module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 763,
"column": 4
} | {
"line": 772,
"column": 32
} | {
"line": 773,
"column": 2
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nF : Type v\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nι : Type u_2\nE : ι → Type u_3\ninst✝² : (i : ι) → NormedAddCommGroup (E i)\ninst✝¹ : (i : ι) → NormedSpace 𝕜 (E i)\ninst✝ : Fintype ι\nf : ContinuousMultilinearMap 𝕜 E ... | [] | · have hkf : k ∉ Set.range (Equiv.embeddingFinSucc n ι e).1 := by
contrapose hke
rw [Equiv.embeddingFinSucc_fst] at hke
exact Set.range_comp_subset_range _ _ hke
simp only [hke, hkf, ↓reduceDIte, Pi.compRightL,
ContinuousLinearMap.coe_mk', LinearMap.coe_mk, AddHom.coe_mk]
rw ... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Module.Alternating.Basic | {
"line": 586,
"column": 6
} | {
"line": 586,
"column": 47
} | {
"line": 587,
"column": 6
} | [
{
"pp": "𝕜 : Type u\nn : ℕ\nE : Type wE\nF : Type wF\nG : Type wG\nι : Type v\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : SeminormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SeminormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : SeminormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : ... | [
"𝕜 : Type u\nn : ℕ\nE : Type wE\nF : Type wF\nG : Type wG\nι : Type v\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : SeminormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\ninst✝⁵ : SeminormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\ninst✝³ : SeminormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\ninst✝¹ : Fintype ι\nι... | simp only [coe_mk, MultilinearMap.coe_mk] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 303,
"column": 6
} | {
"line": 303,
"column": 15
} | {
"line": 303,
"column": 16
} | [
{
"pp": "case e'_9\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\n𝔸 : Type u_3\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nc : 𝕜 → 𝔸\nc' : 𝔸\nhc : HasDerivWithinAt c c' s x\nd : 𝔸\n⊢ c' * d = c' * d + c x * 0",
"ppTerm": "?e'_9",
"assigned": true,
"usedConstant... | [
"case e'_9\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\n𝔸 : Type u_3\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nc : 𝕜 → 𝔸\nc' : 𝔸\nhc : HasDerivWithinAt c c' s x\nd : 𝔸\n⊢ c' * d = c' * d + 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Deriv.Mul | {
"line": 316,
"column": 6
} | {
"line": 316,
"column": 15
} | {
"line": 316,
"column": 16
} | [
{
"pp": "case e'_9\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\n𝔸 : Type u_3\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nc : 𝕜 → 𝔸\nc' : 𝔸\nhc : HasStrictDerivAt c c' x\nd : 𝔸\n⊢ c' * d = c' * d + c x * 0",
"ppTerm": "?e'_9",
"assigned": true,
"usedConstants": [
"E... | [
"case e'_9\n𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\n𝔸 : Type u_3\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕜 𝔸\nc : 𝕜 → 𝔸\nc' : 𝔸\nhc : HasStrictDerivAt c c' x\nd : 𝔸\n⊢ c' * d = c' * d + 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Deriv.Add | {
"line": 182,
"column": 79
} | {
"line": 183,
"column": 40
} | {
"line": 185,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : 𝕜 → F\na b : 𝕜\n⊢ DifferentiableAt 𝕜 f a ↔ DifferentiableAt 𝕜 (fun x ↦ f (x + b)) (a - b)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"NormedCom... | [] | by
simp [differentiableAt_comp_add_const] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 76
} | {
"line": 170,
"column": 0
} | [
{
"pp": "case inr\nk : Type u_1\nE : Type u_2\ninst✝⁷ : Field k\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module k E\ninst✝⁴ : LinearOrder k\ninst✝³ : IsStrictOrderedRing k\ninst✝² : PartialOrder E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : PosSMulMono k E\nf : k → E\nx y : k\ns : Set k\nhf : StrictMonoOn f s\nhx : x ∈ s... | [] | exact slope_comm f x y ▸ (slope_pos_iff_of_le hxy.le).mpr (hf hy hx hxy) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 76
} | {
"line": 170,
"column": 0
} | [
{
"pp": "case inr\nk : Type u_1\nE : Type u_2\ninst✝⁷ : Field k\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module k E\ninst✝⁴ : LinearOrder k\ninst✝³ : IsStrictOrderedRing k\ninst✝² : PartialOrder E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : PosSMulMono k E\nf : k → E\nx y : k\ns : Set k\nhf : StrictMonoOn f s\nhx : x ∈ s... | [] | exact slope_comm f x y ▸ (slope_pos_iff_of_le hxy.le).mpr (hf hy hx hxy) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Slope | {
"line": 168,
"column": 4
} | {
"line": 168,
"column": 76
} | {
"line": 170,
"column": 0
} | [
{
"pp": "case inr\nk : Type u_1\nE : Type u_2\ninst✝⁷ : Field k\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module k E\ninst✝⁴ : LinearOrder k\ninst✝³ : IsStrictOrderedRing k\ninst✝² : PartialOrder E\ninst✝¹ : IsOrderedAddMonoid E\ninst✝ : PosSMulMono k E\nf : k → E\nx y : k\ns : Set k\nhf : StrictMonoOn f s\nhx : x ∈ s... | [] | exact slope_comm f x y ▸ (slope_pos_iff_of_le hxy.le).mpr (hf hy hx hxy) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Uniqueness | {
"line": 217,
"column": 2
} | {
"line": 217,
"column": 52
} | {
"line": 219,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nU : Set E\nhf : AnalyticOnNhd 𝕜 f U\nhU : IsPreconnected U\nz₀ : E\nh₀ : z₀ ∈ U\nhfz₀ : f =ᶠ[𝓝 ... | [] | exact UniformSpace.Completion.coe_injective F this | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.DSlope | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 38
} | {
"line": 113,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\na : 𝕜\ns : Set 𝕜\nh : s ∈ 𝓝 a\nhc : ContinuousOn f s\nhd : DifferentiableAt 𝕜 f a\nx : 𝕜\nhx : x ∈ s\n⊢ ContinuousWithinAt (dslope f a) s x",
"ppTerm": "?m.72"... | [
"case inl\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\ns : Set 𝕜\nhc : ContinuousOn f s\nx : 𝕜\nhx : x ∈ s\nh : s ∈ 𝓝 x\nhd : DifferentiableAt 𝕜 f x\n⊢ ContinuousWithinAt (dslope f x) s x",
"case inr\n𝕜 : Type u_1\nE :... | rcases eq_or_ne x a with (rfl | hne) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Calculus.Deriv.Slope | {
"line": 167,
"column": 2
} | {
"line": 168,
"column": 70
} | {
"line": 169,
"column": 2
} | [
{
"pp": "𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\ninst✝ : OrderTopology 𝕜\ng : 𝕜 → 𝕜\ng' : 𝕜\nhx : AccPt x (𝓟 s)\nhd : HasDerivWithinAt g g' s x\nhg : MonotoneOn g s\nthis : (𝓝[s \\ {x}] x).NeBot\n⊢ 0 ≤ g'",
"ppTerm... | [
"𝕜 : Type u\ninst✝³ : NontriviallyNormedField 𝕜\nx : 𝕜\ns : Set 𝕜\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsStrictOrderedRing 𝕜\ninst✝ : OrderTopology 𝕜\ng : 𝕜 → 𝕜\ng' : 𝕜\nhx : AccPt x (𝓟 s)\nhd : HasDerivWithinAt g g' s x\nhg : MonotoneOn g s\nthis : (𝓝[s \\ {x}] x).NeBot\nh'g : MonotoneOn g (insert x s)\n⊢... | have h'g : MonotoneOn g (insert x s) :=
hg.insert_of_continuousWithinAt hx.clusterPt hd.continuousWithinAt | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.FDeriv.Mul | {
"line": 604,
"column": 2
} | {
"line": 606,
"column": 69
} | {
"line": 608,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nι : Type u_5\n𝔸' : Type u_7\ninst✝² : NormedCommRing 𝔸'\ninst✝¹ : NormedAlgebra 𝕜 𝔸'\nu : Finset ι\ng : ι → E → 𝔸'\ng' : ι → E →L[𝕜] 𝔸'\ninst✝ : DecidableEq ι\nx : E\nhg : ... | [] | simpa [← Finset.prod_attach u] using .congr_fderiv
