module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.Calculus.MeanValue | {
"line": 148,
"column": 6
} | {
"line": 148,
"column": 86
} | [
{
"pp": "case hB'\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 ... | exact (hB' x hx).add (((hasDerivWithinAt_id x (Ici x)).sub_const a).const_mul r) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.MeanValue | {
"line": 663,
"column": 4
} | {
"line": 663,
"column": 39
} | [
{
"pp": "case h.hbc.hab\nE : 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\nf' : E → E ... | exact norm_sub_le_of_mem_segment hy | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 247,
"column": 2
} | {
"line": 247,
"column": 59
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nx : E\nn : ℕ∞ω\nf : E → F... | rcases hf.contDiffOn' hi (by simp) with ⟨U, hU, hxU, hfU⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Calculus.ContDiff.Comp | {
"line": 628,
"column": 2
} | {
"line": 645,
"column": 34
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nx₀ : E\nm n : ℕ∞ω\nf : E ... | have : ∀ k : ℕ, k ≤ m → ContDiffWithinAt 𝕜 k (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀ := by
intro k hkm
obtain ⟨v, hv, -, f', hvf', hf'⟩ :=
(hf.of_le <| by grw [hkm, hmn]).hasFDerivWithinAt_nhds (by simp) (hg.of_le hkm) hgt
refine hf'.congr_of_eventuallyEq_insert ?_
filter_upwards [hv, ht]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Comp | {
"line": 628,
"column": 2
} | {
"line": 645,
"column": 34
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\ns : Set E\nx₀ : E\nm n : ℕ∞ω\nf : E ... | have : ∀ k : ℕ, k ≤ m → ContDiffWithinAt 𝕜 k (fun x => fderivWithin 𝕜 (f x) t (g x)) s x₀ := by
intro k hkm
obtain ⟨v, hv, -, f', hvf', hf'⟩ :=
(hf.of_le <| by grw [hkm, hmn]).hasFDerivWithinAt_nhds (by simp) (hg.of_le hkm) hgt
refine hf'.congr_of_eventuallyEq_insert ?_
filter_upwards [hv, ht]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 606,
"column": 4
} | {
"line": 606,
"column": 68
} | [
{
"pp": "case inr\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 t : Set E\nf : E → F\nn : ℕ∞ω\nhf : ContDiffOn 𝕜 n f s\nhf' : ContDiffOn 𝕜 n f t\nhs : IsOpen[... | exact (hf' x hx).contDiffAt (ht.mem_nhds hx) |>.contDiffWithinAt | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 606,
"column": 4
} | {
"line": 606,
"column": 68
} | [
{
"pp": "case inr\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 t : Set E\nf : E → F\nn : ℕ∞ω\nhf : ContDiffOn 𝕜 n f s\nhf' : ContDiffOn 𝕜 n f t\nhs : IsOpen[... | exact (hf' x hx).contDiffAt (ht.mem_nhds hx) |>.contDiffWithinAt | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Basic | {
"line": 606,
"column": 4
} | {
"line": 606,
"column": 68
} | [
{
"pp": "case inr\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 t : Set E\nf : E → F\nn : ℕ∞ω\nhf : ContDiffOn 𝕜 n f s\nhf' : ContDiffOn 𝕜 n f t\nhs : IsOpen[... | exact (hf' x hx).contDiffAt (ht.mem_nhds hx) |>.contDiffWithinAt | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 557,
"column": 12
} | {
"line": 557,
"column": 30
} | [
{
"pp": "case h.e'_3.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 ... | Nat.sub_add_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 573,
"column": 12
} | {
"line": 573,
"column": 30
} | [
{
"pp": "case h.e'_4.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 ... | Nat.sub_add_cancel | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.CauSeqFilter | {
"line": 39,
"column": 6
} | {
"line": 39,
"column": 34
} | [
{
"pp": "case a\nβ : Type v\ninst✝¹ : NormedRing β\nhn : IsAbsoluteValue norm\nf : CauSeq β norm\ninst✝ : IsComplete β norm\ns : Set β\nos : IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s\nlfs : f.lim ∈ s\nε : ℝ\nhε : ε > 0\nhεs : Metric.ball f.lim ε ⊆ s\nN : ℕ\nhN : ∀ j ≥ N, ‖↑(const norm f.lim ... | rw [dist_comm, dist_eq_norm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 719,
"column": 14
} | {
"line": 720,
"column": 26
} | [
{
"pp": "case neg.emb.refine_2.inl.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 : S... | simp only [this, val_cast, val_succ, cast_mk, cases_succ', comp_apply, succ_mk,
succ_pred] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 774,
"column": 2
} | {
"line": 778,
"column": 38
} | [
{
"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) → ContinuousMultilinearMap 𝕜 (fun i ↦ E) F\nm : Fi... | ext d
by_cases h : d = m
· rw [h]
simp [applyOrderedFinpartition]
· simp [h, applyOrderedFinpartition] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 774,
"column": 2
} | {
"line": 778,
"column": 38
} | [
{
"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) → ContinuousMultilinearMap 𝕜 (fun i ↦ E) F\nm : Fi... | ext d
by_cases h : d = m
· rw [h]
simp [applyOrderedFinpartition]
· simp [h, applyOrderedFinpartition] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.FaaDiBruno | {
"line": 945,
"column": 4
} | {
"line": 945,
"column": 74
} | [
{
"pp": "case refine_1\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... | simpa [mul_assoc] using H₁.norm_left.mul <| H₅.const_mul_left c.length | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.Order.ExtendFrom | {
"line": 31,
"column": 2
} | {
"line": 31,
"column": 31
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : DenselyOrdered α\ninst✝² : OrderTopology α\ninst✝¹ : TopologicalSpace β\nf : α → β\na b : α\nla lb : β\ninst✝ : RegularSpace β\nhf : ContinuousOn f (Ioo a b)\nha : Tendsto f (𝓝[>] a) (𝓝 la)\nhb : Tends... | apply continuousOn_extendFrom | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Order.ExtendFrom | {
"line": 43,
"column": 2
} | {
"line": 51,
"column": 50
