module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.Asymptotics.TVS | {
"line": 776,
"column": 49
} | {
"line": 776,
"column": 81
} | {
"line": 777,
"column": 6
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nf : α → E\ng : α → F\nl : Filter α\nc : 𝕜\nhc : 1 < ‖c‖₊\nhc₀ : 0 < ‖c‖₊\nh :\n ∀ (i... | [] | simp only [div_eq_mul_inv]; ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 800,
"column": 4
} | {
"line": 800,
"column": 20
} | {
"line": 801,
"column": 4
} | [
{
"pp": "case mp\nα : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nf : α → E\ng : α → F\nl : Filter α\nc : 𝕜\nhc : 1 < ‖c‖₊\nh : ∀ (i : ℝ), 0 <... | [
"case mp\nα : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nf : α → E\ng : α → F\nl : Filter α\nc : 𝕜\nhc : 1 < ‖c‖₊\nh : ∀ (i : ℝ), 0 < i → ∃ j, 0 ... | norm_cast at hr₀ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticNorm_cast___1 | Lean.Parser.Tactic.tacticNorm_cast__ |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 816,
"column": 32
} | {
"line": 816,
"column": 48
} | {
"line": 817,
"column": 4
} | [
{
"pp": "case mpr\nα : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nf : α → E\ng : α → F\nl : Filter α\nc : 𝕜\nhc : 1 < ‖c‖₊\nhc₀ : 0 < ‖c‖₊\nC... | [
"case mpr\nα : Type u_1\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NormedSpace 𝕜 F\nf : α → E\ng : α → F\nl : Filter α\nc : 𝕜\nhc : 1 < ‖c‖₊\nhc₀ : 0 < ‖c‖₊\nC : ℝ≥0\nhC :... | norm_cast at hr₀ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticNorm_cast___1 | Lean.Parser.Tactic.tacticNorm_cast__ |
Mathlib.Analysis.Normed.Module.Multilinear.Curry | {
"line": 243,
"column": 2
} | {
"line": 243,
"column": 59
} | {
"line": 245,
"column": 0
} | [
{
"pp": "𝕜 : Type u\nn : ℕ\nEi : Fin n.succ → Type wEi\nG : Type wG\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : (i : Fin n.succ) → NormedAddCommGroup (Ei i)\ninst✝² : (i : Fin n.succ) → NormedSpace 𝕜 (Ei i)\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nf : ContinuousMultilinearMap 𝕜 Ei G\nm : ... | [] | rw [uncurryRight_apply, curryRight_apply, snoc_init_self] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 53
} | {
"line": 204,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedRing E\ninst✝ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\nr : ℝ≥0\nhr : r ≠ 0\nhc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 ↑r)\nr' : ℝ≥0\nhr' : ↑r' < ↑r⁻¹\n⊢ ↑r' ≤ (ofScalars E c).radius",
"ppTerm": "?m.45",
"assigne... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedRing E\ninst✝ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\nr : ℝ≥0\nhr : r ≠ 0\nhc : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop (𝓝 ↑r)\nr' : ℝ≥0\nhr' : r' * r < 1\n⊢ ↑r' ≤ (ofScalars E c).radius"
] | rw [coe_lt_coe, NNReal.lt_inv_iff_mul_lt hr] at hr' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Basic | {
"line": 266,
"column": 39
} | {
"line": 275,
"column": 68
} | {
"line": 277,
"column": 0
} | [
{
"pp": "𝕜 : 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\nf g : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nh : HasFPowerSeriesWithinAt f p s x\nh' :... | [] | by
rcases h with ⟨r, hr⟩
obtain ⟨ε, εpos, hε⟩ : ∃ ε > 0, Metric.eball x ε ∩ s ⊆ {y | g y = f y} :=
EMetric.mem_nhdsWithin_iff.1 h'
let r' := min r ε
refine ⟨r', ?_⟩
have := hr.of_le (r' := r') (by simp [r', εpos, hr.r_pos]) (min_le_left _ _)
apply this.congr _ h''
intro z hz
exact hε ⟨Metric.eball_s... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.OfScalars | {
"line": 284,
"column": 20
} | {
"line": 284,
"column": 41
} | {
"line": 284,
"column": 42
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing E\ninst✝¹ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\ninst✝ : NormOneClass E\nhc✝ : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop atTop\nr : ℝ≥0\nhr : ↑r < (ofScalars E c).radius\nthis : 0 < r\nn : ℕ\nhc : 2 * ↑r⁻¹ ≤ ‖c n.succ‖ / ... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NormedRing E\ninst✝¹ : NormedAlgebra 𝕜 E\nc : ℕ → 𝕜\ninst✝ : NormOneClass E\nhc✝ : Tendsto (fun n ↦ ‖c n.succ‖ / ‖c n‖) atTop atTop\nr : ℝ≥0\nhr : ↑r < (ofScalars E c).radius\nthis : 0 < r\nn : ℕ\nhc : 2 * ↑r⁻¹ ≤ ‖c n.succ‖ / ‖c n‖\nhn : ... | div_mul_cancel_left₀, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.Basic | {
"line": 468,
"column": 32
} | {
"line": 470,
"column": 27
} | {
"line": 472,
"column": 0
} | [
{
"pp": "𝕜 : 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\nf g : E → F\ns : Set E\nx : E\nhf : AnalyticWithinAt 𝕜 f s x\nhs : g =ᶠ[𝓝[s] x] f\nhx : g x = f x\n⊢ Analy... | [] | by
rcases hf with ⟨p, hp⟩
exact ⟨p, hp.congr hs hx⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.Basic | {
"line": 475,
"column": 2
} | {
"line": 476,
"column": 49
} | {
"line": 478,
"column": 0
} | [
{
"pp": "𝕜 : 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\nf g : E → F\ns : Set E\nx : E\nhf : AnalyticWithinAt 𝕜 f s x\nhs : g =ᶠ[𝓝[insert x s] x] f\n⊢ AnalyticWith... | [] | apply hf.congr_of_eventuallyEq (nhdsWithin_mono x (subset_insert x s) hs)
apply mem_of_mem_nhdsWithin (mem_insert x s) hs | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Basic | {
"line": 475,
"column": 2
} | {
"line": 476,
"column": 49
} | {
"line": 478,
"column": 0
} | [
{
"pp": "𝕜 : 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\nf g : E → F\ns : Set E\nx : E\nhf : AnalyticWithinAt 𝕜 f s x\nhs : g =ᶠ[𝓝[insert x s] x] f\n⊢ AnalyticWith... | [] | apply hf.congr_of_eventuallyEq (nhdsWithin_mono x (subset_insert x s) hs)
apply mem_of_mem_nhdsWithin (mem_insert x s) hs | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Basic | {
"line": 640,
"column": 8
} | {
"line": 640,
"column": 33
} | {
"line": 641,
"column": 8
} | [
{
"pp": "𝕜 : 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\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\ny : E\nhf : HasFPowerSeriesWithin... | [
"𝕜 : 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\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\ny : E\nhf : HasFPowerSeriesWithinOnBall f p s... | rw [sum_Ico_eq_sum_range] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Basic | {
"line": 639,
"column": 56
} | {
"line": 642,
"column": 23
} | {
"line": 643,
"column": 4
} | [
{
"pp": "𝕜 : 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\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\ny : E\nhf : HasFPowerSeriesWithin... | [] | by
rw [sum_Ico_eq_sum_range]
congr with i
rw [add_comm k] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.Composition | {
"line": 687,
"column": 2
} | {
"line": 687,
"column": 45
} | {
"line": 689,
"column": 2
} | [
{
"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\nq : FormalMultilinearSeries 𝕜 F G\n... | [
"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\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMu... | rw [Finset.range_eq_Ico, Finset.sum_sigma'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Inverse | {
"line": 132,
"column": 4
} | {
"line": 150,
"column": 14
} | {
"line": 151,
"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\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nx : E\nh : p 1 = (continuousMultilinearCurryFin1 𝕜 E F)... | [
"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\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nx : E\nh : p 1 = (continuousMultilinearCurryFin1 𝕜 E F).symm ↑i\nn✝... | have D :
(p.leftInv i x (n + 2) fun j : Fin (n + 2) => p 1 fun _ => v j) =
-∑ c ∈ {c : Composition (n + 2) | c.length < n + 2}.toFinset,
(p.leftInv i x c.length) (p.applyComposition c v) := by
simp only [leftInv, _root_.neg_apply, neg_inj, _root_.sum_apply]
convert!