(hasStrictFDerivAt_finsetProd.comp x <| hasStrictFDerivAt_pi.mpr fun i ↦ hg i i.prop)
(by ext; simp [Finset.prod_erase_attach (g · x), ← u.sum_attach]) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.FDeriv.Mul | {
"line": 604,
"column": 2
} | {
"line": 606,
"column": 69
} | {
"line": 608,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nι : Type u_5\n𝔸' : Type u_7\ninst✝² : NormedCommRing 𝔸'\ninst✝¹ : NormedAlgebra 𝕜 𝔸'\nu : Finset ι\ng : ι → E → 𝔸'\ng' : ι → E →L[𝕜] 𝔸'\ninst✝ : DecidableEq ι\nx : E\nhg : ... | [] | simpa [← Finset.prod_attach u] using .congr_fderiv
(hasStrictFDerivAt_finsetProd.comp x <| hasStrictFDerivAt_pi.mpr fun i ↦ hg i i.prop)
(by ext; simp [Finset.prod_erase_attach (g · x), ← u.sum_attach]) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Mul | {
"line": 604,
"column": 2
} | {
"line": 606,
"column": 69
} | {
"line": 608,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nι : Type u_5\n𝔸' : Type u_7\ninst✝² : NormedCommRing 𝔸'\ninst✝¹ : NormedAlgebra 𝕜 𝔸'\nu : Finset ι\ng : ι → E → 𝔸'\ng' : ι → E →L[𝕜] 𝔸'\ninst✝ : DecidableEq ι\nx : E\nhg : ... | [] | simpa [← Finset.prod_attach u] using .congr_fderiv
(hasStrictFDerivAt_finsetProd.comp x <| hasStrictFDerivAt_pi.mpr fun i ↦ hg i i.prop)
(by ext; simp [Finset.prod_erase_attach (g · x), ← u.sum_attach]) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.Deriv.Inv | {
"line": 168,
"column": 75
} | {
"line": 170,
"column": 20
} | {
"line": 172,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝² : NontriviallyNormedField 𝕜\nx : 𝕜\n𝕜' : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nc d : 𝕜 → 𝕜'\nc' d' : 𝕜'\nhc : HasDerivAt c c' x\nhd : HasDerivAt d d' x\nhx : d x ≠ 0\n⊢ HasDerivAt (fun y ↦ c y / d y) ((c' * d x - c x * d') / d x ^ 2) x",... | [] | by
rw [← hasDerivWithinAt_univ] at *
exact hc.div hd hx | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 99,
"column": 6
} | {
"line": 99,
"column": 73
} | {
"line": 99,
"column": 73
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\np : FormalMultilinearSeries 𝕜 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhp : HasFPowerSeriesAt f p z₀\nh : p ≠ 0\n⊢ (swap dslope z₀)^[p.order] f z₀ ≠ 0",
"ppTerm": "?m.62",
"assigned": t... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\np : FormalMultilinearSeries 𝕜 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nhp : HasFPowerSeriesAt f p z₀\nh : p ≠ 0\n⊢ (fslope^[p.order] p 0) 1 ≠ 0"
] | ← coeff_zero (has_fpower_series_iterate_dslope_fslope p.order hp) 1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.RCLike | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 58
} | {
"line": 116,
"column": 2
} | [
{
"pp": "E : Type u_4\nF : Type u_5\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\ns : Set E\nx : E\nhf : ContDiffWithinAt ℝ 1 f s x\nhs : Convex ℝ s\nt : Set E\nhst : t ∈ 𝓝[insert x s] x\np : E → FormalMultilinearSeries ℝ E F\nhp : ... | [
"E : Type u_4\nF : Type u_5\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : E → F\ns : Set E\nx : E\nhf : ContDiffWithinAt ℝ 1 f s x\nhs : Convex ℝ s\nt : Set E\nhst : t ∈ 𝓝[insert x s] x\np : E → FormalMultilinearSeries ℝ E F\nhp : HasFTaylorSe... | rcases Metric.mem_nhdsWithin_iff.mp hst with ⟨ε, ε0, hε⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 145,
"column": 21
} | {
"line": 145,
"column": 30
} | {
"line": 145,
"column": 31
} | [
{
"pp": "case ha\nf : ℝ → ℝ\na b : ℝ\nhf : ContinuousOn f (Icc a b)\nB B' : ℝ → ℝ\nha : f a ≤ B a\nhB : ContinuousOn B (Icc a b)\nhB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x\nbound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r\nx : ℝ\nhx : x ∈ Icc a b\nr : ℝ\nhr : r ... | [
"case ha\nf : ℝ → ℝ\na b : ℝ\nhf : ContinuousOn f (Icc a b)\nB B' : ℝ → ℝ\nha : f a ≤ B a\nhB : ContinuousOn B (Icc a b)\nhB' : ∀ x ∈ Ico a b, HasDerivWithinAt B (B' x) (Ici x) x\nbound : ∀ x ∈ Ico a b, ∀ (r : ℝ), B' x < r → ∃ᶠ (z : ℝ) in 𝓝[>] x, slope f x z < r\nx : ℝ\nhx : x ∈ Icc a b\nr : ℝ\nhr : r > 0\n⊢ f a ≤... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 560,
"column": 2
} | {
"line": 561,
"column": 43
} | {
"line": 562,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\n𝕜 : Type u_3\nG : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : E → G\ns : Set E\nx y : E\nhs : Convex ℝ s\nh... | [
"E : Type u_1\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace ℝ E\n𝕜 : Type u_3\nG : Type u_4\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : IsRCLikeNormedField 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : E → G\ns : Set E\nx y : E\nhs : Convex ℝ s\nhf : Differen... | have bound : ∀ x ∈ s, ‖fderivWithin 𝕜 f s x‖ ≤ 0 := fun x hx => by
simp only [hf' x hx, norm_zero, le_rfl] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 819,
"column": 65
} | {
"line": 821,
"column": 28
} | {
"line": 823,
"column": 2
} | [
{
"pp": "𝕜 : Type u_3\ninst✝⁴ : RCLike 𝕜\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nH : Type u_5\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nf : G → H\nf' : G → G →L[𝕜] H\nx : G\nhder : ∀ᶠ (y : G) in 𝓝 x, HasFDerivAt f (f' y) y\nhcont : ContinuousAt f' x\nc : ℝ\nh... | [] | by
rw [← dist_eq_norm]
exact le_of_lt (hε H').2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.Comp | {
"line": 703,
"column": 8
} | {
"line": 703,
"column": 22
} | {
"line": 703,
"column": 23
} | [
{
"pp": "case succ\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx₀ : E\nn : ℕ∞ω\nhf : ContDiffWithinAt 𝕜 n f s x₀\nhs : UniqueDiffOn 𝕜 s\... | [
"case succ\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx₀ : E\nn : ℕ∞ω\nhf : ContDiffWithinAt 𝕜 n f s x₀\nhs : UniqueDiffOn 𝕜 s\nhx₀s : x₀ ∈... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 240,
"column": 2
} | {
"line": 240,
"column": 41
} | {
"line": 241,
"column": 2
} | [
{
"pp": "n : ℕ\nc : OrderedFinpartition n\ninst✝ : NeZero n\n⊢ c.emb (c.index 0) 0 ≤ 0",
"ppTerm": "?m.45",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"OrderedFinpartition.invEmbedding",
"PartialOrder.toPreorder",
"Preorder.toLE",
"OrderedFinpartit... | [
"n : ℕ\nc : OrderedFinpartition n\ninst✝ : NeZero n\n⊢ c.emb (c.index 0) 0 ≤ c.emb (c.index 0) (c.invEmbedding 0)"
] | conv_rhs => rw [← c.emb_invEmbedding 0] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 307,
"column": 14
} | {
"line": 307,
"column": 25
} | {
"line": 308,
"column": 4
} | [
{
"pp": "case zero\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\nq :... | [] | simp at hij | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 307,
"column": 14
} | {
"line": 307,
"column": 25
} | {
"line": 308,
"column": 4
} | [
{
"pp": "case zero\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\nq :... | [] | simp at hij | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 307,
"column": 14
} | {
"line": 307,
"column": 25
} | {
"line": 308,
"column": 4
} | [
{
"pp": "case zero\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\nq :... | [] | simp at hij | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 411,
"column": 6
} | {
"line": 423,
"column": 51
} | {
"line": 425,
"column": 0
} | [
{
"pp": "case mpr.refine_3\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\nhn : n ≠ ∞\nh'n : n + 1 ≠ ∞\nu : Set E\nhu : u ∈ 𝓝[ins... | [] | intro h i
simp only [WithTop.add_eq_top, WithTop.one_ne_top, or_false] at h