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : DenselyOrdered α\ninst✝² : OrderTopology α\ninst✝¹ : TopologicalSpace β\nf : α → β\na b : α\nla lb : β\ninst✝ : RegularSpace β\nhf : ContinuousOn f (uIoo a b)\nha : Tendsto f (𝓝[uIoo a b] a) (𝓝 la)\nhb : Tendsto... | by_cases! hab : a = b
· simp [hab]
obtain hab' | hba' := hab.lt_or_gt
· simp only [hab', uIoo_of_lt, nhdsWithin_Ioo_eq_nhdsGT, nhdsWithin_Ioo_eq_nhdsLT,
uIcc_of_lt] at ha hb hf ⊢
exact continuousOn_Icc_extendFrom_Ioo hf ha hb
· simp only [hba', uIoo_of_gt, nhdsWithin_Ioo_eq_nhdsGT, nhdsWithin_Ioo_eq_n... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.ExtendFrom | {
"line": 43,
"column": 2
} | {
"line": 51,
"column": 50
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : DenselyOrdered α\ninst✝² : OrderTopology α\ninst✝¹ : TopologicalSpace β\nf : α → β\na b : α\nla lb : β\ninst✝ : RegularSpace β\nhf : ContinuousOn f (uIoo a b)\nha : Tendsto f (𝓝[uIoo a b] a) (𝓝 la)\nhb : Tendsto... | by_cases! hab : a = b
· simp [hab]
obtain hab' | hba' := hab.lt_or_gt
· simp only [hab', uIoo_of_lt, nhdsWithin_Ioo_eq_nhdsGT, nhdsWithin_Ioo_eq_nhdsLT,
uIcc_of_lt] at ha hb hf ⊢
exact continuousOn_Icc_extendFrom_Ioo hf ha hb
· simp only [hba', uIoo_of_gt, nhdsWithin_Ioo_eq_nhdsGT, nhdsWithin_Ioo_eq_n... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.ExtendFrom | {
"line": 58,
"column": 2
} | {
"line": 58,
"column": 31
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : DenselyOrdered α\ninst✝² : OrderTopology α\ninst✝¹ : TopologicalSpace β\nf : α → β\na b : α\nla : β\ninst✝ : RegularSpace β\nhf : ContinuousOn f (Ioo a b)\nha : Tendsto f (𝓝[>] a) (𝓝 la)\nhab : a < b\n... | apply continuousOn_extendFrom | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Calculus.Deriv.MeanValue | {
"line": 339,
"column": 4
} | {
"line": 339,
"column": 21
} | [
{
"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⊢ deriv f x < C",
"usedConstants": []
}
] | exact lt_hf' x hx | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 132,
"column": 2
} | {
"line": 139,
"column": 29
} | [
{
"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\nL : E →L[𝕜] F\nr ε : ℝ\n⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (A f L r ε... | rw [Metric.isOpen_iff]
rintro x ⟨r', r'_mem, hr'⟩
obtain ⟨s, s_gt, s_lt⟩ : ∃ s : ℝ, r / 2 < s ∧ s < r' := exists_between r'_mem.1
have : s ∈ Ioc (r / 2) r := ⟨s_gt, le_of_lt (s_lt.trans_le r'_mem.2)⟩
refine ⟨r' - s, by linarith, fun x' hx' => ⟨s, this, ?_⟩⟩
have B : ball x' s ⊆ ball x r' := ball_subset (le_of... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 132,
"column": 2
} | {
"line": 139,
"column": 29
} | [
{
"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\nL : E →L[𝕜] F\nr ε : ℝ\n⊢ IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] (A f L r ε... | rw [Metric.isOpen_iff]
rintro x ⟨r', r'_mem, hr'⟩
obtain ⟨s, s_gt, s_lt⟩ : ∃ s : ℝ, r / 2 < s ∧ s < r' := exists_between r'_mem.1
have : s ∈ Ioc (r / 2) r := ⟨s_gt, le_of_lt (s_lt.trans_le r'_mem.2)⟩
refine ⟨r' - s, by linarith, fun x' hx' => ⟨s, this, ?_⟩⟩
have B : ball x' s ⊆ ball x r' := ball_subset (le_of... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 356,
"column": 4
} | {
"line": 356,
"column": 42
} | [
{
"pp": "x : ℝ\nh : |x| < 1\n⊢ Tendsto (fun n ↦ ∑ i ∈ Finset.range n, x ^ (i + 1) / (↑i + 1)) atTop (𝓝 (-log (1 - x)))",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHDiv",
"Real.instZero",
"congrArg",
"Real.in... | rw [tendsto_iff_norm_sub_tendsto_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 389,
"column": 4
} | {
"line": 389,
"column": 91
} | [
{
"pp": "x : ℝ\nh : |x| < 1\nterm : ℕ → ℝ := fun n ↦ -1 * ((-x) ^ (n + 1) / (↑n + 1)) + x ^ (n + 1) / (↑n + 1)\nh_term_eq_goal : (term ∘ fun x ↦ 2 * x) = fun k ↦ 2 * (1 / (2 * ↑k + 1)) * x ^ (2 * k + 1)\n⊢ HasSum term (log (1 + x) - log (1 - x))",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
... | have h₁ := (hasSum_pow_div_log_of_abs_lt_one (Eq.trans_lt (abs_neg x) h)).mul_left (-1) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 371,
"column": 4
} | {
"line": 373,
"column": 56
} | [
{
"pp": "case insert\n𝕜 : Type u_1\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\nn : ℕ∞ω\nι : Type u_3\nf : ι → E → F\nt : Set E\nx : E\ni : ι\ns : Finset ι\nis : i ∉ s\nIH : (∀ ... | simp only [is, Finset.sum_insert, not_false_iff]
exact (h _ (Finset.mem_insert_self i s)).add
(IH fun j hj => h _ (Finset.mem_insert_of_mem hj)) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 371,
"column": 4
} | {
"line": 373,
"column": 56
} | [
{
"pp": "case insert\n𝕜 : Type u_1\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\nn : ℕ∞ω\nι : Type u_3\nf : ι → E → F\nt : Set E\nx : E\ni : ι\ns : Finset ι\nis : i ∉ s\nIH : (∀ ... | simp only [is, Finset.sum_insert, not_false_iff]
exact (h _ (Finset.mem_insert_self i s)).add
(IH fun j hj => h _ (Finset.mem_insert_of_mem hj)) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 407,
"column": 53
} | {
"line": 409,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\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\nι : Type u_3\nf : ι → E → F\nu : Finset ι\ni : ℕ\nx : E\nhs : UniqueDiffOn 𝕜 s\nhx : x ∈ s\nh : ∀ ... | by
convert! iteratedFDerivWithin_sum_apply hs hx h
rw [Finset.sum_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 870,
"column": 2
} | {
"line": 870,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nF : Type uF\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\nn : ℕ∞ω\ninst✝ : CompleteSpace E\ne : E ≃L[𝕜] F\na✝ : Nontrivial E\nO₁ : (E →L[𝕜] E) → F →L[𝕜] E := fun f ... | convert! contDiffAt_ringInverse 𝕜 (1 : (E →L[𝕜] E)ˣ) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 278,
"column": 4
} | {
"line": 278,
"column": 32
} | [
{
"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 ... | rw [dist_comm, dist_eq_norm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 486,
