module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.FieldTheory.Minpoly.IsIntegrallyClosed | {
"line": 166,
"column": 36
} | {
"line": 166,
"column": 48
} | {
"line": 166,
"column": 48
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : CommRing S\ninst✝⁴ : IsDomain R\ninst✝³ : Algebra R S\ninst✝² : IsIntegrallyClosed R\ninst✝¹ : IsDomain S\ninst✝ : IsTorsionFree R S\ns : S\np : R[X]\nhp : (Polynomial.aeval s) p = 0\nh₀ : p ≠ 0\npmin : ∀ (q : R[X]), q.Monic → (Polynomial.aeval ... | [] | by simp [h₀] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.AlgebraicGeometry.EllipticCurve.Jacobian.Formula | {
"line": 411,
"column": 17
} | {
"line": 411,
"column": 26
} | {
"line": 411,
"column": 27
} | [
{
"pp": "F : Type u\ninst✝ : Field F\nP Q : Fin 3 → F\nhQz : Q z = 0\n⊢ -((P y * 0 ^ 3 - Q y * P z ^ 3) / (P z * 0)) = 0",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"instHDiv",
"HMul.hMul",
"MulZeroClass.toMul",
"AddGr... | [
"F : Type u\ninst✝ : Field F\nP Q : Fin 3 → F\nhQz : Q z = 0\n⊢ -((P y * 0 ^ 3 - Q y * P z ^ 3) / 0) = 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Trace.Defs | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 95
} | {
"line": 115,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : StrongRankCondition R\ninst✝ : Free R S\nx : R\nH : ∃ s, Nonempty (Basis (↥s) R S)\n⊢ (trace R S) ((algebraMap R S) x) = finrank R S • x",
"ppTerm": "?pos✝",
"assigned": true,
"use... | [] | rw [trace_algebraMap_of_basis H.choose_spec.some, finrank_eq_card_basis H.choose_spec.some] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Trace.Defs | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 95
} | {
"line": 115,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : StrongRankCondition R\ninst✝ : Free R S\nx : R\nH : ∃ s, Nonempty (Basis (↥s) R S)\n⊢ (trace R S) ((algebraMap R S) x) = finrank R S • x",
"ppTerm": "?pos✝",
"assigned": true,
"use... | [] | rw [trace_algebraMap_of_basis H.choose_spec.some, finrank_eq_card_basis H.choose_spec.some] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Trace.Defs | {
"line": 114,
"column": 4
} | {
"line": 114,
"column": 95
} | {
"line": 115,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\nS : Type u_2\ninst✝⁴ : CommRing R\ninst✝³ : CommRing S\ninst✝² : Algebra R S\ninst✝¹ : StrongRankCondition R\ninst✝ : Free R S\nx : R\nH : ∃ s, Nonempty (Basis (↥s) R S)\n⊢ (trace R S) ((algebraMap R S) x) = finrank R S • x",
"ppTerm": "?pos✝",
"assigned": true,
"use... | [] | rw [trace_algebraMap_of_basis H.choose_spec.some, finrank_eq_card_basis H.choose_spec.some] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Trace.Basic | {
"line": 68,
"column": 88
} | {
"line": 69,
"column": 81
} | {
"line": 71,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommRing R\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\nh : PowerBasis R S\n⊢ (LinearMap.BilinForm.toMatrix h.basis) (traceForm R S) = of fun i j ↦ (trace R S) (h.gen ^ (↑i + ↑j))",
"ppTerm": "?m.66",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | by
ext; rw [traceForm_toMatrix, of_apply, pow_add, h.basis_eq_pow, h.basis_eq_pow] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Trace.Defs | {
"line": 162,
"column": 28
} | {
"line": 162,
"column": 63
} | {
"line": 164,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Free R S\ninst✝² : Free R T\ninst✝¹ : Module.Finite R S\ninst✝ : Module.Finite R T\np : S × T\n⊢ (trace R (S × T)) p = ((trace R S).coprod (trace... | [] | rw [coprod_apply, trace_prod_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Trace.Defs | {
"line": 162,
"column": 28
} | {
"line": 162,
"column": 63
} | {
"line": 164,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Free R S\ninst✝² : Free R T\ninst✝¹ : Module.Finite R S\ninst✝ : Module.Finite R T\np : S × T\n⊢ (trace R (S × T)) p = ((trace R S).coprod (trace... | [] | rw [coprod_apply, trace_prod_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Trace.Defs | {
