module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 508,
"column": 53
} | {
"line": 509,
"column": 70
} | {
"line": 511,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nf : α → ℝ\nhf : Measurable f\n⊢ MeasurableSet {p | p.2 = f p.1}",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Icc_self",
"Real.partialOrder",
"Real",
"MeasurableSet",
"congrArg",
"S... | [] | by
simpa using measurableSet_region_between_cc hf hf MeasurableSet.univ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 526,
"column": 6
} | {
"line": 526,
"column": 51
} | {
"line": 527,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ\ns : Set α\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nh : (fun x ↦ volume {a | x ∈ s ∧ a ∈ Ioo (f x) (g x)}) = s.indicator fun x ↦ ofReal (g x - f x)\n⊢ ∫⁻ (x : α), volume (Prod.mk x ⁻¹' regionBetween f g s) ∂μ = ∫⁻ (... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ\ns : Set α\nhf : Measurable f\nhg : Measurable g\nhs : MeasurableSet s\nh : (fun x ↦ volume {a | x ∈ s ∧ a ∈ Ioo (f x) (g x)}) = s.indicator fun x ↦ ofReal (g x - f x)\n⊢ ∫⁻ (x : α), volume {a | x ∈ s ∧ a ∈ Ioo (f x) (g x)} ∂μ = ∫⁻ (y : α) in s, o... | dsimp only [regionBetween, preimage_setOf_eq] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 201,
"column": 55
} | {
"line": 207,
"column": 98
} | {
"line": 209,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : AffineSubspace ℝ E\nhs : s ≠ ⊤\n⊢ μ ↑s = 0",
"ppTerm": "?m.25",
"assigned": true,
"usedCon... | [] | by
rcases s.eq_bot_or_nonempty with (rfl | hne)
· rw [AffineSubspace.bot_coe, measure_empty]
rw [Ne, ← AffineSubspace.direction_eq_top_iff_of_nonempty hne] at hs
rcases hne with ⟨x, hx : x ∈ s⟩
simpa only [AffineSubspace.coe_direction_eq_vsub_set_right hx, vsub_eq_sub, sub_eq_add_neg,
image_add_right, neg... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 591,
"column": 4
} | {
"line": 591,
"column": 74
} | {
"line": 592,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ\nf_mble : AEMeasurable f μ\ng_mble : AEMeasurable g μ\ns : Set α\ns_mble : NullMeasurableSet s μ\n⊢ NullMeasurableSet (fun p ↦ Real.le✝ (f p.1) p.2) (μ.prod volume)",
"ppTerm": "?refine_1",
"assigned": true,
... | [
"case refine_1\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ\nf_mble : AEMeasurable f μ\ng_mble : AEMeasurable g μ\ns : Set α\ns_mble : NullMeasurableSet s μ\n⊢ NullMeasurableSet {p | f p.1 ≤ p.2} (μ.prod volume)"
] | change NullMeasurableSet {p : α × ℝ | f p.fst ≤ p.snd} (μ.prod volume) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.MeasureTheory.Measure.Lebesgue.Basic | {
"line": 606,
"column": 4
} | {
"line": 606,
"column": 74
} | {
"line": 607,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ\nf_mble : AEMeasurable f μ\ng_mble : AEMeasurable g μ\ns : Set α\ns_mble : NullMeasurableSet s μ\n⊢ NullMeasurableSet (fun p ↦ Real.le✝ (f p.1) p.2) (μ.prod volume)",
"ppTerm": "?refine_1",
"assigned": true,
... | [
"case refine_1\nα : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\nf g : α → ℝ\nf_mble : AEMeasurable f μ\ng_mble : AEMeasurable g μ\ns : Set α\ns_mble : NullMeasurableSet s μ\n⊢ NullMeasurableSet {p | f p.1 ≤ p.2} (μ.prod volume)"
] | change NullMeasurableSet {p : α × ℝ | f p.fst ≤ p.snd} (μ.prod volume) | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 248,
"column": 2
} | {
"line": 248,
"column": 79
} | {
"line": 249,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = fi... | [
"E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nf : E →ₗ[ℝ] E\nι : Type := Fin (finrank ℝ E)\nthis✝ : FiniteDimensional ℝ (ι → ℝ)\nthis : finrank ℝ E = finrank ℝ (ι →... | have Ce : Continuous e := (e : E →ₗ[ℝ] ι → ℝ).continuous_of_finiteDimensional | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 394,
"column": 2
} | {
"line": 394,
"column": 33
} | {
"line": 395,
"column": 2
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nhs : NullMeasurableSet s μ\nr : ℝ\nhs' : s.Nonempty\n⊢ NullMeasurableSet (r • s) μ",
... | [
"case inr.inl\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\ns : Set E\nhs : NullMeasurableSet s μ\nhs' : s.Nonempty\n⊢ NullMeasurableSet (0 • s) μ",
"case inr.inr... | obtain rfl | hr := eq_or_ne r 0 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Measure.Lebesgue.EqHaar | {
"line": 520,
"column": 2
} | {
"line": 520,
"column": 33
} | {
"line": 521,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nx : E\nr : ℝ\nhr : r ≠ 0\n⊢ μ (sphere x r) = 0",
"ppTerm": "?m.22",
"assigned": true,
"usedCon... | [
"case inl\nE : Type u_1\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℝ E\ninst✝³ : MeasurableSpace E\ninst✝² : BorelSpace E\ninst✝¹ : FiniteDimensional ℝ E\nμ : Measure E\ninst✝ : μ.IsAddHaarMeasure\nx : E\nr : ℝ\nhr : r ≠ 0\nh : r < 0\n⊢ μ (sphere x r) = 0",
"case inr\nE : Type u_1\ninst✝⁵ : NormedAddCom... | rcases hr.lt_or_gt with (h | h) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.BoxIntegral.Box.SubboxInduction | {
"line": 135,
"column": 14
} | {
"line": 135,
"column": 22
} | {
"line": 136,
"column": 4
} | [
{
"pp": "case zero\nι : Type u_1\np : Box ι → Prop\nI : Box ι\nH_nhds :\n ∀ z ∈ Box.Icc I,\n ∃ U ∈ 𝓝[Box.Icc I] z,\n ∀ J ≤ I,\n ∀ (m : ℕ),\n z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J\nhpI : ¬p I\ns : Box ι → Set ι\nhs : ∀... | [] | simp [J] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.BoxIntegral.Box.SubboxInduction | {
"line": 135,
"column": 14
} | {
"line": 135,
"column": 22
} | {
"line": 136,
"column": 4
} | [
{
"pp": "case zero\nι : Type u_1\np : Box ι → Prop\nI : Box ι\nH_nhds :\n ∀ z ∈ Box.Icc I,\n ∃ U ∈ 𝓝[Box.Icc I] z,\n ∀ J ≤ I,\n ∀ (m : ℕ),\n z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J\nhpI : ¬p I\ns : Box ι → Set ι\nhs : ∀... | [] | simp [J] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.Box.SubboxInduction | {
"line": 135,
"column": 14
} | {
"line": 135,
"column": 22
} | {
"line": 136,
"column": 4
} | [
{
"pp": "case zero\nι : Type u_1\np : Box ι → Prop\nI : Box ι\nH_nhds :\n ∀ z ∈ Box.Icc I,\n ∃ U ∈ 𝓝[Box.Icc I] z,\n ∀ J ≤ I,\n ∀ (m : ℕ),\n z ∈ Box.Icc J → Box.Icc J ⊆ U → (∀ (i : ι), J.upper i - J.lower i = (I.upper i - I.lower i) / 2 ^ m) → p J\nhpI : ¬p I\ns : Box ι → Set ι\nhs : ∀... | [] | simp [J] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 239,
"column": 4
} | {
"line": 242,
"column": 51
} | {
"line": 243,
"column": 2
} | [
{
"pp": "case mp\nι : Type u_1\nI : Box ι\nπ₁ π₂ : Prepartition I\n⊢ π₁ ≤ π₂ → (∀ J ∈ π₁, ∀ J' ∈ π₂, (↑J ∩ ↑J').Nonempty → J ≤ J') ∧ π₁.iUnion ⊆ π₂.iUnion",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"BoxIntegral.Prepartition.eq_of_mem_of_mem",
"B... | [] | refine fun H => ⟨fun J hJ J' hJ' Hne => ?_, iUnion_mono H⟩
rcases H hJ with ⟨J'', hJ'', Hle⟩
rcases Hne with ⟨x, hx, hx'⟩
rwa [π₂.eq_of_mem_of_mem hJ' hJ'' hx' (Hle hx)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 239,
"column": 4
} | {
"line": 242,
"column": 51
} | {
"line": 243,
"column": 2
} | [
{
"pp": "case mp\nι : Type u_1\nI : Box ι\nπ₁ π₂ : Prepartition I\n⊢ π₁ ≤ π₂ → (∀ J ∈ π₁, ∀ J' ∈ π₂, (↑J ∩ ↑J').Nonempty → J ≤ J') ∧ π₁.iUnion ⊆ π₂.iUnion",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"BoxIntegral.Prepartition.eq_of_mem_of_mem",
