module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Geometry.Manifold.IsManifold.Basic | {
"line": 829,
"column": 45
} | {
"line": 833,
"column": 17
} | {
"line": 835,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nm n : ℕ∞ω\nhmn : m ≤ n\ninst✝ : IsMan... | [] | by
have : HasGroupoid M (contDiffGroupoid m I) :=
hasGroupoid_of_le (G₁ := contDiffGroupoid n I) (by infer_instance)
(contDiffGroupoid_le hmn)
exact mk' I m M | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 902,
"column": 2
} | {
"line": 904,
"column": 44
} | {
"line": 906,
"column": 0
} | [
{
"pp": "E : Type u_8\n𝕜 : Type u_9\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_10\ninst✝⁴ : TopologicalSpace H\nM : Type u_11\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nI : ModelWithCorners 𝕜 E H\ninst✝¹ : LocallyCompactSpace 𝕜\nin... | [] | have : ProperSpace E := FiniteDimensional.proper 𝕜 E
have : LocallyCompactSpace H := I.locallyCompactSpace
exact ChartedSpace.locallyCompactSpace H M | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.IsManifold.ExtChartAt | {
"line": 902,
"column": 2
} | {
"line": 904,
"column": 44
} | {
"line": 906,
"column": 0
} | [
{
"pp": "E : Type u_8\n𝕜 : Type u_9\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : NormedSpace 𝕜 E\nH : Type u_10\ninst✝⁴ : TopologicalSpace H\nM : Type u_11\ninst✝³ : TopologicalSpace M\ninst✝² : ChartedSpace H M\nI : ModelWithCorners 𝕜 E H\ninst✝¹ : LocallyCompactSpace 𝕜\nin... | [] | have : ProperSpace E := FiniteDimensional.proper 𝕜 E
have : LocallyCompactSpace H := I.locallyCompactSpace
exact ChartedSpace.locallyCompactSpace H M | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.ContMDiff.Basic | {
"line": 428,
"column": 2
} | {
"line": 430,
"column": 84
} | {
"line": 431,
"column": 2
} | [
{
"pp": "case right\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ne : M → H\nh : IsOpenEmbedding e\nn : ℕ∞ω\ninst✝ : N... | [
"case right\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ne : M → H\nh : IsOpenEmbedding e\nn : ℕ∞ω\ninst✝ : Nonempty M\nt... | · rw [I.right_inv]
apply mem_of_subset_of_mem _ hz.1
exact letI := h.singletonChartedSpace; extChartAt_target_subset_range (I := I) x | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.ContMDiff.Basic | {
"line": 448,
"column": 4
} | {
"line": 448,
"column": 57
} | {
"line": 449,
"column": 2
} | [
{
"pp": "case left\n𝕜 : Type u_1\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹ : TopologicalSpace M\ne : M → H\nh : IsOpenEmbedding e\nn : ℕ∞ω\ninst✝ : No... | [] | exact (h.toOpenPartialHomeomorph e).continuousOn_symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 779,
"column": 6
} | {
"line": 779,
"column": 57
} | {
"line": 779,
"column": 57
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : ChartedSpace H M\nE' : Type u_5\ninst✝⁶ : NormedAddC... | [] | rw [extChartAt_to_inv]; apply mem_extChartAt_source | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.ContMDiff.Defs | {
"line": 779,
"column": 6
} | {
"line": 779,
"column": 57
} | {
"line": 779,
"column": 57
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹¹ : NormedAddCommGroup E\ninst✝¹⁰ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁹ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁸ : TopologicalSpace M\ninst✝⁷ : ChartedSpace H M\nE' : Type u_5\ninst✝⁶ : NormedAddC... | [] | rw [extChartAt_to_inv]; apply mem_extChartAt_source | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.Algebra.Monoid | {
"line": 98,
"column": 6
} | {
"line": 98,
"column": 24
} | {
"line": 98,
"column": 24
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹¹ : TopologicalSpace H\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nn : ℕ∞ω\nG : Type u_4\ninst✝⁸ : Mul G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : ChartedSpace H G\nE' : Type... | [
"𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹¹ : TopologicalSpace H\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nn : ℕ∞ω\nG : Type u_4\ninst✝⁸ : Mul G\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : ChartedSpace H G\nE' : Type u_5\ninst✝⁵... | contMDiff_zero_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.Algebra.LieGroup | {
"line": 111,
"column": 6
} | {
"line": 111,
"column": 24
} | {
"line": 111,
"column": 24
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹¹ : TopologicalSpace H\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nn : ℕ∞ω\nG : Type u_4\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : ChartedSpace H G\ninst✝⁶ : Group G\nE' : Ty... | [
"𝕜 : Type u_1\ninst✝¹² : NontriviallyNormedField 𝕜\nH : Type u_2\ninst✝¹¹ : TopologicalSpace H\nE : Type u_3\ninst✝¹⁰ : NormedAddCommGroup E\ninst✝⁹ : NormedSpace 𝕜 E\nI : ModelWithCorners 𝕜 E H\nn : ℕ∞ω\nG : Type u_4\ninst✝⁸ : TopologicalSpace G\ninst✝⁷ : ChartedSpace H G\ninst✝⁶ : Group G\nE' : Type u_5\ninst... | contMDiff_zero_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.ContMDiff.Atlas | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 87
} | {
"line": 176,
"column": 2
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nn : ℕ∞ω\ninst✝ : IsMan... | [
"case refine_3\n𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝³ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝² : TopologicalSpace M\ninst✝¹ : ChartedSpace H M\nn : ℕ∞ω\ninst✝ : IsManifold I n M\... | · apply hφ.comp (he''.comp (I.contMDiffOn_symm.mono (by simp)) (by grind)) (by grind) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.VectorBundle.FiberwiseLinear | {
"line": 132,
"column": 4
} | {
"line": 132,
"column": 16
} | {
"line": 133,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nEB : Type u_4\ninst✝³ : NormedAddCommGroup EB\ninst✝² : NormedSpace 𝕜 EB\nHB : Type u_5\ninst✝¹ : TopologicalSpace HB\ninst✝ : ChartedS... | [
"𝕜 : Type u_1\nB : Type u_2\nF : Type u_3\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup F\ninst✝⁴ : NormedSpace 𝕜 F\nEB : Type u_4\ninst✝³ : NormedAddCommGroup EB\ninst✝² : NormedSpace 𝕜 EB\nHB : Type u_5\ninst✝¹ : TopologicalSpace HB\ninst✝ : ChartedSpace HB B\nI... | intro p q hq | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.VectorBundle.Basic | {
"line": 328,
"column": 6
} | {
"line": 328,
"column": 30
} | {
"line": 328,
"column": 31
} | [
{
"pp": "R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁹ : Semiring R\ninst✝⁸ : TopologicalSpace F\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : TopologicalSpace (TotalSpace F E)\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module R F\ninst✝³ : (x : B) → AddCommMonoid (E x)\ninst✝² : (x : B) → Module R (E x)... | [
"R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁹ : Semiring R\ninst✝⁸ : TopologicalSpace F\ninst✝⁷ : TopologicalSpace B\ninst✝⁶ : TopologicalSpace (TotalSpace F E)\ninst✝⁵ : AddCommMonoid F\ninst✝⁴ : Module R F\ninst✝³ : (x : B) → AddCommMonoid (E x)\ninst✝² : (x : B) → Module R (E x)\ne e' : Tri... | e.mk_coordChangeL e' hb, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.VectorBundle.Basic | {
"line": 403,
"column": 6
} | {
"line": 407,
"column": 41
} | {
"line": 407,
"column": 42
} | [
{
"pp": "R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁹ : NontriviallyNormedField R\ninst✝⁸ : (x : B) → AddCommMonoid (E x)\ninst✝⁷ : (x : B) → Module R (E x)\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace R F\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace (TotalSpace F E)\nin... | [] | rw [e.coe_linearMapAt b]