"line": 162,
"column": 28
} | {
"line": 162,
"column": 63
} | {
"line": 164,
"column": 0
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nT : Type u_3\ninst✝⁸ : CommRing R\ninst✝⁷ : CommRing S\ninst✝⁶ : CommRing T\ninst✝⁵ : Algebra R S\ninst✝⁴ : Algebra R T\ninst✝³ : Free R S\ninst✝² : Free R T\ninst✝¹ : Module.Finite R S\ninst✝ : Module.Finite R T\np : S × T\n⊢ (trace R (S × T)) p = ((trace R S).coprod (trace... | [] | rw [coprod_apply, trace_prod_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Trace.Basic | {
"line": 244,
"column": 30
} | {
"line": 244,
"column": 47
} | {
"line": 244,
"column": 48
} | [
{
"pp": "K : Type u_4\nL : Type u_5\ninst✝¹¹ : Field K\ninst✝¹⁰ : Field L\ninst✝⁹ : Algebra K L\nF : Type u_6\ninst✝⁸ : Field F\ninst✝⁷ : Algebra L F\ninst✝⁶ : Algebra K F\ninst✝⁵ : IsScalarTower K L F\nE : Type u_7\ninst✝⁴ : Field E\ninst✝³ : Algebra K E\ninst✝² : IsAlgClosed E\ninst✝¹ : FiniteDimensional K F\... | [
"K : Type u_4\nL : Type u_5\ninst✝¹¹ : Field K\ninst✝¹⁰ : Field L\ninst✝⁹ : Algebra K L\nF : Type u_6\ninst✝⁸ : Field F\ninst✝⁷ : Algebra L F\ninst✝⁶ : Algebra K F\ninst✝⁵ : IsScalarTower K L F\nE : Type u_7\ninst✝⁴ : Field E\ninst✝³ : Algebra K E\ninst✝² : IsAlgClosed E\ninst✝¹ : FiniteDimensional K F\ninst✝ : Alg... | Finset.sum_sigma, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Trace.Basic | {
"line": 313,
"column": 26
} | {
"line": 313,
"column": 35
} | {
"line": 313,
"column": 36
} | [
{
"pp": "case neg.succ\nK : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nH : ¬Algebra.IsSeparable K L\np : ℕ\nhp : ExpChar K p\nthis : p ≠ 0\nx : L\nh₀ : FiniteDimensional K L\nhx : ¬IsSeparable K x\ng : K[X]\nhg₁ : g.Separable\nn : ℕ\nhg₂ : (expand K (p ^ (n + 1))) g = minpo... | [
"case neg.succ\nK : Type u_4\nL : Type u_5\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nH : ¬Algebra.IsSeparable K L\np : ℕ\nhp : ExpChar K p\nthis : p ≠ 0\nx : L\nh₀ : FiniteDimensional K L\nhx : ¬IsSeparable K x\ng : K[X]\nhg₁ : g.Separable\nn : ℕ\nhg₂ : (expand K (p ^ (n + 1))) g = minpoly K x\n⊢ 0 ... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.OpenSubgroup | {
"line": 540,
"column": 2
} | {
"line": 544,
"column": 16
} | {
"line": 545,
"column": 2
} | [
{
"pp": "case h\nG : Type u_2\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : CompactSpace G\nW : Set G\nWClopen : IsClopen W\neinW : 1 ∈ W\nV : Set G\nhV : mulInvClosureNhd V W\nS : Subgroup G := { carrier := ⋃ n, V ^ (n + 1), mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }... | [
"case h\nG : Type u_2\ninst✝³ : Group G\ninst✝² : TopologicalSpace G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : CompactSpace G\nW : Set G\nWClopen : IsClopen W\neinW : 1 ∈ W\nV : Set G\nhV : mulInvClosureNhd V W\nS : Subgroup G := { carrier := ⋃ n, V ^ (n + 1), mul_mem' := ⋯, one_mem' := ⋯, inv_mem' := ⋯ }\nthis✝ : Is... | have (n : ℕ) : V ^ (n + 1) ⊆ W * V ^ (n + 1) := by
intro x xin
rw [Set.mem_mul]
use 1, einW, x, xin
rw [one_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 73,
"column": 6
} | {
"line": 73,
"column": 12
} | {
"line": 74,
"column": 6
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ γ₀ : (ofClass v).ValueGroup₀ˣ\nh : γ₀ * γ₀ ≤ γ\n⊢ ∃ j,\n ↑(v.ltAddSubgroup ((Units.map ↑embedding) j)) * ↑(v.ltAddSubgroup ((Units.map ↑embedding) j)) ⊆\n ↑(v.ltAddSubgroup ((Units.map ↑em... | [
"case h\nR : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nv : Valuation R Γ₀\nγ γ₀ : (ofClass v).ValueGroup₀ˣ\nh : γ₀ * γ₀ ≤ γ\n⊢ ↑(v.ltAddSubgroup ((Units.map ↑embedding) γ₀)) * ↑(v.ltAddSubgroup ((Units.map ↑embedding) γ₀)) ⊆\n ↑(v.ltAddSubgroup ((Units.map ↑embedding) γ))"
... | use γ₀ | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Topology.Algebra.GroupCompletion | {
"line": 124,
"column": 10
} | {
"line": 124,
"column": 56
} | {
"line": 124,
"column": 57
} | [
{
"pp": "M : Type u_1\nR : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝² : UniformSpace α\ninst✝¹ : AddGroup α\ninst✝ : IsUniformAddGroup α\nn : ℕ\na✝ : Completion α\na : α\n⊢ (n + 1) • ↑a = n • ↑a + ↑a",