match i with
| 0 =>
simp only [FormalMultilinearSeries.unshift]
apply AnalyticOnNhd.comp_analyticOn _ ((hf h).mono inter_subset_right)
(Set.mapsTo_univ _ _)
exact LinearIsometryEquiv.analyticO... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 411,
"column": 6
} | {
"line": 423,
"column": 51
} | {
"line": 425,
"column": 0
} | [
{
"pp": "case mpr.refine_3\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nn : ℕ∞ω\nhn : n ≠ ∞\nh'n : n + 1 ≠ ∞\nu : Set E\nhu : u ∈ 𝓝[ins... | [] | intro h i
simp only [WithTop.add_eq_top, WithTop.one_ne_top, or_false] at h
match i with
| 0 =>
simp only [FormalMultilinearSeries.unshift]
apply AnalyticOnNhd.comp_analyticOn _ ((hf h).mono inter_subset_right)
(Set.mapsTo_univ _ _)
exact LinearIsometryEquiv.analyticO... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 542,
"column": 8
} | {
"line": 542,
"column": 47
} | {
"line": 543,
"column": 8
} | [
{
"pp": "case neg.refine_2\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Se... | [
"case neg.refine_2\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\nq : F →... | conv_rhs => rw [← c.emb_invEmbedding 0] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 623,
"column": 4
} | {
"line": 628,
"column": 40
} | {
"line": 629,
"column": 2
} | [
{
"pp": "case mp\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nhx : x ∈ s\n⊢ ContDiffWithinAt 𝕜 0 f s x → ∃ u ∈ 𝓝[s] x, ContinuousOn f ... | [] | intro h
obtain ⟨u, H, p, hp⟩ := h 0 le_rfl
refine ⟨u, ?_, ?_⟩
· simpa [hx] using H
· simp only [Nat.cast_zero, hasFTaylorSeriesUpToOn_zero_iff] at hp
exact hp.1.mono inter_subset_right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 623,
"column": 4
} | {
"line": 628,
"column": 40
} | {
"line": 629,
"column": 2
} | [
{
"pp": "case mp\n𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nx : E\nhx : x ∈ s\n⊢ ContDiffWithinAt 𝕜 0 f s x → ∃ u ∈ 𝓝[s] x, ContinuousOn f ... | [] | intro h
obtain ⟨u, H, p, hp⟩ := h 0 le_rfl
refine ⟨u, ?_, ?_⟩
· simpa [hx] using H
· simp only [Nat.cast_zero, hasFTaylorSeriesUpToOn_zero_iff] at hp
exact hp.1.mono inter_subset_right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Defs | {
"line": 638,
"column": 64
} | {
"line": 677,
"column": 70
} | {
"line": 679,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\ns : Set E\nf : E → F\nn : ℕ∞ω\nh : ContDiffOn 𝕜 n f s\nhs : UniqueDiffOn 𝕜 s\n⊢ HasFTaylorSeriesUpToOn n f (ft... | [] | by
constructor
· intro x _
simp only [ftaylorSeriesWithin, ContinuousMultilinearMap.curry0_apply,
iteratedFDerivWithin_zero_apply]
· intro m hm x hx
have : (m + 1 : ℕ) ≤ n := ENat.add_one_natCast_le_withTop_of_lt hm
rcases (h x hx).of_le this _ le_rfl with ⟨u, hu, p, Hp⟩
rw [insert_eq_of_mem... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 665,
"column": 10
} | {
"line": 665,
"column": 44
} | {
"line": 666,
"column": 10
} | [
{
"pp": "case emb.inl\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\n... | [
"case emb.inl.refine_1\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\nq :... | refine (Fin.heq_fun_iff ?_).mpr ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 669,
"column": 10
} | {
"line": 669,
"column": 44
} | {
"line": 670,
"column": 10
} | [
{
"pp": "case emb.inr\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\n... | [
"case emb.inr.refine_1\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\nq :... | refine (Fin.heq_fun_iff ?_).mpr ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 690,
"column": 8
} | {
"line": 690,
"column": 42
} | {
"line": 691,
"column": 8
} | [
{
"pp": "case pos.emb\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\n... | [
"case pos.emb.refine_1\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\nq :... | refine (Fin.heq_fun_iff ?_).mpr ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 697,
"column": 12
} | {
"line": 697,
"column": 58
} | {
"line": 698,
"column": 12
} | [
{
"pp": "case pos.emb.refine_2.zero\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set ... | [
"case pos.emb.refine_2.zero\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F... | simp only [cases_zero, cast_zero, val_eq_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 708,
"column": 8
} | {
"line": 708,
"column": 29
} | {
"line": 709,
"column": 8
} | [
{
"pp": "case neg.emb\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\n... | [
"case neg.emb\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nt : Set F\nq : F → Form... | refine hfunext rfl ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 774,
"column": 71
} | {
"line": 779,
"column": 38
} | {
"line": 781,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nc : OrderedFinpartition n\np : (i : Fin c.length) → E [×c.partSize i]→L[𝕜] F\nm : Fin c.length\nv : ... | [] | by
ext d
by_cases h : d = m
· rw [h]
simp [applyOrderedFinpartition]
· simp [h, applyOrderedFinpartition] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 948,
"column": 4
} | {
"line": 949,
"column": 49
} | {
"line": 950,
"column": 4
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nα : Type u_5\nH : Ty... | [
"case refine_2\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : NormedSpace 𝕜 F\nG : Type u_4\ninst✝² : NormedAddCommGroup G\ninst✝¹ : NormedSpace 𝕜 G\nα : Type u_5\nH : Type u_6\ninst... | have H₂ : ∀ i, (q₂ · (c.partSize i)) =O[l] (1 : α → ℝ) := fun i ↦
(hq₂_bdd _ <| c.partSize_le i).isBigO_one ℝ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Algebra.Exponential | {
"line": 361,
"column": 4
} | {
"line": 365,
"column": 31
} | {
"line": 366,
"column": 2
} | [
{
"pp": "𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕂\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedRing 𝔹\ninst✝² : NormedAlgebra 𝕂 𝔸\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx : 𝔸\nhx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius\n⊢ exp (-x) * exp x = 1",
"ppTerm"... | [] | have hnx : -x ∈ Metric.eball (0 : 𝔸) (expSeries 𝕂 𝔸).radius := by
rw [Metric.mem_eball, ← neg_zero, edist_neg_neg]
exact hx
rw [← exp_add_of_commute_of_mem_ball (Commute.neg_left <| Commute.refl x) hnx hx,
neg_add_cancel, exp_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Algebra.Exponential | {
"line": 361,
"column": 4
} | {
"line": 365,
"column": 31
} | {
"line": 366,
"column": 2
} | [
{
"pp": "𝕂 : Type u_1\n𝔸 : Type u_2\n𝔹 : Type u_3\ninst✝⁵ : NontriviallyNormedField 𝕂\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedRing 𝔹\ninst✝² : NormedAlgebra 𝕂 𝔸\ninst✝¹ : CompleteSpace 𝔸\ninst✝ : CharZero 𝕂\nx : 𝔸\nhx : x ∈ Metric.eball 0 (expSeries 𝕂 𝔸).radius\n⊢ exp (-x) * exp x = 1",
"ppTerm"... | [] | have hnx : -x ∈ Metric.eball (0 : 𝔸) (expSeries 𝕂 𝔸).radius := by
rw [Metric.mem_eball, ← neg_zero, edist_neg_neg]
exact hx
rw [← exp_add_of_commute_of_mem_ball (Commute.neg_left <| Commute.refl x) hnx hx,
neg_add_cancel, exp_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.CauSeqFilter | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 27
} | {
"line": 81,
"column": 0
} | [
{
"pp": "case right\nβ : Type v\ninst✝ : NormedField β\nf : CauSeq β norm\ns : Set (β × β)\nhs : s ∈ uniformity β\nε : ℝ\nhε : ε > 0\nhεs : ∀ ⦃a b : β⦄, dist a b < ε → (a, b) ∈ s\nN : ℕ\nhN : ∀ j ≥ N, ∀ k ≥ N, ‖↑f j - ↑f k‖ < ε\na b : β\na' : ℕ\nha'1 : a' ≥ N\nha'2 : ↑f a' = a\nb' : ℕ\nhb'1 : b' ≥ N\nhb'2 : ↑f ... | [] | apply hN <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Topology.ExtendFrom | {