classical
refine continuous_if_const _ (fun hb => ?_) fun _ => continuous_zero
exact (e.continuousOn.comp_continuous (FiberBundle.totalSpaceMk_isInducing F E b).continuous
fun x => e.mem_source.mpr hb).snd | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.VectorBundle.Basic | {
"line": 403,
"column": 6
} | {
"line": 407,
"column": 41
} | {
"line": 407,
"column": 42
} | [
{
"pp": "R : Type u_1\nB : Type u_2\nF : Type u_3\nE : B → Type u_4\ninst✝⁹ : NontriviallyNormedField R\ninst✝⁸ : (x : B) → AddCommMonoid (E x)\ninst✝⁷ : (x : B) → Module R (E x)\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace R F\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace (TotalSpace F E)\nin... | [] | rw [e.coe_linearMapAt b]
classical
refine continuous_if_const _ (fun hb => ?_) fun _ => continuous_zero
exact (e.continuousOn.comp_continuous (FiberBundle.totalSpaceMk_isInducing F E b).continuous
fun x => e.mem_source.mpr hb).snd | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 806,
"column": 2
} | {
"line": 807,
"column": 36
} | {
"line": 809,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | [] | rw [← mdifferentiableOn_univ, ← hst]
exact hf.union_of_isOpen hf' hs ht | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 806,
"column": 2
} | {
"line": 807,
"column": 36
} | {
"line": 809,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : NormedAddCom... | [] | rw [← mdifferentiableOn_univ, ← hst]
exact hf.union_of_isOpen hf' hs ht | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 969,
"column": 4
} | {
"line": 974,
"column": 42
} | {
"line": 975,
"column": 2
} | [
{
"pp": "case h₁\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : Nor... | [] | have :
(extChartAt I x).symm ⁻¹' {y | f₁ y = f y} ∈
𝓝[(extChartAt I x).symm ⁻¹' s ∩ range I] (extChartAt I x) x :=
extChartAt_preimage_mem_nhdsWithin h₁
apply Filter.mem_of_superset this fun y => _
simp +contextual only [hx, mfld_simps] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 969,
"column": 4
} | {
"line": 974,
"column": 42
} | {
"line": 975,
"column": 2
} | [
{
"pp": "case h₁\n𝕜 : Type u_1\ninst✝¹⁰ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁹ : NormedAddCommGroup E\ninst✝⁸ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝⁷ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁶ : TopologicalSpace M\ninst✝⁵ : ChartedSpace H M\nE' : Type u_5\ninst✝⁴ : Nor... | [] | have :
(extChartAt I x).symm ⁻¹' {y | f₁ y = f y} ∈
𝓝[(extChartAt I x).symm ⁻¹' s ∩ range I] (extChartAt I x) x :=
extChartAt_preimage_mem_nhdsWithin h₁
apply Filter.mem_of_superset this fun y => _
simp +contextual only [hx, mfld_simps] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 73
} | {
"line": 198,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁰ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nF : Type u_5\ninst✝⁷ : NormedAddC... | [] | · exact Set.compl_subset_iff_union.mp <| Set.compl_subset_compl.mpr ht' | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection | {
"line": 213,
"column": 27
} | {
"line": 213,
"column": 39
} | {
"line": 213,
"column": 39
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁰ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nF : Type u_5\ninst✝⁷ : NormedAddC... | [
"𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁰ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nF : Type u_5\ninst✝⁷ : NormedAddCommGroup F\n... | tsum_eq_sum' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.VectorBundle.ContMDiffSection | {
"line": 222,
"column": 25
} | {
"line": 222,
"column": 37
} | {
"line": 222,
"column": 37
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁰ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nF : Type u_5\ninst✝⁷ : NormedAddC... | [
"𝕜 : Type u_1\ninst✝¹³ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹² : NormedAddCommGroup E\ninst✝¹¹ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹⁰ : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝⁹ : TopologicalSpace M\ninst✝⁸ : ChartedSpace H M\nF : Type u_5\ninst✝⁷ : NormedAddCommGroup F\n... | tsum_eq_sum' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Geometry.Manifold.VectorBundle.Basic | {
"line": 452,
"column": 4
} | {
"line": 452,
"column": 32
} | {
"line": 453,
"column": 4
} | [
{
"pp": "case refine_1\nn : ℕ∞ω\n𝕜 : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B → Type u_6\ninst✝²⁰ : NontriviallyNormedField 𝕜\nEB : Type u_7\ninst✝¹⁹ : NormedAddCommGroup EB\ninst✝¹⁸ : NormedSpace 𝕜 EB\nHB : Type u_8\ninst✝¹⁷ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB ... | [
"case refine_1\nn : ℕ∞ω\n𝕜 : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B → Type u_6\ninst✝²⁰ : NontriviallyNormedField 𝕜\nEB : Type u_7\ninst✝¹⁹ : NormedAddCommGroup EB\ninst✝¹⁸ : NormedSpace 𝕜 EB\nHB : Type u_8\ninst✝¹⁷ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB HB\ninst✝¹⁶ ... | apply contMDiffOn_fst.prodMk | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Manifold.VectorBundle.Basic | {
"line": 455,
"column": 4
} | {
"line": 455,
"column": 32
} | {
"line": 456,
"column": 4
} | [
{
"pp": "case refine_2\nn : ℕ∞ω\n𝕜 : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B → Type u_6\ninst✝²⁰ : NontriviallyNormedField 𝕜\nEB : Type u_7\ninst✝¹⁹ : NormedAddCommGroup EB\ninst✝¹⁸ : NormedSpace 𝕜 EB\nHB : Type u_8\ninst✝¹⁷ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB ... | [
"case refine_2\nn : ℕ∞ω\n𝕜 : Type u_1\nB : Type u_2\nB' : Type u_3\nF : Type u_4\nM : Type u_5\nE : B → Type u_6\ninst✝²⁰ : NontriviallyNormedField 𝕜\nEB : Type u_7\ninst✝¹⁹ : NormedAddCommGroup EB\ninst✝¹⁸ : NormedSpace 𝕜 EB\nHB : Type u_8\ninst✝¹⁷ : TopologicalSpace HB\nIB : ModelWithCorners 𝕜 EB HB\ninst✝¹⁶ ... | apply contMDiffOn_fst.prodMk | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Geometry.Manifold.MFDeriv.Basic | {
"line": 1286,
"column": 77
} | {
"line": 1288,
"column": 28
} | {
"line": 1290,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝¹⁵ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹⁴ : NormedAddCommGroup E\ninst✝¹³ : NormedSpace 𝕜 E\nH : Type u_3\ninst✝¹² : TopologicalSpace H\nI : ModelWithCorners 𝕜 E H\nM : Type u_4\ninst✝¹¹ : TopologicalSpace M\ninst✝¹⁰ : ChartedSpace H M\nE' : Type u_5\ninst✝⁹ : NormedA... | [] | by
rw [← mdifferentiableOn_univ] at hg
exact hg.comp hf (by simp) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.PartitionOfUnity | {
"line": 564,
"column": 4
} | {
"line": 564,
"column": 44
} | {
"line": 565,
"column": 4
} | [
{
"pp": "case hx\nι : Type u\nX : Type v\ninst✝ : TopologicalSpace X\ns : Set X\nf : BumpCovering ι X s\nx : X\nhx : x ∈ s\n⊢ 1 - (f (f.ind x hx)) x = 0",
"ppTerm": "?hx",
"assigned": true,
"usedConstants": [
"Real",
"BumpCovering.ind",
"sub_self",
"congrArg",
"Real.ins... | [
"case hf\nι : Type u\nX : Type v\ninst✝ : TopologicalSpace X\ns : Set X\nf : BumpCovering ι X s\nx : X\nhx : x ∈ s\n⊢ HasFiniteMulSupport fun i ↦ 1 - (f i) x"
] | · simp only [f.ind_apply x hx, sub_self] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 380,
"column": 4
} | {
"line": 380,
"column": 53
} | {
"line": 381,
"column": 2
} | [
{
"pp": "case refine_1\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nH : Type uH\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : FiniteDimensional ℝ E\ns : Set M\nU : M → Set M\ninst✝¹ : T2Space M\ni... | [] | simpa only [SmoothBumpFunction.support_updateRIn] | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 380,