"B... | [] | refine fun H => ⟨fun J hJ J' hJ' Hne => ?_, iUnion_mono H⟩
rcases H hJ with ⟨J'', hJ'', Hle⟩
rcases Hne with ⟨x, hx, hx'⟩
rwa [π₂.eq_of_mem_of_mem hJ' hJ'' hx' (Hle hx)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 350,
"column": 4
} | {
"line": 350,
"column": 39
} | {
"line": 351,
"column": 2
} | [
{
"pp": "case mp\nι : Type u_1\nI : Box ι\nπ : Prepartition I\nπi : (J : Box ι) → Prepartition J\nπi' : Box ι → (J : Box ι) → Prepartition J\nJ J₁ : Box ι\nhJ₁ : J₁ ∈ π\nJ₂ : Box ι\nhJ₂ : J₂ ∈ πi J₁\nhJ : J ∈ πi' J₁ J₂\n⊢ J ∈ πi' (π.biUnionIndex πi J₂) J₂",
"ppTerm": "?mp",
"assigned": true,
"usedCo... | [] | rwa [π.biUnionIndex_of_mem hJ₁ hJ₂] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 572,
"column": 76
} | {
"line": 573,
"column": 54
} | {
"line": 574,
"column": 4
} | [
{
"pp": "ι : Type u_1\nI J J₁ J₂ : Box ι\nπ π₁✝ π₂✝ : Prepartition I\nx : ι → ℝ\nπi πi₁ πi₂ : (J : Box ι) → Prepartition J\nπ₁ π₂ : Prepartition I\nh : Disjoint π₁.iUnion π₂.iUnion\nthis : ∀ J₁ ∈ π₁, ∀ J₂ ∈ π₂, J₁ ≠ J₂ → Disjoint ↑J₁ ↑J₂\n⊢ (↑(π₁.boxes ∪ π₂.boxes)).Pairwise (Disjoint on Box.toSet)",
"ppTerm... | [] | by
simpa [pairwise_union_of_symm, pairwiseDisjoint] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.BoxIntegral.Partition.Basic | {
"line": 583,
"column": 58
} | {
"line": 584,
"column": 72
} | {
"line": 586,
"column": 0
} | [
{
"pp": "ι : Type u_1\nI : Box ι\nπ₁ π₂ : Prepartition I\nh : Disjoint π₁.iUnion π₂.iUnion\n⊢ (π₁.disjUnion π₂ h).iUnion = π₁.iUnion ∪ π₂.iUnion",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Real",
"Finset.instUnion",
"BoxIntegral.Prepartition",
"Iff.of_eq",
... | [] | by
simp [disjUnion, Prepartition.iUnion, iUnion_or, iUnion_union_distrib] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Module.ZLattice.Basic | {
"line": 557,
"column": 4
} | {
"line": 557,
"column": 81
} | {
"line": 558,
"column": 4
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝⁹ : NormedField K\ninst✝⁸ : LinearOrder K\ninst✝⁷ : IsStrictOrderedRing K\ninst✝⁶ : HasSolidNorm K\ninst✝⁵ : FloorRing K\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\ninst✝² : FiniteDimensional K E\ninst✝¹ : ProperSpace E\nL : Submodule ℤ E\ni... | [
"case refine_1\nK : Type u_1\ninst✝⁹ : NormedField K\ninst✝⁸ : LinearOrder K\ninst✝⁷ : IsStrictOrderedRing K\ninst✝⁶ : HasSolidNorm K\ninst✝⁵ : FloorRing K\nE : Type u_2\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace K E\ninst✝² : FiniteDimensional K E\ninst✝¹ : ProperSpace E\nL : Submodule ℤ E\ninst✝ : Discr... | have : Fintype t := Set.Finite.fintype ((Set.range b).toFinite.subset ht_inc) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.BoxIntegral.Partition.Split | {
"line": 254,
"column": 24
} | {
"line": 254,
"column": 94
} | {
"line": 256,
"column": 0
} | [
{
"pp": "case insert\nι : Type u_1\nI : Box ι\nπ : Prepartition I\np : ι × ℝ\ns : Finset (ι × ℝ)\na✝ : p ∉ s\nihp : π ⊓ splitMany I s = π.biUnion fun J ↦ splitMany J s\n⊢ π ⊓ splitMany I (insert p s) = π.biUnion fun J ↦ splitMany J (insert p s)",
"ppTerm": "?insert",
"assigned": true,
"usedConstants... | [] | simp_rw [splitMany_insert, ← inf_assoc, ihp, inf_split, biUnion_assoc] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.BoxIntegral.Partition.Split | {
"line": 254,
"column": 24
} | {
"line": 254,
"column": 94
} | {
"line": 256,
"column": 0
} | [
{
"pp": "case insert\nι : Type u_1\nI : Box ι\nπ : Prepartition I\np : ι × ℝ\ns : Finset (ι × ℝ)\na✝ : p ∉ s\nihp : π ⊓ splitMany I s = π.biUnion fun J ↦ splitMany J s\n⊢ π ⊓ splitMany I (insert p s) = π.biUnion fun J ↦ splitMany J (insert p s)",
"ppTerm": "?insert",
"assigned": true,
"usedConstants... | [] | simp_rw [splitMany_insert, ← inf_assoc, ihp, inf_split, biUnion_assoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.Partition.Split | {
"line": 254,
"column": 24
} | {
"line": 254,
"column": 94
} | {
"line": 256,
"column": 0
} | [
{
"pp": "case insert\nι : Type u_1\nI : Box ι\nπ : Prepartition I\np : ι × ℝ\ns : Finset (ι × ℝ)\na✝ : p ∉ s\nihp : π ⊓ splitMany I s = π.biUnion fun J ↦ splitMany J s\n⊢ π ⊓ splitMany I (insert p s) = π.biUnion fun J ↦ splitMany J (insert p s)",
"ppTerm": "?insert",
"assigned": true,
"usedConstants... | [] | simp_rw [splitMany_insert, ← inf_assoc, ihp, inf_split, biUnion_assoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Oscillation | {
"line": 141,
"column": 4
} | {
"line": 145,
"column": 73
} | {
"line": 146,
"column": 4
} | [
{
"pp": "case pos\nE : Type u\nF : Type v\ninst✝¹ : PseudoEMetricSpace F\ninst✝ : PseudoEMetricSpace E\nK : Set E\nf : E → F\nD : Set E\nε : ℝ≥0∞\ncomp : IsCompact K\nhK : ∀ x ∈ K, oscillationWithin f D x < ε\nS : ℝ → Set E := fun r ↦ {x | ∃ a > r, ediam (f '' (eball x (ENNReal.ofReal a) ∩ D)) ≤ ε}\nS_open : ∀ ... | [
"case neg\nE : Type u\nF : Type v\ninst✝¹ : PseudoEMetricSpace F\ninst✝ : PseudoEMetricSpace E\nK : Set E\nf : E → F\nD : Set E\nε : ℝ≥0∞\ncomp : IsCompact K\nhK : ∀ x ∈ K, oscillationWithin f D x < ε\nS : ℝ → Set E := fun r ↦ {x | ∃ a > r, ediam (f '' (eball x (ENNReal.ofReal a) ∩ D)) ≤ ε}\nS_open : ∀ r > 0, IsOpe... | · use Tfin.isWF.min T_nonempty, Tb (Tfin.isWF.min_mem T_nonempty)
intro x hx
obtain ⟨r, hr⟩ := mem_iUnion.1 (hT hx)
simp only [mem_iUnion, exists_prop] at hr
exact (S_antitone _ r (IsWF.min_le Tfin.isWF T_nonempty hr.1)) hr.2 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.BoxIntegral.UnitPartition | {
"line": 190,
"column": 2
} | {
"line": 190,
"column": 94
} | {
"line": 191,
"column": 2
} | [
{
"pp": "ι : Type u_1\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : Fintype ι\nν : ι → ℤ\n⊢ Metric.diam (Set.univ.pi fun i ↦ Set.Icc ((box n ν).lower i) ((box n ν).upper i)) ≤ 1 / ↑n",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Real.partialOrder",
"Real... | [
"ι : Type u_1\nn : ℕ\ninst✝¹ : NeZero n\ninst✝ : Fintype ι\nν : ι → ℤ\ni : ι\n⊢ Metric.ediam (Set.Icc ((box n ν).lower i) ((box n ν).upper i)) ≤ ENNReal.ofReal (1 / ↑n)"
] | refine ENNReal.toReal_le_of_le_ofReal (by positivity) <| Metric.ediam_pi_le_of_le fun i ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.Orientation | {
"line": 214,
"column": 2
} | {
"line": 214,
"column": 83
} | {
"line": 215,
"column": 2
} | [
{
"pp": "case h\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_3\ninst✝ : IsEmpty ι\nx : M [⋀^ι]→ₗ[R] R\nhx : x ≠ 0\nh : LinearIndependent R ![x, AlternatingMap.constLinearEquivOfIsEmpty 1]\n⊢ Fal... | [
"case h\nR : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\nM : Type u_2\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nι : Type u_3\ninst✝ : IsEmpty ι\nx : M [⋀^ι]→ₗ[R] R\nhx : x ≠ 0\nh : LinearIndependent R ![x, AlternatingMap.constLinearEquivOfIsEmpty 1]\nf : M [⋀^ι]→ₗ[R] ... | set f : (M [⋀^ι]→ₗ[R] R) ≃ₗ[R] R := AlternatingMap.constLinearEquivOfIsEmpty.symm | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 87,
"column": 2
} | {
"line": 91,
"column": 28
} | {
"line": 93,
"column": 0