"line": 86,
"column": 53
} | {
"line": 86,
"column": 67
} | {
"line": 86,
"column": 67
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : RegularSpace Y\nf : X → Y\nA : Set X\nhA : Dense A\nhf : ∀ (x : X), ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)\n⊢ ∀ x ∈ univ, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)",
"ppTerm": "?m.40",
"assigned": true,
"usedConsta... | [] | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.ExtendFrom | {
"line": 86,
"column": 53
} | {
"line": 86,
"column": 67
} | {
"line": 86,
"column": 67
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : RegularSpace Y\nf : X → Y\nA : Set X\nhA : Dense A\nhf : ∀ (x : X), ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)\n⊢ ∀ x ∈ univ, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)",
"ppTerm": "?m.40",
"assigned": true,
"usedConsta... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ExtendFrom | {
"line": 86,
"column": 53
} | {
"line": 86,
"column": 67
} | {
"line": 86,
"column": 67
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : RegularSpace Y\nf : X → Y\nA : Set X\nhA : Dense A\nhf : ∀ (x : X), ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)\n⊢ ∀ x ∈ univ, ∃ y, Tendsto f (𝓝[A] x) (𝓝 y)",
"ppTerm": "?m.40",
"assigned": true,
"usedConsta... | [] | simpa using hf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 633,
"column": 37
} | {
"line": 633,
"column": 58
} | {
"line": 635,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nF : Type uF\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn : ℕ∞ω\nA : Type u_4\ninst✝⁴ : NormedRing A\ninst✝³ : NormedAlgebra 𝕜 A\ninst✝² : Module A F\ninst✝¹ : IsSc... | [] | exact hf.smul_const v | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.Deriv.MeanValue | {
"line": 185,
"column": 6
} | {
"line": 185,
"column": 93
} | {
"line": 186,
"column": 6
} | [
{
"pp": "f : ℝ → ℝ\na : ℝ\nhf : ∀ t ∈ atTop, ∃ i, a < i ∧ MapsTo (derivWithin f (Ioi a)) (Ioo a i) t\nhcont_at_a : ContinuousWithinAt f (Ici a) a\nhdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))\n⊢ ∃ i, a < i ∧ ∀ ⦃x : ℝ⦄, x ∈ Ioo a i → ∀ x_1 ∈ Ioc a x, max (derivWithin f (Ioi a) a + 1) 0 < ... | [
"f : ℝ → ℝ\na : ℝ\nhf : ∀ t ∈ atTop, ∃ i, a < i ∧ MapsTo (derivWithin f (Ioi a)) (Ioo a i) t\nhcont_at_a : ContinuousWithinAt f (Ici a) a\nhdiff : Tendsto (slope f a) (𝓝[>] a) (𝓝 (derivWithin f (Ioi a) a))\nb : ℝ\nhab : a < b\nhb : MapsTo (derivWithin f (Ioi a)) (Ioo a b) (Ioi (max (derivWithin f (Ioi a) a + 1) 0... | obtain ⟨b, hab, hb⟩ := hf (Ioi (max (derivWithin f (Ioi a) a + 1) 0)) (Ioi_mem_atTop _) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.LocallyConvex.Polar | {
"line": 121,
"column": 6
} | {
"line": 121,
"column": 18
} | {
"line": 121,
"column": 18
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\ns : Set E\nx : E\nhx : x ∈ s\ny : F\nhy : y ∈ B.polar s\n⊢ ‖(B.flip y) x‖ ≤ 1",
"ppTerm": "?m.66",
"ass... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedCommRing 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\ns : Set E\nx : E\nhx : x ∈ s\ny : F\nhy : y ∈ B.polar s\n⊢ ‖(B x) y‖ ≤ 1"
] | B.flip_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Deriv.MeanValue | {
"line": 339,
"column": 4
} | {
"line": 339,
"column": 38
} | {
"line": 340,
"column": 4
} | [
{
"pp": "D : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : DifferentiableOn ℝ f (interior D)\nC : ℝ\nlt_hf' : ∀ x ∈ interior D, deriv f x < C\nx✝ : ℝ\nhx✝ : x✝ ∈ D\ny : ℝ\nhy : y ∈ D\nhxy : x✝ < y\nx : ℝ\nhx : x ∈ interior D\n⊢ -C < deriv (fun y ↦ -f y) x",
"ppTerm": "?m.80",
"assigned... | [
"D : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : DifferentiableOn ℝ f (interior D)\nC : ℝ\nlt_hf' : ∀ x ∈ interior D, deriv f x < C\nx✝ : ℝ\nhx✝ : x✝ ∈ D\ny : ℝ\nhy : y ∈ D\nhxy : x✝ < y\nx : ℝ\nhx : x ∈ interior D\n⊢ deriv f x < C"
] | rw [deriv.fun_neg, neg_lt_neg_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 237,
"column": 12
} | {
"line": 237,
"column": 26
} | {
"line": 237,
"column": 27
} | [
{
"pp": "case e_a.e_f\nx : ℝ\nh : |x| < 1\nn : ℕ\nF : ℝ → ℝ := fun x ↦ ∑ i ∈ Finset.range n, x ^ (i + 1) / (↑i + 1) + log (1 - x)\nF' : ℝ → ℝ := fun x ↦ -x ^ n / (1 - x)\ny : ℝ\nhy : y ∈ Set.Ioo (-1) 1\nthis : HasDerivAt F (∑ i ∈ Finset.range n, ↑(i + 1) * y ^ i / (↑i + 1) + -1 / (1 - y)) y\ni : ℕ\n⊢ y ^ i = ↑(... | [
"case e_a.e_f\nx : ℝ\nh : |x| < 1\nn : ℕ\nF : ℝ → ℝ := fun x ↦ ∑ i ∈ Finset.range n, x ^ (i + 1) / (↑i + 1) + log (1 - x)\nF' : ℝ → ℝ := fun x ↦ -x ^ n / (1 - x)\ny : ℝ\nhy : y ∈ Set.Ioo (-1) 1\nthis : HasDerivAt F (∑ i ∈ Finset.range n, ↑(i + 1) * y ^ i / (↑i + 1) + -1 / (1 - y)) y\ni : ℕ\n⊢ y ^ i = (↑i + 1) * y ^... | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.Polar | {
"line": 164,
"column": 4
} | {
"line": 164,
"column": 51
} | {
"line": 165,
"column": 4
} | [
{
"pp": "case mp\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\nS : Type u_4\ninst✝¹ : SetLike S E\ninst✝ : SMulMemClass S 𝕜 E\nm : S\ny : F\nhy : y ∈ B.po... | [
"case mp\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid F\ninst✝³ : Module 𝕜 E\ninst✝² : Module 𝕜 F\nB : E →ₗ[𝕜] F →ₗ[𝕜] 𝕜\nS : Type u_4\ninst✝¹ : SetLike S E\ninst✝ : SMulMemClass S 𝕜 E\nm : S\ny : F\nhy : y ∈ B.polar ↑m\nx : ... | rw [← one_div, le_div_iff₀ (norm_pos_iff.2 hr)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.Deriv.MeanValue | {
"line": 481,
"column": 92
} | {
"line": 483,
"column": 71
} | {
"line": 485,
"column": 0
} | [
{
"pp": "D : Set ℝ\nhD : Convex ℝ D\nf : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : DifferentiableOn ℝ f (interior D)\nhf'_nonpos : ∀ x ∈ interior D, deriv f x ≤ 0\nx : ℝ\nhx : x ∈ D\ny : ℝ\nhy : y ∈ D\nhxy : x ≤ y\n⊢ f y ≤ f x",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Real.instLE",
... | [] | by
simpa only [zero_mul, sub_nonpos] using
hD.image_sub_le_mul_sub_of_deriv_le hf hf' hf'_nonpos x hx y hy hxy | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Cone.Extension | {
"line": 93,
"column": 4
} | {
"line": 93,
"column": 20
} | {
"line": 94,
"column": 4
} | [
{
"pp": "case refine_2\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : ConvexCone ℝ E\nf : E →ₗ.[ℝ] ℝ\nnonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x\ndense : ∀ (y : E), ∃ x, ↑x + y ∈ s\nhdom : f.domain ≠ ⊤\ny : E\nhy : y ∉ f.domain\nc : ℝ\nle_c : ∀ (x : ↥f.domain), -↑x - y ∈ s → ↑f x ≤ c\nc_le :... | [
"case refine_2\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : ConvexCone ℝ E\nf : E →ₗ.[ℝ] ℝ\nnonneg : ∀ (x : ↥f.domain), ↑x ∈ s → 0 ≤ ↑f x\ndense : ∀ (y : E), ∃ x, ↑x + y ∈ s\nhdom : f.domain ≠ ⊤\ny : E\nhy : y ∉ f.domain\nc : ℝ\nle_c : ∀ (x : ↥f.domain), -↑x - y ∈ s → ↑f x ≤ c\nc_le : ∀ (x : ↥f.d... | simp only at hzs | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 338,
"column": 2
} | {
"line": 338,
"column": 45
} | {
"line": 339,
"column": 2
} | [
{
"pp": "x : ℝ\nh₀ : 0 ≤ x\nh : x < 1\nn : ℕ\nF : ℝ → ℝ := fun x ↦ 1 / 2 * log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1)\nF' : ℝ → ℝ := fun y ↦ (y ^ 2) ^ n / (1 - y ^ 2)\nA : ∀ y ∈ Set.Icc 0 x, HasDerivAt F (F' y) y\ny : ℝ\nhy : y ∈ interior (Set.Icc 0 x)\n⊢ 0 ≤ F' y",