"column": 75
} | {
"line": 509,
"column": 24
} | [
{
"pp": "F : Type u_1\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nf : ℝ → F\nε : ℝ\nhε : 0 < ε\nx : ℝ\nhx : DifferentiableWithinAt ℝ f (Ici x) x\n⊢ ∃ R > 0, ∀ r ∈ Ioo 0 R, x ∈ A f (derivWithin f (Ici x) x) r ε",
"usedConstants": [
"half_lt_self",
"Filter.instMembership",
"Math... | by
have := hx.hasDerivWithinAt
simp_rw [hasDerivWithinAt_iff_isLittleO, isLittleO_iff] at this
rcases mem_nhdsGE_iff_exists_Ico_subset.1 (this (half_pos hε)) with ⟨m, xm, hm⟩
refine ⟨m - x, by linarith [show x < m from xm], fun r hr => ?_⟩
have : r ∈ Ioc (r / 2) r := ⟨half_lt_self hr.1, le_rfl⟩
refine ⟨r, t... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Gauge | {
"line": 121,
"column": 86
} | {
"line": 123,
"column": 38
} | [
{
"pp": "E : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\ns : Set E\nsymmetric : ∀ x ∈ s, -x ∈ s\nx : E\n⊢ gauge s (-x) = gauge s x",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"gauge_def'",
"Eq.mpr",
"NegZeroClass.toNeg",
"SubtractionMonoid.toInvolutiveNeg",
... | by
have : ∀ x, -x ∈ s ↔ x ∈ s := fun x => ⟨fun h => by simpa using symmetric _ h, symmetric x⟩
simp_rw [gauge_def', smul_neg, this] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Gauge | {
"line": 282,
"column": 39
} | {
"line": 291,
"column": 24
} | [
{
"pp": "E : 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 : Module α E\ninst✝² : SMulCommClass α ℝ ℝ\ninst✝¹ : IsScalarTower α ℝ ℝ\ninst✝... | by
rw [← gauge_smul_left_of_nonneg (abs_nonneg a)]
obtain h | h := abs_choice a
· rw [h]
· rw [h, Set.neg_smul_set, ← Set.smul_set_neg]
congr
ext y
refine ⟨symmetric _, fun hy => ?_⟩
rw [← neg_neg y]
exact symmetric _ hy | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 305,
"column": 4
} | {
"line": 305,
"column": 86
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ns t : Set E\ninst✝⁴ : RCLike 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\nhs : Convex ℝ s\nht : Convex ℝ t\nh... | exact hfne <| (StrongDual.extendRCLikeₗ (𝕜 := 𝕜)).injective (by simpa using hzero) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 305,
"column": 4
} | {
"line": 305,
"column": 86
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ns t : Set E\ninst✝⁴ : RCLike 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\nhs : Convex ℝ s\nht : Convex ℝ t\nh... | exact hfne <| (StrongDual.extendRCLikeₗ (𝕜 := 𝕜)).injective (by simpa using hzero) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 305,
"column": 4
} | {
"line": 305,
"column": 86
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ns t : Set E\ninst✝⁴ : RCLike 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\nhs : Convex ℝ s\nht : Convex ℝ t\nh... | exact hfne <| (StrongDual.extendRCLikeₗ (𝕜 := 𝕜)).injective (by simpa using hzero) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 333,
"column": 4
} | {
"line": 333,
"column": 86
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\nx : E\ninst✝⁴ : RCLike 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\nA : Set E\nhA : Convex ℝ A\nhxA : x ∉ int... | exact hfne <| (StrongDual.extendRCLikeₗ (𝕜 := 𝕜)).injective (by simpa using hzero) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 333,
"column": 4
} | {
"line": 333,
"column": 86
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\nx : E\ninst✝⁴ : RCLike 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\nA : Set E\nhA : Convex ℝ A\nhxA : x ∉ int... | exact hfne <| (StrongDual.extendRCLikeₗ (𝕜 := 𝕜)).injective (by simpa using hzero) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 333,
"column": 4
} | {
"line": 333,
"column": 86
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\nx : E\ninst✝⁴ : RCLike 𝕜\ninst✝³ : Module 𝕜 E\ninst✝² : IsScalarTower ℝ 𝕜 E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul 𝕜 E\nA : Set E\nhA : Convex ℝ A\nhxA : x ∉ int... | exact hfne <| (StrongDual.extendRCLikeₗ (𝕜 := 𝕜)).injective (by simpa using hzero) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 607,
"column": 4
} | {
"line": 607,
"column": 32
} | [
{
"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 / ... | rw [dist_comm, dist_eq_norm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 115,
"column": 8
} | {
"line": 115,
"column": 72
} | [
{
"pp": "case pos.refine_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nε : ℝ≥0∞\nε0 : ε ≠ 0\nf : α →ₛ ℝ≥0 := piecewise s hs (const α c) (const α 0)\nh : ¬∫⁻ (x : α), ↑(f x) ∂μ = ∞... | simp only [lintegral_const, zero_mul, zero_le, ENNReal.coe_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 115,
"column": 8
} | {
"line": 115,
"column": 72
} | [
{
"pp": "case pos.refine_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nε : ℝ≥0∞\nε0 : ε ≠ 0\nf : α →ₛ ℝ≥0 := piecewise s hs (const α c) (const α 0)\nh : ¬∫⁻ (x : α), ↑(f x) ∂μ = ∞... | simp only [lintegral_const, zero_mul, zero_le, ENNReal.coe_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 115,
"column": 8
} | {
"line": 115,
"column": 72
} | [
{
"pp": "case pos.refine_2\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nc : ℝ≥0\ns : Set α\nhs : MeasurableSet s\nε : ℝ≥0∞\nε0 : ε ≠ 0\nf : α →ₛ ℝ≥0 := piecewise s hs (const α c) (const α 0)\nh : ¬∫⁻ (x : α), ↑(f x) ∂μ = ∞... | simp only [lintegral_const, zero_mul, zero_le, ENNReal.coe_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 154,
"column": 4
} | {
"line": 154,
"column": 43
} | [
{
"pp": "case h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ⇑f₁) (Function.support ⇑f₂)\nh₁ :\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 → ∃ g, (∀ (x : α), f₁ x ≤ g x) ∧ LowerSemico... | conv_lhs => rw [← ENNReal.add_halves ε] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convLHS_1 | Mathlib.Tactic.Conv.convLHS |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 217,