(sum_to... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Analytic.Basic | {
"line": 886,
"column": 2
} | {
"line": 886,
"column": 81
} | {
"line": 888,
"column": 0
} | [
{
"pp": "𝕜 : 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\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\nhf : HasFPowerSeriesWithinOnBall ... | [] | exact le_principal_iff.2 (inter_mem_nhdsWithin _ (Metric.eball_mem_nhds _ r'0)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Analytic.Within | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 70
} | {
"line": 216,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : E → F\ns : Set E\nx : E\nhf : AnalyticWithinAt 𝕜 f s x\n⊢ ∀ᶠ (y : E) in 𝓝[s]... | [
"𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : CompleteSpace F\nf : E → F\ns : Set E\nx : E\nhf : AnalyticWithinAt 𝕜 f s x\ng : E → F\nhfg : f =ᶠ[𝓝[insert x... | obtain ⟨g, hfg, hga⟩ := analyticWithinAt_iff_exists_analyticAt.mp hf | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Analytic.Constructions | {
"line": 536,
"column": 2
} | {
"line": 536,
"column": 65
} | {
"line": 538,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nι : Type u_9\ninst✝² : Fintype ι\ne : E\nFm : ι → Type u_10\ninst✝¹ : (i : ι) → NormedAddCommGroup (Fm i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (Fm i)\nf : (i : ι) → E → Fm i\ns : Set... | [] | exact ⟨r, (hasFPowerSeriesWithinOnBall_pi_iff hr.r_pos).1 hr i⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Analytic.Constructions | {
"line": 946,
"column": 4
} | {
"line": 946,
"column": 39
} | {
"line": 947,
"column": 4
} | [
{
"pp": "case inr\n𝕜 : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\nA : Type u_7\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : HasSummableGeomSeries A\nz : Aˣ\nhA : Nontrivial A\n⊢ AnalyticAt 𝕜 Ring.inverse ↑z",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"NormedR... | [
"case inr\n𝕜 : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\nA : Type u_7\ninst✝² : NormedRing A\ninst✝¹ : NormedAlgebra 𝕜 A\ninst✝ : HasSummableGeomSeries A\nz : Aˣ\nhA : Nontrivial A\nf1 : A → A := fun a ↦ a * z.inv\n⊢ AnalyticAt 𝕜 Ring.inverse ↑z"
] | let f1 : A → A := fun a ↦ a * z.inv | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Calculus.FDeriv.Add | {
"line": 705,
"column": 2
} | {
"line": 705,
"column": 70
} | {
"line": 707,
"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 g : E → F\nhg : Differentiable 𝕜 g\n⊢ Differentiable 𝕜 (f - g) ↔ Differentiable 𝕜 f",
"ppTerm": "?m... | [] | simp only [sub_eq_add_neg, differentiable_neg_iff, hg, add_iff_left] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.FDeriv.Add | {
"line": 705,
"column": 2
} | {
"line": 705,
"column": 70
} | {
"line": 707,
"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 g : E → F\nhg : Differentiable 𝕜 g\n⊢ Differentiable 𝕜 (f - g) ↔ Differentiable 𝕜 f",
"ppTerm": "?m... | [] | simp only [sub_eq_add_neg, differentiable_neg_iff, hg, add_iff_left] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Add | {
"line": 705,
"column": 2
} | {
"line": 705,
"column": 70
} | {
"line": 707,
"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 g : E → F\nhg : Differentiable 𝕜 g\n⊢ Differentiable 𝕜 (f - g) ↔ Differentiable 𝕜 f",
"ppTerm": "?m... | [] | simp only [sub_eq_add_neg, differentiable_neg_iff, hg, add_iff_left] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Add | {
"line": 927,
"column": 79
} | {
"line": 928,
"column": 47
} | {
"line": 930,
"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\nx a : E\n⊢ DifferentiableAt 𝕜 (fun x ↦ f (x - a)) x ↔ DifferentiableAt 𝕜 f (x - a)",
"ppTer... | [] | by
simp [DifferentiableAt, hasFDerivAt_comp_sub] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.FDeriv.Equiv | {
"line": 218,
"column": 2
} | {
"line": 224,
"column": 90
} | {
"line": 226,
"column": 0
} | [
{
"pp": "𝕜 : 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\niso : E ≃L[𝕜] F\nf : F → G\ns : Set... | [] | by_cases h : DifferentiableWithinAt 𝕜 f s (iso x)
· exact (iso.comp_right_hasFDerivWithinAt_iff.2 h.hasFDerivWithinAt).fderivWithin hxs
· have : ¬DifferentiableWithinAt 𝕜 (f ∘ iso) (iso ⁻¹' s) x := by
intro h'
exact h (iso.comp_right_differentiableWithinAt_iff.1 h')
rw [fderivWithin_zero_of_not_di... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.FDeriv.Equiv | {
"line": 218,
"column": 2
} | {
"line": 224,
"column": 90
} | {
"line": 226,
"column": 0
} | [
{
"pp": "𝕜 : 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\niso : E ≃L[𝕜] F\nf : F → G\ns : Set... | [] | by_cases h : DifferentiableWithinAt 𝕜 f s (iso x)
· exact (iso.comp_right_hasFDerivWithinAt_iff.2 h.hasFDerivWithinAt).fderivWithin hxs
· have : ¬DifferentiableWithinAt 𝕜 (f ∘ iso) (iso ⁻¹' s) x := by
intro h'
exact h (iso.comp_right_differentiableWithinAt_iff.1 h')
rw [fderivWithin_zero_of_not_di... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Calculus.FDeriv.Equiv | {
"line": 339,
"column": 6
} | {
"line": 339,
"column": 52
} | {
"line": 339,
"column": 52
} | [
{
"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\nf' : E →L[𝕜] F\nx : E\ns : Set E\nc : F\nh : HasFDerivWithinAt f f' s x\nhf' : ∃ C, Antilipschit... | [
"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\nf' : E →L[𝕜] F\nx : E\ns : Set E\nc : F\nh : HasFDerivWithinAt f f' s x\nhf' : ∃ C, AntilipschitzWith C ⇑f'\... | ← eventually_map (m := f) (P := fun z ↦ z ≠ c) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Completion | {
"line": 139,
"column": 2
} | {
"line": 141,
"column": 20
} | {
"line": 143,
"column": 0
} | [
{
"pp": "case h\nα : Type u\ninst✝ : PseudoMetricSpace α\ns : Set (Completion α × Completion α)\n⊢ Directed (fun x1 x2 ↦ x1 ≥ x2) fun ε ↦ 𝓟 {p | dist p.1 p.2 < ↑ε}",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"Real",
"Preorder.toLT",
"Latti... | [] | · rintro ⟨r, hr⟩ ⟨p, hp⟩
use ⟨min r p, lt_min hr hp⟩
simp +contextual | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.FDeriv.CompCLM | {
"line": 108,
"column": 84
} | {
"line": 111,
"column": 18
} | {
"line": 113,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nG : Type u_4\ninst✝³ : NormedAddCommGroup G\ninst✝² : NormedSpace 𝕜 G\nx : E\ns : Set E\nH : Type u_5\ninst✝¹ : NormedAddCommGroup H\ninst✝ : NormedSpace 𝕜 H\nc : E → G →L[𝕜] H... | [] | by
-- `by exact` to solve unification issues.
exact (isBoundedBilinearMap_apply.hasFDerivAt (c x, u x)).comp_hasFDerivWithinAt x
(hc.prodMk hu) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 759,
"column": 2
} | {
"line": 760,
"column": 79
} | {
"line": 762,
"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\nf : E → F\nn : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUpTo n f p\nhn : n ≠ 0\nx : E\n⊢... | [] | rw [← hasFDerivWithinAt_univ]
exact (hasFTaylorSeriesUpToOn_univ_iff.2 h).hasFDerivWithinAt hn (mem_univ _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 759,
"column": 2
} | {
"line": 760,
"column": 79
} | {
"line": 762,
"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\nf : E → F\nn : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUpTo n f p\nhn : n ≠ 0\nx : E\n⊢... | [] | rw [← hasFDerivWithinAt_univ]
exact (hasFTaylorSeriesUpToOn_univ_iff.2 h).hasFDerivWithinAt hn (mem_univ _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 159,
"column": 96
} | {
"line": 160,
"column": 81
} | {
"line": 162,
"column": 0
} | [
{
"pp": "a b : ℝ\n⊢ frontier {z | a ≤ z.re ∧ b ≤ z.im} = {z | a ≤ z.re ∧ z.im = b ∨ z.re = a ∧ b ≤ z.im}",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"NormedCommRing.toSeminormedCommRing",
"frontier",
"Real.partialOrder",
"Real",
... | [] | by