"ppTe... | [
"x : ℝ\nh₀ : 0 ≤ x\nh : x < 1\nn : ℕ\nF : ℝ → ℝ := fun x ↦ 1 / 2 * log ((1 + x) / (1 - x)) - ∑ i ∈ Finset.range n, x ^ (2 * i + 1) / (2 * ↑i + 1)\nF' : ℝ → ℝ := fun y ↦ (y ^ 2) ^ n / (1 - y ^ 2)\nA : ∀ y ∈ Set.Icc 0 x, HasDerivAt F (F' y) y\ny : ℝ\nhy : 0 < y ∧ y < x\n⊢ 0 ≤ F' y"
] | simp only [interior_Icc, Set.mem_Ioo] at hy | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Geometry.Convex.Cone.Basic | {
"line": 542,
"column": 16
} | {
"line": 542,
"column": 41
} | {
"line": 543,
"column": 12
} | [
{
"pp": "𝕜 : Type u_1\nM : Type u_4\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup M\ninst✝ : Module 𝕜 M\ns : Set M\nx : M\nhs : Convex 𝕜 s\nhx : x ∈ hull 𝕜 s\ny₁ : M\nr₁ : 𝕜\nhr₁ : 0 < r₁\nhy₁ : y₁ ∈ r₁ • s\ny₂ : M\nr₂ : 𝕜\nhr₂ : 0 < r₂\nhy₂ : y₂ ∈ r₂ ... | [] | exact add_mem_add hy₁ hy₂ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.LocallyConvex.AbsConvexOpen | {
"line": 113,
"column": 4
} | {
"line": 113,
"column": 78
} | {
"line": 114,
"column": 4
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : Module 𝕜 E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyCon... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁹ : RCLike 𝕜\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : Module 𝕜 E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : IsScalarTower ℝ 𝕜 E\ninst✝³ : ContinuousSMul ℝ E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpace 𝕜 E\ns : Set E\nh... | convert! (gaugeSeminormFamily _ _).basisSets_singleton_mem ⟨s, hs⟩ one_pos | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.Convex.Gauge | {
"line": 141,
"column": 4
} | {
"line": 148,
"column": 16
} | {
"line": 149,
"column": 2
} | [
{
"pp": "case refine_1\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\na : ℝ\nhs₁ : Convex ℝ s\nhs₀ : 0 ∈ s\nhs₂ : Absorbent ℝ s\nha : 0 ≤ a\nx : E\nh : gauge s x ≤ a\nr : ℝ\nhr : a < r\n⊢ x ∈ r • s",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | have hr' := ha.trans_lt hr
rw [mem_smul_set_iff_inv_smul_mem₀ hr'.ne']
obtain ⟨δ, δ_pos, hδr, hδ⟩ := exists_lt_of_gauge_lt hs₂ (h.trans_lt hr)
suffices (r⁻¹ * δ) • δ⁻¹ • x ∈ s by rwa [smul_smul, mul_inv_cancel_right₀ δ_pos.ne'] at this
rw [mem_smul_set_iff_inv_smul_mem₀ δ_pos.ne'] at hδ
refine hs₁.s... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Gauge | {
"line": 141,
"column": 4
} | {
"line": 148,
"column": 16
} | {
"line": 149,
"column": 2
} | [
{
"pp": "case refine_1\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\na : ℝ\nhs₁ : Convex ℝ s\nhs₀ : 0 ∈ s\nhs₂ : Absorbent ℝ s\nha : 0 ≤ a\nx : E\nh : gauge s x ≤ a\nr : ℝ\nhr : a < r\n⊢ x ∈ r • s",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | have hr' := ha.trans_lt hr
rw [mem_smul_set_iff_inv_smul_mem₀ hr'.ne']
obtain ⟨δ, δ_pos, hδr, hδ⟩ := exists_lt_of_gauge_lt hs₂ (h.trans_lt hr)
suffices (r⁻¹ * δ) • δ⁻¹ • x ∈ s by rwa [smul_smul, mul_inv_cancel_right₀ δ_pos.ne'] at this
rw [mem_smul_set_iff_inv_smul_mem₀ δ_pos.ne'] at hδ
refine hs₁.s... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 287,
"column": 4
} | {
"line": 287,
"column": 16
} | {
"line": 288,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nc : 𝕜\nhc : 1 < ‖c‖\nx ... | [
"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nK : Set (E →L[𝕜] F)\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nc : 𝕜\nhc : 1 < ‖c‖\nx : E\nhx : x ... | intro e p hp | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Analysis.Convex.Gauge | {
"line": 222,
"column": 2
} | {
"line": 223,
"column": 40
} | {
"line": 224,
"column": 2
} | [
{
"pp": "case refine_1\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\na : ℝ\nhs₀ : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃a : ℝ⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s\nr : ℝ\nhr : r > 0\nb : ℝ\nhb : 0 < b\nx : E\nhx' : x ∈ s\nhs₂ : Absorbs ℝ s {(fun x ↦ b • x) x}\nhx : (fun x ↦ b • x) x ∉ a • s\nh : ∀ (c : ℝ), r ≤ ... | [
"case refine_2\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\na : ℝ\nhs₀ : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃a : ℝ⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s\nr : ℝ\nhr : r > 0\nb : ℝ\nhb : 0 < b\nx : E\nhx' : x ∈ s\nhs₂ : Absorbs ℝ s {(fun x ↦ b • x) x}\nhx : (fun x ↦ b • x) x ∉ a • s\nh : ∀ (c : ℝ), r ≤ ‖c‖ → {(fun ... | · rw [← div_eq_inv_mul]
exact div_le_one_of_le₀ hba.le ha.le | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Gauge | {
"line": 225,
"column": 20
} | {
"line": 225,
"column": 47
} | {
"line": 225,
"column": 47
} | [
{
"pp": "case refine_2\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\na : ℝ\nhs₀ : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃a : ℝ⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s\nr : ℝ\nhr : r > 0\nb : ℝ\nhb : 0 < b\nx : E\nhx' : x ∈ s\nhs₂ : Absorbs ℝ s {(fun x ↦ b • x) x}\nhx : (fun x ↦ b • x) x ∉ a • s\nh : ∀ (c : ℝ), r ≤ ... | [
"case refine_2\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\na : ℝ\nhs₀ : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃a : ℝ⦄, 0 ≤ a → a ≤ 1 → a • x ∈ s\nr : ℝ\nhr : r > 0\nb : ℝ\nhb : 0 < b\nx : E\nhx' : x ∈ s\nhs₂ : Absorbs ℝ s {(fun x ↦ b • x) x}\nhx : (fun x ↦ b • x) x ∉ a • s\nh : ∀ (c : ℝ), r ≤ ‖c‖ → {(fun ... | mul_inv_cancel_left₀ ha.ne' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.RCLike.Extend | {
"line": 54,
"column": 2
} | {
"line": 55,
"column": 23
} | {
"line": 56,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : IsScalarTower ℝ 𝕜 F\nfr : StrongDual ℝ F\nx : F\n⊢ ‖fr.extendRCLike x‖ ≤ ‖fr‖ * ‖x‖",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
... | [
"𝕜 : Type u_1\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : SeminormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : IsScalarTower ℝ 𝕜 F\nfr : StrongDual ℝ F\nx : F\n⊢ ∀ (x : F), |↑fr x| ≤ (‖fr‖₊ • normSeminorm 𝕜 F) x"
] | refine Module.Dual.norm_extendRCLike_le_seminorm (p := ‖fr‖₊ • normSeminorm 𝕜 F)
fr.toLinearMap ?_ x | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Convex.Gauge | {
"line": 250,
"column": 52
} | {
"line": 250,
"column": 63
} | {
"line": 251,
"column": 6
} | [
{
"pp": "case inr.mp\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module ℝ E\nα : Type u_3\ninst✝⁶ : Field α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : IsStrictOrderedRing α\ninst✝³ : MulActionWithZero α ℝ\ninst✝² : IsStrictOrderedModule α ℝ\ninst✝¹ : MulActionWithZero α E\ninst✝ : IsScalarTower α ℝ (Set E)\ns : Set... | [
"case inr.mp\nE : Type u_2\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module ℝ E\nα : Type u_3\ninst✝⁶ : Field α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : IsStrictOrderedRing α\ninst✝³ : MulActionWithZero α ℝ\ninst✝² : IsStrictOrderedModule α ℝ\ninst✝¹ : MulActionWithZero α E\ninst✝ : IsScalarTower α ℝ (Set E)\ns : Set E\na : α\nh... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Gauge | {
"line": 274,
"column": 20
} | {
"line": 274,
"column": 31
} | {
"line": 274,
"column": 32
} | [
{
"pp": "case inr.mp\nE : Type u_2\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : Module ℝ E\nα : Type u_3\ninst✝⁸ : Field α\ninst✝⁷ : LinearOrder α\ninst✝⁶ : IsStrictOrderedRing α\ninst✝⁵ : MulActionWithZero α ℝ\ninst✝⁴ : IsStrictOrderedModule α ℝ\ninst✝³ : MulActionWithZero α E\ninst✝² : SMulCommClass α ℝ ℝ\ninst✝¹ : Is... | [