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nI : Box ι\nπ : TaggedPrepartition I\nf₁ f₂ : (ι → ℝ) → E\nvol₁ vol₂ : ι →ᵇᵃ[⊤] E →L[ℝ] F\nhf : EqOn f₁ f₂ (Box.Icc I)\nhvol : EqOn ⇑vol₁ ⇑vol₂ ↑π.boxes\n⊢... | [] | unfold integralSum
refine Finset.sum_congr rfl (fun J hJ ↦ ?_)
congr 1
· exact hvol hJ
exact hf (π.tag_mem_Icc J) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 87,
"column": 2
} | {
"line": 91,
"column": 28
} | {
"line": 93,
"column": 0
} | [
{
"pp": "ι : Type u\nE : Type v\nF : Type w\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace ℝ E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace ℝ F\nI : Box ι\nπ : TaggedPrepartition I\nf₁ f₂ : (ι → ℝ) → E\nvol₁ vol₂ : ι →ᵇᵃ[⊤] E →L[ℝ] F\nhf : EqOn f₁ f₂ (Box.Icc I)\nhvol : EqOn ⇑vol₁ ⇑vol₂ ↑π.boxes\n⊢... | [] | unfold integralSum
refine Finset.sum_congr rfl (fun J hJ ↦ ?_)
congr 1
· exact hvol hJ
exact hf (π.tag_mem_Icc J) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.BoxIntegral.UnitPartition | {
"line": 407,
"column": 40
} | {
"line": 408,
"column": 54
} | {
"line": 408,
"column": 54
} | [
{
"pp": "case refine_3\nι : Type u_1\nn : ℕ\ninst✝¹ : NeZero n\ns : Set (ι → ℝ)\nF : (ι → ℝ) → ℝ\ninst✝ : Fintype ι\nB : Box ι\nhB : hasIntegralVertices B\nhs₀ : s ⊆ ↑B\nthis : Fintype ↑(s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))\nI : Box ι\nhI : I ∈ (prepartition n B).boxes ∧ (prepartition n B).tag... | [
"case refine_3\nι : Type u_1\nn : ℕ\ninst✝¹ : NeZero n\ns : Set (ι → ℝ)\nF : (ι → ℝ) → ℝ\ninst✝ : Fintype ι\nB : Box ι\nhB : hasIntegralVertices B\nhs₀ : s ⊆ ↑B\nthis : Fintype ↑(s ∩ (↑n)⁻¹ • ↑(span ℤ (Set.range ⇑(Pi.basisFun ℝ ι))))\nI : Box ι\nhI : I ∈ (prepartition n B).boxes ∧ (prepartition n B).tag I ∈ s\n⊢ ta... | prepartition_tag n
(mem_admissibleIndex_of_mem_box n hB (hs₀ hI.2)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 356,
"column": 2
} | {
"line": 359,
"column": 30
} | {
"line": 361,
"column": 0
} | [
{
"pp": "ι : Type u\nI : Box ι\ninst✝¹ : Fintype ι\nl : IntegrationParams\ng : (ι → ℝ) → ℝ\nhg : ∀ x ∈ Box.Icc I, 0 ≤ g x\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\n⊢ 0 ≤ integral I l g μ.toBoxAdditive.toSMul",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Real.instIsOr... | [] | by_cases hgi : Integrable I l g μ.toBoxAdditive.toSMul
· refine ge_of_tendsto' hgi.hasIntegral fun π => sum_nonneg fun J _ => ?_
exact mul_nonneg ENNReal.toReal_nonneg (hg _ <| π.tag_mem_Icc _)
· rw [integral, dif_neg hgi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.BoxIntegral.Basic | {
"line": 356,
"column": 2
} | {
"line": 359,
"column": 30
} | {
"line": 361,
"column": 0
} | [
{
"pp": "ι : Type u\nI : Box ι\ninst✝¹ : Fintype ι\nl : IntegrationParams\ng : (ι → ℝ) → ℝ\nhg : ∀ x ∈ Box.Icc I, 0 ≤ g x\nμ : Measure (ι → ℝ)\ninst✝ : IsLocallyFiniteMeasure μ\n⊢ 0 ≤ integral I l g μ.toBoxAdditive.toSMul",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Real.instIsOr... | [] | by_cases hgi : Integrable I l g μ.toBoxAdditive.toSMul
· refine ge_of_tendsto' hgi.hasIntegral fun π => sum_nonneg fun J _ => ?_
exact mul_nonneg ENNReal.toReal_nonneg (hg _ <| π.tag_mem_Icc _)
· rw [integral, dif_neg hgi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.SpecificAsymptotics | {
"line": 43,
"column": 39
} | {
"line": 46,
"column": 66
} | {
"line": 48,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : NormedRing R\ninst✝ : NormMulClass R\np q : ℕ\nhpq : p < q\n⊢ (fun x ↦ x ^ p) =o[cobounded R] fun x ↦ x ^ q",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"one_pow",
"Eq.mpr",
"MulOne.toOne",
"SeminormedRing.toNorm",
"NormedRin... | [] | by
rw [← Nat.add_sub_of_le hpq.le]
simpa [pow_add] using (isBigO_refl (· ^ p) (cobounded R)).mul_isLittleO
((isLittleO_const_id_cobounded 1).pow (Nat.sub_pos_of_lt hpq)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.SpecificAsymptotics | {
"line": 177,
"column": 2
} | {
"line": 186,
"column": 47
} | {
"line": 188,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nu : ℕ → E\nl : E\nh : Tendsto u atTop (𝓝 l)\n⊢ Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, u i) atTop (𝓝 l)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMon... | [] | rw [← tendsto_sub_nhds_zero_iff, ← isLittleO_one_iff ℝ]
have := Asymptotics.isLittleO_sum_range_of_tendsto_zero (tendsto_sub_nhds_zero_iff.2 h)
apply ((isBigO_refl (fun n : ℕ => (n : ℝ)⁻¹) atTop).smul_isLittleO this).congr' _ _
· filter_upwards [Ici_mem_atTop 1] with n npos
have nposℝ : (0 : ℝ) < n := Nat.cas... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.SpecificAsymptotics | {
"line": 177,
"column": 2
} | {
"line": 186,
"column": 47
} | {
"line": 188,
"column": 0
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nu : ℕ → E\nl : E\nh : Tendsto u atTop (𝓝 l)\n⊢ Tendsto (fun n ↦ (↑n)⁻¹ • ∑ i ∈ Finset.range n, u i) atTop (𝓝 l)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMon... | [] | rw [← tendsto_sub_nhds_zero_iff, ← isLittleO_one_iff ℝ]
have := Asymptotics.isLittleO_sum_range_of_tendsto_zero (tendsto_sub_nhds_zero_iff.2 h)
apply ((isBigO_refl (fun n : ℕ => (n : ℝ)⁻¹) atTop).smul_isLittleO this).congr' _ _
· filter_upwards [Ici_mem_atTop 1] with n npos
have nposℝ : (0 : ℝ) < n := Nat.cas... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Module.ZLattice.Covolume | {
"line": 200,
"column": 34
} | {
"line": 200,
"column": 51
} | {
"line": 200,
"column": 52
} | [
{
"pp": "E : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : InnerProductSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\nL : Submodule ℤ E\ninst✝² : DiscreteTopology ↥L\ninst✝¹ : IsZLattice ℝ L\nι : Type u_3\ninst✝ : Fintype ι\nb : Basis ι ℤ ↥L\ns : Set E\nhs : Nu... | [
"E : Type u_2\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : InnerProductSpace ℝ E\ninst✝⁵ : FiniteDimensional ℝ E\ninst✝⁴ : MeasurableSpace E\ninst✝³ : BorelSpace E\nL : Submodule ℤ E\ninst✝² : DiscreteTopology ↥L\ninst✝¹ : IsZLattice ℝ L\nι : Type u_3\ninst✝ : Fintype ι\nb : Basis ι ℤ ↥L\ns : Set E\nhs : NullMeasurable... | Fintype.card_fin, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.PSeries | {
"line": 236,
"column": 2
} | {
"line": 242,
"column": 26
} | {
"line": 244,
"column": 0
} | [
{
"pp": "f : ℕ → ℝ\nh_nonneg : ∀ (n : ℕ), 0 ≤ f n\nh_mono : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m\n⊢ (Summable fun k ↦ 2 ^ k * f (2 ^ k)) ↔ Summable f",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"pow_pos",
"N... | [] | have h_succ_diff : SuccDiffBounded 2 (2 ^ ·) := by
intro n
simp [pow_succ, mul_two, two_mul]
convert!
summable_schlomilch_iff_of_nonneg h_nonneg h_mono (pow_pos zero_lt_two)
(pow_right_strictMono₀ one_lt_two) two_ne_zero h_succ_diff
simp [pow_succ, mul_two] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.PSeries | {
"line": 236,
"column": 2
} | {
"line": 242,
"column": 26
} | {
"line": 244,
"column": 0
} | [
{
"pp": "f : ℕ → ℝ\nh_nonneg : ∀ (n : ℕ), 0 ≤ f n\nh_mono : ∀ ⦃m n : ℕ⦄, 0 < m → m ≤ n → f n ≤ f m\n⊢ (Summable fun k ↦ 2 ^ k * f (2 ^ k)) ↔ Summable f",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"pow_pos",
"N... | [] | have h_succ_diff : SuccDiffBounded 2 (2 ^ ·) := by
intro n
simp [pow_succ, mul_two, two_mul]
convert!