"ppTerm": "?m.466",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"UniformSpace.Com... | [] | rw [← coe_smul, succ_nsmul, coe_add, coe_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.GroupCompletion | {
"line": 124,
"column": 10
} | {
"line": 124,
"column": 56
} | {
"line": 124,
"column": 57
} | [
{
"pp": "M : Type u_1\nR : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝² : UniformSpace α\ninst✝¹ : AddGroup α\ninst✝ : IsUniformAddGroup α\nn : ℕ\na✝ : Completion α\na : α\n⊢ (n + 1) • ↑a = n • ↑a + ↑a",
"ppTerm": "?m.466",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"UniformSpace.Com... | [] | rw [← coe_smul, succ_nsmul, coe_add, coe_smul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.GroupCompletion | {
"line": 124,
"column": 10
} | {
"line": 124,
"column": 56
} | {
"line": 124,
"column": 57
} | [
{
"pp": "M : Type u_1\nR : Type u_2\nα : Type u_3\nβ : Type u_4\ninst✝² : UniformSpace α\ninst✝¹ : AddGroup α\ninst✝ : IsUniformAddGroup α\nn : ℕ\na✝ : Completion α\na : α\n⊢ (n + 1) • ↑a = n • ↑a + ↑a",
"ppTerm": "?m.466",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"UniformSpace.Com... | [] | rw [← coe_smul, succ_nsmul, coe_add, coe_smul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Valued.ValuationTopology | {
"line": 299,
"column": 39
} | {
"line": 302,
"column": 40
} | {
"line": 304,
"column": 0
} | [
{
"pp": "R : Type u\ninst✝¹ : Ring R\nΓ₀ : Type v\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\n_i : Valued R Γ₀\nr : (ofClass v).ValueGroup₀\n⊢ IsClosed[_i.toTopologicalSpace] {x | v.restrict x = r}",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoi... | [] | by
rcases eq_or_ne r 0 with rfl | hr
· simpa using isClosed_closedBall R 0
exact isClopen_sphere _ hr |>.isClosed | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 625,
"column": 4
} | {
"line": 625,
"column": 60
} | {
"line": 625,
"column": 60
} | [
{
"pp": "case mk\nR : Type u_1\ninst✝¹ : Semiring R\ninst✝ : ValuativeRel R\nx x' y y' z : R\nb c : ValueGroupWithZero R\nhbc : b ≤ c\na₁ : R\na₂ : ↥(posSubmonoid R)\nhab : ValueGroupWithZero.mk a₁ a₂ ≤ b\n⊢ ValueGroupWithZero.mk a₁ a₂ ≤ c",
"ppTerm": "?mk",
"assigned": true,
"usedConstants": [
... | [
"case mk.mk\nR : Type u_1\ninst✝¹ : Semiring R\ninst✝ : ValuativeRel R\nx x' y y' z : R\nc : ValueGroupWithZero R\na₁ : R\na₂ : ↥(posSubmonoid R)\nb₁ : R\nb₂ : ↥(posSubmonoid R)\nhbc : ValueGroupWithZero.mk b₁ b₂ ≤ c\nhab : ValueGroupWithZero.mk a₁ a₂ ≤ ValueGroupWithZero.mk b₁ b₂\n⊢ ValueGroupWithZero.mk a₁ a₂ ≤ c... | induction b using ValueGroupWithZero.ind with | mk b₁ b₂
=> _ | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.RingTheory.Valuation.ValuativeRel.Basic | {
"line": 1194,
"column": 14
} | {
"line": 1194,
"column": 45
} | {
"line": 1195,
"column": 2
} | [
{
"pp": "R✝ : Type u_1\ninst✝⁵ : Semiring R✝\ninst✝⁴ : ValuativeRel R✝\nR : Type u_2\nΓ : Type u_3\ninst✝³ : Ring R\ninst✝² : ValuativeRel R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nv : Valuation R Γ\ninst✝ : v.Compatible\n⊢ ValueGroupWithZero.lift (fun r s ↦ (restrict₀ (ofClass v)) r / (restrict₀ (ofClass v... | [] | by simp [ValueGroup₀.restrict₀] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Valued.WithVal | {
"line": 541,
"column": 4
} | {
"line": 541,
"column": 49
} | {
"line": 542,
"column": 2
} | [
{
"pp": "R : Type u_4\nΓ₀ : Type u_5\nΓ₀' : Type u_6\ninst✝² : Ring R\ninst✝¹ : LinearOrderedCommGroupWithZero Γ₀\ninst✝ : LinearOrderedCommGroupWithZero Γ₀'\nv : Valuation R Γ₀\nw : Valuation R Γ₀'\nhval : Valued R Γ₀'\nhv : Valued.v = w\nh : v.IsEquiv w\nγ : (ofClass Valued.v).ValueGroup₀ˣ\nr s : R\nhr₀ : 0 <... | [] | simp [restrict_pos_iff, h.pos_iff, ← hv, hs₀] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 359,
"column": 2
} | {
"line": 363,
"column": 26
} | {
"line": 364,
"column": 2
} | [
{