"case inr.mp\nE : Type u_2\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : Module ℝ E\nα : Type u_3\ninst✝⁸ : Field α\ninst✝⁷ : LinearOrder α\ninst✝⁶ : IsStrictOrderedRing α\ninst✝⁵ : MulActionWithZero α ℝ\ninst✝⁴ : IsStrictOrderedModule α ℝ\ninst✝³ : MulActionWithZero α E\ninst✝² : SMulCommClass α ℝ ℝ\ninst✝¹ : IsScalarTower ... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Gauge | {
"line": 278,
"column": 19
} | {
"line": 278,
"column": 30
} | {
"line": 278,
"column": 31
} | [
{
"pp": "case inr.mpr\nE : Type u_2\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : Module ℝ E\nα : Type u_3\ninst✝⁸ : Field α\ninst✝⁷ : LinearOrder α\ninst✝⁶ : IsStrictOrderedRing α\ninst✝⁵ : MulActionWithZero α ℝ\ninst✝⁴ : IsStrictOrderedModule α ℝ\ninst✝³ : MulActionWithZero α E\ninst✝² : SMulCommClass α ℝ ℝ\ninst✝¹ : I... | [
"case inr.mpr\nE : Type u_2\ninst✝¹⁰ : AddCommGroup E\ninst✝⁹ : Module ℝ E\nα : Type u_3\ninst✝⁸ : Field α\ninst✝⁷ : LinearOrder α\ninst✝⁶ : IsStrictOrderedRing α\ninst✝⁵ : MulActionWithZero α ℝ\ninst✝⁴ : IsStrictOrderedModule α ℝ\ninst✝³ : MulActionWithZero α E\ninst✝² : SMulCommClass α ℝ ℝ\ninst✝¹ : IsScalarTower... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Gauge | {
"line": 600,
"column": 2
} | {
"line": 601,
"column": 27
} | {
"line": 603,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : Set E\nr : ℝ\nx : E\nhs : Absorbent ℝ s\nhr : 0 ≤ r\nhsr : s ⊆ closedBall 0 r\n⊢ ‖x‖ / r ≤ gauge s x",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
... | [] | rw [← gauge_closedBall hr]
exact gauge_mono hs hsr _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Gauge | {
"line": 600,
"column": 2
} | {
"line": 601,
"column": 27
} | {
"line": 603,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : Set E\nr : ℝ\nx : E\nhs : Absorbent ℝ s\nhr : 0 ≤ r\nhsr : s ⊆ closedBall 0 r\n⊢ ‖x‖ / r ≤ gauge s x",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
... | [] | rw [← gauge_closedBall hr]
exact gauge_mono hs hsr _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 616,
"column": 4
} | {
"line": 616,
"column": 16
} | {
"line": 617,
"column": 4
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nx : ℝ\nhx : x ∈ D f K\nn : ℕ → ℕ\nL : ℕ → ℕ → ℕ → F\nhn :\n ∀ (e p q : ℕ),\n n e ≤ p →\n n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / ... | [
"F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nK : Set F\nhK : IsComplete K\nP : ∀ {n : ℕ}, 0 < (1 / 2) ^ n\nx : ℝ\nhx : x ∈ D f K\nn : ℕ → ℕ\nL : ℕ → ℕ → ℕ → F\nhn :\n ∀ (e p q : ℕ),\n n e ≤ p →\n n e ≤ q → L e p q ∈ K ∧ x ∈ A f (L e p q) ((1 / 2) ^ p) ((1 / 2) ^ e) ∩ A ... | intro e p hp | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 56,
"column": 31
} | {
"line": 56,
"column": 42
} | {
"line": 56,
"column": 43
} | [
{
"pp": "case h_ind\nX : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : MeasurableSpace X\nμ : Measure X\n𝕜 : Type u_6\n𝕜' : Type u_7\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : RCLike 𝕜'\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜' F\ninst✝⁴ : Normed... | [
"case h_ind\nX : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝¹¹ : MeasurableSpace X\nμ : Measure X\n𝕜 : Type u_6\n𝕜' : Type u_7\ninst✝¹⁰ : RCLike 𝕜\ninst✝⁹ : RCLike 𝕜'\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜' F\ninst✝⁴ : NormedSpace ℝ F\nσ... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 154,
"column": 9
} | {
"line": 154,
"column": 32
} | {
"line": 154,
"column": 32
} | [
{
"pp": "case refine_3\nα : Type u_1\nE : Type u_2\ninst✝² : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : ℕ → Set α\nf : α → E\nhsm : ∀ (i : ℕ), MeasurableSet (s i)\nh_anti : Antitone s\nhfi : IntegrableOn f (s 0) μ\nbound : α → ℝ := (s 0).indicator fun a ↦ ‖f a‖... | [
"case refine_3\nα : Type u_1\nE : Type u_2\ninst✝² : MeasurableSpace α\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : ℕ → Set α\nf : α → E\nhsm : ∀ (i : ℕ), MeasurableSet (s i)\nh_anti : Antitone s\nhfi : IntegrableOn f (s 0) μ\nbound : α → ℝ := (s 0).indicator fun a ↦ ‖f a‖\nh_int_eq :... | (h_anti zero_le).subset | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 564,
"column": 32
} | {
"line": 564,
"column": 37
} | {
"line": 564,
"column": 37
} | [
{
"pp": "E : Type u_5\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nh₁f : ∀ (x : ℝ), f x = f (-x)\nh₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x\nt✝ : ℝ\nht : ‖f (min 0 t✝)‖ₑ ≠ ∞\nh : t✝ < 0\nt : ℝ\n| f (-t)",
"ppTerm": "?m.86",
"assigned": true,
"usedConstants": [
"Real",
"congrA... | [
"E : Type u_5\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nh₁f : ∀ (x : ℝ), f x = f (-x)\nh₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x\nt✝ : ℝ\nht : ‖f (min 0 t✝)‖ₑ ≠ ∞\nh : t✝ < 0\nt : ℝ\n| f t"
] | ← h₁f | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 591,
"column": 32
} | {
"line": 591,
"column": 37
} | {
"line": 591,
"column": 37
} | [
{
"pp": "E : Type u_5\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nh₁f : ∀ (x : ℝ), -f x = f (-x)\nh₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x\nt✝ : ℝ\nht : ‖f (min 0 t✝)‖ₑ ≠ ∞\nh : t✝ < 0\nt : ℝ\n| f (-t)",
"ppTerm": "?m.90",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg"... | [
"E : Type u_5\ninst✝ : NormedAddCommGroup E\nf : ℝ → E\nh₁f : ∀ (x : ℝ), -f x = f (-x)\nh₂f : ∀ (x : ℝ), 0 < x → IntervalIntegrable f volume 0 x\nt✝ : ℝ\nht : ‖f (min 0 t✝)‖ₑ ≠ ∞\nh : t✝ < 0\nt : ℝ\n| -f t"
] | ← h₁f | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 749,
"column": 50
} | {
"line": 749,
"column": 88
} | {
"line": 751,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\na b : ℝ\nf : ℝ → E\nμ : Measure ℝ\nh : a ≤ b\n⊢ ∫ (x : ℝ) in Ι a b, ‖f x‖ ∂μ = ∫ (x : ℝ) in a..b, ‖f x‖ ∂μ",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Set.Ioc",
... | [] | by rw [uIoc_of_le h, integral_of_le h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 866,
"column": 74
} | {
"line": 866,
"column": 84
} | {
"line": 866,
"column": 84
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_5\nF : Type u_6\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\nμ : Measure ℝ\ninst✝³ : RCLike 𝕜\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\na b : ℝ\nφ : ℝ → F →L[𝕜] E\nhφ : IntervalIntegrable φ μ a b\nv : F\n⊢ ((if a ≤ b th... | [] | smul_apply | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.NonIntegrable | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 27
} | {
"line": 113,
"column": 4
} | [
{
"pp": "E : Type u_1\nF : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\nf : ℝ → E\ng : ℝ → F\nk : Set ℝ\nl : Filter ℝ\ninst✝¹ : l.NeBot\ninst✝ : TendstoIxxClass Icc l l\nhl : k ∈ l\nhd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x\nhf : Tendsto (fun x ↦ ‖f x‖) l ... | [
"E : Type u_1\nF : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : NormedAddCommGroup F\nf : ℝ → E\ng : ℝ → F\nk : Set ℝ\nl : Filter ℝ\ninst✝¹ : l.NeBot\ninst✝ : TendstoIxxClass Icc l l\nhl : k ∈ l\nhd : ∀ᶠ (x : ℝ) in l, DifferentiableAt ℝ f x\nhf : Tendsto (fun x ↦ ‖f x‖) l atTop\nhfg :... | rw [← isBigO_norm_norm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.NonIntegrable | {
"line": 191,
"column": 81
} | {
"line": 198,
"column": 65
} | {
"line": 200,
"column": 0
} | [