summable_schlomilch_iff_of_nonneg h_nonneg h_mono (pow_pos zero_lt_two)
(pow_right_strictMono₀ one_lt_two) two_ne_zero h_succ_diff
simp [pow_succ, mul_two] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.MonoidAlgebra.Ideal | {
"line": 51,
"column": 8
} | {
"line": 51,
"column": 28
} | {
"line": 51,
"column": 28
} | [
{
"pp": "case mpr\nk : Type u_1\nG : Type u_3\ninst✝¹ : Monoid G\ninst✝ : Semiring k\ns : Set G\nx : k[G]\nRHS : Ideal k[G] :=\n { carrier := {p | ∀ m ∈ p.coeff.support, ∃ m' ∈ s, ∃ d, m = d * m'}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\nhx : x ∈ RHS\n⊢ x ∈ Ideal.span (⇑(of k G) '' s)",
"ppTerm": ... | [
"case mpr\nk : Type u_1\nG : Type u_3\ninst✝¹ : Monoid G\ninst✝ : Semiring k\ns : Set G\nx : k[G]\nRHS : Ideal k[G] :=\n { carrier := {p | ∀ m ∈ p.coeff.support, ∃ m' ∈ s, ∃ d, m = d * m'}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\nhx : x ∈ RHS\n⊢ x.coeff.sum single ∈ Ideal.span (⇑(of k G) '' s)"
] | ← x.sum_coeff_single | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MonoidAlgebra.Ideal | {
"line": 89,
"column": 8
} | {
"line": 89,
"column": 28
} | {
"line": 89,
"column": 28
} | [
{
"pp": "case mpr\nk : Type u_1\nA : Type u_2\ninst✝¹ : AddMonoid A\ninst✝ : Semiring k\ns : Set A\nx : k[A]\nRHS : Ideal k[A] :=\n { carrier := {p | ∀ m ∈ p.coeff.support, ∃ m' ∈ s, ∃ d, m = d + m'}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\nhx : x ∈ RHS\n⊢ x ∈ Ideal.span (of' k A '' s)",
"ppTerm":... | [
"case mpr\nk : Type u_1\nA : Type u_2\ninst✝¹ : AddMonoid A\ninst✝ : Semiring k\ns : Set A\nx : k[A]\nRHS : Ideal k[A] :=\n { carrier := {p | ∀ m ∈ p.coeff.support, ∃ m' ∈ s, ∃ d, m = d + m'}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\nhx : x ∈ RHS\n⊢ x.coeff.sum single ∈ Ideal.span (of' k A '' s)"
] | ← x.sum_coeff_single | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.MvPolynomial.Coeff | {
"line": 96,
"column": 2
} | {
"line": 97,
"column": 55
} | {
"line": 99,
"column": 0
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\nd : σ →₀ ℕ\nn : ℕ\n⊢ coeff d ((∑ i, X i) ^ n) = ↑(if (d.sum fun x m ↦ m) = n then d.multinomial else 0)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"one_pow",
"NonAssocSemiring.toAddCommMonoidW... | [] | have : (∑ i, X i : MvPolynomial σ R) = ∑ i, (1 : σ → R) i • X i := by simp
simp [this, coeff_linearCombination_X_pow_of_fintype] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.MvPolynomial.Coeff | {
"line": 96,
"column": 2
} | {
"line": 97,
"column": 55
} | {
"line": 99,
"column": 0
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : Fintype σ\nd : σ →₀ ℕ\nn : ℕ\n⊢ coeff d ((∑ i, X i) ^ n) = ↑(if (d.sum fun x m ↦ m) = n then d.multinomial else 0)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"one_pow",
"NonAssocSemiring.toAddCommMonoidW... | [] | have : (∑ i, X i : MvPolynomial σ R) = ∑ i, (1 : σ → R) i • X i := by simp
simp [this, coeff_linearCombination_X_pow_of_fintype] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.GameAdd | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 26
} | {
"line": 194,
"column": 2
} | [
{
"pp": "case intro.intro.h.inl.fst\nα : Type u_1\nrα : α → α → Prop\na✝ a : α\nh✝ : ∀ (y : α), rα y a → Acc rα y\niha : ∀ (y : α), rα y a → ∀ {b : α}, Acc rα b → Acc (Sym2.GameAdd rα) s(y, b)\nb✝ b : α\nhb : ∀ (y : α), rα y b → Acc rα y\nihb : ∀ (y : α), rα y b → Acc (Sym2.GameAdd rα) s(a, y)\nc : α\nrc : rα c... | [] | exact iha c rc ⟨b, hb⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.GameAdd | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 26
} | {
"line": 194,
"column": 2
} | [
{
"pp": "case intro.intro.h.inl.fst\nα : Type u_1\nrα : α → α → Prop\na✝ a : α\nh✝ : ∀ (y : α), rα y a → Acc rα y\niha : ∀ (y : α), rα y a → ∀ {b : α}, Acc rα b → Acc (Sym2.GameAdd rα) s(y, b)\nb✝ b : α\nhb : ∀ (y : α), rα y b → Acc rα y\nihb : ∀ (y : α), rα y b → Acc (Sym2.GameAdd rα) s(a, y)\nc : α\nrc : rα c... | [] | exact iha c rc ⟨b, hb⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.GameAdd | {
"line": 193,
"column": 4
} | {
"line": 193,
"column": 26
} | {
"line": 194,
"column": 2
} | [
{
"pp": "case intro.intro.h.inl.fst\nα : Type u_1\nrα : α → α → Prop\na✝ a : α\nh✝ : ∀ (y : α), rα y a → Acc rα y\niha : ∀ (y : α), rα y a → ∀ {b : α}, Acc rα b → Acc (Sym2.GameAdd rα) s(y, b)\nb✝ b : α\nhb : ∀ (y : α), rα y b → Acc rα y\nihb : ∀ (y : α), rα y b → Acc (Sym2.GameAdd rα) s(a, y)\nc : α\nrc : rα c... | [] | exact iha c rc ⟨b, hb⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.DFinsupp.WellFounded | {
"line": 135,
"column": 2
} | {
"line": 135,
"column": 30
} | {
"line": 135,
"column": 30
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\nhbot : ∀ ⦃i : ι⦄ ⦃a : α i⦄, ¬s i a 0\nhs : ∀ (i : ι), WellFounded (s i)\ninst✝ : DecidableEq ι\ni : ι\nhi : Acc (rᶜ ⊓ fun x1 x2 ↦ x1 ≠ x2) i\n⊢ ∀ (a : α i), Acc (DFinsupp.Lex r s) (single i ... | [
"case intro\nι : Type u_1\nα : ι → Type u_2\ninst✝¹ : (i : ι) → Zero (α i)\nr : ι → ι → Prop\ns : (i : ι) → α i → α i → Prop\nhbot : ∀ ⦃i : ι⦄ ⦃a : α i⦄, ¬s i a 0\nhs : ∀ (i : ι), WellFounded (s i)\ninst✝ : DecidableEq ι\ni✝ i : ι\nh✝ : ∀ (y : ι), (rᶜ ⊓ fun x1 x2 ↦ x1 ≠ x2) y i → Acc (rᶜ ⊓ fun x1 x2 ↦ x1 ≠ x2) y\ni... | induction hi with | _ i _ ih
=> _ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Algebra.MvPolynomial.NoZeroDivisors | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 22
} | {
"line": 90,
"column": 2
} | [
{
"pp": "R : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\nf g : MvPolynomial σ R\nh : f ∣ g\nhg : g ≠ 0\n⊢ f.totalDegree ≤ g.totalDegree",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Nat.instMulZeroClass",
"AddMonoidAlgebra.semiring",
"Sem... | [
"R : Type u_1\nσ : Type u_2\ninst✝¹ : CommSemiring R\ninst✝ : NoZeroDivisors R\nf r : MvPolynomial σ R\nhg : f * r ≠ 0\n⊢ f.totalDegree ≤ (f * r).totalDegree"
] | obtain ⟨r, rfl⟩ := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Data.Finsupp.MonomialOrder.DegLex | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 47
} | {
"line": 242,
"column": 2
} | [
{
"pp": "α : Type u_1\n⊢ degLex.toSyn (single 0 1 + single 1 1) < degLex.toSyn (single 0 2)",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"Finsupp.instAddZeroClass",
"Eq.mpr",
"MonomialOrder.linearOrderSyn",
"Nat.instMulZeroClass",
"Preorder.toLT",
"Eq... | [
"α : Type u_1\n⊢ degree (single 0 1 + single 1 1) < degree (ofDegLex (toDegLex (single 0 2))) ∨\n degree (single 0 1 + single 1 1) = degree (ofDegLex (toDegLex (single 0 2))) ∧\n toLex (single 0 1 + single 1 1) < toLex (ofDegLex (toDegLex (single 0 2)))"
] | rw [degLex_lt_iff, lt_iff, ofDegLex_toDegLex] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.Antidiag.FinsuppEquiv | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 89
} | {
"line": 81,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝ : DecidableEq ι\ns : Finset ι\nn : ℕ\n⊢ #(s.finsuppAntidiag n) = (#s).multichoose n",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"congrArg",
"Finset",
"Sym.instFintype",
"AddMonoid.toAddZeroClass",
"Finset.finsuppAntidiag",
... | [] | simp [card_eq_of_equiv_fintype (finsuppAntidiagEquiv s n), Sym.card_sym_eq_multichoose] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Antidiag.FinsuppEquiv | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 89
} | {
"line": 81,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝ : DecidableEq ι\ns : Finset ι\nn : ℕ\n⊢ #(s.finsuppAntidiag n) = (#s).multichoose n",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"congrArg",
"Finset",
"Sym.instFintype",
"AddMonoid.toAddZeroClass",
"Finset.finsuppAntidiag",
... | [] | simp [card_eq_of_equiv_fintype (finsuppAntidiagEquiv s n), Sym.card_sym_eq_multichoose] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Antidiag.FinsuppEquiv | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 89
} | {
"line": 81,
"column": 0
} | [
{
"pp": "ι : Type u_1\ninst✝ : DecidableEq ι\ns : Finset ι\nn : ℕ\n⊢ #(s.finsuppAntidiag n) = (#s).multichoose n",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"congrArg",
"Finset",
"Sym.instFintype",
"AddMonoid.toAddZeroClass",
"Finset.finsuppAntidiag",
... | [] | simp [card_eq_of_equiv_fintype (finsuppAntidiagEquiv s n), Sym.card_sym_eq_multichoose] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 555,
"column": 41
} | {
"line": 560,
"column": 10
} | {
"line": 562,
"column": 0
