)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Polynomial.Basic | {
"line": 132,
"column": 47
} | {
"line": 132,
"column": 63
} | [
{
"pp": "p : ℚ[X]\np_irr : Irreducible p\np_deg : Nat.Prime p.natDegree\np_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2\n⊢ Fintype.card ↥(p.aroots ℂ).toFinset = p.natDegree",
"usedConstants": [
"Multiset.toFinset",
"Eq.mpr",
"Complex.commRing",
"congrArg",
... | Fintype.card_coe | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Real.Embedding | {
"line": 43,
"column": 4
} | {
"line": 43,
"column": 26
} | [
{
"pp": "M : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : LinearOrder M\ninst✝¹ : IsOrderedAddMonoid M\ninst✝ : One M\nu v : ℚ\nx y : M\nhu : u.num • 1 < u.den • x\nhv : v.num • 1 < v.den • y\n⊢ (u.num * ↑v.den) • 1 < (↑u.den * ↑v.den) • x",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings | {
"line": 237,
"column": 36
} | {
"line": 237,
"column": 47
} | [
{
"pp": "case a\nK : Type u_1\ninst✝¹ : Field K\nk : Type u_2\ninst✝ : Field k\nf : k →+* K\nφ : K →+* ℂ\nhφ : IsReal φ\nx : k\n⊢ (star (φ.comp f)) x = (φ.comp f) x",
"usedConstants": [
"RingHom",
"id",
"Field.toSemifield",
"RingHom.comp",
"RingHom.instFunLike",
"Semifiel... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Polynomial.Basic | {
"line": 144,
"column": 4
} | {
"line": 146,
"column": 11
} | [
{
"pp": "case h1\np : ℚ[X]\np_irr : Irreducible p\np_deg : Nat.Prime p.natDegree\np_roots : Fintype.card ↑(p.rootSet ℂ) = Fintype.card ↑(p.rootSet ℝ) + 2\nh1 : Fintype.card ↑(p.rootSet ℂ) = p.natDegree\nconj' : p.Gal := (restrict p ℂ) (AlgEquiv.restrictScalars ℚ conjAe)\nx : Equiv.Perm ↑(p.rootSet ℂ)\n⊢ p.natDe... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings | {
"line": 286,
"column": 46
} | {
"line": 286,
"column": 57
} | [
{
"pp": "K : Type u_1\ninst✝² : Field K\nk : Type u_2\ninst✝¹ : Field k\ninst✝ : Algebra k K\nφ : K →+* ℂ\nσ : Gal(K/k)\nhσ : IsConj φ σ\nx : K\n⊢ (conjugate φ) x = (φ.comp ↑σ.symm) x",
"usedConstants": [
"NumberField.ComplexEmbedding.conjugate",
"AlgEquiv.symm",
"RingHom",
"id",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings | {
"line": 298,
"column": 2
} | {
"line": 298,
"column": 49
} | [
{
"pp": "case a\nK : Type u_1\ninst✝² : Field K\nk : Type u_2\ninst✝¹ : Field k\ninst✝ : Algebra k K\nφ : K →+* ℂ\nσ : Gal(K/k)\nhσ : IsConj φ σ\nν : Gal(K/k)\nx✝ : K\n⊢ (conjugate (φ.comp ↑ν)) x✝ = ((φ.comp ↑ν).comp ↑(ν⁻¹ * σ * ν)) x✝",
"usedConstants": [
"Eq.mpr",
"AlgEquiv.instEquivLike",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings | {
"line": 302,
"column": 36
} | {
"line": 302,
"column": 47
} | [
{
"pp": "case h\nK : Type u_1\ninst✝² : Field K\nk : Type u_2\ninst✝¹ : Field k\ninst✝ : Algebra k K\nφ : K →+* ℂ\nσ : Gal(K/k)\nhσ : IsConj φ σ\nhσ' : σ ≠ 1\na✝ : K\n⊢ (σ ^ 2) a✝ = 1 a✝",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"AlgEqui... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 50,
"column": 39
} | {
"line": 50,
"column": 50
} | [
{
"pp": "M : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : LinearOrder M\ninst✝¹ : IsOrderedAddMonoid M\ninst✝ : One M\nu v : ℚ\nx y : M\nhu : u.num • 1 < u.den • x\nhv : v.num • 1 < v.den • y\nhu' : (u.num * ↑v.den) • 1 < (↑u.den * ↑v.den) • x\n⊢ 0 < ↑(u + v).den",
"usedConstants": [
"Eq.mpr",
"P... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 63,
"column": 2
} | {
"line": 63,
"column": 13
} | [
{
"pp": "M : Type u_1\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : LinearOrder M\ninst✝³ : IsOrderedAddMonoid M\ninst✝² : One M\ninst✝¹ : ZeroLEOneClass M\ninst✝ : NeZero 1\nnum : ℤ\nden : ℕ\nx : M\nh : num • 1 ≤ den • x\nn : ℤ\nhn : x ≤ n • 1\n⊢ den • x ≤ ↑den • n • 1",
"usedConstants": [
"Eq.mpr",
"inst... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 94,
"column": 48
} | {
"line": 94,
"column": 59
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : M\nn : ℕ\nhn : x ≤ n • 1\nnum : ℤ\nden : ℕ\nden_nz✝ : den ≠ 0\nreduced✝ : num.natAbs.Coprime den\nthis : num • 1 < den •... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 96,
"column": 36
} | {
"line": 96,
"column": 47