"column": 4
} | {
"line": 380,
"column": 53
} | {
"line": 381,
"column": 2
} | [
{
"pp": "case refine_1\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nH : Type uH\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : FiniteDimensional ℝ E\ns : Set M\nU : M → Set M\ninst✝¹ : T2Space M\ni... | [] | simpa only [SmoothBumpFunction.support_updateRIn] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 380,
"column": 4
} | {
"line": 380,
"column": 53
} | {
"line": 381,
"column": 2
} | [
{
"pp": "case refine_1\nE : Type uE\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace ℝ E\nH : Type uH\ninst✝⁵ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁴ : TopologicalSpace M\ninst✝³ : ChartedSpace H M\ninst✝² : FiniteDimensional ℝ E\ns : Set M\nU : M → Set M\ninst✝¹ : T2Space M\ni... | [] | simpa only [SmoothBumpFunction.support_updateRIn] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.AEEqOfIntegral | {
"line": 140,
"column": 6
} | {
"line": 140,
"column": 88
} | {
"line": 141,
"column": 6
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable f μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → 0 ≤ ∫ (x : α) in s, f x ∂μ\nb : ℝ\nhb_neg : b < 0\ns : Set α := {x | f x ≤ b}\nhs : NullMeasurableSet s μ\nmus : μ s < ∞\n⊢ ∫ (x : α) in s, f x ∂μ ≤ ∫ (x : α) in s, ... | [
"α : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nf : α → ℝ\nhf : Integrable f μ\nhf_zero : ∀ (s : Set α), MeasurableSet s → μ s < ∞ → 0 ≤ ∫ (x : α) in s, f x ∂μ\nb : ℝ\nhb_neg : b < 0\ns : Set α := {x | f x ≤ b}\nhs : NullMeasurableSet s μ\nmus : μ s < ∞\n⊢ f ≤ᵐ[μ.restrict s] fun x ↦ b"
] | refine setIntegral_mono_ae_restrict hf.integrableOn (integrableOn_const mus.ne) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Manifold.PartitionOfUnity | {
"line": 784,
"column": 12
} | {
"line": 784,
"column": 26
} | {
"line": 784,
"column": 26
} | [
{
"pp": "case neg\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\nH : Type uH\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : IsManifold I ∞ M\ninst✝¹ : SigmaCompactSpac... | [
"case neg\nE : Type uE\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace ℝ E\nH : Type uH\ninst✝⁶ : TopologicalSpace H\nI : ModelWithCorners ℝ E H\nM : Type uM\ninst✝⁵ : TopologicalSpace M\ninst✝⁴ : ChartedSpace H M\ninst✝³ : FiniteDimensional ℝ E\ninst✝² : IsManifold I ∞ M\ninst✝¹ : SigmaCompactSpace M\ninst✝ :... | notMem_support | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.TangentCone.Seq | {
"line": 129,
"column": 38
} | {
"line": 129,
"column": 57
} | {
"line": 129,
"column": 58
} | [
{
"pp": "case mp.inr\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\nx y : E\nhy₀ : y ≠ 0\nc : ℕ → 𝕜\nd : ℕ → E\nhds : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s\nhd₀' : ∀ᶠ (n : ℕ) in atTop, d n ≠ 0\nhd₀ : Tendsto (fun x ↦ ‖d x‖) atTo... | [
"case mp.inr\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\nx y : E\nhy₀ : y ≠ 0\nc : ℕ → 𝕜\nd : ℕ → E\nhds : ∀ᶠ (n : ℕ) in atTop, x + d n ∈ s\nhd₀' : ∀ᶠ (n : ℕ) in atTop, d n ≠ 0\nhd₀ : Tendsto (fun x ↦ ‖d x‖) atTop (𝓝 0)\nhc... | ← Filter.comap_inv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Rademacher | {
"line": 223,
"column": 55
} | {
"line": 223,
"column": 74
} | {
"line": 223,
"column": 75
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nC : ℝ≥0\nf : E → ℝ\nμ : Measure E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\nhf : LipschitzWith C f\nι : Type u_3\ns : Finset ι\na : ι → ℝ\nv : ι → E\ng : E → ℝ\ng... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\nC : ℝ≥0\nf : E → ℝ\nμ : Measure E\ninst✝¹ : FiniteDimensional ℝ E\ninst✝ : μ.IsAddHaarMeasure\nhf : LipschitzWith C f\nι : Type u_3\ns : Finset ι\na : ι → ℝ\nv : ι → E\ng : E → ℝ\ng_smooth : Co... | integral_const_mul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable | {
"line": 79,
"column": 12
} | {
"line": 79,
"column": 41
} | {
"line": 79,
"column": 42
} | [
{
"pp": "f : ℝ → ℝ\na b : ℝ\nhab : a ≤ b\nhf : MonotoneOn f (Icc a b)\ng : ℝ → ℝ := fun x ↦ f (max a (min x b))\nhg : Monotone g\nhfg : EqOn (deriv f) (deriv g) (Ioo a b)\nG : ℝ → ℝ → ℝ := fun c x ↦ slope g x (x + c)\nG_integrable : ∀ (n : ℕ), Integrable (G (↑n)⁻¹) (volume.restrict (Icc a b))\nn : ℕ\n⊢ ∫ (x : ℝ... | [
"f : ℝ → ℝ\na b : ℝ\nhab : a ≤ b\nhf : MonotoneOn f (Icc a b)\ng : ℝ → ℝ := fun x ↦ f (max a (min x b))\nhg : Monotone g\nhfg : EqOn (deriv f) (deriv g) (Ioo a b)\nG : ℝ → ℝ → ℝ := fun c x ↦ slope g x (x + c)\nG_integrable : ∀ (n : ℕ), Integrable (G (↑n)⁻¹) (volume.restrict (Icc a b))\nn : ℕ\n⊢ ∫ (t : ℝ) in Ioc a b... | integral_Icc_eq_integral_Ioc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.IntervalIntegral.DerivIntegrable | {
"line": 134,
"column": 8
} | {
"line": 134,
"column": 37
} | {
"line": 135,
"column": 8
} | [
{
"pp": "case right\nf : ℝ → ℝ\na b : ℝ\nhf : MonotoneOn f (Icc a b)\nhab : a ≤ b\nG : ℕ → ℝ → ℝ\nhGf : ∀ᵐ (x : ℝ), x ∈ uIcc a b → Tendsto (fun n ↦ G n x) atTop (𝓝 (deriv f x))\nhG : ∀ (n : ℕ), AEStronglyMeasurable (G n) (volume.restrict (uIcc a b))\nhG' : liminf (fun n ↦ ∫⁻ (x : ℝ) in uIcc a b, ‖G n x‖ₑ) atTo... | [
"case right\nf : ℝ → ℝ\na b : ℝ\nhf : MonotoneOn f (Icc a b)\nhab : a ≤ b\nG : ℕ → ℝ → ℝ\nhGf : ∀ᵐ (x : ℝ), x ∈ uIcc a b → Tendsto (fun n ↦ G n x) atTop (𝓝 (deriv f x))\nhG : ∀ (n : ℕ), AEStronglyMeasurable (G n) (volume.restrict (uIcc a b))\nhG' : liminf (fun n ↦ ∫⁻ (x : ℝ) in uIcc a b, ‖G n x‖ₑ) atTop ≤ ENNReal.... | integral_Icc_eq_integral_Ioc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Taylor | {
"line": 90,
"column": 57
} | {
"line": 90,
"column": 83
} | {
"line": 90,
"column": 83
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nn : ℕ\ns : Set ℝ\nx₀ x : ℝ\n⊢ (PolynomialModule.eval x) (taylorWithin f n s x₀) +\n (PolynomialModule.eval x)\n ((PolynomialModule.comp (Polynomial.X - Polynomial.C x₀))\n (PolynomialModule.single ℝ (n... | [
"E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\nn : ℕ\ns : Set ℝ\nx₀ x : ℝ\n⊢ (PolynomialModule.eval x) (taylorWithin f n s x₀) +\n (PolynomialModule.eval (Polynomial.eval x (Polynomial.X - Polynomial.C x₀)))\n (PolynomialModule.single ℝ (n + 1) (taylorCoeffWithin f (n ... | PolynomialModule.comp_eval | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Calculus.Taylor | {
"line": 142,
"column": 6
} | {
"line": 142,
"column": 20
} | {
"line": 142,
"column": 21
} | [
{
"pp": "t x : ℝ\nn : ℕ\n⊢ HasDerivAt (fun y ↦ (-y + x) ^ (n + 1)) (-(↑n + 1) * (-t + x) ^ n) t",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"Eq.mpr",
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
"MulOne.... | [
"t x : ℝ\nn : ℕ\n⊢ HasDerivAt (fun y ↦ (-y + x) ^ (n + 1)) (-1 * (↑n + 1) * (-t + x) ^ n) t"
] | ← neg_one_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Taylor | {
"line": 142,
"column": 21
} | {
"line": 142,
"column": 39
} | {
"line": 142,
"column": 40
} | [
{
"pp": "t x : ℝ\nn : ℕ\n⊢ HasDerivAt (fun y ↦ (-y + x) ^ (n + 1)) (-1 * (↑n + 1) * (-t + x) ^ n) t",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"NegZeroCl... | [
"t x : ℝ\nn : ℕ\n⊢ HasDerivAt (fun y ↦ (-y + x) ^ (n + 1)) ((↑n + 1) * -1 * (-t + x) ^ n) t"
] | mul_comm (-1 : ℝ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Taylor | {
"line": 142,
"column": 51
} | {
"line": 142,
"column": 69
} | {