"pp": "case h\nΓ : Type u_1\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nK : Type u_2\ninst✝ : Field K\nv : Valuation K Γ\nhv : v.IsRankOneDiscrete\nr : ↥v.valuationSubring\nhr : r ≠ 0\nπ : v.Uniformizer\nhr₀ : v ↑r ≠ 0\nvr : Γˣ := Units.mk0 (v ↑r) hr₀\nhvr_def : vr = Units.mk0 (v ↑r) hr₀\nm : ℤ\nhm : Units.mk... | [
"case h\nΓ : Type u_1\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nK : Type u_2\ninst✝ : Field K\nv : Valuation K Γ\nhv : v.IsRankOneDiscrete\nr : ↥v.valuationSubring\nhr : r ≠ 0\nπ : v.Uniformizer\nhr₀ : v ↑r ≠ 0\nvr : Γˣ := Units.mk0 (v ↑r) hr₀\nhvr_def : vr = Units.mk0 (v ↑r) hr₀\nm : ℤ\nhm : Units.mk0 (v ↑π.val)... | have ha₀ : (↑a : K) ≠ 0 := by
simp only [zpow_neg, ne_eq, mul_eq_zero, inv_eq_zero, ZeroMemClass.coe_eq_zero, not_or, ha]
refine ⟨?_, hr⟩
rw [hn, zpow_natCast, pow_eq_zero_iff', not_and_or]
exact Or.inl π.ne_zero | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Valuation.Discrete.Basic | {
"line": 376,
"column": 4
} | {
"line": 376,
"column": 26
} | {
"line": 377,
"column": 2
} | [
{
"pp": "case hJ\nΓ : Type u_1\ninst✝¹ : LinearOrderedCommGroupWithZero Γ\nK : Type u_2\ninst✝ : Field K\nv : Valuation K Γ\nhv : v.IsRankOneDiscrete\nπ : v.Uniformizer\nh : IsUnit π.val\n⊢ False",
"ppTerm": "?hJ",
"assigned": true,
"usedConstants": [
"Field.toDivisionRing",
"DivisionRin... | [] | apply π.2.not_isUnit h | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.RingTheory.Valuation.Discrete.IsDiscreteValuationRing | {
"line": 161,
"column": 2
} | {
"line": 162,
"column": 68
} | {
"line": 164,
"column": 0
} | [
{
"pp": "case neg\nA : Type u_1\ninst✝² : CommRing A\ninst✝¹ : IsDomain A\ninst✝ : IsDiscreteValuationRing A\nϖ : A\nhϖ : Irreducible ϖ\nn : ℕ\nu : Aˣ\nhx : ¬↑u * ϖ ^ n = 0\nthis : (maximalIdeal A).intValuation ↑u = 1\n⊢ (maximalIdeal A).intValuation (↑u * ϖ ^ n) = (ENat.recTopCoe 0 (fun x ↦ ↑(ofAdd ↑x)) ((addV... | [] | simp [(maximalIdeal A).intValuation_singleton hϖ.ne_zero
hϖ.maximalIdeal_eq, hϖ, this, WithZero.exp_eq_coe_ofAdd (n : ℤ)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 235,
"column": 4
} | {
"line": 243,
"column": 47
} | {
"line": 245,
"column": 0
} | [
{
"pp": "case neg\nG : Type u_2\ninst✝³ : AddCommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedAddMonoid G\ninst✝ : Archimedean G\nH : ∀ (x : G), ¬IsLeast {y | 0 < y} x\n⊢ Nonempty (G ≃+o ℤ) ∨ DenselyOrdered G",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.add... | [] | refine Or.inr ⟨?_⟩
intro x y hxy
specialize H (y - x)
obtain ⟨z, hz⟩ : ∃ z : G, 0 < z ∧ z < y - x := by
contrapose! H
refine ⟨by simp [hxy], fun _ ↦ H _⟩
refine ⟨x + z, ?_, ?_⟩
· simp [hz.left]
· simpa [lt_sub_iff_add_lt'] using hz.right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.GroupTheory.ArchimedeanDensely | {
"line": 235,
"column": 4
} | {
"line": 243,
"column": 47
} | {
"line": 245,
"column": 0
} | [
{
"pp": "case neg\nG : Type u_2\ninst✝³ : AddCommGroup G\ninst✝² : LinearOrder G\ninst✝¹ : IsOrderedAddMonoid G\ninst✝ : Archimedean G\nH : ∀ (x : G), ¬IsLeast {y | 0 < y} x\n⊢ Nonempty (G ≃+o ℤ) ∨ DenselyOrdered G",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"IsRightCancelAdd.add... | [] | refine Or.inr ⟨?_⟩
intro x y hxy
specialize H (y - x)
obtain ⟨z, hz⟩ : ∃ z : G, 0 < z ∧ z < y - x := by
contrapose! H
refine ⟨by simp [hxy], fun _ ↦ H _⟩
refine ⟨x + z, ?_, ?_⟩
· simp [hz.left]
· simpa [lt_sub_iff_add_lt'] using hz.right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 118,
"column": 14
} | {
"line": 118,
"column": 23
} | {
"line": 118,
"column": 24
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nx y : R\nhx : ¬x = 0\nhy : y = 0\n⊢ (if x * 0 = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {x * 0})).factors))) =\n (if x = 0 then 0 else exp (-↑((Associates.mk v.a... | [
"case pos\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nx y : R\nhx : ¬x = 0\nhy : y = 0\n⊢ (if 0 = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Associates.mk (Ideal.span {0})).factors))) =\n (if x = 0 then 0 else exp (-↑((Associates.mk v.asIdeal).count (Assoc... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 267,