{
"pp": "a b c : ℝ\n⊢ IntervalIntegrable (fun x ↦ (x - c)⁻¹) volume a b ↔ a = b ∨ c ∉ [[a, b]]",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Real",
"Continuous.continuousOn",
"Real.lattice",
"Real.instZero",
... | [] | by
constructor
· refine fun h => or_iff_not_imp_left.2 fun hne hc => ?_
exact not_intervalIntegrable_of_sub_inv_isBigO_punctured (isBigO_refl _ _) hne hc h
· rintro (rfl | h₀)
· exact IntervalIntegrable.refl
refine ((continuous_sub_right c).continuousOn.inv₀ ?_).intervalIntegrable
exact fun x hx =... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Haar.Quotient | {
"line": 162,
"column": 2
} | {
"line": 162,
"column": 9
} | {
"line": 163,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹⁴ : Group G\ninst✝¹³ : MeasurableSpace G\ninst✝¹² : TopologicalSpace G\ninst✝¹¹ : IsTopologicalGroup G\ninst✝¹⁰ : BorelSpace G\ninst✝⁹ : PolishSpace G\nΓ : Subgroup G\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Mea... | [
"G : Type u_1\ninst✝¹⁴ : Group G\ninst✝¹³ : MeasurableSpace G\ninst✝¹² : TopologicalSpace G\ninst✝¹¹ : IsTopologicalGroup G\ninst✝¹⁰ : BorelSpace G\ninst✝⁹ : PolishSpace G\nΓ : Subgroup G\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\nν : Measure G\ninst... | ext U _ | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.MeasureTheory.Integral.DominatedConvergence | {
"line": 518,
"column": 6
} | {
"line": 518,
"column": 99
} | {
"line": 519,
"column": 6
} | [
{
"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... | [
"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] (Function.uncu... | filter_upwards [(tendsto_order.1 I).1 _ a_lt.2, (tendsto_order.1 J).2 _ lt_b.2] with δ hδ h'δ | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Measure.Haar.Quotient | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 51
} | {
"line": 226,
"column": 2
} | [
{
"pp": "G : Type u_1\ninst✝¹⁴ : Group G\ninst✝¹³ : MeasurableSpace G\ninst✝¹² : TopologicalSpace G\ninst✝¹¹ : IsTopologicalGroup G\ninst✝¹⁰ : BorelSpace G\ninst✝⁹ : PolishSpace G\nΓ : Subgroup G\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\ninst✝⁵ ... | [
"G : Type u_1\ninst✝¹⁴ : Group G\ninst✝¹³ : MeasurableSpace G\ninst✝¹² : TopologicalSpace G\ninst✝¹¹ : IsTopologicalGroup G\ninst✝¹⁰ : BorelSpace G\ninst✝⁹ : PolishSpace G\nΓ : Subgroup G\ninst✝⁸ : Γ.Normal\ninst✝⁷ : T2Space (G ⧸ Γ)\ninst✝⁶ : SecondCountableTopology (G ⧸ Γ)\nμ : Measure (G ⧸ Γ)\ninst✝⁵ : Countable ... | obtain ⟨K⟩ := PositiveCompacts.nonempty' (α := G) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 114,
"column": 20
} | {
"line": 114,
"column": 85
} | {
"line": 114,
"column": 86
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nx : AddCircle T\nε : ℝ\nhT' : |T| = T\nI : Set ℝ := Ioc (-(T / 2)) (T / 2)\nh₁ : ε < T / 2 → Metric.closedBall 0 ε ∩ I = Metric.closedBall 0 ε\n| if ε < T / 2 then Metric.closedBall 0 ε else I",
"ppTerm": "?m.184",
"assigned": true,
"usedConstants": [
"Real",... | [
"T : ℝ\nhT : Fact (0 < T)\nx : AddCircle T\nε : ℝ\nhT' : |T| = T\nI : Set ℝ := Ioc (-(T / 2)) (T / 2)\nh₁ : ε < T / 2 → Metric.closedBall 0 ε ∩ I = Metric.closedBall 0 ε\n| if ε < T / 2 then Metric.closedBall 0 ε ∩ I else I"
] | ← if_ctx_congr (Iff.rfl : ε < T / 2 ↔ ε < T / 2) h₁ fun _ => rfl, | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 55
} | {
"line": 123,
"column": 0
} | [
{
"pp": "case neg\nT : ℝ\nhT : Fact (0 < T)\nx : AddCircle T\nε : ℝ\nhT' : |T| = T\nI : Set ℝ := Ioc (-(T / 2)) (T / 2)\nh₁ : ε < T / 2 → Metric.closedBall 0 ε ∩ I = Metric.closedBall 0 ε\nh₂ : QuotientAddGroup.mk ⁻¹' Metric.closedBall 0 ε ∩ I = if ε < T / 2 then Metric.closedBall 0 ε else I\nhε : ¬ε < T / 2\n⊢... | [] | simp [I, hε, min_eq_left (by linarith : T ≤ 2 * ε)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 55
} | {
"line": 123,
"column": 0
} | [
{
"pp": "case neg\nT : ℝ\nhT : Fact (0 < T)\nx : AddCircle T\nε : ℝ\nhT' : |T| = T\nI : Set ℝ := Ioc (-(T / 2)) (T / 2)\nh₁ : ε < T / 2 → Metric.closedBall 0 ε ∩ I = Metric.closedBall 0 ε\nh₂ : QuotientAddGroup.mk ⁻¹' Metric.closedBall 0 ε ∩ I = if ε < T / 2 then Metric.closedBall 0 ε else I\nhε : ¬ε < T / 2\n⊢... | [] | simp [I, hε, min_eq_left (by linarith : T ≤ 2 * ε)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 55
} | {
"line": 123,
"column": 0
} | [
{
"pp": "case neg\nT : ℝ\nhT : Fact (0 < T)\nx : AddCircle T\nε : ℝ\nhT' : |T| = T\nI : Set ℝ := Ioc (-(T / 2)) (T / 2)\nh₁ : ε < T / 2 → Metric.closedBall 0 ε ∩ I = Metric.closedBall 0 ε\nh₂ : QuotientAddGroup.mk ⁻¹' Metric.closedBall 0 ε ∩ I = if ε < T / 2 then Metric.closedBall 0 ε else I\nhε : ¬ε < T / 2\n⊢... | [] | simp [I, hε, min_eq_left (by linarith : T ≤ 2 * ε)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 313,
"column": 4
} | {
"line": 313,
"column": 26
} | {
"line": 314,
"column": 4
} | [
{
"pp": "case e'_7\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 : (max... | [
"case e'_7\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 : (max a₁ a₂ - t) ... | simp [a, Nat.cast_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 505,
"column": 6
} | {
"line": 513,
"column": 14
} | {
"line": 514,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR C : ℝ\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C\nθ₀ : ℝ\nhmem : θ₀ ∈ Ioc 0 (2 * π)\nhlt : ‖f (circleMap c R θ₀)‖ < C\n⊢ ∫ (θ : ℝ) in 0..2 * π, ‖deriv (circleMap c R) θ • f (ci... | [] | simp only [deriv_circleMap, norm_smul, norm_mul, norm_circleMap_zero, abs_of_pos hR, norm_I,
mul_one]
refine intervalIntegral.integral_lt_integral_of_continuousOn_of_le_of_exists_lt
Real.two_pi_pos ?_ continuousOn_const (fun θ _ => ?_) ⟨θ₀, Ioc_subset_Icc_self hmem, ?_⟩
· exact continuousO... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 505,
"column": 6
} | {
"line": 513,
"column": 14
} | {
"line": 514,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR C : ℝ\nhR : 0 < R\nhc : ContinuousOn f (sphere c R)\nhf : ∀ z ∈ sphere c R, ‖f z‖ ≤ C\nθ₀ : ℝ\nhmem : θ₀ ∈ Ioc 0 (2 * π)\nhlt : ‖f (circleMap c R θ₀)‖ < C\n⊢ ∫ (θ : ℝ) in 0..2 * π, ‖deriv (circleMap c R) θ • f (ci... | [] | simp only [deriv_circleMap, norm_smul, norm_mul, norm_circleMap_zero, abs_of_pos hR, norm_I,
mul_one]
refine intervalIntegral.integral_lt_integral_of_continuousOn_of_le_of_exists_lt
Real.two_pi_pos ?_ continuousOn_const (fun θ _ => ?_) ⟨θ₀, Ioc_subset_Icc_self hmem, ?_⟩
· exact continuousO... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 608,
"column": 73
} | {
"line": 609,
"column": 57
} | {
"line": 610,
"column": 4
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nc : ℂ\nR : ℝ\nn : ℕ\n⊢ ‖cauchyPowerSeries f c R n‖ = (2 * π)⁻¹ * ‖∮ (z : ℂ) in C(c, R), (z - c)⁻¹ ^ n • (z - c)⁻¹ • f z‖",
"ppTerm": "?m.105",
"assigned": true,
"usedConstants": [
"ContinuousMultilinearMa... | [] | by
simp [cauchyPowerSeries, norm_smul, Real.pi_pos.le] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 32
} | {
"line": 224,
"column": 0
} | [
{
"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 only [hF'] using! Hi.neg | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 198,
"column": 45
} | {
"line": 208,
"column": 70
} | {
"line": 210,
"column": 0
} | [
{