"line": 142,
"column": 70
} | [
{
"pp": "t x : ℝ\nn : ℕ\n⊢ HasDerivAt (fun y ↦ (-y + x) ^ (n + 1)) ((↑n + 1) * (-1 * (-t + x) ^ n)) t",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"NegZero... | [
"t x : ℝ\nn : ℕ\n⊢ HasDerivAt (fun y ↦ (-y + x) ^ (n + 1)) ((↑n + 1) * ((-t + x) ^ n * -1)) t"
] | mul_comm (-1 : ℝ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.Taylor | {
"line": 288,
"column": 2
} | {
"line": 288,
"column": 23
} | {
"line": 291,
"column": 0
} | [
{
"pp": "f : ℝ → ℝ\nx₀ : ℝ\nn : ℕ\ns : Set ℝ\nhs : Convex ℝ s\nhx₀s : x₀ ∈ s\nhf : ContDiffOn ℝ (↑n) f s\nx : ℝ\n⊢ (f x - taylorWithinEval f n s x₀ x) / (x - x₀) ^ n = ((x - x₀) ^ n)⁻¹ • (f x - taylorWithinEval f n s x₀ x)",
"ppTerm": "?m.116",
"assigned": true,
"usedConstants": [
"taylorWithi... | [] | simp [div_eq_inv_mul] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Module.Ray | {
"line": 77,
"column": 45
} | {
"line": 79,
"column": 26
} | {
"line": 81,
"column": 0
} | [
{
"pp": "F : Type u_2\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nx y : F\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ SameRay ℝ x y ↔ ‖x‖⁻¹ • x = ‖y‖⁻¹ • y",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"GroupWithZ... | [] | by
rw [inv_smul_eq_iff₀, smul_comm, eq_comm, inv_smul_eq_iff₀, sameRay_iff_norm_smul_eq] <;>
rwa [norm_ne_zero_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.StrictConvexSpace | {
"line": 116,
"column": 2
} | {
"line": 116,
"column": 73
} | {
"line": 118,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nh : ∀ (x y : E), ‖x‖ = 1 → ‖y‖ = 1 → x ≠ y → ∃ a b, 0 ≤ a ∧ 0 ≤ b ∧ a + b = 1 ∧ ‖a • x + b • y‖ ≠ 1\nx : E\nhx : ‖x‖ = 1\ny : E\nhy : ‖y‖ = 1\nhne : x ≠ y\na b : ℝ\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhne' : ‖a • x + b • y‖ ≠ 1\... | [] | exact ⟨_, ⟨a, b, ha, hb, hab, rfl⟩, mt mem_sphere_zero_iff_norm.1 hne'⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.Convex | {
"line": 45,
"column": 4
} | {
"line": 45,
"column": 11
} | {
"line": 46,
"column": 4
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nε : ℝ\nhε : 0 < ε\nx : F\nhx : ‖x‖ = 1\ny : F\nhy : ‖y‖ = 1\nhxy : ε ≤ ‖x - y‖\n⊢ 2 * (1 * ... | [
"case refine_2\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : RCLike 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : InnerProductSpace 𝕜 E\ninst✝¹ : SeminormedAddCommGroup F\ninst✝ : InnerProductSpace ℝ F\nε : ℝ\nhε : 0 < ε\nx : F\nhx : ‖x‖ = 1\ny : F\nhy : ‖y‖ = 1\nhxy : ε ≤ ‖x - y‖\n⊢ 4 - ‖x - y‖ ^ 2 ≤ 4 -... | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Convex.Uniform | {
"line": 110,
"column": 4
} | {
"line": 111,
"column": 89
} | {
"line": 112,
"column": 2
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : UniformConvexSpace E\nε : ℝ\ninst✝ : NormedSpace ℝ E\nhε : 0 < ε\nr : ℝ\nhr : r ≤ 0\n⊢ ∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ ≤ r → ∀ ⦃y : E⦄, ‖y‖ ≤ r → ε ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 * r - δ",
"ppTerm": "?inl",
"assigned": true,
"usedCons... | [] | exact ⟨1, one_pos, fun x hx y hy h => (hε.not_ge <|
h.trans <| (norm_sub_le _ _).trans <| add_nonpos (hx.trans hr) (hy.trans hr)).elim⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Uniform | {
"line": 110,
"column": 4
} | {
"line": 111,
"column": 89
} | {
"line": 112,
"column": 2
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : UniformConvexSpace E\nε : ℝ\ninst✝ : NormedSpace ℝ E\nhε : 0 < ε\nr : ℝ\nhr : r ≤ 0\n⊢ ∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ ≤ r → ∀ ⦃y : E⦄, ‖y‖ ≤ r → ε ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 * r - δ",
"ppTerm": "?inl",
"assigned": true,
"usedCons... | [] | exact ⟨1, one_pos, fun x hx y hy h => (hε.not_ge <|
h.trans <| (norm_sub_le _ _).trans <| add_nonpos (hx.trans hr) (hy.trans hr)).elim⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Uniform | {
"line": 110,
"column": 4
} | {
"line": 111,
"column": 89
} | {
"line": 112,
"column": 2
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : UniformConvexSpace E\nε : ℝ\ninst✝ : NormedSpace ℝ E\nhε : 0 < ε\nr : ℝ\nhr : r ≤ 0\n⊢ ∃ δ, 0 < δ ∧ ∀ ⦃x : E⦄, ‖x‖ ≤ r → ∀ ⦃y : E⦄, ‖y‖ ≤ r → ε ≤ ‖x - y‖ → ‖x + y‖ ≤ 2 * r - δ",
"ppTerm": "?inl",
"assigned": true,
"usedCons... | [] | exact ⟨1, one_pos, fun x hx y hy h => (hε.not_ge <|
h.trans <| (norm_sub_le _ _).trans <| add_nonpos (hx.trans hr) (hy.trans hr)).elim⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.SpecificFunctions.Deriv | {
"line": 152,
"column": 2
} | {
"line": 153,
"column": 73
} | {
"line": 154,
"column": 2
} | [
{
"pp": "⊢ ContinuousOn (fun x ↦ √x * log x) (Ioi 1)",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real",
"Set.Ioi",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Continuous.continuousOn",
"CommRing.toNonUnitalCommRing",
... | [
"x : ℝ\nhx : x ∈ Ioi 1\n⊢ deriv^[2] (fun x ↦ √x * log x) x < 0"
] | · exact continuous_sqrt.continuousOn.mul
(continuousOn_log.mono fun x hx => ne_of_gt (zero_lt_one.trans hx)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.SpecificFunctions.Deriv | {
"line": 163,
"column": 2
} | {
"line": 163,
"column": 86
} | {
"line": 164,
"column": 2
} | [
{
"pp": "⊢ StrictConcaveOn ℝ (Icc 0 π) sin",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"IsOrderedModule.toPosSMulMono",
"Real.partialOrder",
"Real",
"Semiring.toModule",
"Real.pi",
"Real.instZero",
"IsStrictOrderedModule.toIsOrderedModule",
... | [
"x : ℝ\nhx : x ∈ interior (Icc 0 π)\n⊢ deriv^[2] sin x < 0"
] | apply strictConcaveOn_of_deriv2_neg (convex_Icc _ _) continuousOn_sin fun x hx => ?_ | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Complex | {
"line": 37,
"column": 4
} | {
"line": 37,
"column": 11
} | {
"line": 38,
"column": 2
} | [
{
"pp": "θ : ℂ\n⊢ cexp (θ * I - -θ * I) = -1 ↔ cexp (2 * θ * I) = -1",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddCommMo... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Convex.Deriv | {
"line": 102,
"column": 4
} | {
"line": 106,
"column": 63
} | {
"line": 107,
"column": 4
} | [
{
"pp": "case neg\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : ℝ\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : ℝ\nhxa : x < a\nhaw : a < w\nb : ℝ\nhwb : w < b\nhby : b < y\nha : f w - f x < deriv f a * (w - x)\nhb : f y - f w < deriv f b ... | [
"case neg\nx y : ℝ\nf : ℝ → ℝ\nhf : ContinuousOn f (Icc x y)\nhxy : x < y\nhf'_mono : StrictMonoOn (deriv f) (Ioo x y)\nw : ℝ\nhw : deriv f w = 0\nhxw : x < w\nhwy : w < y\na : ℝ\nhxa : x < a\nhaw : a < w\nb : ℝ\nhwb : w < b\nhby : b < y\nha : f w - f x < deriv f a * (w - x)\nhb : f y - f w < deriv f b * (y - w)\nt... | have : deriv f a * (w - x) < deriv f b * (w - x) := by
apply mul_lt_mul _ le_rfl (sub_pos.2 hxw) _
· exact hf'_mono ⟨hxa, haw.trans hwy⟩ ⟨hxw.trans hwb, hby⟩ (haw.trans hwb)
· rw [← hw]
exact (hf'_mono ⟨hxw, hwy⟩ ⟨hxw.trans hwb, hby⟩ hwb).le | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.Deriv | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 93
} | {
"line": 238,
"column": 2
} | [
{
"pp": "D : Set ℝ\nhD : Convex ℝ D\nf f' f'' : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x\nhf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x\nhf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x\nthis : EqOn (deriv f) f' (interior D)\n⊢ ConvexOn ℝ D f",
... | [