"column": 8
} | {
"line": 217,
"column": 59
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : μ.WeaklyRegular\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\nw : α → ℝ≥0\nwpos : ∀ (x : α), 0 < w x\nwmeas : Measurable w\nwint : ∫... | rw [lintegral_add_right _ wmeas.coe_nnreal_ennreal] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 217,
"column": 8
} | {
"line": 217,
"column": 59
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : μ.WeaklyRegular\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\nw : α → ℝ≥0\nwpos : ∀ (x : α), 0 < w x\nwmeas : Measurable w\nwint : ∫... | rw [lintegral_add_right _ wmeas.coe_nnreal_ennreal] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 217,
"column": 8
} | {
"line": 217,
"column": 59
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : BorelSpace α\nμ : Measure α\ninst✝¹ : μ.WeaklyRegular\ninst✝ : SigmaFinite μ\nf : α → ℝ≥0\nfmeas : Measurable f\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\nw : α → ℝ≥0\nwpos : ∀ (x : α), 0 < w x\nwmeas : Measurable w\nwint : ∫... | rw [lintegral_add_right _ wmeas.coe_nnreal_ennreal] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Operator.CompleteCodomain | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 97
} | [
{
"pp": "case intro\n𝕜 : Type u_1\nF : Type u_3\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nι : Type u_4\ninst✝⁴ : Finite ι\nM : ι → Type u_5\ninst✝³ : (i : ι) → NormedAddCommGroup (M i)\ninst✝² : (i : ι) → NormedSpace 𝕜 (M i)\ninst✝¹ : ∀ (i : ι), Separating... | have : Tendsto (fun n ↦ g n m) atTop (𝓝 (a m)) := ((continuous_eval_const _).tendsto _).comp ha | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 345,
"column": 4
} | {
"line": 346,
"column": 35
} | [
{
"pp": "case add\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ⇑f₁) (Function.support ⇑f₂)\nh₁ :\n ∫⁻ (x : α), ↑(f₁ x) ∂μ ≠ ∞ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 → ∃ g, (∀ (... | rcases h₁ (ENNReal.add_ne_top.1 A).1 (ENNReal.half_pos ε0).ne' with
⟨g₁, f₁_le_g₁, g₁cont, g₁int⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 355,
"column": 4
} | {
"line": 355,
"column": 43
} | [
{
"pp": "case h.e'_4\nα : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf₁ f₂ : α →ₛ ℝ≥0\na✝ : Disjoint (Function.support ⇑f₁) (Function.support ⇑f₂)\nh₁ :\n ∫⁻ (x : α), ↑(f₁ x) ∂μ ≠ ∞ →\n ∀ {ε : ℝ≥0∞},\n ε ≠ 0 → ∃ g, (... | conv_lhs => rw [← ENNReal.add_halves ε] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convLHS_1 | Mathlib.Tactic.Conv.convLHS |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 75
} | [
{
"pp": "case pos\nX : Type u_1\nE : Type u_3\ninst✝⁶ : MeasurableSpace X\nμ : Measure X\n𝕜 : Type u_6\ninst✝⁵ : RCLike 𝕜\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace ℝ E\nH : Type u_8\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nφ : X → H →L[𝕜] E\nφ_int : Inte... | · exact ((ContinuousLinearMap.apply 𝕜 E v).integral_comp_comm φ_int).symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.ContinuousMap.ContinuousMapZero | {
"line": 200,
"column": 6
} | {
"line": 200,
"column": 33
} | [
{
"pp": "X : Type u_1\nR : Type u_2\ninst✝³ : Zero R\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace R\ns : Set X\ninst✝ : Zero ↑s\nf : X → R\ng : C(↑s, R)₀\nx : ↑s\nhf : ContinuousOn f s\nhf₀ : f ↑0 = 0\n⊢ (mkD (s.restrict f) g) x = f ↑x",
"usedConstants": [
"Eq.mpr",
"congrArg",
... | mkD_of_continuousOn hf hf₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 266,
"column": 4
} | {
"line": 267,
"column": 78
} | [
{
"pp": "case pos.refine_1.hfi\nX : Type u_1\nE : Type u_3\ninst✝² : MeasurableSpace X\nμ : Measure X\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : X → ℝ≥0\nf_meas : Measurable f\ng : X → E\nhE : CompleteSpace E\nhg : Integrable g (μ.withDensity fun x ↦ ↑(f x))\nc : E\ns : Set X\ns_meas : Measura... | · refine ⟨f_meas.coe_nnreal_real.aemeasurable.aestronglyMeasurable, ?_⟩
simpa [withDensity_apply _ s_meas, hasFiniteIntegral_iff_enorm] using hs | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 936,
"column": 6
} | {
"line": 936,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nF : Type u_3\ninst✝⁸ : NormedAddCommGroup F\ninst✝⁷ : NormedSpace 𝕜 F\nα : Type u_4\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : MeasurableSpace α\ninst✝⁴ : OpensMeasurableSpace α\ninst✝³ : CompleteSpace F\ninst✝² : LocallyCompactSpace 𝕜\ninst✝¹ : Measura... | rintro - ⟨p, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 393,
"column": 2
} | {
"line": 394,
"column": 83
} | [
{
"pp": "case inr\nε : Type u_3\ninst✝² : TopologicalSpace ε\ninst✝¹ : ENormedAddMonoid ε\nf : ℝ → ε\na b : ℝ\ninst✝ : PseudoMetrizableSpace ε\nhf : IntegrableOn f [[a, b]] volume\nc : ℝ\nh : ‖f (min a b)‖ₑ ≠ ∞\nh' : ‖f (c * min (a / c) (b / c))‖ₑ ≠ ∞\nhc : c ≠ 0\n⊢ IntegrableOn (fun x ↦ f (c * x)) [[a / c, b /... | have A : MeasurableEmbedding fun x => x * c⁻¹ :=
(Homeomorph.mulRight₀ _ (inv_ne_zero hc)).isClosedEmbedding.measurableEmbedding | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 431,
"column": 2
} | {
"line": 431,
"column": 45
} | [
{
"pp": "ε : Type u_3\ninst✝² : TopologicalSpace ε\ninst✝¹ : ENormedAddMonoid ε\nf : ℝ → ε\na✝ b✝ : ℝ\ninst✝ : PseudoMetrizableSpace ε\nc a b : ℝ\nhf : IntegrableOn f [[a, b]] volume\nh : ‖f (min a b)‖ₑ ≠ ∞\nh' : ‖f (min (a - c) (b - c) + c)‖ₑ ≠ ∞\nhab : a ≤ b\nA : MeasurableEmbedding fun x ↦ x + c\n⊢ Integrabl... | rw [← map_add_right_eq_self volume c] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 519,
"column": 2
} | {
"line": 520,
"column": 79
} | [
{