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommSemiring R\nf : MvPolynomial σ R\nn : ℕ\nhf : m.leadingCoeff f ^ n ≠ 0\n⊢ m.degree (f ^ n) = n • m.degree f",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MonomialOrder.linearOrderSyn",
"Nat... | [] | by
apply m.toSyn.injective
apply le_antisymm (m.degree_pow_le n)
apply le_degree
rw [mem_support_iff, coeff_pow_nsmul_degree]
exact hf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Squarefree | {
"line": 60,
"column": 8
} | {
"line": 60,
"column": 47
} | {
"line": 60,
"column": 47
} | [
{
"pp": "case neg\nn p : ℕ\nhn : ∀ (x : ℕ), emultiplicity x n ≤ 1 ∨ IsUnit x\nhn' : n ≠ 0\nhp : ¬Prime p\n⊢ n.factorization p ≤ 1",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instMulZeroClass",
"congrArg",
"id",
"... | [
"case neg\nn p : ℕ\nhn : ∀ (x : ℕ), emultiplicity x n ≤ 1 ∨ IsUnit x\nhn' : n ≠ 0\nhp : ¬Prime p\n⊢ 0 ≤ 1"
] | factorization_eq_zero_of_not_prime _ hp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Nat.Squarefree | {
"line": 88,
"column": 47
} | {
"line": 88,
"column": 86
} | {
"line": 88,
"column": 86
} | [
{
"pp": "case neg\nn m : ℕ\nhn : Squarefree n\nhm : Squarefree m\nh : ∀ (p : ℕ), Prime p → (p ∣ n ↔ p ∣ m)\np : ℕ\nhp : ¬Prime p\n⊢ 0 = m.factorization p",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"Eq.mpr",
"Nat.instMulZeroClass",
"congrA... | [
"case neg\nn m : ℕ\nhn : Squarefree n\nhm : Squarefree m\nh : ∀ (p : ℕ), Prime p → (p ∣ n ↔ p ∣ m)\np : ℕ\nhp : ¬Prime p\n⊢ 0 = 0"
] | factorization_eq_zero_of_not_prime _ hp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPolynomial.MonomialOrder | {
"line": 1001,
"column": 52
} | {
"line": 1002,
"column": 55
} | {
"line": 1004,
"column": 0
} | [
{
"pp": "σ : Type u_1\nm : MonomialOrder σ\nR : Type u_2\ninst✝ : CommRing R\nf g : MvPolynomial σ R\nh : m.degree f = m.degree g\nhs : m.sPolynomial f g ≠ 0\n⊢ m.toSyn (m.degree (m.sPolynomial f g)) < m.toSyn (m.degree f)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by
simpa [h] using m.degree_sPolynomial_lt_sup_degree hs | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Nat.Squarefree | {
"line": 285,
"column": 2
} | {
"line": 296,
"column": 19
} | {
"line": 298,
"column": 0
} | [
{
"pp": "n : ℕ\nh0 : n ≠ 0\n⊢ ∀ x ∈ (normalizedFactors n).toFinset.powerset.val,\n ∀ y ∈ (normalizedFactors n).toFinset.powerset.val, x.val.prod = y.val.prod → x = y",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"UniqueFactorizationMonoid.normalizedFactors",
"Multiset.toFi... | [] | · intro x hx y hy h
rw [← Finset.val_inj, ← Multiset.rel_eq, ← associated_eq_eq]
rw [← Finset.mem_def, Finset.mem_powerset] at hx hy
apply UniqueFactorizationMonoid.factors_unique _ _ (associated_iff_eq.2 h)
· intro z hz
apply irreducible_of_normalized_factor z
· rw [← Multiset.mem_toFinset]... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.NumberTheory.ArithmeticFunction.Zeta | {
"line": 118,
"column": 2
} | {
"line": 118,
"column": 35
} | {
"line": 120,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : SemigroupWithZero R\nf₁ f₂ f₃ : ArithmeticFunction R\nx✝ : ℕ\n⊢ ((f₁.pmul f₂).pmul f₃) x✝ = (f₁.pmul (f₂.pmul f₃)) x✝",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"ArithmeticFunction.pmul",
"Semigroup.toMul",
"HMul.hMul",
"Arithmeti... | [] | simp only [pmul_apply, mul_assoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Antidiag.Nat | {
"line": 72,
"column": 75
} | {
"line": 72,
"column": 87
} | {
"line": 73,
"column": 6
} | [
{
"pp": "case pos\nd n : ℕ\nf : Fin d → ℕ\nh : 0 < n\n⊢ (∃ x, ↑(∏ x_1, x x_1) = ↑⟨n, h⟩ ∧ (fun x_1 ↦ ↑(x x_1)) = f) ↔ ∏ i, f i = n ∧ n ≠ 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"PNat.val",
"Finset.univ",
"Exists",
"id",
"Ne",
"instOfNatNat",
... | [
"case pos\nd n : ℕ\nf : Fin d → ℕ\nh : 0 < n\n⊢ (∃ x, ↑(∏ x_1, x x_1) = n ∧ (fun x_1 ↦ ↑(x x_1)) = f) ↔ ∏ i, f i = n ∧ n ≠ 0"
] | PNat.mk_coe, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Order.Antidiag.Nat | {
"line": 132,
"column": 2
} | {
"line": 133,
"column": 72
} | {
"line": 134,
"column": 2
} | [
{
"pp": "case mp\nd n : ℕ\ni : Fin d\nhd : d ≠ 1\nk : ℕ\n⊢ (∃ a ∈ d.finMulAntidiag n, a i = k) → k ∣ n ∧ ¬n = 0",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Nat.mem_finMulAntidiag",
"Dvd.dvd",
"Finset.univ",
"Nat.dvd_of_mem_finMulAntidiag",
"Finset",
"... | [
"case mpr\nd n : ℕ\ni : Fin d\nhd : d ≠ 1\nk : ℕ\n⊢ k ∣ n ∧ ¬n = 0 → ∃ a ∈ d.finMulAntidiag n, a i = k"
] | · rintro ⟨f, hf, rfl⟩
exact ⟨dvd_of_mem_finMulAntidiag hf _, (mem_finMulAntidiag.mp hf).2⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.CompleteField | {
"line": 124,
"column": 2
} | {
"line": 125,
"column": 70
} | {
"line": 127,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝⁶ : Field α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : IsStrictOrderedRing α\ninst✝³ : Field β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : Archimedean α\na : α\n⊢ BddAbove (cutMap β a)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
... | [] | obtain ⟨q, hq⟩ := exists_rat_gt a
exact ⟨q, forall_mem_image.2 fun r hr => mod_cast (hq.trans' hr).le⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.CompleteField | {
"line": 124,
"column": 2
} | {
"line": 125,
"column": 70
} | {
"line": 127,
"column": 0
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝⁶ : Field α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : IsStrictOrderedRing α\ninst✝³ : Field β\ninst✝² : LinearOrder β\ninst✝¹ : IsStrictOrderedRing β\ninst✝ : Archimedean α\na : α\n⊢ BddAbove (cutMap β a)",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
... | [] | obtain ⟨q, hq⟩ := exists_rat_gt a
exact ⟨q, forall_mem_image.2 fun r hr => mod_cast (hq.trans' hr).le⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 525,
"column": 4
} | {
"line": 533,
"column": 26
} | {
"line": 534,
"column": 2
} | [
{
"pp": "case i_surj\nR : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : ℕ\ncop : m.Coprime n\n⊢ Set.SurjOn\n (fun x ↦\n match x with\n | ((i, j), k, l) => (i * k, j * l))\n ↑(m.divisorsAntidiagonal ×ˢ n.divisorsAntidiagonal)... | [] | simp only [Set.SurjOn, Set.subset_def, mem_coe, mem_divisorsAntidiagonal, mem_product,
Set.mem_image]
rintro ⟨b1, b2⟩ h
use ((b1.gcd m, b2.gcd m), (b1.gcd n, b2.gcd n))
rw [← cop.gcd_mul _, ← cop.gcd_mul _, ← h.1, gcd_mul_gcd_of_coprime_of_mul_eq_mul cop h.1,
gcd_mul_gcd_of_coprime_of_mul_eq_mul... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.NumberTheory.ArithmeticFunction.Defs | {
"line": 525,
"column": 4
} | {
"line": 533,
"column": 26
} | {
"line": 534,
"column": 2
} | [
{
"pp": "case i_surj\nR : Type u_1\ninst✝ : CommSemiring R\nf g : ArithmeticFunction R\nhf : f.IsMultiplicative\nhg : g.IsMultiplicative\nm n : ℕ\ncop : m.Coprime n\n⊢ Set.SurjOn\n (fun x ↦\n match x with\n | ((i, j), k, l) => (i * k, j * l))\n ↑(m.divisorsAntidiagonal ×ˢ n.divisorsAntidiagonal)... | [] | simp only [Set.SurjOn, Set.subset_def, mem_coe, mem_divisorsAntidiagonal, mem_product,
Set.mem_image]
rintro ⟨b1, b2⟩ h
use ((b1.gcd m, b2.gcd m), (b1.gcd n, b2.gcd n))
rw [← cop.gcd_mul _, ← cop.gcd_mul _, ← h.1, gcd_mul_gcd_of_coprime_of_mul_eq_mul cop h.1,
gcd_mul_gcd_of_coprime_of_mul_eq_mul... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Disjointed | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 63
} | {
"line": 54,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝⁵ : GeneralizedBooleanAlgebra α\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrderBot ι\ninst✝² : Add ι\ninst✝¹ : One ι\ninst✝ : SuccAddOrder ι\nf : ι → α\nhf : Monotone f\ni : ι\n⊢ disjointed f (i + 1) ⊔ f i = f (i + 1)",
"ppTerm": "?m.29",
"assigned": true,
... | [] | simpa only [succ_eq_add_one i] using hf.disjointed_succ_sup i | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Disjointed | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 63
} | {
"line": 54,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝⁵ : GeneralizedBooleanAlgebra α\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrderBot ι\ninst✝² : Add ι\ninst✝¹ : One ι\ninst✝ : SuccAddOrder ι\nf : ι → α\nhf : Monotone f\ni : ι\n⊢ disjointed f (i + 1) ⊔ f i = f (i + 1)",
"ppTerm": "?m.29",
"assigned": true,
... | [] | simpa only [succ_eq_add_one i] using hf.disjointed_succ_sup i | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Disjointed | {
"line": 52,
"column": 2
} | {
"line": 52,
"column": 63
} | {
"line": 54,
"column": 0