simpa only [closure_Ici, frontier_Ici] using! frontier_reProdIm (Ici a) (Ici b) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.ReImTopology | {
"line": 198,
"column": 74
} | {
"line": 199,
"column": 73
} | {
"line": 201,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_2\nf : ι → α → ℂ\np : Filter ι\ng : α → ℂ\nK : Set α\nhf : TendstoUniformlyOn f g p K\n⊢ TendstoUniformlyOn (fun n x ↦ (f n x).im) (fun y ↦ (g y).im) p K",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Re... | [] | by
apply UniformContinuous.comp_tendstoUniformlyOn uniformContinuous_im hf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Complex.CircleMap | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 78
} | {
"line": 78,
"column": 2
} | [
{
"pp": "R₁ R₂ θ₁ θ₂ : ℝ\n⊢ circleMap 0 R₁ θ₁ / circleMap 0 R₂ θ₂ = circleMap 0 (R₁ / R₂) (θ₁ - θ₂)",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"congrArg",
... | [
"R₁ R₂ θ₁ θ₂ : ℝ\n⊢ ↑R₁ * cexp (↑θ₁ * I) / (↑R₂ * cexp (↑θ₂ * I)) = ↑R₁ / ↑R₂ * (cexp (↑θ₁ * I) / cexp (↑θ₂ * I))"
] | simp only [circleMap_zero, ofReal_div, ofReal_sub, sub_mul, Complex.exp_sub] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Complex.CircleMap | {
"line": 131,
"column": 55
} | {
"line": 141,
"column": 41
} | {
"line": 143,
"column": 0
} | [
{
"pp": "a b R : ℝ\nc : ℂ\nh_R : R ≠ 0\nh_dist : |a - b| < 2 * π\nh : circleMap c R a = circleMap c R b\n⊢ a = b",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Int.... | [] | by
rw [circleMap_eq_circleMap_iff c h_R] at h
obtain ⟨n, hn⟩ := h
simp only [show n * (2 * π * I) = (n * 2 * π) * I by ring, ← add_mul, mul_eq_mul_right_iff,
I_ne_zero, or_false] at hn
norm_cast at hn
simp only [hn, Int.cast_mul, Int.cast_ofNat, mul_assoc, add_sub_cancel_left, abs_mul,
Nat.abs_ofNat, ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.IsolatedZeros | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 36
} | {
"line": 157,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nm n : ℤ\nhm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ m • g z\nhn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in ... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nz₀ : 𝕜\nm n : ℤ\nhm : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f z = (z - z₀) ^ m • g z\nhn : ∃ g, AnalyticAt 𝕜 g z₀ ∧ g z₀ ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] z₀, f ... | let ⟨j, hj_an, hj_ne, hj_eq⟩ := hn | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Calculus.LogDeriv | {
"line": 138,
"column": 6
} | {
"line": 138,
"column": 68
} | {
"line": 139,
"column": 4
} | [
{
"pp": "case inr.mp\n𝕜 : Type u_1\n𝕜' : Type u_2\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : NontriviallyNormedField 𝕜'\ninst✝¹ : NormedAlgebra 𝕜 𝕜'\ninst✝ : IsRCLikeNormedField 𝕜\nf g : 𝕜 → 𝕜'\ns : Set 𝕜\nhf : DifferentiableOn 𝕜 f s\nhg : DifferentiableOn 𝕜 g s\nhs2 : IsOpen[PseudoMetricSpace.to... | [] | grind [Pi.mul_apply, Pi.inv_apply, Pi.smul_apply, smul_eq_mul] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Topology.OpenPartialHomeomorph.IsImage | {
"line": 265,
"column": 79
} | {
"line": 267,
"column": 49
} | {
"line": 269,
"column": 0
} | [
{
"pp": "X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\ns : Set X\n⊢ e.restr (e.source ∩ s) = e.restr s",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"interior_inter",
"OpenPartialHomeomorph.open_source",
... | [] | by
refine OpenPartialHomeomorph.ext _ _ (fun x => rfl) (fun x => rfl) ?_
simp [e.open_source.interior_eq, ← inter_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Algebra.Exponential | {
"line": 188,
"column": 2
} | {
"line": 188,
"column": 90
} | {
"line": 189,
"column": 2
} | [
{
"pp": "case pos\n𝔸 : Type u_2\ninst✝² : Ring 𝔸\ninst✝¹ : TopologicalSpace 𝔸\ninst✝ : IsTopologicalRing 𝔸\nh✝ : Nonempty (Algebra ℚ 𝔸)\n⊢ (expSeries ℚ 𝔸).sum 0 = 1",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Analysis.Normed.Algebra.Exponen... | [
"case neg\n𝔸 : Type u_2\ninst✝² : Ring 𝔸\ninst✝¹ : TopologicalSpace 𝔸\ninst✝ : IsTopologicalRing 𝔸\nh✝ : ¬Nonempty (Algebra ℚ 𝔸)\n⊢ 1 = 1"
] | · simp_rw [expSeries_sum_eq, ← expSeries_apply_eq, expSeries_apply_zero, tsum_pi_single] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Normed.Algebra.Exponential | {
"line": 455,
"column": 45
} | {
"line": 455,
"column": 58
} | {
"line": 455,
"column": 59
} | [
{
"pp": "𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕂\ninst✝³ : CharZero 𝕂\ninst✝² : ContinuousSMul ℚ 𝕂\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕂 𝔸\nthis : ∀ {n : ℕ}, ↑n ! ≠ 0\n⊢ Tendsto (fun n ↦ ‖(↑((n + 1) * n !))⁻¹ / (↑n !)⁻¹‖) atTop (𝓝 0)",
"ppTerm": "?m.56",
"assign... | [
"𝕂 : Type u_1\n𝔸 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕂\ninst✝³ : CharZero 𝕂\ninst✝² : ContinuousSMul ℚ 𝕂\ninst✝¹ : NormedRing 𝔸\ninst✝ : NormedAlgebra 𝕂 𝔸\nthis : ∀ {n : ℕ}, ↑n ! ≠ 0\n⊢ Tendsto (fun n ↦ ‖(↑(n + 1) * ↑n !)⁻¹ / (↑n !)⁻¹‖) atTop (𝓝 0)"
] | Nat.cast_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.Exponential | {
"line": 274,
"column": 4
} | {
"line": 274,
"column": 24
} | {
"line": 275,
"column": 4
} | [
{
"pp": "𝕂 : Type u_1\n𝕊 : Type u_2\n𝔸 : Type u_3\ninst✝⁹ : NontriviallyNormedField 𝕂\ninst✝⁸ : CharZero 𝕂\ninst✝⁷ : NormedCommRing 𝕊\ninst✝⁶ : NormedRing 𝔸\ninst✝⁵ : NormedSpace 𝕂 𝕊\ninst✝⁴ : NormedAlgebra 𝕂 𝔸\ninst✝³ : Algebra 𝕊 𝔸\ninst✝² : ContinuousSMul 𝕊 𝔸\ninst✝¹ : IsScalarTower 𝕂 𝕊 𝔸\ni... | [
"𝕂 : Type u_1\n𝕊 : Type u_2\n𝔸 : Type u_3\ninst✝⁹ : NontriviallyNormedField 𝕂\ninst✝⁸ : CharZero 𝕂\ninst✝⁷ : NormedCommRing 𝕊\ninst✝⁶ : NormedRing 𝔸\ninst✝⁵ : NormedSpace 𝕂 𝕊\ninst✝⁴ : NormedAlgebra 𝕂 𝔸\ninst✝³ : Algebra 𝕊 𝔸\ninst✝² : ContinuousSMul 𝕊 𝔸\ninst✝¹ : IsScalarTower 𝕂 𝕊 𝔸\ninst✝ : Compl... | rw [← zero_smul 𝕊 x] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Calculus.LocalExtr.Basic | {
"line": 177,
"column": 10
} | {
"line": 177,
"column": 31
} | {
"line": 177,
"column": 31
} | [
{
"pp": "case hy\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nf' : StrongDual ℝ E\na : E\nh : IsLocalMin f a\nhf : HasFDerivAt f f' a\ny : E\n⊢ y ∈ posTangentConeAt univ a",
"ppTerm": "?hy",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"... | [
"case hy\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nf' : StrongDual ℝ E\na : E\nh : IsLocalMin f a\nhf : HasFDerivAt f f' a\ny : E\n⊢ y ∈ univ"
] | posTangentConeAt_univ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.LocalExtr.Basic | {
"line": 177,
"column": 10
} | {
"line": 177,
"column": 31
} | {
"line": 177,
"column": 31
} | [
{
"pp": "case hy'\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nf' : StrongDual ℝ E\na : E\nh : IsLocalMin f a\nhf : HasFDerivAt f f' a\ny : E\n⊢ -y ∈ posTangentConeAt univ a",
"ppTerm": "?hy'",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClas... | [
"case hy'\nE : Type u\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : E → ℝ\nf' : StrongDual ℝ E\na : E\nh : IsLocalMin f a\nhf : HasFDerivAt f f' a\ny : E\n⊢ -y ∈ univ"
] | posTangentConeAt_univ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.ContDiff.Operations | {
"line": 91,
"column": 4
} | {
"line": 91,
"column": 33
} | {
"line": 92,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ns : Set E\nx : E\nn : ℕ∞ω\nι : Type u_3\ninst✝² : Fintype ι\nF' : ι → Type u_5\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nΦ : E → (i : ... | [
"𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\ns : Set E\nx : E\nn : ℕ∞ω\nι : Type u_3\ninst✝² : Fintype ι\nF' : ι → Type u_5\ninst✝¹ : (i : ι) → NormedAddCommGroup (F' i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (F' i)\nΦ : E → (i : ι) → F' i\np... | choose u hux p hp h'p using h | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 185,