"pp": "E : Type u_5\ninst✝⁴ : NormedAddCommGroup E\nμ : Measure ℝ\ninst✝³ : IsLocallyFiniteMeasure μ\ninst✝² : ConditionallyCompleteLinearOrder E\ninst✝¹ : OrderTopology E\ninst✝ : SecondCountableTopology E\nu : ℝ → E\na b : ℝ\nhu : MonotoneOn u [[a, b]]\n⊢ IntervalIntegrable u μ a b",
"usedConstants": [
... | rw [intervalIntegrable_iff]
exact (hu.integrableOn_isCompact isCompact_uIcc).mono_set Ioc_subset_Icc_self | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 519,
"column": 2
} | {
"line": 520,
"column": 79
} | [
{
"pp": "E : Type u_5\ninst✝⁴ : NormedAddCommGroup E\nμ : Measure ℝ\ninst✝³ : IsLocallyFiniteMeasure μ\ninst✝² : ConditionallyCompleteLinearOrder E\ninst✝¹ : OrderTopology E\ninst✝ : SecondCountableTopology E\nu : ℝ → E\na b : ℝ\nhu : MonotoneOn u [[a, b]]\n⊢ IntervalIntegrable u μ a b",
"usedConstants": [
... | rw [intervalIntegrable_iff]
exact (hu.integrableOn_isCompact isCompact_uIcc).mono_set Ioc_subset_Icc_self | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus | {
"line": 403,
"column": 34
} | {
"line": 403,
"column": 64
} | [
{
"pp": "ι : Type u_1\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nca cb : E\nla la' lb lb' : Filter ℝ\nlt : Filter ι\nμ : Measure ℝ\nua va ub vb : ι → ℝ\ninst✝² : IsLocallyFiniteMeasure μ\ninst✝¹ : FTCFilter a la la'\ninst✝ : FTCFilter b lb lb'\nhab : IntervalInte... | haveI := FTCFilter.meas_gen lb | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHaveI___1 | Lean.Parser.Tactic.tacticHaveI__ |
Mathlib.MeasureTheory.Measure.Haar.Quotient | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 73
} | [
{
"pp": "G : Type u_1\ninst✝¹² : Group G\ninst✝¹¹ : MeasurableSpace G\ninst✝¹⁰ : TopologicalSpace G\ninst✝⁹ : IsTopologicalGroup G\ninst✝⁸ : BorelSpace G\ninst✝⁷ : PolishSpace G\nΓ : Subgroup G\ninst✝⁶ : Γ.Normal\ninst✝⁵ : T2Space (G ⧸ Γ)\ninst✝⁴ : SecondCountableTopology (G ⧸ Γ)\ninst✝³ : Countable ↥Γ\nν : Mea... | have c_ne_top : c ≠ ∞ := measure_inter_ne_top_of_right_ne_top h𝓕_finite | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1169,
"column": 6
} | {
"line": 1169,
"column": 34
} | [
{
"pp": "E : Type u_5\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\na✝ b✝ : ℝ\nf : ℝ → E\nμ : Measure ℝ\ninst✝ : NoAtoms μ\na b : ℝ\nhf : IntegrableOn f (Iio b) μ\nhg : IntegrableOn f (Iio a) μ\nhab : a ≤ b\n⊢ ∫ (x : ℝ) in Iio b, f x ∂μ - ∫ (x : ℝ) in Iio a, f x ∂μ = ∫ (x : ℝ) in a..b, f x ∂μ",
... | integral_Iio_sub_Iio hf hab, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 55,
"column": 2
} | {
"line": 59,
"column": 51
} | [
{
"pp": "T : ℝ\nhT : 0 < T\nt : ℝ\nμ : Measure ℝ\n⊢ IsAddFundamentalDomain (↥(zmultiples T).op) (Ioc t (t + T)) μ",
"usedConstants": [
"Iff.mpr",
"Int.cast",
"Eq.mpr",
"Set.Ioc",
"nullMeasurableSet_Ioc",
"zsmul_eq_mul",
"instClosedIicTopology",
"Real.partialOr... | refine IsAddFundamentalDomain.mk' nullMeasurableSet_Ioc fun x => ?_
have : Bijective (codRestrict (fun n : ℤ => n • T) (AddSubgroup.zmultiples T) _) :=
(Equiv.ofInjective (fun n : ℤ => n • T) (zsmul_left_strictMono hT).injective).bijective
refine (AddSubgroup.equivOp _).bijective.comp this |>.existsUnique_iff.2... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 55,
"column": 2
} | {
"line": 59,
"column": 51
} | [
{
"pp": "T : ℝ\nhT : 0 < T\nt : ℝ\nμ : Measure ℝ\n⊢ IsAddFundamentalDomain (↥(zmultiples T).op) (Ioc t (t + T)) μ",
"usedConstants": [
"Iff.mpr",
"Int.cast",
"Eq.mpr",
"Set.Ioc",
"nullMeasurableSet_Ioc",
"zsmul_eq_mul",
"instClosedIicTopology",
"Real.partialOr... | refine IsAddFundamentalDomain.mk' nullMeasurableSet_Ioc fun x => ?_
have : Bijective (codRestrict (fun n : ℤ => n • T) (AddSubgroup.zmultiples T) _) :=
(Equiv.ofInjective (fun n : ℤ => n • T) (zsmul_left_strictMono hT).injective).bijective
refine (AddSubgroup.equivOp _).bijective.comp this |>.existsUnique_iff.2... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 133,
"column": 46
} | {
"line": 133,
"column": 62
} | [
{
"pp": "c : ℂ\nR θ : ℝ\n⊢ deriv (circleMap c R) θ = 0 ↔ R = 0",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"NormedCommRing.toNormedRing",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"False",
"Real",
"HMul.hMul",
"Real... | simp [I_ne_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 133,
"column": 46
} | {
"line": 133,
"column": 62
} | [
{
"pp": "c : ℂ\nR θ : ℝ\n⊢ deriv (circleMap c R) θ = 0 ↔ R = 0",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"NormedCommRing.toNormedRing",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"False",
"Real",
"HMul.hMul",
"Real... | simp [I_ne_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 133,
"column": 46
} | {
"line": 133,
"column": 62
} | [
{
"pp": "c : ℂ\nR θ : ℝ\n⊢ deriv (circleMap c R) θ = 0 ↔ R = 0",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"NormedCommRing.toNormedRing",
"InnerProductSpace.toNormedSpace",
"NormedCommRing.toSeminormedCommRing",
"False",
"Real",
"HMul.hMul",
"Real... | simp [I_ne_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.CircleIntegral | {
"line": 307,
"column": 6
} | {
"line": 307,
"column": 33
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nc : ℂ\nR : ℝ\nf₁ f₂ : ℂ → E\nhf : f₁ =ᶠ[codiscreteWithin (sphere c |R|)] f₂\nhf₁ : CircleIntegrable f₁ c R\nhR : ¬R = 0\n⊢ (fun θ ↦ f₁ (circleMap c R θ)) =ᶠ[codiscreteWithin (Ι 0 (2 * π))] fun θ ↦ f₂ (circleMap c R θ)",
"usedConstants": [
"Filter.in... | eventuallyEq_iff_exists_mem | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 233,
"column": 2
} | {
"line": 235,
"column": 72
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\np q : F[X]\nhpq : p ∣ q\nhq : q ≠ 0\n⊢ Function.Surjective ⇑(restrictDvd hpq)",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Dvd.dvd",
"MonoidHom.instFunLike",
"CommRing.toNonUnitalCommRing",
"Algebra.algebraMap",
"MonoidHom",
... | haveI := Fact.mk <|
(SplittingField.splits q).of_dvd (map_ne_zero hq) ((map_dvd_map' _).mpr hpq)