"pp": "F : Type u_1\ninst✝³ : Field F\np : F[X]\nE : Type u_2\ninst✝² : Field E\ninst✝¹ : Algebra F E\ninst✝ : Fact (map (algebraMap F E) p).Splits\n⊢ Function.Injective ⇑(galActionHom p E)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonoidHom.instMonoidHomClas... | [] | by
rw [injective_iff_map_eq_one]
intro ϕ hϕ
ext (x hx)
have key := Equiv.Perm.ext_iff.mp hϕ (rootsEquivRoots p E ⟨x, hx⟩)
change
rootsEquivRoots p E (ϕ • (rootsEquivRoots p E).symm (rootsEquivRoots p E ⟨x, hx⟩)) =
rootsEquivRoots p E ⟨x, hx⟩
at key
rw [Equiv.symm_apply_apply] at key
exact Su... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 453,
"column": 16
} | {
"line": 453,
"column": 51
} | {
"line": 455,
"column": 0
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nh0✝ : 0 ≤ R\nf : ℂ → E\nc : ℂ\ns : Set ℂ\nhs : s.Countable\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ z ∈ ball c R \\ s, DifferentiableAt ℂ f z\nhE : CompleteSpace E\nh0 : 0 < R\n⊢ (2 * ↑π * I) • (c - c) • f c = 0",
"ppTe... | [] | rw [sub_self, zero_smul, smul_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 453,
"column": 16
} | {
"line": 453,
"column": 51
} | {
"line": 455,
"column": 0
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nh0✝ : 0 ≤ R\nf : ℂ → E\nc : ℂ\ns : Set ℂ\nhs : s.Countable\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ z ∈ ball c R \\ s, DifferentiableAt ℂ f z\nhE : CompleteSpace E\nh0 : 0 < R\n⊢ (2 * ↑π * I) • (c - c) • f c = 0",
"ppTe... | [] | rw [sub_self, zero_smul, smul_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.CauchyIntegral | {
"line": 453,
"column": 16
} | {
"line": 453,
"column": 51
} | {
"line": 455,
"column": 0
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nh0✝ : 0 ≤ R\nf : ℂ → E\nc : ℂ\ns : Set ℂ\nhs : s.Countable\nhc : ContinuousOn f (closedBall c R)\nhd : ∀ z ∈ ball c R \\ s, DifferentiableAt ℂ f z\nhE : CompleteSpace E\nh0 : 0 < R\n⊢ (2 * ↑π * I) • (c - c) • f c = 0",
"ppTe... | [] | rw [sub_self, zero_smul, smul_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 240,
"column": 6
} | {
"line": 241,
"column": 35
} | {
"line": 242,
"column": 6
} | [
{
"pp": "E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nn : ℕ\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHc : ContinuousOn f (Box.Icc I)\nHd : ∀ x ∈ Box.Ioo I... | [
"E : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nn : ℕ\nI : Box (Fin (n + 1))\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHc : ContinuousOn f (Box.Icc I)\nHd : ∀ x ∈ Box.Ioo I \\ s, HasFD... | rw [Box.Icc_def, Real.volume_Icc_pi_toReal ((J k).face i).lower_le_upper,
← le_div_iff₀ (hvol_pos _)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Polynomial.Vieta | {
"line": 84,
"column": 22
} | {
"line": 84,
"column": 36
} | {
"line": 84,
"column": 37
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\ns : Multiset R\nk : ℕ\nx : Multiset R\nhx : x ∈ powersetCard k s\n⊢ (prod ∘ map Neg.neg) x = (map (fun i ↦ Function.const R (-1) i * i) x).prod",
"ppTerm": "?m.97",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"HMul... | [
"R : Type u_1\ninst✝ : CommRing R\ns : Multiset R\nk : ℕ\nx : Multiset R\nhx : x ∈ powersetCard k s\n⊢ (prod ∘ map Neg.neg) x = (map ?m.101 x).prod",
"R : Type u_1\ninst✝ : CommRing R\ns : Multiset R\nk : ℕ\nx : Multiset R\nhx : x ∈ powersetCard k s\n⊢ ∀ x_1 ∈ x, Function.const R (-1) x_1 * x_1 = ?m.101 x_1",
"... | map_congr rfl, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPolynomial.Symmetric.Defs | {
"line": 292,
"column": 4
} | {
"line": 292,
"column": 60
} | {
"line": 294,
"column": 0
} | [
{
"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! powersetCard_sup _ _ (Nat.lt_of_succ_le hn) | Lean.Elab.Tactic.Simpa.evalSimpaUsingBang | Lean.Parser.Tactic.simpaUsingBang |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 280,
"column": 4
} | {
"line": 280,
"column": 45
} | {
"line": 281,
"column": 4
} | [
{
"pp": "case inl\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHc : ContinuousOn f (Set.Icc a... | [
"case inl\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nn : ℕ\na b : Fin (n + 1) → ℝ\nhle : a ≤ b\nf : (Fin (n + 1) → ℝ) → Fin (n + 1) → E\nf' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] Fin (n + 1) → E\ns : Set (Fin (n + 1) → ℝ)\nhs : s.Countable\nHc : ContinuousOn f (Set.Icc a b)\nHd : ∀ ... | rw [this, setIntegral_empty, sum_eq_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Real.Embedding | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 24
} | {
"line": 93,
"column": 2
} | [
{
"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\nn : ℕ\nhn : x ≤ n • 1\n⊢ ∀ x_1 ∈ {r | r.num • 1 < r.den • x}, x_1 ≤ ↑n",
"ppTerm": "?h",
"assigned": ... | [
"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\nn : ℕ\nhn : x ≤ n • 1\nnum : ℤ\nden : ℕ\nden_nz✝ : den ≠ 0\nreduced✝ : num.natAbs.Coprime den\n⊢ { num := num, den := den... | intro ⟨num, den, _, _⟩ | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Data.Real.Embedding | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 24
} | {
"line": 93,
"column": 2
} | [
{
"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\nn : ℕ\nhn : x ≤ n • 1\n⊢ ∀ x_1 ∈ {r | r.num • 1 < r.den • x}, x_1 ≤ ↑n",
"ppTerm": "?h",
"assigned": ... | [
"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\nn : ℕ\nhn : x ≤ n • 1\nnum : ℤ\nden : ℕ\nden_nz✝ : den ≠ 0\nreduced✝ : num.natAbs.Coprime den\n⊢ { num := num, den := den... | intro ⟨num, den, _, _⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Data.Real.Embedding | {
"line": 111,
"column": 2
} | {
"line": 147,
"column": 44
} | {
"line": 149,
"column": 0
} | [
{
"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\n⊢ ratLt (x + y) = ratLt x + ratLt y",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"In... | [] | ext a
rw [Set.mem_add]
constructor
· /- Given `a ∈ ratLt 1 (x + y)`, find `u ∈ ratLt 1 x`, `v ∈ ratLt 1 y`
such that `u + v = a`.
In a naive attempt, one can take the denominator `d` of `a`,
and find the largest `u = p / d < x / 1`.
However, `d` could be too "coarse", and `v = a - u` could... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Real.Embedding | {
"line": 111,
"column": 2
} | {
"line": 147,
"column": 44
} | {
"line": 149,
"column": 0
} | [
{
"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\n⊢ ratLt (x + y) = ratLt x + ratLt y",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"In... | [] | ext a
rw [Set.mem_add]
constructor
· /- Given `a ∈ ratLt 1 (x + y)`, find `u ∈ ratLt 1 x`, `v ∈ ratLt 1 y`
such that `u + v = a`.
In a naive attempt, one can take the denominator `d` of `a`,
and find the largest `u = p / d < x / 1`.
However, `d` could be too "coarse", and `v = a - u` could... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Ultra | {
"line": 285,
"column": 62
} | {
"line": 288,
"column": 92
} | {
"line": 290,
"column": 0
} | [
{
"pp": "M : Type u_1\nι : Type u_2\ninst✝² : SeminormedCommGroup M\ninst✝¹ : IsUltrametricDist M\nt : Finset ι\ninst✝ : Nonempty ι\nf : ι → M\n⊢ ∃ i, (t.Nonempty → i ∈ t) ∧ ‖∏ j ∈ t, f j‖ ≤ ‖f i‖",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Norm.norm",
"False",
"Real... | [] | by
rcases t.eq_empty_or_nonempty with rfl | ht
· simp
exact (fun ⟨i, h, h'⟩ => ⟨i, fun _ ↦ h, h'⟩) <| exists_norm_finsetProd_le_of_nonempty ht f | [anonymous] | Lean.Parser.Term.byTactic |
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