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝⁵ : GeneralizedBooleanAlgebra α\ninst✝⁴ : LinearOrder ι\ninst✝³ : LocallyFiniteOrderBot ι\ninst✝² : Add ι\ninst✝¹ : One ι\ninst✝ : SuccAddOrder ι\nf : ι → α\nhf : Monotone f\ni : ι\n⊢ disjointed f (i + 1) ⊔ f i = f (i + 1)",
"ppTerm": "?m.29",
"assigned": true,
... | [] | simpa only [succ_eq_add_one i] using hf.disjointed_succ_sup i | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 536,
"column": 21
} | {
"line": 536,
"column": 34
} | {
"line": 536,
"column": 34
} | [
{
"pp": "M : Type u_1\ninst✝⁵ : CommGroup M\ninst✝⁴ : LinearOrder M\ninst✝³ : IsOrderedMonoid M\na b : M\nN : Type u_2\ninst✝² : CommGroup N\ninst✝¹ : LinearOrder N\ninst✝ : IsOrderedMonoid N\nf : M →*o N\nh : mk a ≤ mk b\n⊢ (orderHom f) (mk a) ≤ mk (f b)",
"ppTerm": "?m.42",
"assigned": true,
"used... | [
"M : Type u_1\ninst✝⁵ : CommGroup M\ninst✝⁴ : LinearOrder M\ninst✝³ : IsOrderedMonoid M\na b : M\nN : Type u_2\ninst✝² : CommGroup N\ninst✝¹ : LinearOrder N\ninst✝ : IsOrderedMonoid N\nf : M →*o N\nh : mk a ≤ mk b\n⊢ (orderHom f) (mk a) ≤ (orderHom f) (mk b)"
] | ← orderHom_mk | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 568,
"column": 4
} | {
"line": 570,
"column": 39
} | {
"line": 572,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝³ : CommGroup M\ninst✝² : LinearOrder M\ninst✝¹ : IsOrderedMonoid M\na b : M\nα : Type u_2\ninst✝ : PartialOrder α\nf : M → α\nh : ∀ (a b : M), mk a ≤ mk b → f a ≤ f b\nA B : MulArchimedeanClass M\nhle : A ≤ B\n⊢ lift f ⋯ A ≤ lift f ⋯ B",
"ppTerm": "?m.54",
"assigned": true,
... | [] | induction A using ind with | mk a
induction B using ind with | mk b
simpa using h a b (mk_le_mk.mp hle) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Archimedean.Class | {
"line": 568,
"column": 4
} | {
"line": 570,
"column": 39
} | {
"line": 572,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝³ : CommGroup M\ninst✝² : LinearOrder M\ninst✝¹ : IsOrderedMonoid M\na b : M\nα : Type u_2\ninst✝ : PartialOrder α\nf : M → α\nh : ∀ (a b : M), mk a ≤ mk b → f a ≤ f b\nA B : MulArchimedeanClass M\nhle : A ≤ B\n⊢ lift f ⋯ A ≤ lift f ⋯ B",
"ppTerm": "?m.54",
"assigned": true,
... | [] | induction A using ind with | mk a
induction B using ind with | mk b
simpa using h a b (mk_le_mk.mp hle) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.Basic | {
"line": 159,
"column": 4
} | {
"line": 163,
"column": 93
} | {
"line": 165,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\ninst✝ : PartialOrder Γ'\nx : R⟦Γ'⟧⟦Γ⟧\n⊢ (Function.support fun g ↦ (x.coeff g.1).coeff g.2).IsPWO",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Set.IsPWO",
"Eq.mpr",
... | [] | refine Set.PartiallyWellOrderedOn.subsetProdLex ?_ ?_
· refine Set.IsPWO.mono x.isPWO_support' ?_
simp_rw [Set.image_subset_iff, support_subset_iff, Set.mem_preimage, Function.mem_support]
exact fun _ ↦ ne_zero_of_coeff_ne_zero
· exact fun a => by simpa [Function.mem_support, ne_eq] using! (x.coeff ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Basic | {
"line": 159,
"column": 4
} | {
"line": 163,
"column": 93
} | {
"line": 165,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\ninst✝ : PartialOrder Γ'\nx : R⟦Γ'⟧⟦Γ⟧\n⊢ (Function.support fun g ↦ (x.coeff g.1).coeff g.2).IsPWO",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"Set.IsPWO",
"Eq.mpr",
... | [] | refine Set.PartiallyWellOrderedOn.subsetProdLex ?_ ?_
· refine Set.IsPWO.mono x.isPWO_support' ?_
simp_rw [Set.image_subset_iff, support_subset_iff, Set.mem_preimage, Function.mem_support]
exact fun _ ↦ ne_zero_of_coeff_ne_zero
· exact fun a => by simpa [Function.mem_support, ne_eq] using! (x.coeff ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.Basic | {
"line": 489,
"column": 2
} | {
"line": 489,
"column": 23
} | {
"line": 490,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\ng : Γ\nr : R\ng' : Γ'\n⊢ (embDomain f ((single g) r)).coeff g' = ((single (f g)) r).coeff g'",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"ZeroHom.fu... | [
"case pos\nΓ : Type u_1\nΓ' : Type u_2\nR : Type u_3\ninst✝² : PartialOrder Γ\ninst✝¹ : Zero R\ninst✝ : PartialOrder Γ'\nf : Γ ↪o Γ'\ng : Γ\nr : R\ng' : Γ'\nh : g' = f g\n⊢ (embDomain f ((single g) r)).coeff g' = ((single (f g)) r).coeff g'",
"case neg\nΓ : Type u_1\nΓ' : Type u_2\nR : Type u_3\ninst✝² : PartialO... | by_cases h : g' = f g | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.RingTheory.HahnSeries.Addition | {
"line": 460,
"column": 20
} | {
"line": 462,
"column": 19
} | {
"line": 463,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nS : Type u_4\nU : Type u_5\nV✝ : Type u_6\nα : Type u_7\ninst✝³ : PartialOrder Γ\nV : Type u_8\ninst✝² : Monoid R\ninst✝¹ : AddMonoid V\ninst✝ : DistribMulAction R V\nx✝² : R\nx✝¹ x✝ : V⟦Γ⟧\n⊢ x✝² • (x✝¹ + x✝) = x✝² • x✝¹ + x✝² • x✝",
"ppTerm": "?m.72",
... | [] | by
ext
simp [smul_add] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Order.Monoid.Associated | {
"line": 24,
"column": 24
} | {
"line": 24,
"column": 45
} | {
"line": 24,
"column": 45
} | [
{
"pp": "M : Type u_1\ninst✝ : CommMonoidWithZero M\n⊢ ∀ (a b : Associates M), a ≤ b → ∀ (c : Associates M), a * c ≤ b * c",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Semigroup.toMul",
"Associates.instCommMonoid",
"HMul.hMul",
... | [
"M : Type u_1\ninst✝ : CommMonoidWithZero M\na : Associates M\nd : Associates M\nc : Associates M\n⊢ a * c ≤ a * d * c"
] | rintro a _ ⟨d, rfl⟩ c | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder | {
"line": 53,
"column": 11
} | {
"line": 53,
"column": 40
} | {
"line": 53,
"column": 40
} | [
{
"pp": "M : Type u_1\ninst✝⁴ : CancelCommMonoid M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedMonoid M\ninst✝¹ : LocallyFiniteOrder M\ninst✝ : ExistsMulOfLE M\na b c d : M\nh₁ : a * c ≤ a * c * d\nh₂ : a * c * d < b * c\n⊢ a * d * c = a * c * d",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants"... | [] | simp_rw [mul_assoc, mul_comm] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder | {
"line": 53,
"column": 11
} | {
"line": 53,
"column": 40
} | {
"line": 53,
"column": 40
} | [
{
"pp": "M : Type u_1\ninst✝⁴ : CancelCommMonoid M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedMonoid M\ninst✝¹ : LocallyFiniteOrder M\ninst✝ : ExistsMulOfLE M\na b c d : M\nh₁ : a * c ≤ a * c * d\nh₂ : a * c * d < b * c\n⊢ a * d * c = a * c * d",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants"... | [] | simp_rw [mul_assoc, mul_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder | {
"line": 53,
"column": 11
} | {
"line": 53,
"column": 40
} | {
"line": 53,
"column": 40
} | [
{
"pp": "M : Type u_1\ninst✝⁴ : CancelCommMonoid M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedMonoid M\ninst✝¹ : LocallyFiniteOrder M\ninst✝ : ExistsMulOfLE M\na b c d : M\nh₁ : a * c ≤ a * c * d\nh₂ : a * c * d < b * c\n⊢ a * d * c = a * c * d",
"ppTerm": "?m.103",
"assigned": true,
"usedConstants"... | [] | simp_rw [mul_assoc, mul_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder | {
"line": 107,
"column": 4
} | {
"line": 110,
"column": 28
} | {
"line": 112,
"column": 0
} | [
{
"pp": "M : Type u_1\nG : Type u_2\ninst✝⁷ : AddCancelCommMonoid M\ninst✝⁶ : LinearOrder M\ninst✝⁵ : IsOrderedAddMonoid M\ninst✝⁴ : LocallyFiniteOrder M\ninst✝³ : AddCommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedAddMonoid G\ninst✝ : LocallyFiniteOrder G\na b : G\nhab : a ≤ b\n⊢ (↑(addMonoidHom G)).toF... | [] | obtain ⟨b, rfl⟩ := add_left_surjective a b
replace hab : 0 ≤ b := by simpa using hab
suffices 0 ≤ addMonoidHom G b by simpa
simp [addMonoidHom, hab] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Monoid.LocallyFiniteOrder | {
"line": 107,
"column": 4
} | {
"line": 110,
"column": 28
} | {
"line": 112,
"column": 0
} | [
{
"pp": "M : Type u_1\nG : Type u_2\ninst✝⁷ : AddCancelCommMonoid M\ninst✝⁶ : LinearOrder M\ninst✝⁵ : IsOrderedAddMonoid M\ninst✝⁴ : LocallyFiniteOrder M\ninst✝³ : AddCommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedAddMonoid G\ninst✝ : LocallyFiniteOrder G\na b : G\nhab : a ≤ b\n⊢ (↑(addMonoidHom G)).toF... | [] | obtain ⟨b, rfl⟩ := add_left_surjective a b
replace hab : 0 ≤ b := by simpa using hab
suffices 0 ≤ addMonoidHom G b by simpa
simp [addMonoidHom, hab] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 63
} | {
"line": 81,
"column": 4
} | [
{