simpa only [restrictDvd_def, dif_neg hq] using restrict_surjective _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 233,
"column": 2
} | {
"line": 235,
"column": 72
} | [
{
"pp": "F : Type u_1\ninst✝ : Field F\np q : F[X]\nhpq : p ∣ q\nhq : q ≠ 0\n⊢ Function.Surjective ⇑(restrictDvd hpq)",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Dvd.dvd",
"MonoidHom.instFunLike",
"CommRing.toNonUnitalCommRing",
"Algebra.algebraMap",
"MonoidHom",
... | haveI := Fact.mk <|
(SplittingField.splits q).of_dvd (map_ne_zero hq) ((map_dvd_map' _).mpr hpq)
simpa only [restrictDvd_def, dif_neg hq] using restrict_surjective _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 291,
"column": 4
} | {
"line": 291,
"column": 80
} | [
{
"pp": "case inr\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... | have hlt : ∀ i, a i < b i := fun i => (hle i).lt_of_ne fun hi => hne ⟨i, hi⟩ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Maps.Proper.CompactlyGenerated | {
"line": 46,
"column": 2
} | {
"line": 52,
"column": 35
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactlyCoherentSpace Y\nf : X → Y\n⊢ IsProperMap f ↔ Continuous f ∧ Tendsto f (cocompact X) (cocompact Y)",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Co... | simp_rw [isProperMap_iff_isCompact_preimage,
hasBasis_cocompact.tendsto_right_iff, ← mem_preimage, eventually_mem_set, preimage_compl]
refine and_congr_right fun f_cont ↦
⟨fun H K hK ↦ (H hK).compl_mem_cocompact, fun H K hK ↦ ?_⟩
rcases mem_cocompact.mp (H K hK) with ⟨K', hK', hK'y⟩
exact hK'.of_isClosed_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Maps.Proper.CompactlyGenerated | {
"line": 46,
"column": 2
} | {
"line": 52,
"column": 35
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : T2Space Y\ninst✝ : CompactlyCoherentSpace Y\nf : X → Y\n⊢ IsProperMap f ↔ Continuous f ∧ Tendsto f (cocompact X) (cocompact Y)",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"Co... | simp_rw [isProperMap_iff_isCompact_preimage,
hasBasis_cocompact.tendsto_right_iff, ← mem_preimage, eventually_mem_set, preimage_compl]
refine and_congr_right fun f_cont ↦
⟨fun H K hK ↦ (H hK).compl_mem_cocompact, fun H K hK ↦ ?_⟩
rcases mem_cocompact.mp (H K hK) with ⟨K', hK', hK'y⟩
exact hK'.of_isClosed_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Polynomial | {
"line": 112,
"column": 4
} | {
"line": 112,
"column": 46
} | [
{
"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... | refine .atTop_of_add_const (abv (-f a)) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Complex.Polynomial.Basic | {
"line": 110,
"column": 13
} | {
"line": 110,
"column": 15
} | [
{
"pp": "p : ℚ[X]\nhp : ¬p = 0\ninj : Function.Injective ⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)\na : Finset ℂ := (p.rootSet ℂ).toFinset\nb : Finset ℂ := Finset.image (⇑(IsScalarTower.toAlgHom ℚ ℝ ℂ)) (p.rootSet ℝ).toFinset\nc : Finset ℂ :=\n Finset.image (fun a ↦ ↑a) ((galActionHom p ℂ) ((restrict p ℂ) (AlgEquiv.rest... | c, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 84
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\ninst✝³ : NumberField K\nA : Type u_2\ninst✝² : NormedField A\ninst✝¹ : IsAlgClosed A\ninst✝ : NormedAlgebra ℚ A\nB : ℝ\n⊢ {x | IsIntegral ℤ x ∧ ∀ (φ : K →+* A), ‖φ x‖ ≤ B}.Finite",
"usedConstants": [
"Real.instIsOrderedRing",
"Real.partialOrder",
"R... | let C := Nat.ceil (max B 1 ^ finrank ℚ K * (finrank ℚ K).choose (finrank ℚ K / 2)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Topology.MetricSpace.Ultra.Basic | {
"line": 81,
"column": 2
} | {
"line": 87,
"column": 69
} | [
{
"pp": "X : Type u_1\ninst✝¹ : PseudoMetricSpace X\ninst✝ : IsUltrametricDist X\nx y : X\nr s : ℝ\n⊢ ball x r ⊆ ball y s ∨ ball y s ⊆ ball x r ∨ Disjoint (ball x r) (ball y s)",
"usedConstants": [
"Eq.mpr",
"Set.disjoint_or_nonempty_inter",
"Real.instLE",
"Real",
"Preorder.toL... | wlog! hrs : r ≤ s generalizing x y r s
· rw [disjoint_comm, ← or_assoc, or_comm (b := _ ⊆ _), or_assoc]
exact this y x s r hrs.le
· refine Set.disjoint_or_nonempty_inter (ball x r) (ball y s) |>.symm.imp (fun h ↦ ?_) (Or.inr ·)
obtain ⟨hxz, hyz⟩ := (Set.mem_inter_iff _ _ _).mp h.some_mem
have hx := ball... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Ultra.Basic | {
"line": 81,
"column": 2
} | {
"line": 87,
"column": 69
} | [
{
"pp": "X : Type u_1\ninst✝¹ : PseudoMetricSpace X\ninst✝ : IsUltrametricDist X\nx y : X\nr s : ℝ\n⊢ ball x r ⊆ ball y s ∨ ball y s ⊆ ball x r ∨ Disjoint (ball x r) (ball y s)",
"usedConstants": [
"Eq.mpr",
"Set.disjoint_or_nonempty_inter",
"Real.instLE",
"Real",
"Preorder.toL... | wlog! hrs : r ≤ s generalizing x y r s
· rw [disjoint_comm, ← or_assoc, or_comm (b := _ ⊆ _), or_assoc]
exact this y x s r hrs.le
· refine Set.disjoint_or_nonempty_inter (ball x r) (ball y s) |>.symm.imp (fun h ↦ ?_) (Or.inr ·)
obtain ⟨hxz, hyz⟩ := (Set.mem_inter_iff _ _ _).mp h.some_mem
have hx := ball... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.LexOrder | {
"line": 132,
"column": 4
} | {
"line": 134,
"column": 13
} | [
{
"pp": "case inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝² : Semiring R\ninst✝¹ : LinearOrder σ\ninst✝ : WellFoundedGT σ\nφ ψ : MvPowerSeries σ R\np q : σ →₀ ℕ\nhp : φ.lexOrder = ↑(toLex p)\nhq : ψ.lexOrder = ↑(toLex q)\nu v : σ →₀ ℕ\nh' : (u, v) ≠ (p, q)\nh : u + v = p + q\nh'' : u < p\n⊢ (coeff (u, v).1) φ * (... | rw [coeff_eq_zero_of_lt_lexOrder (d := u), zero_mul]
rw [hp]
norm_cast | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.LexOrder | {
"line": 132,
"column": 4
} | {
"line": 134,
"column": 13
} | [
{