"column": 50
} | {
"line": 185,
"column": 89
} | {
"line": 186,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\nf g : 𝕜 → F\nhf : ContDiffWithinAt 𝕜 (↑n) f s x\nhg : ContDiffWithinAt 𝕜 (↑n) g s x\n⊢ iteratedDerivWithin n (f + f... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nn : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\nf g : 𝕜 → F\nhf : ContDiffWithinAt 𝕜 (↑n) f s x\nhg : ContDiffWithinAt 𝕜 (↑n) g s x\n⊢ iteratedDerivWithin n f s x + iteratedDe... | iteratedDerivWithin_add hx h hf hg.neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 265,
"column": 8
} | {
"line": 265,
"column": 75
} | {
"line": 266,
"column": 8
} | [
{
"pp": "case succ\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\ninst✝¹ : IsBou... | [
"case succ\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\ninst✝¹ : IsBoundedSMul 𝔸 ... | simp only [Nat.choose_succ_succ', add_smul, Finset.sum_add_distrib] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.IteratedDeriv.Lemmas | {
"line": 266,
"column": 8
} | {
"line": 266,
"column": 40
} | {
"line": 267,
"column": 8
} | [
{
"pp": "case succ\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\ninst✝¹ : IsBou... | [
"case succ\n𝕜 : Type u_1\ninst✝⁷ : NontriviallyNormedField 𝕜\nF : Type u_2\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\nn✝ : ℕ\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\nh : UniqueDiffOn 𝕜 s\n𝔸 : Type u_5\ninst✝⁴ : NormedRing 𝔸\ninst✝³ : NormedAlgebra 𝕜 𝔸\ninst✝² : Module 𝔸 F\ninst✝¹ : IsBoundedSMul 𝔸 ... | nth_rw 3 [Finset.sum_range_succ] | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rw______1 | Mathlib.Tactic.tacticNth_rw_____ |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 211,
"column": 4
} | {
"line": 216,
"column": 73
} | {
"line": 218,
"column": 0
} | [
{
"pp": "case refine_2\nα : 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 : Measura... | [] | calc
(∫⁻ x : α, g x ∂μ) ≤ (∫⁻ x : α, f x + w x ∂μ) + ε / 2 := gint
_ = ((∫⁻ x : α, f x ∂μ) + ∫⁻ x : α, w x ∂μ) + ε / 2 := by
rw [lintegral_add_right _ wmeas.coe_nnreal_ennreal]
_ ≤ (∫⁻ x : α, f x ∂μ) + ε / 2 + ε / 2 := by grw [wint]
_ = (∫⁻ x : α, f x ∂μ) + ε := by rw [add_assoc, ENNReal... | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 211,
"column": 4
} | {
"line": 216,
"column": 73
} | {
"line": 218,
"column": 0
} | [
{
"pp": "case refine_2\nα : 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 : Measura... | [] | calc
(∫⁻ x : α, g x ∂μ) ≤ (∫⁻ x : α, f x + w x ∂μ) + ε / 2 := gint
_ = ((∫⁻ x : α, f x ∂μ) + ∫⁻ x : α, w x ∂μ) + ε / 2 := by
rw [lintegral_add_right _ wmeas.coe_nnreal_ennreal]
_ ≤ (∫⁻ x : α, f x ∂μ) + ε / 2 + ε / 2 := by grw [wint]
_ = (∫⁻ x : α, f x ∂μ) + ε := by rw [add_assoc, ENNReal... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 211,
"column": 4
} | {
"line": 216,
"column": 73
} | {
"line": 218,
"column": 0
} | [
{
"pp": "case refine_2\nα : 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 : Measura... | [] | calc
(∫⁻ x : α, g x ∂μ) ≤ (∫⁻ x : α, f x + w x ∂μ) + ε / 2 := gint
_ = ((∫⁻ x : α, f x ∂μ) + ∫⁻ x : α, w x ∂μ) + ε / 2 := by
rw [lintegral_add_right _ wmeas.coe_nnreal_ennreal]
_ ≤ (∫⁻ x : α, f x ∂μ) + ε / 2 + ε / 2 := by grw [wint]
_ = (∫⁻ x : α, f x ∂μ) + ε := by rw [add_assoc, ENNReal... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.Deriv | {
"line": 247,
"column": 2
} | {
"line": 252,
"column": 52
} | {
"line": 254,
"column": 2
} | [
{
"pp": "x : 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)\nA : ∀ y ∈ Set.Ioo (-1) 1, HasDerivAt F (F' y) y\nB : ∀ y ∈ Set.Icc (-|x|) |x|, |F' y| ≤ |x| ^ n / (1 - |x|)\n⊢ |∑ i ∈ Finset.range n, x ^ (i + 1) / (↑i + 1... | [
"x : 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)\nA : ∀ y ∈ Set.Ioo (-1) 1, HasDerivAt F (F' y) y\nB : ∀ y ∈ Set.Icc (-|x|) |x|, |F' y| ≤ |x| ^ n / (1 - |x|)\nC : ‖F x - F 0‖ ≤ |x| ^ n / (1 - |x|) * ‖x - 0‖\n⊢ |∑ i ∈ ... | have C : ‖F x - F 0‖ ≤ |x| ^ n / (1 - |x|) * ‖x - 0‖ := by
refine Convex.norm_image_sub_le_of_norm_hasDerivWithin_le
(fun y hy ↦ (A _ ?_).hasDerivWithinAt) B (convex_Icc _ _) ?_ ?_
· exact Icc_subset_Ioo (neg_lt_neg h) h hy
· simp
· simp [le_abs_self x, neg_le.mp (neg_le_abs x)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 241,
"column": 2
} | {
"line": 251,
"column": 86
} | {
"line": 253,
"column": 0
} | [
{
"pp": "case refine_2\nα : 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 : AEMeasurable f μ\nε : ℝ≥0∞\nε0 : ε ≠ 0\nthis : ε / 2 ≠ 0\ng0 : α → ℝ≥0∞\nf_lt_g0 : ∀ (x : α), ↑(AEMeasurable.... | [] | · calc
∫⁻ x, g0 x + g1 x ∂μ = (∫⁻ x, g0 x ∂μ) + ∫⁻ x, g1 x ∂μ :=
lintegral_add_left g0_cont.measurable _
_ ≤ (∫⁻ x, f x ∂μ) + ε / 2 + (0 + ε / 2) := by
refine add_le_add ?_ ?_
· convert! g0_int using 2
exact lintegral_congr_ae (fmeas.ae_eq_mk.fun_comp _)
· convert! ... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 224,
"column": 4
} | {
"line": 224,
"column": 41
} | {
"line": 225,
"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 ... | rcases mem_iUnion.1 this with ⟨n, hn⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Convex.Gauge | {
"line": 104,
"column": 2
} | {
"line": 106,
"column": 79
} | {
"line": 108,
"column": 0
} | [
{
"pp": "case inr\nE : Type u_2\ninst✝¹ : AddCommGroup E\ninst✝ : Module ℝ E\nx : E\nhx : x ≠ 0\n⊢ sInf {r | r ∈ Ioi 0 ∧ r⁻¹ • x ∈ 0} = 0 x",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"False",
"Real",
... | [] | · simp only [mem_zero, Pi.zero_apply, inv_eq_zero, smul_eq_zero]
convert! Real.sInf_empty
exact eq_empty_iff_forall_notMem.2 fun r hr => hr.2.elim (ne_of_gt hr.1) hx | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Integral.Bochner.VitaliCaratheodory | {
"line": 401,
"column": 2
} | {
"line": 403,
"column": 20
} | {
"line": 404,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x ↦ ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\nIf : ∫⁻ (x : α), ↑(f x) ∂μ < ∞\ng : α → ℝ≥0\ngf : ∀ (x : α), g x ≤ f x\ngcont : UpperSemiconti... | [
"α : Type u_1\ninst✝³ : TopologicalSpace α\ninst✝² : MeasurableSpace α\ninst✝¹ : BorelSpace α\nμ : Measure α\ninst✝ : μ.WeaklyRegular\nf : α → ℝ≥0\nfint : Integrable (fun x ↦ ↑(f x)) μ\nε : ℝ≥0\nεpos : 0 < ↑ε\nIf : ∫⁻ (x : α), ↑(f x) ∂μ < ∞\ng : α → ℝ≥0\ngf : ∀ (x : α), g x ≤ f x\ngcont : UpperSemicontinuous g\ngin... | have Ig : (∫⁻ x, g x ∂μ) < ∞ := by
refine lt_of_le_of_lt (lintegral_mono fun x => ?_) If
simpa using gf x | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.LocallyConvex.SeparatingDual | {
"line": 76,
"column": 2
} | {
"line": 76,
"column": 44
} | {
"line": 78,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝⁵ : Ring R\ninst✝⁴ : AddCommGroup V\ninst✝³ : TopologicalSpace V\ninst✝² : TopologicalSpace R\ninst✝¹ : Module R V\ninst✝ : SeparatingDual R V\nx y : V\nh : x ≠ y\nf : StrongDual R V\nhf : f (x - y) ≠ 0\n⊢ ∃ f, f x ≠ f y",
"ppTerm": "?m.39",
"assigned": true,
... | [] | exact ⟨f, by simpa [sub_ne_zero] using hf⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.LocallyConvex.Separation | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 28
} | {
"line": 204,
"column": 2
} | [
{
"pp": "case inr.inl\nE : Type u_2\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ns : Set E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nhs : s.Nonempty\nht₁ : Convex ℝ ∅\nht₂ : IsClosed ∅\ndisj... | [
"case inr.inr\nE : Type u_2\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ns t : Set E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\nhs₁ : Convex ℝ s\nhs₂ : IsCompact s\nht₁ : Convex ℝ t\nht₂ : IsClosed t\ndisj : Disjoint s t\nhs : s.Non... | · exact ⟨0, 1, 2, by simp⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Calculus.FDeriv.Measurable | {