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : M\nn : ℕ\nhn : x ≤ n • 1\nnum : ℤ\nden : ℕ\nden_nz✝ : den ≠ 0\nreduced✝ : num.natAbs.Coprime den\nh : num • 1 < den • x\... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 102,
"column": 29
} | {
"line": 102,
"column": 40
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : M\nhneg : x < 0\nn : ℕ\nhn : -x - x ≤ n • 1\nthis : -(n • 1) < x\n⊢ Rat.ofInt (-↑n) ∈ ratLt x",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Analysis.Complex.Polynomial.Basic | {
"line": 202,
"column": 70
} | {
"line": 202,
"column": 81
} | [
{
"pp": "p : ℝ[X]\nhp : Irreducible p\nz : ℂ\nhz : (aeval z) p = 0\n⊢ C p.leadingCoeff⁻¹ ≠ 0",
"usedConstants": [
"Eq.mpr",
"inv_eq_zero._simp_1",
"Polynomial.C",
"GroupWithZero.toMonoidWithZero",
"RingHom.instRingHomClass",
"Real",
"GroupWithZero.toDivisionMonoid",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 103,
"column": 41
} | {
"line": 103,
"column": 52
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : M\nhneg : x < 0\nn : ℕ\nhn : -x - x ≤ n • 1\n⊢ -x < -x - x",
"usedConstants": [
"IsRightCancelAdd.addRightStri... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 15
} | [
{
"pp": "case h\nM : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : M\nhxpos : 0 < x\nn : ℕ\nhn : 1 ≤ n • x\n⊢ { num := 1, den := n + 1, den_nz := ⋯, reduced := ⋯ } ∈ ratLt x",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Ultra.Basic | {
"line": 63,
"column": 8
} | {
"line": 63,
"column": 25
} | [
{
"pp": "case inl\nX : Type u_1\ninst✝¹ : PseudoMetricSpace X\ninst✝ : IsUltrametricDist X\nx y z : X\nh✝ : dist x y ≠ dist y z\nh : dist x y < dist y z\n⊢ max (dist x y) (dist y z) ≤ dist x z",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"congrArg",
"PartialOrder... | max_eq_right h.le | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Ultra.Basic | {
"line": 65,
"column": 4
} | {
"line": 65,
"column": 44
} | [
{
"pp": "case inl\nX : Type u_1\ninst✝¹ : PseudoMetricSpace X\ninst✝ : IsUltrametricDist X\nx y z : X\nh✝ : dist x y ≠ dist y z\nh : dist x y < dist y z\n⊢ ¬dist y z ≤ dist y x",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"_private.Mathlib.Topology.MetricSpace.Ultra.Basic.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Ultra.Basic | {
"line": 68,
"column": 4
} | {
"line": 68,
"column": 44
} | [
{
"pp": "case inr\nX : Type u_1\ninst✝¹ : PseudoMetricSpace X\ninst✝ : IsUltrametricDist X\nx y z : X\nh✝ : dist x y ≠ dist y z\nh : dist y z < dist x y\n⊢ ¬dist y x ≤ dist y z",
"usedConstants": [
"Eq.mpr",
"Real",
"Preorder.toLT",
"_private.Mathlib.Topology.MetricSpace.Ultra.Basic.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Topology.MetricSpace.Ultra.Basic | {
"line": 71,
"column": 18
} | {
"line": 71,
"column": 47
} | [
{
"pp": "X : Type u_1\ninst✝¹ : PseudoMetricSpace X\ninst✝ : IsUltrametricDist X\nx y z : X\nr s : ℝ\np : X → Prop\nx✝² x✝¹ x✝ : Subtype p\n⊢ dist x✝² x✝ ≤ max (dist x✝² x✝¹) (dist x✝¹ x✝)",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"PartialOrder.toPreorder",
"Preorde... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 131,
"column": 6
} | {
"line": 131,
"column": 17
} | [
{
"pp": "case h.mp.refine_1\nM : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx y : M\na : ℚ\nh : a.num • 1 < a.den • (x + y)\nk : ℕ\nhk : 1 + 1 ≤ k • (a.den • (x + y) - a.num • 1)\n... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Real.Embedding | {
"line": 138,
"column": 62
} | {
"line": 138,
"column": 73
} | [
{
"pp": "M : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx y : M\na : ℚ\nh : a.num • 1 < a.den • (x + y)\nk : ℕ\nhk : 1 + 1 ≤ k • (a.den • (x + y) - a.num • 1)\nhk0 : k ≠ 0\nhka0 : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.RingTheory.Valuation.RankOne | {
"line": 88,
"column": 6
} | {
"line": 88,
"column": 63
} | [
{
"pp": "R : Type u_1\nΓ₀ : Type u_2\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\ninst✝ : v.IsNontrivial\na✝ : MulArchimedean (ValueGroup₀ v)\nf : Additive (ValueGroup₀ v)ˣ →+o ℝ\nhf : Injective ⇑f\ne : (ValueGroup₀ v)ˣ →* Multiplicative ℝ := AddMonoidHom.toMultiplicativeRig... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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