"column": 65
} | {
"line": 267,
"column": 94
} | {
"line": 268,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nr : R\nn : ℕ\n⊢ exp (-↑n) ≤ v.intValuation r ↔ emultiplicity v.asIdeal (Ideal.span {r}) < ↑(n + 1)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Int.instAddCommMonoid",... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nr : R\nn : ℕ\n⊢ exp (-↑n) ≤ v.intValuation r ↔ ¬v.asIdeal ^ (n + 1) ∣ Ideal.span {r}"
] | emultiplicity_lt_iff_not_dvd, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 267,
"column": 2
} | {
"line": 269,
"column": 41
} | {
"line": 270,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nr : R\nn : ℕ\n⊢ exp (-↑n) ≤ v.intValuation r ↔ emultiplicity v.asIdeal (Ideal.span {r}) ≤ ↑n",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"WithZero.exp_ad... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nr : R\nn : ℕ\n⊢ exp (-↑n) ≤ v.intValuation r ↔ exp (-↑n) < v.intValuation r * exp 1"
] | rw [← ENat.lt_coe_add_one_iff, ← ENat.coe_one, ← ENat.coe_add, emultiplicity_lt_iff_not_dvd,
← intValuation_le_pow_iff_dvd, not_le, Nat.cast_add, Nat.cast_one, neg_add, exp_add,
exp_neg 1, mul_inv_lt_iff₀ (by simp)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 292,
"column": 2
} | {
"line": 292,
"column": 78
} | {
"line": 293,
"column": 2
} | [
{
"pp": "case h\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nhv : Irreducible (Associates.mk v.asIdeal)\nhlt : v.asIdeal ^ 2 < v.asIdeal\nπ : R\nmem : π ∈ v.asIdeal\nnotMem : π ∉ v.asIdeal ^ 2\nhπ : Associates.mk (Ideal.span {π}) ≠ 0\n⊢ (Associates.mk v.asIdeal).count... | [
"case h\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nhv : Irreducible (Associates.mk v.asIdeal)\nhlt : v.asIdeal ^ 2 < v.asIdeal\nπ : R\nmem : Associates.mk v.asIdeal ≤ Associates.mk (Ideal.span {π})\nnotMem : ¬Associates.mk (v.asIdeal ^ 2) ≤ Associates.mk (Ideal.span {π}... | rw [← Ideal.dvd_span_singleton, ← Associates.mk_le_mk_iff_dvd] at mem notMem | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 478,
"column": 2
} | {
"line": 478,
"column": 15
} | {
"line": 479,
"column": 2
} | [
{
"pp": "S : Type u_2\nT : Type u_3\ninst✝¹ : Semiring S\ninst✝ : Semiring T\nf : S →+* T\nhf : Function.Injective ⇑f\n⊢ Function.Injective ⇑(map f)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"RingHom",
"MvPowerSeries.instSemiring",
"PowerSeries.map",
"Unit",
... | [
"S : Type u_2\nT : Type u_3\ninst✝¹ : Semiring S\ninst✝ : Semiring T\nf : S →+* T\nhf : Function.Injective ⇑f\nu v : S⟦X⟧\nhuv : (map f) u = (map f) v\n⊢ u = v"
] | intro u v huv | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 519,
"column": 2
} | {
"line": 519,
"column": 26
} | {
"line": 521,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\np : R⟦X⟧\nT : Subring R\nhp : ∀ (n : ℕ), (coeff n) p ∈ T\nn : ℕ\n⊢ ↑((coeff n) (p.toSubring T hp)) = (coeff n) p",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"Subring.instSetLike",
"Ring.toNonA... | [] | rw [toSubring, coeff_mk] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 519,
"column": 2
} | {
"line": 519,
"column": 26
} | {
"line": 521,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\np : R⟦X⟧\nT : Subring R\nhp : ∀ (n : ℕ), (coeff n) p ∈ T\nn : ℕ\n⊢ ↑((coeff n) (p.toSubring T hp)) = (coeff n) p",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"Subring.instSetLike",
"Ring.toNonA... | [] | rw [toSubring, coeff_mk] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Basic | {
"line": 519,
"column": 2
} | {
"line": 519,
"column": 26
} | {
"line": 521,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\np : R⟦X⟧\nT : Subring R\nhp : ∀ (n : ℕ), (coeff n) p ∈ T\nn : ℕ\n⊢ ↑((coeff n) (p.toSubring T hp)) = (coeff n) p",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"Subring.instSetLike",