"pp": "case mp\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrder Γ\ninst✝¹ : Zero R\ninst✝ : LinearOrder R\nx : Lex R⟦Γ⟧\nhpos : 0 < (ofLex x).leadingCoeff\nhne : ofLex x ≠ 0\n⊢ ∃ i, (∀ j < i, (ofLex 0).coeff j = (ofLex x).coeff j) ∧ (ofLex 0).coeff i < (ofLex x).coeff i",
"ppTerm": "?mp",
"assigned":... | [
"case mp\nΓ : Type u_1\nR : Type u_2\ninst✝² : LinearOrder Γ\ninst✝¹ : Zero R\ninst✝ : LinearOrder R\nx : Lex R⟦Γ⟧\nhpos : 0 < (ofLex x).leadingCoeff\nhne : ofLex x ≠ 0\nhtop : (ofLex x).orderTop ≠ ⊤\n⊢ ∃ i, (∀ j < i, (ofLex 0).coeff j = (ofLex x).coeff j) ∧ (ofLex 0).coeff i < (ofLex x).coeff i"
] | have htop : (ofLex x).orderTop ≠ ⊤ := orderTop_ne_top.2 hne | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 106,
"column": 4
} | {
"line": 106,
"column": 34
} | {
"line": 107,
"column": 4
} | [
{
"pp": "K : Type u_1\ninst✝⁸ : DivisionRing K\ninst✝⁷ : LinearOrder K\ninst✝⁶ : IsOrderedRing K\ninst✝⁵ : Archimedean K\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedAddMonoid M\ninst✝¹ : Module K M\ninst✝ : IsOrderedModule K M\nu : ArchimedeanStrata K M\nc : FiniteArchimede... | [
"K : Type u_1\ninst✝⁸ : DivisionRing K\ninst✝⁷ : LinearOrder K\ninst✝⁶ : IsOrderedRing K\ninst✝⁵ : Archimedean K\nM : Type u_2\ninst✝⁴ : AddCommGroup M\ninst✝³ : LinearOrder M\ninst✝² : IsOrderedAddMonoid M\ninst✝¹ : Module K M\ninst✝ : IsOrderedModule K M\nu : ArchimedeanStrata K M\nc : FiniteArchimedeanClass M\na... | rw [← u.ball_sup_stratum_eq c] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 284,
"column": 6
} | {
"line": 284,
"column": 88
} | {
"line": 285,
"column": 6
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝³ : LinearOrder Γ\ninst✝² : LinearOrder R\ninst✝¹ : AddCommGroup R\ninst✝ : IsOrderedAddMonoid R\nx : Lex (Γ × FiniteArchimedeanClass R)\nx✝¹ x✝ : { a // a ≠ 0 }\na : R\nha : a ≠ 0\nb : R\nhb : b ≠ 0\nh : FiniteArchimedeanClass.mk ↑⟨a, ha⟩ ⋯ ≤ FiniteArchimedeanClass.mk ... | [
"Γ : Type u_1\nR : Type u_2\ninst✝³ : LinearOrder Γ\ninst✝² : LinearOrder R\ninst✝¹ : AddCommGroup R\ninst✝ : IsOrderedAddMonoid R\nx : Lex (Γ × FiniteArchimedeanClass R)\nx✝¹ x✝ : { a // a ≠ 0 }\na : R\nha : a ≠ 0\nb : R\nhb : b ≠ 0\nh : FiniteArchimedeanClass.mk ↑⟨a, ha⟩ ⋯ ≤ FiniteArchimedeanClass.mk ↑⟨b, hb⟩ ⋯\n... | rw [FiniteArchimedeanClass.mk_le_mk, archimedeanClassMk_le_archimedeanClassMk_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 291,
"column": 6
} | {
"line": 291,
"column": 88
} | {
"line": 292,
"column": 6
} | [
{
"pp": "case inl.mk.mk\nΓ : Type u_1\nR : Type u_2\ninst✝³ : LinearOrder Γ\ninst✝² : LinearOrder R\ninst✝¹ : AddCommGroup R\ninst✝ : IsOrderedAddMonoid R\na✝ b✝ : Lex (Γ × FiniteArchimedeanClass R)\nx✝¹ : Γ × FiniteArchimedeanClass R\nao : Γ\nx✝ : Γ × FiniteArchimedeanClass R\nbo : Γ\na : R\nha : a ≠ 0\nb : R\... | [
"case inl.mk.mk\nΓ : Type u_1\nR : Type u_2\ninst✝³ : LinearOrder Γ\ninst✝² : LinearOrder R\ninst✝¹ : AddCommGroup R\ninst✝ : IsOrderedAddMonoid R\na✝ b✝ : Lex (Γ × FiniteArchimedeanClass R)\nx✝¹ : Γ × FiniteArchimedeanClass R\nao : Γ\nx✝ : Γ × FiniteArchimedeanClass R\nbo : Γ\na : R\nha : a ≠ 0\nb : R\nhb : b ≠ 0\... | rw [FiniteArchimedeanClass.mk_le_mk, archimedeanClassMk_le_archimedeanClassMk_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 352,
"column": 2
} | {
"line": 354,
"column": 63
} | {
"line": 356,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝⁵ : LinearOrder Γ\ninst✝⁴ : LinearOrder R\ninst✝³ : AddCommGroup R\ninst✝² : IsOrderedAddMonoid R\ninst✝¹ : Archimedean R\ninst✝ : Nontrivial R\nx : Lex R⟦Γ⟧\n⊢ (archimedeanClassOrderIsoWithTop Γ R) (ArchimedeanClass.mk x) = (ofLex x).orderTop",
"ppTerm": "?m.25",
... | [] | unfold archimedeanClassOrderIsoWithTop
obtain rfl | h := eq_or_ne x 0 <;>
simp [FiniteArchimedeanClass.withTopOrderIso_symm_apply, *] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 352,
"column": 2
} | {
"line": 354,
"column": 63
} | {
"line": 356,
"column": 0
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝⁵ : LinearOrder Γ\ninst✝⁴ : LinearOrder R\ninst✝³ : AddCommGroup R\ninst✝² : IsOrderedAddMonoid R\ninst✝¹ : Archimedean R\ninst✝ : Nontrivial R\nx : Lex R⟦Γ⟧\n⊢ (archimedeanClassOrderIsoWithTop Γ R) (ArchimedeanClass.mk x) = (ofLex x).orderTop",
"ppTerm": "?m.25",
... | [] | unfold archimedeanClassOrderIsoWithTop
obtain rfl | h := eq_or_ne x 0 <;>
simp [FiniteArchimedeanClass.withTopOrderIso_symm_apply, *] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 367,
"column": 4
} | {
"line": 371,
"column": 35
} | {
"line": 372,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝⁶ : LinearOrder Γ\ninst✝⁵ : LinearOrder R\ninst✝⁴ : Ring R\ninst✝³ : AddCommMonoid Γ\ninst✝² : IsOrderedCancelAddMonoid Γ\ninst✝¹ : IsOrderedRing R\ninst✝ : NoZeroDivisors R\na : Lex R⟦Γ⟧\nha : 0 ≤ a\nb c : Lex R⟦Γ⟧\nhbc : b ≤ c\n⊢ a * b ≤ a * c",
"ppTerm": "?m.20",... | [] | rw [← sub_nonneg] at hbc ⊢
rw [← mul_sub, ← leadingCoeff_nonneg_iff, ofLex_mul, leadingCoeff_mul]
apply mul_nonneg
· simpa
· rwa [leadingCoeff_nonneg_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.HahnSeries.Lex | {
"line": 367,
"column": 4
} | {
"line": 371,
"column": 35
} | {
"line": 372,
"column": 2
} | [
{
"pp": "Γ : Type u_1\nR : Type u_2\ninst✝⁶ : LinearOrder Γ\ninst✝⁵ : LinearOrder R\ninst✝⁴ : Ring R\ninst✝³ : AddCommMonoid Γ\ninst✝² : IsOrderedCancelAddMonoid Γ\ninst✝¹ : IsOrderedRing R\ninst✝ : NoZeroDivisors R\na : Lex R⟦Γ⟧\nha : 0 ≤ a\nb c : Lex R⟦Γ⟧\nhbc : b ≤ c\n⊢ a * b ≤ a * c",
"ppTerm": "?m.20",... | [] | rw [← sub_nonneg] at hbc ⊢
rw [← mul_sub, ← leadingCoeff_nonneg_iff, ofLex_mul, leadingCoeff_mul]
apply mul_nonneg
· simpa
· rwa [leadingCoeff_nonneg_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.Ordering.Basic | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 9
} | {
"line": 142,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nh : IsUnit 2\nx a : R\na✝¹ : a ∈ P\na✝ : -a ∈ P\nhalf : R\nh2 : 2 * half = 1\ny : R := (1 + x) * half\nz : R := (1 - x) * half\n⊢ (y ^ 2 - z ^ 2) * a ∈ P ∧ -((y ^ 2 - z ^ 2) * a) ∈ P",
"ppTerm": "?m.129",
"assigned": true,
"usedConsta... | [
"R : Type u_1\ninst✝ : CommRing R\nP : RingPreordering R\nh : IsUnit 2\nx a : R\na✝¹ : a ∈ P\na✝ : -a ∈ P\nhalf : R\nh2 : 2 * half = 1\ny : R := (1 + x) * half\nz : R := (1 - x) * half\n⊢ y ^ 2 * a - z ^ 2 * a ∈ P ∧ -(y ^ 2 * a) + z ^ 2 * a ∈ P"
] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Algebra.Order.Ring.Ordering.Basic | {
"line": 167,
"column": 35
} | {
"line": 167,
"column": 64
} | {
"line": 169,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nP✝ : RingPreordering R\nF : Type u_2\ninst✝ : Field F\nP : RingPreordering F\n⊢ P.support.IsPrime",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsDomain.to_noZeroDivisors",
"Semiring.toModule",
"congrArg",
... | [] | simpa using Ideal.isPrime_bot | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Algebra.Order.Ring.Ordering.Basic | {
"line": 167,
"column": 35
} | {
"line": 167,
"column": 64
} | {
"line": 169,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nP✝ : RingPreordering R\nF : Type u_2\ninst✝ : Field F\nP : RingPreordering F\n⊢ P.support.IsPrime",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsDomain.to_noZeroDivisors",
"Semiring.toModule",
"congrArg",
... | [] | simpa using Ideal.isPrime_bot | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Ring.Ordering.Basic | {
"line": 167,
"column": 35
} | {
"line": 167,
"column": 64
} | {
"line": 169,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\nP✝ : RingPreordering R\nF : Type u_2\ninst✝ : Field F\nP : RingPreordering F\n⊢ P.support.IsPrime",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsDomain.to_noZeroDivisors",
"Semiring.toModule",
"congrArg",
... | [] | simpa using Ideal.isPrime_bot | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 340,
"column": 2
} | {
"line": 341,
"column": 70
} | {
"line": 342,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹¹ : DivisionRing K\ninst✝¹⁰ : LinearOrder K\ninst✝⁹ : IsOrderedRing K\ninst✝⁸ : Archimedean K\nM : Type u_2\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : LinearOrder M\ninst✝⁵ : IsOrderedAddMonoid M\ninst✝⁴ : Module K M\ninst✝³ : IsOrderedModule K M\nR : Type u_3\ninst✝² : AddCommGroup R\ninst... | [
"K : Type u_1\ninst✝¹¹ : DivisionRing K\ninst✝¹⁰ : LinearOrder K\ninst✝⁹ : IsOrderedRing K\ninst✝⁸ : Archimedean K\nM : Type u_2\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : LinearOrder M\ninst✝⁵ : IsOrderedAddMonoid M\ninst✝⁴ : Module K M\ninst✝³ : IsOrderedModule K M\nR : Type u_3\ninst✝² : AddCommGroup R\ninst✝¹ : LinearO... | let f' : Π₀ (i : FiniteArchimedeanClass M), seed.stratum i :=