"pp": "case inr.inl\nσ : Type u_1\nR : Type u_2\ninst✝² : Semiring R\ninst✝¹ : LinearOrder σ\ninst✝ : WellFoundedGT σ\nφ ψ : MvPowerSeries σ R\np q : σ →₀ ℕ\nhp : φ.lexOrder = ↑(toLex p)\nhq : ψ.lexOrder = ↑(toLex q)\nu v : σ →₀ ℕ\nh' : (u, v) ≠ (p, q)\nh : u + v = p + q\nh'' : u < p\n⊢ (coeff (u, v).1) φ * (... | rw [coeff_eq_zero_of_lt_lexOrder (d := u), zero_mul]
rw [hp]
norm_cast | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.LexOrder | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 43
} | [
{
"pp": "case inr.inr\nσ : Type u_1\nR : Type u_2\ninst✝³ : Semiring R\ninst✝² : LinearOrder σ\ninst✝¹ : WellFoundedGT σ\ninst✝ : NoZeroDivisors R\nφ ψ : MvPowerSeries σ R\nhφ : φ ≠ 0\nhψ : ψ ≠ 0\np : σ →₀ ℕ\nhp : φ.lexOrder = ↑(toLex p)\nq : σ →₀ ℕ\nhq : ψ.lexOrder = ↑(toLex q)\n⊢ (φ * ψ).lexOrder = φ.lexOrder... | apply le_antisymm _ (lexOrder_mul_ge φ ψ) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 151,
"column": 6
} | {
"line": 170,
"column": 70
} | [
{
"pp": "case pos\nσ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nw : σ → ℕ\nf g : MvPowerSeries σ R\nhf : weightedOrder w f < ⊤\nhg : weightedOrder w g < ⊤\n⊢ weightedOrder w (f * g) ≤ weightedOrder w f + weightedOrder w g",
"usedConstants": [
"Eq.mpr",
"False",
... | let p := (f.weightedOrder w).toNat
have hp : p = f.weightedOrder w := by
simpa only [p, ENat.coe_toNat_eq_self, ← lt_top_iff_ne_top]
let q := (g.weightedOrder w).toNat
have hq : q = g.weightedOrder w := by
simpa only [q, ENat.coe_toNat_eq_self, ← lt_top_iff_ne_top]
have : f.weigh... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 151,
"column": 6
} | {
"line": 170,
"column": 70
} | [
{
"pp": "case pos\nσ : Type u_1\nR : Type u_2\ninst✝¹ : Semiring R\ninst✝ : NoZeroDivisors R\nw : σ → ℕ\nf g : MvPowerSeries σ R\nhf : weightedOrder w f < ⊤\nhg : weightedOrder w g < ⊤\n⊢ weightedOrder w (f * g) ≤ weightedOrder w f + weightedOrder w g",
"usedConstants": [
"Eq.mpr",
"False",
... | let p := (f.weightedOrder w).toNat
have hp : p = f.weightedOrder w := by
simpa only [p, ENat.coe_toNat_eq_self, ← lt_top_iff_ne_top]
let q := (g.weightedOrder w).toNat
have hq : q = g.weightedOrder w := by
simpa only [q, ENat.coe_toNat_eq_self, ← lt_top_iff_ne_top]
have : f.weigh... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Inverse | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 32
} | [
{
"pp": "σ : Type u_1\nk : Type u_3\ninst✝ : Field k\nr : k\nφ : MvPowerSeries σ k\n⊢ (r • φ)⁻¹ = r⁻¹ • φ⁻¹",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssoc... | simp [smul_eq_C_mul, mul_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.MvPowerSeries.Inverse | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 32
} | [
{
"pp": "σ : Type u_1\nk : Type u_3\ninst✝ : Field k\nr : k\nφ : MvPowerSeries σ k\n⊢ (r • φ)⁻¹ = r⁻¹ • φ⁻¹",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssoc... | simp [smul_eq_C_mul, mul_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.Inverse | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 32
} | [
{
"pp": "σ : Type u_1\nk : Type u_3\ninst✝ : Field k\nr : k\nφ : MvPowerSeries σ k\n⊢ (r • φ)⁻¹ = r⁻¹ • φ⁻¹",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"NonAssocSemiring.toAddCommMonoidWithOne",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssoc... | simp [smul_eq_C_mul, mul_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.ModelsWithJ | {
"line": 118,
"column": 35
} | {
"line": 120,
"column": 41
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nW : WeierstrassCurve R\nj✝ : R\ninst✝ : W.IsElliptic\nj : R\nh1 : Fact (IsUnit j)\nh2 : Fact (IsUnit (j - 1728))\n⊢ (ofJNe0Or1728 j).IsElliptic",
"usedConstants": [
"Eq.mpr",
"WeierstrassCurve.Δ",
"WeierstrassCurve.ofJNe0Or1728_Δ",
"HMul.hM... | by
rw [isElliptic_iff, ofJNe0Or1728_Δ]
exact (h1.out.pow 2).mul (h2.out.pow 9) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.ModelsWithJ | {
"line": 189,
"column": 19
} | {
"line": 189,
"column": 45
} | [
{
"pp": "case neg\nF : Type u_2\ninst✝¹ : Field F\nj : F\ninst✝ : DecidableEq F\nh0 : j = 0\nh3 : ¬3 = 0\nthis : Fact (IsUnit 3)\n⊢ (ofJ 0).j = 0",
"usedConstants": [
"Eq.mpr",
"WeierstrassCurve.j.congr_simp",
"WeierstrassCurve.IsElliptic",
"congrArg",
"WeierstrassCurve.j",
... | ofJ_0_of_three_ne_zero h3, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.AlgebraicGeometry.EllipticCurve.ModelsWithJ | {
"line": 194,
"column": 4
} | {
"line": 196,
"column": 59
} | [
{
"pp": "case neg\nF : Type u_2\ninst✝¹ : Field F\nj : F\ninst✝ : DecidableEq F\nh0 : ¬j = 0\nh1728 : ¬j = 1728\n⊢ (ofJ j).j = j",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"WeierstrassCurve.j.congr_simp",
"W... | · have := Fact.mk (Ne.isUnit h0)
have := Fact.mk (sub_ne_zero.2 h1728).isUnit
simp_rw [ofJ_ne_0_ne_1728 j h0 h1728, ofJNe0Or1728_j] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | {
"line": 125,
"column": 9
} | {
"line": 125,
"column": 28
} | [
{
"pp": "case h\nR : Type r\nP : Fin 3 → R\nn : Fin (Nat.succ 0).succ.succ\n⊢ ![P x, P y, P z] n = P n",
"usedConstants": [
"Fintype.elems",
"Nat.le_refl",
"HEq.refl",
"Finset",
"List.Mem.tail",
"False.elim",
"noConfusion_of_Nat",
"Membership.mem",
"Fin.... | fin_cases n <;> rfl | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {
"line": 457,
"column": 6
} | {
"line": 458,
"column": 62
} | [
{
"pp": "case neg\nF : Type u\ninst✝ : Field F\nW : Projective F\nP : Fin 3 → F\nu : F\nhu : IsUnit u\nhP : W.Nonsingular P\nhPz : ¬P z = 0\n⊢ toAffine W (u • P) = toAffine W P",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"WeierstrassCurve.Projective.toAffine",
"instHSMul",
"IsDom... | rw [toAffine_of_Z_ne_zero ((nonsingular_smul P hu).mpr hP) <| mul_ne_zero hu.ne_zero hPz,
toAffine_of_Z_ne_zero hP hPz, Affine.Point.some.injEq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 58,
"column": 59
} | {
"line": 61,
"column": 66
} | [
{