"line": 554,
"column": 4
} | {
"line": 554,
"column": 41
} | {
"line": 555,
"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\ne : ℕ\nthis : x ∈ ⋃ n, ⋂ p, ⋂ (_ : p ≥ n), ⋂ q, ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)\n⊢ ∃ n, ∀ (p q : ℕ), n ≤ p... | [
"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\ne : ℕ\nthis : x ∈ ⋃ n, ⋂ p, ⋂ (_ : p ≥ n), ⋂ q, ⋂ (_ : q ≥ n), B f K ((1 / 2) ^ p) ((1 / 2) ^ q) ((1 / 2) ^ e)\nn : ℕ\nhn : x ∈ ⋂ p, ⋂ (_ : p ≥ n), ⋂... | rcases mem_iUnion.1 this with ⟨n, hn⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 91
} | {
"line": 64,
"column": 2
} | [
{
"pp": "case h_closed\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 : Nor... | [] | exact isClosed_eq L.continuous_integral_comp_L1 (L.continuous.comp continuous_integral) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 91
} | {
"line": 64,
"column": 2
} | [
{
"pp": "case h_closed\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 : Nor... | [] | exact isClosed_eq L.continuous_integral_comp_L1 (L.continuous.comp continuous_integral) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 91
} | {
"line": 64,
"column": 2
} | [
{
"pp": "case h_closed\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 : Nor... | [] | exact isClosed_eq L.continuous_integral_comp_L1 (L.continuous.comp continuous_integral) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 286,
"column": 4
} | {
"line": 293,
"column": 52
} | {
"line": 295,
"column": 0
} | [
{
"pp": "case pos.refine_4\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))\n⊢ ∀ ⦃f_1 g : X → E⦄,\n f_1 =ᵐ[μ.wit... | [] | intro u v huv _ hu
rw [← integral_congr_ae huv, hu]
apply integral_congr_ae
filter_upwards [(ae_withDensity_iff f_meas.coe_nnreal_ennreal).1 huv] with x hx
rcases eq_or_ne (f x) 0 with (h'x | h'x)
· simp only [h'x, zero_smul]
· rw [hx _]
simpa only [Ne, ENNReal.coe_eq_zero] using h'x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Bochner.ContinuousLinearMap | {
"line": 286,
"column": 4
} | {
"line": 293,
"column": 52
} | {
"line": 295,
"column": 0
} | [
{
"pp": "case pos.refine_4\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))\n⊢ ∀ ⦃f_1 g : X → E⦄,\n f_1 =ᵐ[μ.wit... | [] | intro u v huv _ hu
rw [← integral_congr_ae huv, hu]
apply integral_congr_ae
filter_upwards [(ae_withDensity_iff f_meas.coe_nnreal_ennreal).1 huv] with x hx
rcases eq_or_ne (f x) 0 with (h'x | h'x)
· simp only [h'x, zero_smul]
· rw [hx _]
simpa only [Ne, ENNReal.coe_eq_zero] using h'x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 408,
"column": 9
} | {
"line": 408,
"column": 31
} | {
"line": 408,
"column": 31
} | [
{
"pp": "case e'_6\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\nA : MeasurableEmbedding fun x ↦ x * c⁻¹\nx✝ : ... | [
"case e'_6\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\nA : MeasurableEmbedding fun x ↦ x * c⁻¹\nx✝ : ℝ\n⊢ f (c * ... | simp only [comp_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Filter.AtTopBot.Floor | {
"line": 34,
"column": 13
} | {
"line": 34,
"column": 26
} | {
"line": 34,
"column": 27
} | [
{
"pp": "case calc_2\nK : Type u_1\ninst✝³ : Ring K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : FloorSemiring K\na c : K\nd n : ℕ\nh : ⌈|a|⌉₊ * ⌈|c|⌉₊ ^ n < (n - d)!\n⊢ ↑⌈|a|⌉₊ * ↑⌈|c|⌉₊ ^ n = ↑(⌈|a|⌉₊ * ⌈|c|⌉₊ ^ n)",
"ppTerm": "?calc_2",
"assigned": true,
"usedConstants": [
... | [
"case calc_2\nK : Type u_1\ninst✝³ : Ring K\ninst✝² : LinearOrder K\ninst✝¹ : IsStrictOrderedRing K\ninst✝ : FloorSemiring K\na c : K\nd n : ℕ\nh : ⌈|a|⌉₊ * ⌈|c|⌉₊ ^ n < (n - d)!\n⊢ ↑⌈|a|⌉₊ * ↑⌈|c|⌉₊ ^ n = ↑⌈|a|⌉₊ * ↑(⌈|c|⌉₊ ^ n)"
] | Nat.cast_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Basic | {
"line": 1351,
"column": 4
} | {
"line": 1351,
"column": 55
} | {
"line": 1351,
"column": 55
} | [
{
"pp": "f g : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\nhab : a ≤ b\nhfi : IntervalIntegrable f μ a b\nhgi : IntervalIntegrable g μ a b\nhle : f ≤ᵐ[μ.restrict (Ioc a b)] g\nhlt : (μ.restrict (Ioc a b)) {x | f x < g x} ≠ 0\n⊢ 0 < ∫ (x : ℝ) in Ioc a b, g x - f x ∂μ",
"ppTerm": "?m.71",
"assigned": true,
"usedCo... | [
"f g : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\nhab : a ≤ b\nhfi : IntervalIntegrable f μ a b\nhgi : IntervalIntegrable g μ a b\nhle : f ≤ᵐ[μ.restrict (Ioc a b)] g\nhlt : (μ.restrict (Ioc a b)) {x | f x < g x} ≠ 0\n⊢ 0 < (μ.restrict (Ioc a b)) (support fun x ↦ g x - f x)",
"case hf\nf g : ℝ → ℝ\na b : ℝ\nμ : Measure ℝ\nhab... | MeasureTheory.integral_pos_iff_support_of_nonneg_ae | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Order.Floor | {
"line": 153,
"column": 23
} | {
"line": 153,
"column": 33
} | {
"line": 153,
"column": 33
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : Ring α\ninst✝⁴ : LinearOrder α\ninst✝³ : FloorRing α\ninst✝² : TopologicalSpace α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : OrderClosedTopology α\nn : ℤ\n⊢ pure (IntCast.intCast (n - 1)) ≤ 𝓝[≤] (↑n - ↑1)",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
... | [
"α : Type u_1\ninst✝⁵ : Ring α\ninst✝⁴ : LinearOrder α\ninst✝³ : FloorRing α\ninst✝² : TopologicalSpace α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : OrderClosedTopology α\nn : ℤ\n⊢ pure (IntCast.intCast (n - 1)) ≤ 𝓝[≤] ↑(n - 1)"
] | ← cast_sub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 170,
"column": 2
} | {
"line": 171,
"column": 57
} | {
"line": 172,
"column": 2
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nt : ℝ\nf : AddCircle T → ℝ≥0∞\nm : MeasurableSet (Ioc t (t + T))\nthis :\n ∫⁻ (a : AddCircle T), f a ∂Measure.map (⇑(measurableEquivIoc T t).symm) volume =\n ∫⁻ (a : ↑(Ioc t (t + T))), f ((measurableEquivIoc T t).symm a)\n⊢ ∫⁻ (a : ℝ) in Ioc t (t + T), f ↑a = ∫⁻ (b : AddCi... | [
"T : ℝ\nhT : Fact (0 < T)\nt : ℝ\nf : AddCircle T → ℝ≥0∞\nm : MeasurableSet (Ioc t (t + T))\nthis : ∫⁻ (a : AddCircle T), f a ∂Measure.map (fun x ↦ ↑↑x) volume = ∫⁻ (a : ↑(Ioc t (t + T))), f ↑↑a\n⊢ ∫⁻ (a : ℝ) in Ioc t (t + T), f ↑a = ∫⁻ (b : AddCircle T), f b"
] | simp only [measurableEquivIoc, equivIoc, QuotientAddGroup.equivIocMod, MeasurableEquiv.symm_mk,
MeasurableEquiv.coe_mk, Equiv.coe_fn_symm_mk] at this | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.IntervalIntegral.Periodic | {
"line": 192,
"column": 2
} | {
"line": 193,
"column": 57
} | {
"line": 194,
"column": 2
} | [
{
"pp": "T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nt : ℝ\nf : AddCircle T → E\nm : MeasurableSet (Ioc t (t + T))\nthis :\n ∫ (y : AddCircle T), f y ∂Measure.map (⇑(measurableEquivIoc T t).symm) volume =\n ∫ (x : ↑(Ioc t (t + T))), f ((measurableEquivIoc ... | [
"T : ℝ\nhT : Fact (0 < T)\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nt : ℝ\nf : AddCircle T → E\nm : MeasurableSet (Ioc t (t + T))\nthis : ∫ (y : AddCircle T), f y ∂Measure.map (fun x ↦ ↑↑x) volume = ∫ (x : ↑(Ioc t (t + T))), f ↑↑x\n⊢ ∫ (a : ℝ) in Ioc t (t + T), f ↑a = ∫ (b : AddCircle T... | simp only [measurableEquivIoc, equivIoc, QuotientAddGroup.equivIocMod, MeasurableEquiv.symm_mk,
MeasurableEquiv.coe_mk, Equiv.coe_fn_symm_mk] at this | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Integral.DivergenceTheorem | {
"line": 129,
"column": 4
} | {
"line": 129,
"column": 87
} | {
"line": 130,
"column": 4
} | [
{
"pp": "case refine_1\nE : 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 : ∀ (i : Fin (n + 1)), ContinuousOn... | [
"case refine_1\nE : 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 : ∀ (i : Fin (n + 1)), ContinuousOn (fun y ↦ f ... | have := Box.continuousOn_face_Icc (Hc i) (Set.right_mem_Icc.2 (I.lower_le_upper i)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 199,