"Ring.toNonA... | [] | rw [toSubring, coeff_mk] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 604,
"column": 8
} | {
"line": 604,
"column": 24
} | {
"line": 604,
"column": 24
} | [
{
"pp": "case mp.hnc\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\ns : σ\nn : ℕ\nφ : MvPowerSeries σ R\nm j : σ →₀ ℕ\nhij : (single s n, j) ∈ antidiagonal m\n⊢ n ≤ m s",
"ppTerm": "?mp.hnc",
"assigned": true,
"usedConstants": [
"Finsupp.instHasAntidiagonal",
"Nat.instMulZeroClass",
... | [
"case mp.hnc\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\ns : σ\nn : ℕ\nφ : MvPowerSeries σ R\nm j : σ →₀ ℕ\nhij : (single s n, j).1 + (single s n, j).2 = m\n⊢ n ≤ m s"
] | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 605,
"column": 15
} | {
"line": 605,
"column": 33
} | {
"line": 605,
"column": 34
} | [
{
"pp": "case mp.hnc\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\ns : σ\nn : ℕ\nφ : MvPowerSeries σ R\nm j : σ →₀ ℕ\nhij : (single s n, j).1 + (single s n, j).2 = m\n⊢ n ≤ ((single s n, j).1 + (single s n, j).2) s",
"ppTerm": "?mp.hnc",
"assigned": true,
"usedConstants": [
"Finsupp.instFun... | [
"case mp.hnc\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\ns : σ\nn : ℕ\nφ : MvPowerSeries σ R\nm j : σ →₀ ℕ\nhij : (single s n, j).1 + (single s n, j).2 = m\n⊢ n ≤ (single s n, j).1 s + (single s n, j).2 s"
] | Finsupp.add_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 615,
"column": 12
} | {
"line": 615,
"column": 28
} | {
"line": 615,
"column": 28
} | [
{
"pp": "case pos.h₀\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\ns : σ\nn : ℕ\nφ : MvPowerSeries σ R\nh : ∀ (m : σ →₀ ℕ), m s < n → (coeff m) φ = 0\nm : σ →₀ ℕ\nH : m - single s n + single s n = m\ni j : σ →₀ ℕ\nhij : (i, j) ∈ antidiagonal m\nhne : (i, j) ≠ (single s n, m - single s n)\n⊢ ((coeff (i, j).1)... | [
"case pos.h₀\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\ns : σ\nn : ℕ\nφ : MvPowerSeries σ R\nh : ∀ (m : σ →₀ ℕ), m s < n → (coeff m) φ = 0\nm : σ →₀ ℕ\nH : m - single s n + single s n = m\ni j : σ →₀ ℕ\nhij : (i, j).1 + (i, j).2 = m\nhne : (i, j) ≠ (single s n, m - single s n)\n⊢ ((coeff (i, j).1) (X s ^ n) *... | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 631,
"column": 12
} | {
"line": 631,
"column": 28
} | {
"line": 631,
"column": 28
} | [
{
"pp": "case neg\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\ns : σ\nn : ℕ\nφ : MvPowerSeries σ R\nh : ∀ (m : σ →₀ ℕ), m s < n → (coeff m) φ = 0\nm : σ →₀ ℕ\nH : ¬m - single s n + single s n = m\ni j : σ →₀ ℕ\nhij : (i, j) ∈ antidiagonal m\n⊢ ((coeff (i, j).1) (X s ^ n) * (coeff (i, j).2) fun m ↦ (coeff (m... | [
"case neg\nσ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\ns : σ\nn : ℕ\nφ : MvPowerSeries σ R\nh : ∀ (m : σ →₀ ℕ), m s < n → (coeff m) φ = 0\nm : σ →₀ ℕ\nH : ¬m - single s n + single s n = m\ni j : σ →₀ ℕ\nhij : (i, j).1 + (i, j).2 = m\n⊢ ((coeff (i, j).1) (X s ^ n) * (coeff (i, j).2) fun m ↦ (coeff (m + single s ... | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.MvPowerSeries.Basic | {
"line": 717,
"column": 4
} | {
"line": 717,
"column": 30
} | {
"line": 717,
"column": 30
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommSemiring R\nm : σ →₀ ℕ\na : R\nn : ℕ\n⊢ (monomial (n • m)) (∏ i ∈ range n, a) = (monomial (n • m)) (a ^ n)",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instMulZeroClass",
"instHSMul",
"Semiring.toMod... | [
"σ : Type u_1\nR : Type u_3\ninst✝ : CommSemiring R\nm : σ →₀ ℕ\na : R\nn : ℕ\n⊢ (monomial (n • m)) (a ^ n) = (monomial (n • m)) (a ^ n)"
] | ← Finset.pow_eq_prod_const | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 450,
"column": 8
} | {
"line": 458,
"column": 72
} | {
"line": 460,
"column": 0
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\na b : (ofClass v).ValueGroup₀\nx : K := ⋯.choose\nhx_def : x = ⋯.choose\ny : K := ⋯.choose\nhy_def : y = ⋯.choose\nxy : K := ⋯.choose\nhxy_def : xy = ⋯.choose\nhxy : v xy = embedding a ... | [] | rw [dif_neg, dif_neg, dif_neg]