"pp": "A : Type u_1\ninst✝⁴ : CommRing A\ninst✝³ : IsDedekindDomain A\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra A K\ninst✝ : IsFractionRing A K\nv : HeightOneSpectrum A\nhv : Finite (A ⧸ v.asIdeal)\n⊢ IsPrincipalIdealRing ↥(valuation K v).integer",
"usedConstants": [
"Int.instAddCommGroup",
... | by
rw [(Valuation.integer.integers (v.valuation K)).isPrincipalIdealRing_iff_not_denselyOrdered,
WithZero.denselyOrdered_set_iff_subsingleton]
simpa using (v.valuation K).toMonoidWithZeroHom.range_nontrivial | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyOpen | {
"line": 120,
"column": 4
} | {
"line": 123,
"column": 79
} | [
{
"pp": "case inr\nX✝ Y : Scheme\nf✝ : X✝ ⟶ Y\nX : Scheme\nR : CommRingCat\nf : X ⟶ Spec R\ninst✝ : LocallyOfFinitePresentation f\nhf : GeneralizingMap ⇑f\nthis :\n ∀ {X Y : Scheme} (f : X ⟶ Y) {X : Scheme} (R : CommRingCat) (f : X ⟶ Spec R) [LocallyOfFinitePresentation f],\n GeneralizingMap ⇑f →\n (∃ ... | rw [IsZariskiLocalAtSource.iff_of_openCover (P := topologically IsOpenMap) X.affineCover]
intro i
refine this f _ _ ?_ ⟨_, rfl⟩
exact IsZariskiLocalAtSource.comp (P := topologically GeneralizingMap) hf _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.AlgebraicGeometry.Morphisms.UniversallyOpen | {
"line": 120,
"column": 4
} | {
"line": 123,
"column": 79
} | [
{
"pp": "case inr\nX✝ Y : Scheme\nf✝ : X✝ ⟶ Y\nX : Scheme\nR : CommRingCat\nf : X ⟶ Spec R\ninst✝ : LocallyOfFinitePresentation f\nhf : GeneralizingMap ⇑f\nthis :\n ∀ {X Y : Scheme} (f : X ⟶ Y) {X : Scheme} (R : CommRingCat) (f : X ⟶ Spec R) [LocallyOfFinitePresentation f],\n GeneralizingMap ⇑f →\n (∃ ... | rw [IsZariskiLocalAtSource.iff_of_openCover (P := topologically IsOpenMap) X.affineCover]
intro i
refine this f _ _ ?_ ⟨_, rfl⟩
exact IsZariskiLocalAtSource.comp (P := topologically GeneralizingMap) hf _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.AlgebraicGeometry.Morphisms.SchemeTheoreticallyDominant | {
"line": 97,
"column": 6
} | {
"line": 97,
"column": 39
} | [
{
"pp": "X Y : Scheme\nZ : Scheme\nS : Scheme\nf✝ : X ⟶ S\ng✝ : Y ⟶ S\nf : X ⟶ S\ng : Y ⟶ S\ninst✝² : IsSchemeTheoreticallyDominant f\ninst✝¹ : QuasiCompact f\ninst✝ : Flat g\n⊢ IsSchemeTheoreticallyDominant (pullback.snd f g)",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Limits.pullback",
... | isSchemeTheoreticallyDominant_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Sites.Hypercover.SheafOfTypes | {
"line": 184,
"column": 4
} | {
"line": 184,
"column": 36
} | [
{
"pp": "case refine_1\nC : Type u_1\ninst✝ : Category.{v_1, u_1} C\nX : C\nE : PreOneHypercover X\nF : Cᵒᵖ ⥤ Type u_2\nh₁ : E.IsStronglySheafFor F\nh₂ : ∀ ⦃Y : C⦄ (f : Y ⟶ X), Presieve.IsSeparatedFor F (Sieve.pullback f E.sieve₀).arrows\nS : Sieve X\nH : ∀ (i : E.I₀), Presieve.IsSheafFor F (Sieve.pullback (E.f... | obtain ⟨W, w, u, ⟨i⟩, heq⟩ := hg | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.CategoryTheory.Sites.Preserves | {
"line": 88,
"column": 2
} | {
"line": 88,
"column": 48
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\nF : Cᵒᵖ ⥤ Type w\nα : Type u_1\ninst✝ : Small.{w, u_1} α\nX : α → C\nc : Cofan X\nhc : IsColimit c\nthis : HasCoproduct X\n⊢ have this := ⋯;\n (piComparison F fun x ↦ op (X x)) =\n F.map (opCoproductIsoProduct' hc (productIsProduct fun x ↦ op (X x))).inv ≫\n ... | dsimp only [Equalizer.Presieve.Arrows.forkMap] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.AlgebraicGeometry.Sites.BigZariski | {
"line": 77,
"column": 23
} | {
"line": 77,
"column": 43
} | [
{
"pp": "⊢ forgetToTop.IsContinuous (grothendieckTopology IsOpenImmersion) TopCat.grothendieckTopology",
"usedConstants": [
"Eq.mpr",
"AlgebraicGeometry.Scheme",
"congrArg",
"TopCat.instCategory",
"AlgebraicGeometry.Scheme.grothendieckTopology",
"AlgebraicGeometry.Scheme.... | grothendieckTopology | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Unramified.Basic | {
"line": 95,
"column": 20
} | {
"line": 95,
"column": 21
} | [
{
"pp": "R : Type v\ninst✝⁶ : CommRing R\nA : Type u\ninst✝⁵ : CommRing A\ninst✝⁴ : Algebra R A\ninst✝³ : Small.{w, u} A\nH :\n ∀ ⦃B : Type w⦄ [inst : CommRing B] [inst_1 : Algebra R B] (I : Ideal B),\n I ^ 2 = ⊥ → Function.Injective (Ideal.Quotient.mkₐ R I).comp\nB : Type u\ninst✝² : CommRing B\ninst✝¹ : S... | I | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Unramified.Basic | {
"line": 130,
"column": 14
} | {
"line": 130,
"column": 15
} | [
{
"pp": "case h₁\nR : Type v\ninst✝⁵ : CommRing R\nA : Type u\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nB✝ : Type w\ninst✝² : CommRing B✝\ninst✝¹ : FormallyUnramified R A\nI : Ideal B✝\nhI : IsNilpotent I\nB : Type w\ninst✝ : CommRing B\n⊢ ∀ (I : Ideal B), I ^ 2 = ⊥ → ∀ [inst : Algebra R B], Function.Injectiv... | I | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Unramified.Basic | {
"line": 131,
"column": 14
} | {
"line": 131,
"column": 15
} | [
{
"pp": "case h₂\nR : Type v\ninst✝⁵ : CommRing R\nA : Type u\ninst✝⁴ : CommRing A\ninst✝³ : Algebra R A\nB✝ : Type w\ninst✝² : CommRing B✝\ninst✝¹ : FormallyUnramified R A\nI : Ideal B✝\nhI : IsNilpotent I\nB : Type w\ninst✝ : CommRing B\n⊢ ∀ (I J : Ideal B),\n I ≤ J →\n (∀ [inst : Algebra R B], Functi... | I | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.Unramified.Basic | {
"line": 202,
"column": 14
} | {
"line": 202,
"column": 15
} | [
{
"pp": "R : Type u_1\ninst✝⁷ : CommRing R\nA : Type u_2\nB : Type u_3\ninst✝⁶ : CommRing A\ninst✝⁵ : Algebra R A\ninst✝⁴ : CommRing B\ninst✝³ : Algebra R B\ninst✝² : FormallyUnramified R A\ne : A ≃ₐ[R] B\nC : Type u_3\ninst✝¹ : CommRing C\ninst✝ : Algebra R C\n⊢ ∀ (I : Ideal C), I ^ 2 = ⊥ → Function.Injective ... | I | Lean.Elab.Tactic.evalIntro | ident |
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