"column": 2
} | {
"line": 211,
"column": 80
} | {
"line": 213,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nE : Type u_3\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SFinite ν\nf : α × β → E\nh1f : StronglyMeasurable f\n⊢ HasFiniteIntegral f (μ.prod ν) ↔\n (∀ᵐ (x : α) ∂μ, HasFiniteIntegral (fun y ↦... | [] | simp only [hasFiniteIntegral_iff_enorm, lintegral_prod _ h1f.enorm.aemeasurable]
have (x : _) : ∀ᵐ y ∂ν, 0 ≤ ‖f (x, y)‖ := by filter_upwards with y using norm_nonneg _
simp_rw [integral_eq_lintegral_of_nonneg_ae (this _)
(h1f.norm.comp_measurable measurable_prodMk_left).aestronglyMeasurable,
enorm_eq_ofRe... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.Prod | {
"line": 199,
"column": 2
} | {
"line": 211,
"column": 80
} | {
"line": 213,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nE : Type u_3\ninst✝³ : MeasurableSpace α\ninst✝² : MeasurableSpace β\nμ : Measure α\nν : Measure β\ninst✝¹ : NormedAddCommGroup E\ninst✝ : SFinite ν\nf : α × β → E\nh1f : StronglyMeasurable f\n⊢ HasFiniteIntegral f (μ.prod ν) ↔\n (∀ᵐ (x : α) ∂μ, HasFiniteIntegral (fun y ↦... | [] | simp only [hasFiniteIntegral_iff_enorm, lintegral_prod _ h1f.enorm.aemeasurable]
have (x : _) : ∀ᵐ y ∂ν, 0 ≤ ‖f (x, y)‖ := by filter_upwards with y using norm_nonneg _
simp_rw [integral_eq_lintegral_of_nonneg_ae (this _)
(h1f.norm.comp_measurable measurable_prodMk_left).aestronglyMeasurable,
enorm_eq_ofRe... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 141,
"column": 4
} | {
"line": 141,
"column": 92
} | {
"line": 143,
"column": 0
} | [
{
"pp": "case right\nF : Type u_1\ninst✝² : Field F\np : F[X]\nE : Type u_2\ninst✝¹ : Field E\ninst✝ : Algebra F E\nh : Fact (map (algebraMap F E) p).Splits\ny : ↑(p.rootSet E)\nkey :\n (map (algebraMap F E) p).roots =\n Multiset.map (⇑↑(IsScalarTower.toAlgHom F p.SplittingField E)) (map (algebraMap F p.Spl... | [] | exact ⟨⟨x, (@Multiset.mem_toFinset _ (Classical.decEq _) _ _).mpr hx1⟩, Subtype.ext hx2⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 247,
"column": 2
} | {
"line": 248,
"column": 80
} | {
"line": 249,
"column": 2
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝ : Field F\np q : F[X]\nhpq : p * q = 0\n⊢ Function.Injective ⇑(restrictProd p q)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Inhabited.default",
"MonoidHom.instFunLike",
"HMul.hMul",
"MonoidHom",
"Mon... | [
"case neg\nF : Type u_1\ninst✝ : Field F\np q : F[X]\nhpq : ¬p * q = 0\n⊢ Function.Injective ⇑(restrictProd p q)"
] | · have : Unique (p * q).Gal := by rw [hpq]; infer_instance
exact fun f g _ => Eq.trans (Unique.eq_default f) (Unique.eq_default g).symm | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.FieldTheory.PolynomialGaloisGroup | {
"line": 290,
"column": 2
} | {
"line": 294,
"column": 18
} | {
"line": 296,
"column": 0
} | [
{
"pp": "case hg\nF : Type u_1\ninst✝ : Field F\np₁ q₁ p₂ q₂ : F[X]\nhq₁ : q₁ ≠ 0\nhq₂ : q₂ ≠ 0\nh₁ : (map (algebraMap F q₁.SplittingField) p₁).Splits\nh₂ : (map (algebraMap F q₂.SplittingField) p₂).Splits\n⊢ (map (algebraMap F (q₁ * q₂).SplittingField) p₂).Splits",
"ppTerm": "?hg",
"assigned": true,
... | [] | · rw [←
(SplittingField.lift q₂
((SplittingField.splits _).of_dvd (map_ne_zero (mul_ne_zero hq₁ hq₂))
((map_dvd_map' _).mpr (dvd_mul_left q₂ q₁)))).comp_algebraMap, ← map_map]
exact h₂.map _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.Polynomial | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 32
} | {
"line": 176,
"column": 2
} | [
{
"pp": "F : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : (map f p).Splits\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\n⊢ ‖(map f p).coeff i‖ ≤ B ^ (p.natDegree - i) * ↑(p.natDegree.choose i)",
"ppTerm": "?m.60",
"assigned": true,
... | [
"case inl\nF : Type u_3\nK : Type u_4\ninst✝¹ : CommRing F\ninst✝ : NormedField K\np : F[X]\nf : F →+* K\nB : ℝ\ni : ℕ\nh1 : p.Monic\nh2 : (map f p).Splits\nh3 : ∀ z ∈ (map f p).roots, ‖z‖ ≤ B\nhB : B < 0\n⊢ ‖(map f p).coeff i‖ ≤ B ^ (p.natDegree - i) * ↑(p.natDegree.choose i)",
"case inr\nF : Type u_3\nK : Type ... | obtain hB | hB := lt_or_ge B 0 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.NumberTheory.NumberField.InfinitePlace.Embeddings | {
"line": 64,
"column": 33
} | {
"line": 66,
"column": 26
} | {
"line": 68,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝⁴ : Field K\nA : Type u_2\ninst✝³ : Field A\ninst✝² : CharZero A\ninst✝¹ : NumberField K\ninst✝ : IsAlgClosed A\n⊢ Nonempty (K →+* A)",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instIsTorsionFreeOfIsDomainOfNoZeroSMulDivisors",
"c... | [] | by
rw [← Fintype.card_pos_iff, NumberField.Embeddings.card K A]
exact Module.finrank_pos | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Polynomial.Basic | {
"line": 110,
"column": 2
} | {
"line": 117,
"column": 97
} | {
"line": 118,
"column": 2
} | [
{
"pp": "case neg.refine_2.refine_2.refine_2\np : 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) ((galActionHo... | [
"case neg.refine_2.refine_2.refine_2\np : 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) ((res... | have hc : ∀ z : ℂ, z ∈ c ↔ aeval z p = 0 ∧ z.im ≠ 0 := by
intro z
simp_rw [c, Finset.mem_image]
constructor
· rintro ⟨w, hw, rfl⟩
exact ⟨(mem_rootSet.mp w.2).2, mt (hc0 w).mpr (Equiv.Perm.mem_support.mp hw)⟩
· rintro ⟨hz1, hz2⟩
exact ⟨⟨z, mem_rootSet.mpr ⟨hp, hz1⟩⟩, Equiv.Perm.mem_suppor... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Data.Real.Embedding | {
"line": 47,
"column": 9
} | {
"line": 47,
"column": 22
} | {
"line": 47,
"column": 23
} | [
{
"pp": "M : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : LinearOrder M\ninst✝¹ : IsOrderedAddMonoid M\ninst✝ : One M\nu v : ℚ\nx y : M\nhu : u.num • 1 < u.den • x\nhv : v.num • 1 < v.den • y\nhu' : (u.num * ↑v.den) • 1 < (↑u.den * ↑v.den) • x\nthis : ((u + v).num * ↑u.den * ↑v.den) • 1 < ↑(u + v).den • (↑u.den ... | [
"M : Type u_1\ninst✝³ : AddCommGroup M\ninst✝² : LinearOrder M\ninst✝¹ : IsOrderedAddMonoid M\ninst✝ : One M\nu v : ℚ\nx y : M\nhu : u.num • 1 < u.den • x\nhv : v.num • 1 < v.den • y\nhu' : (u.num * ↑v.den) • 1 < (↑u.den * ↑v.den) • x\nthis : ((u + v).num * ↑u.den * ↑v.den) • 1 < ↑(u + v).den • (↑u.den * ↑v.den) • ... | Nat.cast_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Real.Embedding | {
"line": 107,
"column": 4
} | {
"line": 107,
"column": 76
} | {
"line": 109,
"column": 0
} | [
{
"pp": "case h\nM : Type u_1\ninst✝⁶ : AddCommGroup M\ninst✝⁵ : LinearOrder M\ninst✝⁴ : IsOrderedAddMonoid M\ninst✝³ : One M\ninst✝² : ZeroLEOneClass M\ninst✝¹ : NeZero 1\ninst✝ : Archimedean M\nx : M\nhxpos : 0 < x\nn : ℕ\nhn : 1 ≤ n • x\n⊢ { num := 1, den := n + 1, den_nz := ⋯, reduced := ⋯ } ∈ ratLt x",
... | [] | simpa using hn.trans_lt <| (nsmul_lt_nsmul_iff_left hxpos).mpr (by simp) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 79,
"column": 2
} | {
"line": 94,
"column": 20
} | {
"line": 96,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nφ : MvPowerSeries σ R\nhφ : constantCoeff φ ∈ nonZeroDivisorsLeft R\n⊢ φ ∈ nonZeroDivisorsLeft (MvPowerSeries σ R)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Finsupp.instHasAntidiagonal",
"MvPowerSeries.coeff_zero_e... | [] | intro x hx
ext d
apply WellFoundedLT.induction d
intro e he
rw [map_zero, ← mul_left_mem_nonZeroDivisorsLeft_eq_zero_iff hφ,
← map_zero (f := coeff e), ← hx]
convert! (coeff_mul e φ x).symm
rw [Finset.sum_eq_single (0, e), coeff_zero_eq_constantCoeff]
· rintro ⟨_, u⟩ huv _
suffices u < e by simp o... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.MvPowerSeries.NoZeroDivisors | {