· simp only [← WithZero.coe_mul, MulMemClass.mk_mul_mk, WithZero.coe_inj, Subtype.mk.injEq]
rw [← Units.mk0_mul]
· ext
simp [Units.val_mk0, hx, hy, hxy]
· aesop
· simpa
· simpa
· simp [extensionValuation_apply_coe, ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 450,
"column": 8
} | {
"line": 458,
"column": 72
} | {
"line": 460,
"column": 0
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\na b : (ofClass v).ValueGroup₀\nx : K := ⋯.choose\nhx_def : x = ⋯.choose\ny : K := ⋯.choose\nhy_def : y = ⋯.choose\nxy : K := ⋯.choose\nhxy_def : xy = ⋯.choose\nhxy : v xy = embedding a ... | [] | rw [dif_neg, dif_neg, dif_neg]
· simp only [← WithZero.coe_mul, MulMemClass.mk_mul_mk, WithZero.coe_inj, Subtype.mk.injEq]
rw [← Units.mk0_mul]
· ext
simp [Units.val_mk0, hx, hy, hxy]
· aesop
· simpa
· simpa
· simp [extensionValuation_apply_coe, ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 465,
"column": 4
} | {
"line": 504,
"column": 22
} | {
"line": 505,
"column": 2
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\n⊢ Function.Injective ⇑valueGroup₀_hom_extensionValuation",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Valued.valueGroup₀_equiv_extensionValuatio... | [] | intro a b hab
set x := (restrict₀_surjective (.ofClass hv.v) a).choose with hx_def
have hx := (restrict₀_surjective (.ofClass hv.v) a).choose_spec
set y := (restrict₀_surjective (.ofClass hv.v) b).choose with hy_def
have hy := (restrict₀_surjective (.ofClass hv.v) b).choose_spec
apply_fun embedding ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 465,
"column": 4
} | {
"line": 504,
"column": 22
} | {
"line": 505,
"column": 2
} | [
{
"pp": "case refine_1\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\n⊢ Function.Injective ⇑valueGroup₀_hom_extensionValuation",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Valued.valueGroup₀_equiv_extensionValuatio... | [] | intro a b hab
set x := (restrict₀_surjective (.ofClass hv.v) a).choose with hx_def
have hx := (restrict₀_surjective (.ofClass hv.v) a).choose_spec
set y := (restrict₀_surjective (.ofClass hv.v) b).choose with hy_def
have hy := (restrict₀_surjective (.ofClass hv.v) b).choose_spec
apply_fun embedding ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 525,
"column": 44
} | {
"line": 525,
"column": 67
} | {
"line": 525,
"column": 68
} | [
{
"pp": "K : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\ns : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\nthis : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.map ↑embeddi... | [
"K : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\ns : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\nthis : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.map ↑embedding) γ)}\nx :... | Valuation.restrict_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 531,
"column": 16
} | {
"line": 531,
"column": 39
} | {
"line": 531,
"column": 40
} | [
{
"pp": "case pos\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\ns : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\nthis : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.ma... | [
"case pos\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\ns : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\nthis : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.map ↑embedding... | Valuation.restrict_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Finset.Finsupp | {
"line": 57,
"column": 4
} | {
"line": 59,
"column": 86
} | {
"line": 61,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\nα : Type u_2\ninst✝ : Zero α\ns : Finset ι\nf : ι →₀ α\nt : ι → Finset α\n⊢ (f.support ⊆ s ∧ ∀ i ∈ s, f i ∈ t i) → ∃ a ∈ s.pi t, { toFun := indicator s, inj' := ⋯ } a = f",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Finsu... | [] | refine fun h => ⟨fun i _ => f i, mem_pi.2 h.2, ?_⟩
ext i
exact ite_eq_left_iff.2 fun hi => (notMem_support_iff.1 fun H => hi <| h.1 H).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Finset.Finsupp | {