"case refine_1\nD : Set ℝ\nhD : Convex ℝ D\nf f' f'' : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x\nhf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x\nhf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x\nthis : EqOn (deriv f) f' (interior D)\n⊢ DifferentiableOn ℝ... | refine convexOn_of_deriv2_nonneg hD hf (fun x hx ↦ (hf' _ hx).differentiableWithinAt) ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Convex.Deriv | {
"line": 238,
"column": 2
} | {
"line": 239,
"column": 55
} | {
"line": 240,
"column": 2
} | [
{
"pp": "case refine_1\nD : Set ℝ\nhD : Convex ℝ D\nf f' f'' : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x\nhf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x\nhf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x\nthis : EqOn (deriv f) f' (interior D)\n⊢ Differ... | [
"case refine_2\nD : Set ℝ\nhD : Convex ℝ D\nf f' f'' : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x\nhf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x\nhf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x\nthis : EqOn (deriv f) f' (interior D)\n⊢ ∀ x ∈ interior D, ... | · rw [differentiableOn_congr this]
exact fun x hx ↦ (hf'' _ hx).differentiableWithinAt | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Deriv | {
"line": 241,
"column": 4
} | {
"line": 241,
"column": 22
} | {
"line": 242,
"column": 4
} | [
{
"pp": "case refine_2\nD : Set ℝ\nhD : Convex ℝ D\nf f' f'' : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x\nhf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x\nhf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x\nthis : EqOn (deriv f) f' (interior D)\nx : ℝ\nh... | [
"D : Set ℝ\nhD : Convex ℝ D\nf f' f'' : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x\nhf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x\nhf''₀ : ∀ x ∈ interior D, 0 ≤ f'' x\nthis : EqOn (deriv f) f' (interior D)\nx : ℝ\nhx : x ∈ interior D\n⊢ deriv... | convert hf''₀ _ hx | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
Mathlib.Analysis.Convex.Deriv | {
"line": 254,
"column": 2
} | {
"line": 255,
"column": 55
} | {
"line": 256,
"column": 2
} | [
{
"pp": "case refine_1\nD : Set ℝ\nhD : Convex ℝ D\nf f' f'' : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x\nhf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x\nhf''₀ : ∀ x ∈ interior D, f'' x ≤ 0\nthis : EqOn (deriv f) f' (interior D)\n⊢ Differ... | [
"case refine_2\nD : Set ℝ\nhD : Convex ℝ D\nf f' f'' : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x\nhf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x\nhf''₀ : ∀ x ∈ interior D, f'' x ≤ 0\nthis : EqOn (deriv f) f' (interior D)\n⊢ ∀ x ∈ interior D, ... | · rw [differentiableOn_congr this]
exact fun x hx ↦ (hf'' _ hx).differentiableWithinAt | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Deriv | {
"line": 257,
"column": 4
} | {
"line": 257,
"column": 22
} | {
"line": 258,
"column": 4
} | [
{
"pp": "case refine_2\nD : Set ℝ\nhD : Convex ℝ D\nf f' f'' : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x\nhf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x\nhf''₀ : ∀ x ∈ interior D, f'' x ≤ 0\nthis : EqOn (deriv f) f' (interior D)\nx : ℝ\nh... | [
"D : Set ℝ\nhD : Convex ℝ D\nf f' f'' : ℝ → ℝ\nhf : ContinuousOn f D\nhf' : ∀ x ∈ interior D, HasDerivWithinAt f (f' x) (interior D) x\nhf'' : ∀ x ∈ interior D, HasDerivWithinAt f' (f'' x) (interior D) x\nhf''₀ : ∀ x ∈ interior D, f'' x ≤ 0\nthis : EqOn (deriv f) f' (interior D)\nx : ℝ\nhx : x ∈ interior D\n⊢ deriv... | convert hf''₀ _ hx | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
Mathlib.Analysis.Convex.Deriv | {
"line": 414,
"column": 4
} | {
"line": 414,
"column": 23
} | {
"line": 416,
"column": 0
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nx : 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nhfc : ConvexOn 𝕜 s f\nhxs : x ∈ interior s\ny : 𝕜\nhyx : x < y\nhys : y ∈ s\ny' : 𝕜\nx✝ : y' ∈ slope f x... | [] | simp [hys, hyx.ne'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Convex.Deriv | {
"line": 414,
"column": 4
} | {
"line": 414,
"column": 23
} | {
"line": 416,
"column": 0
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nx : 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nhfc : ConvexOn 𝕜 s f\nhxs : x ∈ interior s\ny : 𝕜\nhyx : x < y\nhys : y ∈ s\ny' : 𝕜\nx✝ : y' ∈ slope f x... | [] | simp [hys, hyx.ne'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Deriv | {
"line": 414,
"column": 4
} | {
"line": 414,
"column": 23
} | {
"line": 416,
"column": 0
} | [
{
"pp": "case refine_2\n𝕜 : Type u_1\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nx : 𝕜\ninst✝¹ : TopologicalSpace 𝕜\ninst✝ : OrderTopology 𝕜\nhfc : ConvexOn 𝕜 s f\nhxs : x ∈ interior s\ny : 𝕜\nhyx : x < y\nhys : y ∈ s\ny' : 𝕜\nx✝ : y' ∈ slope f x... | [] | simp [hys, hyx.ne'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 112,
"column": 2
} | {
"line": 112,
"column": 18
} | {
"line": 113,
"column": 2
} | [
{
"pp": "case inl\n⊢ sin 0 ^ 2 ≤ 0 ^ 2",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"False",
"Real.instLE",
"Real",
"Real.instZero",
"instReflLe",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"Std.le_refl._simp_1",
"Nat.instCharZero",
... | [
"case inr\nx : ℝ\nhx : x ≠ 0\n⊢ sin x ^ 2 ≤ x ^ 2"
] | case inl => simp | Lean.Elab.Tactic.evalCase | Lean.Parser.Tactic.case |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 125,
"column": 2
} | {
"line": 125,
"column": 18
} | {
"line": 126,
"column": 2
} | [
{
"pp": "case inl\n⊢ 1 - 0 ^ 2 / 2 ≤ cos 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"False",
"Real.instLE",
"Real",
"instHDiv",
"Real.instZero",
"Real.cos",
"instReflLe",
"congrArg",
"sub_ze... | [
"case inr\nx : ℝ\nhx : x ≠ 0\n⊢ 1 - x ^ 2 / 2 ≤ cos x"
] | case inl => simp | Lean.Elab.Tactic.evalCase | Lean.Parser.Tactic.case |
Mathlib.Analysis.Convex.Deriv | {
"line": 497,
"column": 4
} | {
"line": 497,
"column": 20
} | {
"line": 498,
"column": 4
} | [
{
"pp": "case inr.refine_1\nS : Set ℝ\nf : ℝ → ℝ\nhfc : ConvexOn ℝ S f\nx : ℝ\nhxs : x ∈ interior S\ny : ℝ\nhys : y ∈ interior S\nhxy✝ : x ≤ y\nhxy : x < y\n⊢ (slope f y '' {y_1 | y_1 ∈ S ∧ y < y_1}).Nonempty",
"ppTerm": "?inr.refine_1",
"assigned": true,
"usedConstants": [
"Real",
"Pseu... | [
"case inr.refine_1\nS : Set ℝ\nf : ℝ → ℝ\nhfc : ConvexOn ℝ S f\nx : ℝ\nhxs : x ∈ interior S\ny : ℝ\nhys : y ∈ interior S\nhxy✝ : x ≤ y\nhxy : x < y\nhys' : y ∈ interior S\n⊢ (slope f y '' {y_1 | y_1 ∈ S ∧ y < y_1}).Nonempty"
] | have hys' := hys | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic | {
"line": 57,
"column": 37
} | {
"line": 58,
"column": 60
} | {
"line": 60,
"column": 0
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedAddCommGroup F\ninst✝¹ : InnerProductSpace ℝ E\ninst✝ : InnerProductSpace ℝ F\nf : E →ₗᵢ[ℝ] F\nu v : E\n⊢ angle (f u) (f v) = angle u v",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"LinearIsometry",... | [] | by
rw [angle, angle, f.inner_map_map, f.norm_map, f.norm_map] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.BorelCaratheodory | {
"line": 46,
"column": 22
} | {
"line": 46,
"column": 29
} | {
"line": 46,
"column": 29
} | [
{
"pp": "M : ℝ\nz w : ℂ\nx✝¹ : M ≠ 0\nx✝ : 2 * ↑M - z ≠ 0\nh : w = z / (2 * ↑M - z)\n⊢ z = z * 2 * ↑M / (2 * ↑M - z + z)",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"Group... | [
"M : ℝ\nz w : ℂ\nx✝¹ : M ≠ 0\nx✝ : 2 * ↑M - z ≠ 0\nh : w = z / (2 * ↑M - z)\n⊢ z = z * ↑M * (↑M)⁻¹"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic | {
"line": 187,
"column": 2
} | {
"line": 187,
"column": 9
} | {
"line": 189,
"column": 0
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\nhx : x ≠ 0\nhy : x + y ≠ 0\n⊢ √(⟪x, x⟫ * (⟪x, x⟫ + ⟪x, y⟫ + (⟪x, y⟫ + ⟪y, y⟫)) - (⟪x, x⟫ + ⟪x, y⟫) ^ 2) = √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ ^ 2)",
"ppTerm": "?m.65",
"assigned": true,
"usedConstants": [
"Mat... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Analysis.Convex.Deriv | {
"line": 965,
"column": 2
} | {
"line": 966,
"column": 75
} | {
"line": 968,
"column": 0
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nx y f' : ℝ\nhfc : StrictConcaveOn ℝ S f\nhx : x ∈ S\nhy : y ∈ S\nhxy : x < y\nhf' : HasDerivWithinAt f f' (Ioi x) x\n⊢ slope f x y < f'",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
... | [] | simpa only [Pi.neg_def, slope_neg, neg_neg] using
neg_lt_neg (hfc.neg.lt_slope_of_hasDerivWithinAt_Ioi hx hy hxy hf'.neg) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Convex.Deriv | {
"line": 965,
"column": 2
} | {
"line": 966,
"column": 75
} | {
"line": 968,
"column": 0
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nx y f' : ℝ\nhfc : StrictConcaveOn ℝ S f\nhx : x ∈ S\nhy : y ∈ S\nhxy : x < y\nhf' : HasDerivWithinAt f f' (Ioi x) x\n⊢ slope f x y < f'",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
... | [] | simpa only [Pi.neg_def, slope_neg, neg_neg] using
neg_lt_neg (hfc.neg.lt_slope_of_hasDerivWithinAt_Ioi hx hy hxy hf'.neg) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Deriv | {
"line": 965,
"column": 2
} | {
"line": 966,
"column": 75
} | {
"line": 968,
"column": 0
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nx y f' : ℝ\nhfc : StrictConcaveOn ℝ S f\nhx : x ∈ S\nhy : y ∈ S\nhxy : x < y\nhf' : HasDerivWithinAt f f' (Ioi x) x\n⊢ slope f x y < f'",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
... | [] | simpa only [Pi.neg_def, slope_neg, neg_neg] using
neg_lt_neg (hfc.neg.lt_slope_of_hasDerivWithinAt_Ioi hx hy hxy hf'.neg) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Schwarz | {
"line": 101,
"column": 6
} | {
"line": 101,
"column": 68
} | {
"line": 102,
"column": 4
} | [
{
"pp": "case hc\nf : ℂ → ℂ\nc z : ℂ\nR₂ : ℝ\nn : ℕ\nhn : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ (w - c) ^ n\nR₁ : ℝ\nhz : z ∈ ball c R₁\nhR₁ : 0 < R₁\nhd : DifferentiableOn ℂ f (closedBall c R₁)\nh_maps : MapsTo f (closedBall c R₁) (closedBall (f c) R₂)\nhne : z ≠ c\ng : ℂ → ℂ := ⋯\ng' : ℂ → ℂ := ⋯\n⊢ closedBall... | [] | exact sdiff_mem_nhdsWithin_compl (closedBall_mem_nhds _ hR₁) _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Complex.Schwarz | {
"line": 101,
"column": 6
} | {
"line": 101,
"column": 68
} | {
"line": 102,
"column": 4
} | [
{
"pp": "case hc\nf : ℂ → ℂ\nc z : ℂ\nR₂ : ℝ\nn : ℕ\nhn : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ (w - c) ^ n\nR₁ : ℝ\nhz : z ∈ ball c R₁\nhR₁ : 0 < R₁\nhd : DifferentiableOn ℂ f (closedBall c R₁)\nh_maps : MapsTo f (closedBall c R₁) (closedBall (f c) R₂)\nhne : z ≠ c\ng : ℂ → ℂ := ⋯\ng' : ℂ → ℂ := ⋯\n⊢ closedBall... | [] | exact sdiff_mem_nhdsWithin_compl (closedBall_mem_nhds _ hR₁) _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Schwarz | {
"line": 101,
"column": 6
} | {
"line": 101,
"column": 68
} | {
"line": 102,
"column": 4
} | [
{
"pp": "case hc\nf : ℂ → ℂ\nc z : ℂ\nR₂ : ℝ\nn : ℕ\nhn : (fun x ↦ f x - f c) =o[𝓝 c] fun w ↦ (w - c) ^ n\nR₁ : ℝ\nhz : z ∈ ball c R₁\nhR₁ : 0 < R₁\nhd : DifferentiableOn ℂ f (closedBall c R₁)\nh_maps : MapsTo f (closedBall c R₁) (closedBall (f c) R₂)\nhne : z ≠ c\ng : ℂ → ℂ := ⋯\ng' : ℂ → ℂ := ⋯\n⊢ closedBall... | [] | exact sdiff_mem_nhdsWithin_compl (closedBall_mem_nhds _ hR₁) _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 214,
"column": 4
} | {
"line": 214,
"column": 34
} | {
"line": 215,
"column": 4
} | [
{
"pp": "case neg.h\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : 𝕜\nf : 𝕜 → E\nh₁ : MeromorphicAt f x\nh₂ : ¬meromorphicOrderAt f x = ⊤\ng : 𝕜 → E\nh₁g : AnalyticAt 𝕜 g x\nh₂g : g x ≠ 0\nh₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ ... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nx : 𝕜\nf : 𝕜 → E\nh₁ : MeromorphicAt f x\nh₂ : ¬meromorphicOrderAt f x = ⊤\ng : 𝕜 → E\nh₁g : AnalyticAt 𝕜 g x\nh₂g : g x ≠ 0\nh₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f x)... | filter_upwards [h₃g] with a ha | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.Meromorphic.Order | {
"line": 120,
"column": 8
} | {
"line": 120,
"column": 20
} | {
"line": 120,
"column": 20
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nn : ℤ\nhf : MeromorphicAt f x\nh✝ : ¬analyticOrderAt (fun z ↦ (z - x) ^ Exists.choose hf • f z) x = ⊤\nm : ℕ\nh : ↑m = analyticOrderAt (fun z ↦ (z - x) ^ Exists... | [
"𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nn : ℤ\nhf : MeromorphicAt f x\nh✝ : ¬analyticOrderAt (fun z ↦ (z - x) ^ Exists.choose hf • f z) x = ⊤\nm : ℕ\nh : ↑m = analyticOrderAt (fun z ↦ (z - x) ^ Exists.choose hf •... | zpow_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 399,
"column": 26
} | {
"line": 399,
"column": 38
} | {
"line": 399,
"column": 38
} | [
{
"pp": "case e_a.e_a\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\ng : 𝕜 → 𝕜\nx : 𝕜\nhg : AnalyticAt 𝕜 g x\nn : ℤ\nq : 𝕜 → E\nhq_an : AnalyticAt 𝕜 q (g x)\nm : ℕ\np : 𝕜 → 𝕜\nhq_ne : ¬q (g x) = 0\nhf : f =ᶠ[𝓝 (g x... | [
"case e_a.e_a\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\ng : 𝕜 → 𝕜\nx : 𝕜\nhg : AnalyticAt 𝕜 g x\nn : ℤ\nq : 𝕜 → E\nhq_an : AnalyticAt 𝕜 q (g x)\nm : ℕ\np : 𝕜 → 𝕜\nhq_ne : ¬q (g x) = 0\nhf : f =ᶠ[𝓝 (g x)] (fun x_1 ... | zpow_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 481,
"column": 6
} | {
"line": 481,
"column": 36
} | {
"line": 482,
"column": 6
} | [
{
"pp": "case neg\n𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\nn : ℤ\nf : 𝕜 → 𝕜\nh₁ : MeromorphicAt f x\nh₂ : ¬meromorphicOrderAt f x = ⊤\ng : 𝕜 → 𝕜\nh₁g : AnalyticAt 𝕜 g x\nh₂g : g x ≠ 0\nh₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f x).untop₀ • g z\n⊢ f ^ n =ᶠ[𝓝[≠] x] fun z ... | [
"𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nx : 𝕜\nn : ℤ\nf : 𝕜 → 𝕜\nh₁ : MeromorphicAt f x\nh₂ : ¬meromorphicOrderAt f x = ⊤\ng : 𝕜 → 𝕜\nh₁g : AnalyticAt 𝕜 g x\nh₂g : g x ≠ 0\nh₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ (meromorphicOrderAt f x).untop₀ • g z\na : 𝕜\nha : f a = (a - x) ^ (meromorphicOrderAt... | filter_upwards [h₃g] with a ha | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.Meromorphic.Order | {
"line": 288,
"column": 75
} | {
"line": 288,
"column": 87
} | {
"line": 288,
"column": 87
} | [
{
"pp": "case coe\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nhf : AnalyticAt 𝕜 f x\na✝ : ℕ\nhn : analyticOrderAt f x = ↑a✝\n⊢ ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ ↑a✝ • g z... | [
"case coe\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nhf : AnalyticAt 𝕜 f x\na✝ : ℕ\nhn : analyticOrderAt f x = ↑a✝\n⊢ ∃ g, AnalyticAt 𝕜 g x ∧ g x ≠ 0 ∧ ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = (z - x) ^ a✝ • g z"
] | zpow_natCast | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 784,
"column": 2
} | {
"line": 791,
"column": 51
} | {
"line": 793,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nU : Set 𝕜\n⊢ toMeromorphicNFOn f U = f ↔ MeromorphicNFOn f U",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"If... | [] | constructor <;> intro h
· rw [h.symm]