"line": 79,
"column": 2
} | {
"line": 94,
"column": 20
} | {
"line": 96,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nφ : MvPowerSeries σ R\nhφ : constantCoeff φ ∈ nonZeroDivisorsLeft R\n⊢ φ ∈ nonZeroDivisorsLeft (MvPowerSeries σ R)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Finsupp.instHasAntidiagonal",
"MvPowerSeries.coeff_zero_e... | [] | intro x hx
ext d
apply WellFoundedLT.induction d
intro e he
rw [map_zero, ← mul_left_mem_nonZeroDivisorsLeft_eq_zero_iff hφ,
← map_zero (f := coeff e), ← hx]
convert! (coeff_mul e φ x).symm
rw [Finset.sum_eq_single (0, e), coeff_zero_eq_constantCoeff]
· rintro ⟨_, u⟩ huv _
suffices u < e by simp o... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Inverse | {
"line": 115,
"column": 10
} | {
"line": 115,
"column": 20
} | {
"line": 115,
"column": 21
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Ring R\nφ : MvPowerSeries σ R\nu : Rˣ\nh : constantCoeff φ = ↑u\nn : σ →₀ ℕ\nthis✝ : DecidableEq (σ →₀ ℕ) := Classical.decEq (σ →₀ ℕ)\nH : ¬n = 0\nthis : (0, n) ∈ antidiagonal n\n⊢ (coeff n) (φ * φ.invOfUnit u) = (coeff n) 1",
"ppTerm": "?m.55",
"assigned": t... | [
"σ : Type u_1\nR : Type u_2\ninst✝ : Ring R\nφ : MvPowerSeries σ R\nu : Rˣ\nh : constantCoeff φ = ↑u\nn : σ →₀ ℕ\nthis✝ : DecidableEq (σ →₀ ℕ) := Classical.decEq (σ →₀ ℕ)\nH : ¬n = 0\nthis : (0, n) ∈ antidiagonal n\n⊢ (coeff n) (φ * φ.invOfUnit u) = if n = 0 then 1 else 0"
] | coeff_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 296,
"column": 2
} | {
"line": 297,
"column": 60
} | {
"line": 299,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝⁶ : Field K\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : Algebra R K\ninst✝³ : IsDedekindDomain R\ninst✝² : IsFractionRing R K\nv : HeightOneSpectrum R\ninst✝¹ : Module.Finite ℤ R\ninst✝ : Module.Free ℤ R\nx : R\n⊢ ‖(embedding v) ((algebraMap R K) x)‖ = 1 ↔ x ∉ v.asIdeal",
"ppTer... | [] | rw [norm_embedding]
exact v.adicAbv_coe_eq_one_iff (one_lt_absNorm_nnreal v) x | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 296,
"column": 2
} | {
"line": 297,
"column": 60
} | {
"line": 299,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝⁶ : Field K\nR : Type u_2\ninst✝⁵ : CommRing R\ninst✝⁴ : Algebra R K\ninst✝³ : IsDedekindDomain R\ninst✝² : IsFractionRing R K\nv : HeightOneSpectrum R\ninst✝¹ : Module.Finite ℤ R\ninst✝ : Module.Free ℤ R\nx : R\n⊢ ‖(embedding v) ((algebraMap R K) x)‖ = 1 ↔ x ∉ v.asIdeal",
"ppTer... | [] | rw [norm_embedding]
exact v.adicAbv_coe_eq_one_iff (one_lt_absNorm_nnreal v) x | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 378,
"column": 14
} | {
"line": 378,
"column": 48
} | {
"line": 380,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : FinitePlace K\nx : K\n| w x",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NumberField.FinitePlace.mk_apply",
"Real",
"NumberField.RingOfIntegers.instIsFractionRing",
"NumberFie... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : FinitePlace K\nx : K\n| ‖(embedding w.maximalIdeal) x‖"
] | rw [← mk_maximalIdeal w, mk_apply] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 378,
"column": 14
} | {
"line": 378,
"column": 48
} | {
"line": 380,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : FinitePlace K\nx : K\n| w x",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NumberField.FinitePlace.mk_apply",
"Real",
"NumberField.RingOfIntegers.instIsFractionRing",
"NumberFie... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : FinitePlace K\nx : K\n| ‖(embedding w.maximalIdeal) x‖"
] | rw [← mk_maximalIdeal w, mk_apply] | Lean.Elab.Tactic.Conv.evalConvSeq1Indented | Lean.Parser.Tactic.Conv.convSeq1Indented |
Mathlib.NumberTheory.NumberField.Completion.FinitePlace | {
"line": 378,
"column": 14
} | {
"line": 378,
"column": 48
} | {
"line": 380,
"column": 0
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : FinitePlace K\nx : K\n| w x",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NumberField.FinitePlace.mk_apply",
"Real",
"NumberField.RingOfIntegers.instIsFractionRing",
"NumberFie... | [
"K : Type u_1\ninst✝¹ : Field K\ninst✝ : NumberField K\nw : FinitePlace K\nx : K\n| ‖(embedding w.maximalIdeal) x‖"
] | rw [← mk_maximalIdeal w, mk_apply] | Lean.Elab.Tactic.Conv.evalConvSeq | Lean.Parser.Tactic.Conv.convSeq |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {
"line": 127,
"column": 91
} | {
"line": 132,
"column": 42
} | {
"line": 134,
"column": 0
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Projective F\nP : Fin 3 → F\nhP : W.Nonsingular P\n⊢ W.Nonsingular (W.neg P)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"NegZeroClass.toNeg",
"WeierstrassCurve.Projective.toAffine",
"Nat.instM... | [] | by
by_cases hPz : P z = 0
· simp only [neg_of_Z_eq_zero hP.left hPz, nonsingular_smul _ (isUnit_Y_of_Z_eq_zero hP hPz).neg,
nonsingular_zero]
· simp only [neg_of_Z_ne_zero hPz, nonsingular_smul _ <| Ne.isUnit hPz,
nonsingular_neg_of_Z_ne_zero hP hPz] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Basic | {
"line": 300,
"column": 6
} | {
"line": 300,
"column": 18
} | {
"line": 300,
"column": 19
} | [
{
"pp": "R : Type r\ninst✝ : CommRing R\nW' : Projective R\n⊢ W'.polynomialX = C W'.a₁ * Y * Z - (C 3 * X ^ 2 + C (2 * W'.a₂) * X * Z + C W'.a₄ * Z ^ 2)",
"ppTerm": "?m.139",
"assigned": true,
"usedConstants": [
"Derivation",
"Finsupp.instAddZeroClass",
"Eq.mpr",
"Nat.instMul... | [
"R : Type r\ninst✝ : CommRing R\nW' : Projective R\n⊢ (pderiv x) W'.polynomial = C W'.a₁ * Y * Z - (C 3 * X ^ 2 + C (2 * W'.a₂) * X * Z + C W'.a₄ * Z ^ 2)"
] | polynomialX, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Point | {
"line": 483,
"column": 6
} | {
"line": 483,
"column": 70
} | {
"line": 483,
"column": 71
} | [
{
"pp": "F : Type u\ninst✝¹ : Field F\nW : Projective F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Nonsingular P\nhQ : W.Nonsingular Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhxy : ¬(P x * Q z = Q x * P z ∧ P y * Q z = W.negY Q * P z)\n⊢ toAffine W\n ![W.toAffine.addX (P x / P z) (Q x / Q z) (W.toAffine.slope ... | [
"F : Type u\ninst✝¹ : Field F\nW : Projective F\ninst✝ : DecidableEq F\nP Q : Fin 3 → F\nhP : W.Nonsingular P\nhQ : W.Nonsingular Q\nhPz : P z ≠ 0\nhQz : Q z ≠ 0\nhxy : ¬(P x * Q z = Q x * P z ∧ P y * Q z = W.negY Q * P z)\n⊢ Affine.Point.some\n (W.toAffine.addX (P x / P z) (Q x / Q z) (W.toAffine.slope (P x /... | toAffine_some <| nonsingular_add_of_Z_ne_zero hP hQ hPz hQz hxy, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.AlgebraicGeometry.GluingOneHypercover | {
"line": 56,
"column": 27
} | {
"line": 56,
"column": 39
} | {
"line": 56,
"column": 39
} | [
{
"pp": "D : GlueData\nY : Scheme\ni : D.openCover.I₀\n⊢ 𝟙 (D.openCover.X i) ≫ { I₀ := D.J, X := D.U, f := D.ι }.f i = D.openCover.f i",
"ppTerm": "?m.142",
"assigned": true,
"usedConstants": [
"AlgebraicGeometry.Scheme.GlueData.ι",
"CategoryTheory.PreZeroHypercover.mk",
"Eq.mpr",... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Unramified.Pi | {
"line": 37,
"column": 24
} | {
"line": 37,
"column": 26
} | {
"line": 37,
"column": 27
} | [
{
"pp": "case intro.mpr\nR : Type u_1\nI : Type u_2\ninst✝⁵ : Finite I\nf : I → Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : (i : I) → CommRing (f i)\ninst✝² : (i : I) → Algebra R (f i)\nval✝ : Fintype I\nH : ∀ (i : I), FormallyUnramified R (f i)\nB : Type (max u_2 u_3)\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nJ :... | [