DFinsupp.mk f.support fun d ↦ if c.val ≤ d.val then 0 else f d.val | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.RingTheory.HahnSeries.Multiplication | {
"line": 1005,
"column": 4
} | {
"line": 1010,
"column": 76
} | {
"line": 1011,
"column": 4
} | [
{
"pp": "case hfg\nΓ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nS : Type u_4\nV : Type u_5\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : IsCancelAdd R\ninst✝ : IsCancelMulZero R\nx : R⟦Γ⟧\nhx : x ≠ 0\ny z : R⟦Γ⟧\nthis✝ : AddC... | [
"Γ : Type u_1\nΓ' : Type u_2\nR : Type u_3\nS : Type u_4\nV : Type u_5\ninst✝⁵ : AddCommMonoid Γ\ninst✝⁴ : LinearOrder Γ\ninst✝³ : IsOrderedCancelAddMonoid Γ\ninst✝² : NonUnitalNonAssocSemiring R\ninst✝¹ : IsCancelAdd R\ninst✝ : IsCancelMulZero R\nx : R⟦Γ⟧\nhx : x ≠ 0\ny z : R⟦Γ⟧\nthis✝ : AddCancelCommMonoid R := {... | · simp +contextual only [mem_union, mem_antidiagonal, mul_eq_mul_right_iff, Prod.mk.injEq,
ne_eq, ← or_and_right, or_false, and_imp, Prod.forall, mem_support, not_and]
rintro b c - hxb hbc hbc'
contrapose! hbc'
rwa [eq_comm, eq_comm (a := c), ← add_eq_add_iff_eq_and_eq
(Set.IsWF.min_le... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 420,
"column": 40
} | {
"line": 420,
"column": 53
} | {
"line": 420,
"column": 53
} | [
{
"pp": "K : Type u_1\ninst✝¹² : DivisionRing K\ninst✝¹¹ : LinearOrder K\ninst✝¹⁰ : IsOrderedRing K\ninst✝⁹ : Archimedean K\nM : Type u_2\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LinearOrder M\ninst✝⁶ : IsOrderedAddMonoid M\ninst✝⁵ : Module K M\ninst✝⁴ : IsOrderedModule K M\nR : Type u_3\ninst✝³ : AddCommGroup R\nins... | [
"K : Type u_1\ninst✝¹² : DivisionRing K\ninst✝¹¹ : LinearOrder K\ninst✝¹⁰ : IsOrderedRing K\ninst✝⁹ : Archimedean K\nM : Type u_2\ninst✝⁸ : AddCommGroup M\ninst✝⁷ : LinearOrder M\ninst✝⁶ : IsOrderedAddMonoid M\ninst✝⁵ : Module K M\ninst✝⁴ : IsOrderedModule K M\nR : Type u_3\ninst✝³ : AddCommGroup R\ninst✝² : Linear... | ← orderHom_mk | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Algebra.Order.Star.Real | {
"line": 35,
"column": 4
} | {
"line": 35,
"column": 22
} | {
"line": 36,
"column": 4
} | [
{
"pp": "case refine_2\nx y : ℝ≥0\n⊢ (∃ s, y = x + star s * s) → x ≤ y",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"NNReal.instCommSemiring",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"HEq.refl",
"PartialOrder.toPreorder",
"Preorder.to... | [
"case refine_2\nx p : ℝ≥0\n⊢ x ≤ x + star p * p"
] | rintro ⟨p, -, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Algebra.Order.Sub.Unbundled.Hom | {
"line": 44,
"column": 19
} | {
"line": 44,
"column": 44
} | {
"line": 44,
"column": 44
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nF : Type u_3\ninst✝¹² : PartialOrder α\ninst✝¹¹ : AddCommSemigroup α\ninst✝¹⁰ : ExistsAddOfLE α\ninst✝⁹ : AddLeftMono α\ninst✝⁸ : Sub α\ninst✝⁷ : OrderedSub α\ninst✝⁶ : PartialOrder β\ninst✝⁵ : AddCommSemigroup β\ninst✝⁴ : Sub β\ninst✝³ : OrderedSub β\ninst✝² : AddLeftReflec... | [
"α : Type u_1\nβ : Type u_2\nF : Type u_3\ninst✝¹² : PartialOrder α\ninst✝¹¹ : AddCommSemigroup α\ninst✝¹⁰ : ExistsAddOfLE α\ninst✝⁹ : AddLeftMono α\ninst✝⁸ : Sub α\ninst✝⁷ : OrderedSub α\ninst✝⁶ : PartialOrder β\ninst✝⁵ : AddCommSemigroup β\ninst✝⁴ : Sub β\ninst✝³ : OrderedSub β\ninst✝² : AddLeftReflectLE β\ninst✝... | ← tsub_add_cancel_of_le h | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.RingTheory.Valuation.ValuationSubring | {
"line": 368,
"column": 6
} | {
"line": 370,
"column": 60
} | {
"line": 370,
"column": 60
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nA R S : ValuationSubring K\nhR : A ≤ R\nhS : A ≤ S\nh : R ≤ S\nx : ↥A\nhx : x ∈ A.idealOfLE S hS\n⊢ R.valuation ↑((A.inclusion R hR) x) < 1",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"ValuationSubring.valuation_lt_one_iff",
"GroupWithZ... | [] | by_contra! c; replace c := monotone_mapOfLE R S h c
rw [(mapOfLE _ _ _).map_one, mapOfLE_valuation_apply] at c
apply not_le_of_gt ((valuation_lt_one_iff S _).1 hx) c | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Valuation.ValuationSubring | {
"line": 368,
"column": 6
} | {
"line": 370,
"column": 60
} | {
"line": 370,
"column": 60
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nA R S : ValuationSubring K\nhR : A ≤ R\nhS : A ≤ S\nh : R ≤ S\nx : ↥A\nhx : x ∈ A.idealOfLE S hS\n⊢ R.valuation ↑((A.inclusion R hR) x) < 1",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"ValuationSubring.valuation_lt_one_iff",
"GroupWithZ... | [] | by_contra! c; replace c := monotone_mapOfLE R S h c
rw [(mapOfLE _ _ _).map_one, mapOfLE_valuation_apply] at c
apply not_le_of_gt ((valuation_lt_one_iff S _).1 hx) c | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Valuation.ValuationSubring | {
"line": 484,
"column": 7
} | {
"line": 484,
"column": 36
} | {
"line": 484,
"column": 36
} | [
{
"pp": "K : Type u\ninst✝ : Field K\nA : ValuationSubring K\nx✝ : K\n⊢ x✝ ∈ A.valuation.valuationSubring ↔ x✝ ∈ A",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"LinearOrderedCommGroupWithZero.toLinearOrderedCommMonoidWithZero",
"InvOneClass.toOne",
"Div... | [
"K : Type u\ninst✝ : Field K\nA : ValuationSubring K\nx✝ : K\n⊢ x✝ ∈ A.valuation.valuationSubring ↔ A.valuation x✝ ≤ 1"
] | rw [← A.valuation_le_one_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Order.Module.HahnEmbedding | {
"line": 680,
"column": 2
} | {
"line": 680,
"column": 37
} | {
"line": 681,
"column": 2
} | [
{
"pp": "K : Type u_1\ninst✝¹³ : DivisionRing K\ninst✝¹² : LinearOrder K\ninst✝¹¹ : IsOrderedRing K\ninst✝¹⁰ : Archimedean K\nM : Type u_2\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : LinearOrder M\ninst✝⁷ : IsOrderedAddMonoid M\ninst✝⁶ : Module K M\ninst✝⁵ : IsOrderedModule K M\nR : Type u_3\ninst✝⁴ : AddCommGroup R\nin... | [
"K : Type u_1\ninst✝¹³ : DivisionRing K\ninst✝¹² : LinearOrder K\ninst✝¹¹ : IsOrderedRing K\ninst✝¹⁰ : Archimedean K\nM : Type u_2\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : LinearOrder M\ninst✝⁷ : IsOrderedAddMonoid M\ninst✝⁶ : Module K M\ninst✝⁵ : IsOrderedModule K M\nR : Type u_3\ninst✝⁴ : AddCommGroup R\ninst✝³ : Linea... | have h := f.eval_eq_truncLT hc this | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Algebra.Pointwise.Stabilizer | {
"line": 233,
"column": 12
} | {
"line": 233,
"column": 62
} | {
"line": 235,
"column": 0
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G α\na : G\ns : Finset α\n⊢ a ∈ stabilizer G ↑s ↔ ∀ ⦃b : α⦄, b ∈ ↑s → a • b ∈ ↑s",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"instHSMul",
"congrArg",
"Finset",
... | [] | simp [-mem_stabilizer_iff, mem_stabilizer_finset'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Pointwise.Stabilizer | {
"line": 233,
"column": 12
} | {
"line": 233,
"column": 62
} | {
"line": 235,
"column": 0
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G α\na : G\ns : Finset α\n⊢ a ∈ stabilizer G ↑s ↔ ∀ ⦃b : α⦄, b ∈ ↑s → a • b ∈ ↑s",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"instHSMul",
"congrArg",
"Finset",
... | [] | simp [-mem_stabilizer_iff, mem_stabilizer_finset'] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Pointwise.Stabilizer | {
"line": 233,
"column": 12
} | {
"line": 233,
"column": 62
} | {
"line": 235,
"column": 0
} | [
{
"pp": "G : Type u_1\nα : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G α\na : G\ns : Finset α\n⊢ a ∈ stabilizer G ↑s ↔ ∀ ⦃b : α⦄, b ∈ ↑s → a • b ∈ ↑s",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"SetLike.mem_coe._simp_1",
"instHSMul",
"congrArg",
"Finset",
... | [] | simp [-mem_stabilizer_iff, mem_stabilizer_finset'] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Ring.StandardPart | {
"line": 292,
"column": 9
} | {
"line": 292,
"column": 56
} | {
"line": 292,
"column": 57
} | [
{
"pp": "K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nx : K\nh : mk x = 0\n⊢ stdPart x ≠ 0",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Classical.ofNonempty",
"Real",
"Eq.ge",
"IsDomain.to_noZeroDivisors",
... | [
"K : Type u_1\ninst✝² : LinearOrder K\ninst✝¹ : Field K\ninst✝ : IsOrderedRing K\nx : K\nh : mk x = 0\n⊢ Classical.ofNonempty (FiniteResidueField.mk (FiniteElement.mk x ⋯)) ≠ 0"
] | stdPart_of_mk_nonneg Classical.ofNonempty h.ge, | Lean.Elab.Tactic.evalRewriteSeq | null |
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