apply meromorphicNFOn_toMeromorphicNFOn
· ext x
by_cases hx : x ∈ U
· simp only [toMeromorphicNFOn, h.meromorphicOn, ↓reduceDIte, hx]
rw [toMeromorphicNFAt_eq_self.2 (h hx)]
· simp [toMeromorphicNFOn, h.meromorphicOn, hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 784,
"column": 2
} | {
"line": 791,
"column": 51
} | {
"line": 793,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nU : Set 𝕜\n⊢ toMeromorphicNFOn f U = f ↔ MeromorphicNFOn f U",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"If... | [] | constructor <;> intro h
· rw [h.symm]
apply meromorphicNFOn_toMeromorphicNFOn
· ext x
by_cases hx : x ∈ U
· simp only [toMeromorphicNFOn, h.meromorphicOn, ↓reduceDIte, hx]
rw [toMeromorphicNFAt_eq_self.2 (h hx)]
· simp [toMeromorphicNFOn, h.meromorphicOn, hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.Order | {
"line": 428,
"column": 2
} | {
"line": 428,
"column": 41
} | {
"line": 429,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\nR : Type u_3\ninst✝⁶ : NormedRing R\ninst✝⁵ : NoZeroDivisors R\ninst✝⁴ : Module R E\ninst✝³ : IsBoundedSMul R E\ninst✝² : Module.IsTorsionFree R E\nx : 𝕜\ninst✝¹ : NormedAlgebra ... | [] | cases h₂f : meromorphicOrderAt f x with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.Analysis.Meromorphic.Order | {
"line": 460,
"column": 4
} | {
"line": 462,
"column": 31
} | {
"line": 463,
"column": 2
} | [
{
"pp": "case empty\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\n𝕜' : Type u_4\ninst✝¹ : NontriviallyNormedField 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nx : 𝕜\nι : Type u_5\nf : ι → 𝕜 → 𝕜'\nhf : ∀ i ∈ ∅, MeromorphicAt (f i) x\n⊢ meromorphicOrderAt (∏ i ∈ ∅, f i) x = ∑ i ∈ ∅, meromorphicOrderAt (f i) x",
... | [] | rw [Finset.prod_empty, Finset.sum_empty, ← WithTop.coe_zero, meromorphicOrderAt_eq_int_iff]
· exact ⟨1, analyticAt_const, by simp⟩
· apply MeromorphicAt.const | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.Order | {
"line": 460,
"column": 4
} | {
"line": 462,
"column": 31
} | {
"line": 463,
"column": 2
} | [
{
"pp": "case empty\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\n𝕜' : Type u_4\ninst✝¹ : NontriviallyNormedField 𝕜'\ninst✝ : NormedAlgebra 𝕜 𝕜'\nx : 𝕜\nι : Type u_5\nf : ι → 𝕜 → 𝕜'\nhf : ∀ i ∈ ∅, MeromorphicAt (f i) x\n⊢ meromorphicOrderAt (∏ i ∈ ∅, f i) x = ∑ i ∈ ∅, meromorphicOrderAt (f i) x",
... | [] | rw [Finset.prod_empty, Finset.sum_empty, ← WithTop.coe_zero, meromorphicOrderAt_eq_int_iff]
· exact ⟨1, analyticAt_const, by simp⟩
· apply MeromorphicAt.const | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.CanonicalDecomposition | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 68
} | {
"line": 182,
"column": 2
} | [
{
"pp": "case pos\nz : ℂ\nR : ℝ\nw : ℂ\nhR : 0 < R\nhw : w = 0\n⊢ meromorphicOrderAt (canonicalFactor R w) 0 ≠ ⊤",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Complex.meromorphicOrderAt_canonicalFactor",
"NormedCommRing.toSeminormedCommRing",
... | [] | simp_all [meromorphicOrderAt_canonicalFactor (mem_ball_self hR)] | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Analysis.Complex.CanonicalDecomposition | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 68
} | {
"line": 182,
"column": 2
} | [
{
"pp": "case pos\nz : ℂ\nR : ℝ\nw : ℂ\nhR : 0 < R\nhw : w = 0\n⊢ meromorphicOrderAt (canonicalFactor R w) 0 ≠ ⊤",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Complex.meromorphicOrderAt_canonicalFactor",
"NormedCommRing.toSeminormedCommRing",
... | [] | simp_all [meromorphicOrderAt_canonicalFactor (mem_ball_self hR)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.CanonicalDecomposition | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 68
} | {
"line": 182,
"column": 2
} | [
{
"pp": "case pos\nz : ℂ\nR : ℝ\nw : ℂ\nhR : 0 < R\nhw : w = 0\n⊢ meromorphicOrderAt (canonicalFactor R w) 0 ≠ ⊤",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Complex.meromorphicOrderAt_canonicalFactor",
"NormedCommRing.toSeminormedCommRing",
... | [] | simp_all [meromorphicOrderAt_canonicalFactor (mem_ball_self hR)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.CanonicalDecomposition | {
"line": 345,
"column": 6
} | {
"line": 345,
"column": 38
} | {
"line": 345,
"column": 39
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nf : ℂ → E\nh₁f : MeromorphicOn f (closedBall 0 R)\nh₂f : ∀ (u : ↑(closedBall 0 R)), meromorphicOrderAt f ↑u ≠ ⊤\nhR : 0 < R\nφ : ℂ → E := (∏ᶠ (c : ℂ), canonicalFactor R c ^ (divisor f (ball 0 R)) c) • f\nhφ : MeromorphicOn φ (... | [
"E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nR : ℝ\nf : ℂ → E\nh₁f : MeromorphicOn f (closedBall 0 R)\nh₂f : ∀ (u : ↑(closedBall 0 R)), meromorphicOrderAt f ↑u ≠ ⊤\nhR : 0 < R\nφ : ℂ → E := (∏ᶠ (c : ℂ), canonicalFactor R c ^ (divisor f (ball 0 R)) c) • f\nhφ : MeromorphicOn φ (closedBall 0... | canonicalDecomposition_aux₂ h₁f, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Isometry | {
"line": 74,
"column": 30
} | {
"line": 74,
"column": 44
} | {
"line": 74,
"column": 45
} | [
{
"pp": "a : Circle\nh : rotation a = conjLIE\nh1 : ↑a = 1\nhI : ↑a * I = -I\n⊢ False",
"ppTerm": "?m.105",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"MulOne.toOne",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"congrAr... | [
"a : Circle\nh : rotation a = conjLIE\nh1 : ↑a = 1\nhI : ↑a * I = -1 * I\n⊢ False"
] | ← neg_one_mul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Meromorphic.Order | {
"line": 706,
"column": 4
} | {
"line": 706,
"column": 40
} | {
"line": 707,
"column": 4
} | [
{
"pp": "case right\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nU : Set 𝕜\nhf : MeromorphicOn f U\n⊢ IsOpen[instTopologicalSpaceSubtype] {u | meromorphicOrderAt f ↑u = ⊤}",
"ppTerm": "?right✝",
"assigned": true,... | [
"case right\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nU : Set 𝕜\nhf : MeromorphicOn f U\n⊢ ∀ x ∈ {u | meromorphicOrderAt f ↑u = ⊤},\n ∃ t ⊆ {u | meromorphicOrderAt f ↑u = ⊤}, IsOpen[instTopologicalSpaceSubtype] t ∧ x ∈... | apply isOpen_iff_forall_mem_open.mpr | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Homotopy.Lifting | {
"line": 229,
"column": 10
} | {
"line": 229,
"column": 23
} | {
"line": 229,
"column": 23
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\np : E → X\ncov : IsCoveringMap p\nγ : C(↑I, X)\ne : E\nγ_0 : γ 0 = p e\nU : X → Set X := fun x ↦ ⋯.choose\nmem_base : ∀ (x : X), x ∈ ⋯.choose\nU_open : ∀ (x : X), IsOpen[inst✝] ⋯.choose\nh✝ : ∀ (x : X), IsOpen[inst✝¹] ... | [
"E : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace E\ninst✝ : TopologicalSpace X\np : E → X\ncov : IsCoveringMap p\nγ : C(↑I, X)\ne : E\nγ_0 : γ 0 = p e\nU : X → Set X := fun x ↦ ⋯.choose\nmem_base : ∀ (x : X), x ∈ ⋯.choose\nU_open : ∀ (x : X), IsOpen[inst✝] ⋯.choose\nh✝ : ∀ (x : X), IsOpen[inst✝¹] (p ⁻¹' ⋯.cho... | convert! cont | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.Complex.Hadamard | {
"line": 115,
"column": 2
} | {
"line": 115,
"column": 71
} | {
"line": 116,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\nz : ℂ\nε : ℝ\nhε : ε > 0\n⊢ ‖↑(ε + sSupNormIm f 0) ^ (z - 1)‖ * ‖↑(ε + sSupNormIm f 1) ^ (-z)‖ =\n (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re)",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"No... | [
"E : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\nz : ℂ\nε : ℝ\nhε : ε > 0\n⊢ (ε + sSupNormIm f 0) ^ (z - 1).re * (ε + sSupNormIm f 1) ^ (-z).re =\n (ε + sSupNormIm f 0) ^ (z.re - 1) * (ε + sSupNormIm f 1) ^ (-z.re)"