"case intro.mpr\nR : Type u_1\nI : Type u_2\ninst✝⁵ : Finite I\nf : I → Type u_3\ninst✝⁴ : CommRing R\ninst✝³ : (i : I) → CommRing (f i)\ninst✝² : (i : I) → Algebra R (f i)\nval✝ : Fintype I\nH : ∀ (i : I), FormallyUnramified R (f i)\nB : Type (max u_2 u_3)\ninst✝¹ : CommRing B\ninst✝ : Algebra R B\nJ : Ideal B\nhJ... | f₂ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.AlgebraicGeometry.EllipticCurve.Projective.Formula | {
"line": 159,
"column": 40
} | {
"line": 159,
"column": 44
} | {
"line": 159,
"column": 45
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nW : Projective F\nP : Fin 3 → F\nhPz : P z ≠ 0\nhy : P y = W.negY P\nhy' : (eval P) W.polynomialY = (P y - W.negY P) * P z\n⊢ W.Equation P ∧ ((eval P) W.polynomialX ≠ 0 ∨ (eval P) W.polynomialY ≠ 0) ↔ W.Equation P ∧ (eval P) W.polynomialX ≠ 0",
"ppTerm": "?m.73",
"a... | [
"F : Type u\ninst✝ : Field F\nW : Projective F\nP : Fin 3 → F\nhPz : P z ≠ 0\nhy : P y = W.negY P\nhy' : (eval P) W.polynomialY = (P y - W.negY P) * P z\n⊢ W.Equation P ∧ ((eval P) W.polynomialX ≠ 0 ∨ (P y - W.negY P) * P z ≠ 0) ↔ W.Equation P ∧ (eval P) W.polynomialX ≠ 0"
] | hy', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Smooth.Fiber | {
"line": 177,
"column": 4
} | {
"line": 178,
"column": 84
} | {
"line": 179,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Flat R S\ninst✝⁹ : CommRing P\ninst✝⁸ : Algebra R P\ninst✝⁷ : Algebra P S\ninst✝⁶ : IsScalarTower R P S\ninst✝⁵ : IsLocalRing R\ninst✝⁴ : IsLocalRing S\ninst✝³ : IsLocalHom (alg... | [
"R : Type u_1\nS : Type u_2\nP : Type u_3\ninst✝¹³ : CommRing R\ninst✝¹² : CommRing S\ninst✝¹¹ : Algebra R S\ninst✝¹⁰ : Module.Flat R S\ninst✝⁹ : CommRing P\ninst✝⁸ : Algebra R P\ninst✝⁷ : Algebra P S\ninst✝⁶ : IsScalarTower R P S\ninst✝⁵ : IsLocalRing R\ninst✝⁴ : IsLocalRing S\ninst✝³ : IsLocalHom (algebraMap R S)... | obtain ⟨a, ha, e⟩ : ∃ a ∈ M, a * f₀ x = 0 := by
simpa [fP, IsLocalization.lift_mk', IsLocalization.map_eq_zero_iff M] using hx | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.RingTheory.AdicCompletion.Algebra | {
"line": 308,
"column": 6
} | {
"line": 309,
"column": 66
} | {
"line": 310,
"column": 6
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nI : Ideal R\nM : Type u_3\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr✝ : AdicCompletion I R\nx✝ : AdicCompletion I M\nm n : ℕ\nhmn : m ≤ n\nr : AdicCauchySequence I R\nx : AdicCauchySequence I M\n⊢ (transitionMap I M hmn) ((fun n... | [
"R : Type u_1\nS : Type u_2\ninst✝³ : CommRing R\ninst✝² : CommRing S\nI : Ideal R\nM : Type u_3\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nr✝ : AdicCompletion I R\nx✝ : AdicCompletion I M\nm n : ℕ\nhmn : m ≤ n\nr : AdicCauchySequence I R\nx : AdicCauchySequence I M\n⊢ ↑r n • Submodule.Quotient.mk (↑x n) = ↑r m ... | simp only [coe_eval, mapQ_eq_factor, mk_apply_coe, mkQ_apply, Ideal.Quotient.mk_eq_mk,
mk_smul_mk, map_smul, mapQ_apply, LinearMap.id_coe, id_eq] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.AdicCompletion.Exactness | {
"line": 54,
"column": 8
} | {
"line": 54,
"column": 29
} | {
"line": 54,
"column": 30
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type w\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nhf : Function.Surjective ⇑f\nx : AdicCauchySequence I N\nn : ℕ\ny yₙ : M\nhy : f y = ↑x (n + 1)\nhyₙ : f yₙ = ↑x n\n⊢ f (yₙ - y... | [
"R : Type u\ninst✝⁴ : CommRing R\nI : Ideal R\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\nN : Type w\ninst✝¹ : AddCommGroup N\ninst✝ : Module R N\nf : M →ₗ[R] N\nhf : Function.Surjective ⇑f\nx : AdicCauchySequence I N\nn : ℕ\ny yₙ : M\nhy : f y = ↑x (n + 1)\nhyₙ : f yₙ = ↑x n\n⊢ f (yₙ - y) ∈ I ^ n • ... | Submodule.map_smul'', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdicCompletion.Algebra | {
"line": 382,
"column": 2
} | {
"line": 382,
"column": 62
} | {
"line": 383,
"column": 2
} | [
{
"pp": "R : Type u_4\nS : Type u_5\ninst✝¹ : NonAssocSemiring R\ninst✝ : CommRing S\nI : Ideal S\nf : (n : ℕ) → R →+* S ⧸ I ^ n\nhf : ∀ {m n : ℕ} (hle : m ≤ n), (Ideal.Quotient.factorPow I hle).comp (f n) = f m\nn : ℕ\nx : R\n⊢ (evalₐ I n) ((liftRingHom I f ⋯) x) = (f n) x",
"ppTerm": "?m.39",
"assigne... | [
"R : Type u_4\nS : Type u_5\ninst✝¹ : NonAssocSemiring R\ninst✝ : CommRing S\nI : Ideal S\nf : (n : ℕ) → R →+* S ⧸ I ^ n\nhf : ∀ {m n : ℕ} (hle : m ≤ n), (Ideal.Quotient.factorPow I hle).comp (f n) = f m\nn : ℕ\nx : R\n⊢ (factor ⋯) ((eval I S n) ((liftRingHom I f ⋯) x)) = (f n) x"
] | rw [← factor_eval_eq_evalₐ I _ (le_of_eq (Ideal.mul_top _))] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.AdicCompletion.Exactness | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 25
} | {
"line": 101,
"column": 26
} | [
{
"pp": "R : Type u\ninst✝⁶ : CommRing R\nI : Ideal R\nM : Type u\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nN : Type u\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R N\nf : M →ₗ[R] N\nhf : Function.Injective ⇑f\nk : ℕ\nhk : ∀ n ≥ k, I ^ n • ⊤ ⊓ f.range =... | [
"R : Type u\ninst✝⁶ : CommRing R\nI : Ideal R\nM : Type u\ninst✝⁵ : AddCommGroup M\ninst✝⁴ : Module R M\nN : Type u\ninst✝³ : AddCommGroup N\ninst✝² : Module R N\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R N\nf : M →ₗ[R] N\nhf : Function.Injective ⇑f\nk : ℕ\nhk : ∀ n ≥ k, I ^ n • ⊤ ⊓ f.range = I ^ (n - k)... | Submodule.map_smul'', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.AdicCompletion.Exactness | {
"line": 143,
"column": 8
} | {
"line": 143,
"column": 29
} | {
"line": 143,
"column": 30
} | [
{
"pp": "R : Type u\ninst✝⁸ : CommRing R\nI : Ideal R\nM : Type u\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN : Type u\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : Type u\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R N\nf : M →ₗ[R] N\ng : N →ₗ[R] ... | [
"R : Type u\ninst✝⁸ : CommRing R\nI : Ideal R\nM : Type u\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : Module R M\nN : Type u\ninst✝⁵ : AddCommGroup N\ninst✝⁴ : Module R N\nP : Type u\ninst✝³ : AddCommGroup P\ninst✝² : Module R P\ninst✝¹ : IsNoetherianRing R\ninst✝ : Module.Finite R N\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nhf : Func... | Submodule.map_smul'', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Extension.Presentation.Submersive | {
"line": 220,
"column": 4
} | {
"line": 220,
"column": 22
} | {
"line": 221,
"column": 4
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nh : Function.Bijective ⇑(algebraMap R S)\n⊢ (algebraMap (ofBijectiveAlgebraMap h).Ring S).mapMatrix (ofBijectiveAlgebraMap h).jacobiMatrix = 1",
"ppTerm": "?m.69",
"assigned": true,
"usedConstants": [
... | [
"R : Type u\nS : Type v\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nh : Function.Bijective ⇑(algebraMap R S)\ni j : PEmpty.{?u.53 + 1}\n⊢ (algebraMap (ofBijectiveAlgebraMap h).Ring S).mapMatrix (ofBijectiveAlgebraMap h).jacobiMatrix i j = 1 i j"
] | ext (i j : PEmpty) | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.RingTheory.Extension.Presentation.Core | {
"line": 160,
"column": 4
} | {
"line": 160,
"column": 54
} | {
"line": 161,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nP : Presentation R S ι σ\nR₀ : Type u_5\ninst✝⁴ : CommRing R₀\ninst✝³ : Algebra R₀ R\ninst✝² : Algebra R₀ S\ninst✝¹ : IsScalarTower R₀ R S\ninst✝ : P.HasCoeffs R₀\n⊢ P.ker ≤\n Ring... | [
"R : Type u_1\nS : Type u_2\nι : Type u_3\nσ : Type u_4\ninst✝⁷ : CommRing R\ninst✝⁶ : CommRing S\ninst✝⁵ : Algebra R S\nP : Presentation R S ι σ\nR₀ : Type u_5\ninst✝⁴ : CommRing R₀\ninst✝³ : Algebra R₀ R\ninst✝² : Algebra R₀ S\ninst✝¹ : IsScalarTower R₀ R S\ninst✝ : P.HasCoeffs R₀\n⊢ Set.range P.relation ⊆\n ↑... | rw [← P.span_range_relation_eq_ker, Ideal.span_le] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
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