] | repeat rw [norm_cpow_eq_rpow_re_of_pos (sSupNormIm_eps_pos f hε _) _] | Lean.Elab.Tactic.evalRepeat | Lean.Parser.Tactic.tacticRepeat_ |
Mathlib.Analysis.Meromorphic.Order | {
"line": 886,
"column": 4
} | {
"line": 888,
"column": 62
} | {
"line": 889,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nx : 𝕜\nf : 𝕜 → E\ng : 𝕜 → 𝕜\nhg : AnalyticAt 𝕜 g x\nhg' : deriv g x ≠ 0\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : CharZero 𝕜\nhf : MeromorphicAt f (g x)\n⊢ meromorphicOr... | [] | have hgo : analyticOrderAt _ x = 1 := hg.analyticOrderAt_sub_eq_one_of_deriv_ne_zero hg'
rw [hf.meromorphicOrderAt_comp hg, hgo] <;>
simp [eventuallyConst_iff_analyticOrderAt_sub_eq_top, hgo] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.Order | {
"line": 886,
"column": 4
} | {
"line": 888,
"column": 62
} | {
"line": 889,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nx : 𝕜\nf : 𝕜 → E\ng : 𝕜 → 𝕜\nhg : AnalyticAt 𝕜 g x\nhg' : deriv g x ≠ 0\ninst✝¹ : CompleteSpace 𝕜\ninst✝ : CharZero 𝕜\nhf : MeromorphicAt f (g x)\n⊢ meromorphicOr... | [] | have hgo : analyticOrderAt _ x = 1 := hg.analyticOrderAt_sub_eq_one_of_deriv_ne_zero hg'
rw [hf.meromorphicOrderAt_comp hg, hgo] <;>
simp [eventuallyConst_iff_analyticOrderAt_sub_eq_top, hgo] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Hadamard | {
"line": 162,
"column": 28
} | {
"line": 162,
"column": 91
} | {
"line": 162,
"column": 91
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nε : ℝ\nhε : ε > 0\nB : ℝ\nhB : ∀ y ∈ norm ∘ f '' verticalClosedStrip 0 1, y ≤ B\nz : ℂ\nhset : z ∈ verticalClosedStrip 0 1\n⊢ ‖f z‖ ∈ norm ∘ f '' verticalClosedStrip 0 1",
"ppTerm": "?m.118",
"assigned": true,
... | [] | simpa [image_congr, mem_image, comp_apply] using ⟨z, hset, rfl⟩ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Complex.Hadamard | {
"line": 162,
"column": 28
} | {
"line": 162,
"column": 91
} | {
"line": 162,
"column": 91
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nε : ℝ\nhε : ε > 0\nB : ℝ\nhB : ∀ y ∈ norm ∘ f '' verticalClosedStrip 0 1, y ≤ B\nz : ℂ\nhset : z ∈ verticalClosedStrip 0 1\n⊢ ‖f z‖ ∈ norm ∘ f '' verticalClosedStrip 0 1",
"ppTerm": "?m.118",
"assigned": true,
... | [] | simpa [image_congr, mem_image, comp_apply] using ⟨z, hset, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Hadamard | {
"line": 162,
"column": 28
} | {
"line": 162,
"column": 91
} | {
"line": 162,
"column": 91
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nε : ℝ\nhε : ε > 0\nB : ℝ\nhB : ∀ y ∈ norm ∘ f '' verticalClosedStrip 0 1, y ≤ B\nz : ℂ\nhset : z ∈ verticalClosedStrip 0 1\n⊢ ‖f z‖ ∈ norm ∘ f '' verticalClosedStrip 0 1",
"ppTerm": "?m.118",
"assigned": true,
... | [] | simpa [image_congr, mem_image, comp_apply] using ⟨z, hset, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 58
} | {
"line": 76,
"column": 0
} | [
{
"pp": "c : ℂ\nr a₁ a₂ b : ℝ\nha₁ : ↑a₁ + ↑b * I ∈ ball c r\nha₂ : ↑a₂ + ↑b * I ∈ ball c r\n⊢ (fun x ↦ ↑x + ↑b * I) '' [[a₁, a₂]] = (↑a₁ + ↑b * I).Rectangle (↑a₂ + ↑b * I)",
"ppTerm": "?m.107",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
"Set.Icc_self",
"Real.partialOrd... | [] | simp [horizontalSegment_eq a₁ a₂ b, ha₁, ha₂, Rectangle] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 58
} | {
"line": 76,
"column": 0
} | [
{
"pp": "case convert_1\nc : ℂ\nr a₁ a₂ b : ℝ\nha₁ : ↑a₁ + ↑b * I ∈ ball c r\nha₂ : ↑a₂ + ↑b * I ∈ ball c r\n⊢ ↑(↑a₁ + ↑b * I).re + ↑(↑a₂ + ↑b * I).im * I ∈ ball c r",
"ppTerm": "?convert_1",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
"NormedCommRing.toSeminormedCommRing",
... | [] | simp [horizontalSegment_eq a₁ a₂ b, ha₁, ha₂, Rectangle] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.HasPrimitives | {
"line": 74,
"column": 2
} | {
"line": 74,
"column": 58
} | {
"line": 76,
"column": 0
} | [
{
"pp": "case convert_2\nc : ℂ\nr a₁ a₂ b : ℝ\nha₁ : ↑a₁ + ↑b * I ∈ ball c r\nha₂ : ↑a₂ + ↑b * I ∈ ball c r\n⊢ ↑(↑a₂ + ↑b * I).re + ↑(↑a₁ + ↑b * I).im * I ∈ ball c r",
"ppTerm": "?convert_2",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
"NormedCommRing.toSeminormedCommRing",
... | [] | simp [horizontalSegment_eq a₁ a₂ b, ha₁, ha₂, Rectangle] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.Hadamard | {
"line": 374,
"column": 6
} | {
"line": 378,
"column": 58
} | {
"line": 379,
"column": 4
} | [
{
"pp": "case mp\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\nl u : ℝ\nhul : l < u\ne : E\n⊢ (∃ x, x.re = 1 ∧ f (↑l + x * (↑u - ↑l)) = e) → ∃ x, x.re = u ∧ f x = e",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Real",
"Complex.mul_re",
"HMul.hMul",
"sub_s... | [] | intro h
obtain ⟨z, hz₁, hz₂⟩ := h
use ↑l + z * (↑u - ↑l)
simp only [add_re, ofReal_re, mul_re, hz₁, sub_re, one_mul, sub_im, ofReal_im, sub_self,
mul_zero, sub_zero, add_sub_cancel, hz₂, and_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Hadamard | {
"line": 374,
"column": 6
} | {
"line": 378,
"column": 58
} | {
"line": 379,
"column": 4
} | [
{
"pp": "case mp\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\nl u : ℝ\nhul : l < u\ne : E\n⊢ (∃ x, x.re = 1 ∧ f (↑l + x * (↑u - ↑l)) = e) → ∃ x, x.re = u ∧ f x = e",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Real",
"Complex.mul_re",
"HMul.hMul",
"sub_s... | [] | intro h
obtain ⟨z, hz₁, hz₂⟩ := h
use ↑l + z * (↑u - ↑l)
simp only [add_re, ofReal_re, mul_re, hz₁, sub_re, one_mul, sub_im, ofReal_im, sub_self,
mul_zero, sub_zero, add_sub_cancel, hz₂, and_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Homotopy.Lifting | {
"line": 601,
"column": 4
} | {
"line": 601,
"column": 51
} | {
"line": 603,
"column": 0
} | [
{
"pp": "case mpr\nE : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\np : E → X\nG : Type u_4\ninst✝¹ : Group G\ninst✝ : MulAction G E\nhp : IsQuotientCoveringMap p G\nx : X\ne : ↑(p ⁻¹' {x})\nγ : FundamentalGroup E ↑e\n⊢ Path.Homotopic.Quotient.map γ { toFun := p, continuous_... | [] | aesop (add simp FundamentalGroup.mapOfEq_apply) | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Homotopy.Lifting | {
"line": 632,
"column": 2
} | {
"line": 635,
"column": 7
} | {
"line": 637,
"column": 0
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\np : E → X\nG : Type u_4\ninst✝¹ : Group G\ninst✝ : MulAction G E\nhp : IsQuotientCoveringMap p G\nx : X\ne : ↑(p ⁻¹' {x})\nγ : FundamentalGroup X x\ng : Gᵐᵒᵖ\n⊢ (hp.fundamentalGroupToMulOpposite e) γ = g ↔ MulOpposite... | [] | rw [fundamentalGroupToMulOpposite, ← MulOpposite.unop_injective.eq_iff, iff_comm, eq_comm,
← hp.fiberEquivGroup_smul_self e]
have := hp.isCancelSMul.right_cancel'
aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Homotopy.Lifting | {
"line": 632,
"column": 2
} | {
"line": 635,
"column": 7
} | {
"line": 637,
"column": 0
} | [
{
"pp": "E : Type u_1\nX : Type u_2\ninst✝³ : TopologicalSpace E\ninst✝² : TopologicalSpace X\np : E → X\nG : Type u_4\ninst✝¹ : Group G\ninst✝ : MulAction G E\nhp : IsQuotientCoveringMap p G\nx : X\ne : ↑(p ⁻¹' {x})\nγ : FundamentalGroup X x\ng : Gᵐᵒᵖ\n⊢ (hp.fundamentalGroupToMulOpposite e) γ = g ↔ MulOpposite... | [] | rw [fundamentalGroupToMulOpposite, ← MulOpposite.unop_injective.eq_iff, iff_comm, eq_comm,
← hp.fiberEquivGroup_smul_self e]
have := hp.isCancelSMul.right_cancel'
aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
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