"line": 57,
"column": 4
} | {
"line": 59,
"column": 86
} | {
"line": 61,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\nα : Type u_2\ninst✝ : Zero α\ns : Finset ι\nf : ι →₀ α\nt : ι → Finset α\n⊢ (f.support ⊆ s ∧ ∀ i ∈ s, f i ∈ t i) → ∃ a ∈ s.pi t, { toFun := indicator s, inj' := ⋯ } a = f",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Finsu... | [] | refine fun h => ⟨fun i _ => f i, mem_pi.2 h.2, ?_⟩
ext i
exact ite_eq_left_iff.2 fun hi => (notMem_support_iff.1 fun H => hi <| h.1 H).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Finset.Finsupp | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 26
} | {
"line": 77,
"column": 0
} | [
{
"pp": "case refine_2\nι : Type u_1\nα : Type u_2\ninst✝ : Zero α\ns : Finset ι\nf : ι →₀ α\nt : ι →₀ Finset α\nht : t.support ⊆ s\ni : ι\nh : f i ∈ t i\nhi : i ∈ f.support\nH : t i = 0\n⊢ f i = 0",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Finsupp.instFunLike",
"cong... | [] | · rwa [H, mem_zero] at h | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 544,
"column": 16
} | {
"line": 544,
"column": 39
} | {
"line": 544,
"column": 40
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\ns : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\nthis : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.ma... | [
"case neg\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\ns : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\nthis : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.map ↑embedding... | Valuation.restrict_def, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 537,
"column": 12
} | {
"line": 551,
"column": 23
} | {
"line": 552,
"column": 6
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\ns : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\nthis : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.ma... | [] | set y := (restrict₀_surjective (.ofClass hv.v) γ).choose with hy_def
have hy := (restrict₀_surjective (.ofClass hv.v) γ).choose_spec
apply_fun embedding at hy
simp only [← hy_def, embedding_restrict₀, coe_ofClass] at hy
simp only [coe_ofClass, extensionValuation_toFun, va... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.Valued.ValuedField | {
"line": 537,
"column": 12
} | {
"line": 551,
"column": 23
} | {
"line": 552,
"column": 6
} | [
{
"pp": "case neg\nK : Type u_1\ninst✝¹ : Field K\nΓ₀ : Type u_2\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nhv : Valued K Γ₀\ns : failed to pretty print expression (use 'set_option pp.rawOnError true' for raw representation)\nthis : (𝓝 0).HasBasis (fun x ↦ True) fun γ ↦ {x | extensionValuation x < ↑((Units.ma... | [] | set y := (restrict₀_surjective (.ofClass hv.v) γ).choose with hy_def
have hy := (restrict₀_surjective (.ofClass hv.v) γ).choose_spec
apply_fun embedding at hy
simp only [← hy_def, embedding_restrict₀, coe_ofClass] at hy
simp only [coe_ofClass, extensionValuation_toFun, va... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.AdicValuation | {
"line": 770,
"column": 2
} | {
"line": 771,
"column": 13
} | {
"line": 772,
"column": 2
} | [
{
"pp": "case pos\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\na : adicCompletion K v\nha : a ∈ adicCompletionIntegers K v\n⊢ ∃ b ∈ R⁰, a * ↑b ∈ adicCompletionIntegers K v",
"ppTerm... | [
"case neg\nR : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : IsDedekindDomain R\nK : Type u_2\ninst✝² : Field K\ninst✝¹ : Algebra R K\ninst✝ : IsFractionRing R K\nv : HeightOneSpectrum R\na : adicCompletion K v\nha : a ∉ adicCompletionIntegers K v\n⊢ ∃ b ∈ R⁰, a * ↑b ∈ adicCompletionIntegers K v"
] | · use 1
simp [ha] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.MvPowerSeries.Order | {
"line": 630,
"column": 2
} | {
"line": 630,
"column": 85
} | {
"line": 631,
"column": 2
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nw : σ → ℕ\nf : MvPowerSeries σ R\np : ℕ\nhf : ↑p = weightedOrder w f\nd : σ →₀ ℕ\nhf' : (coeff d) ((weightedHomogeneousComponent w p) f) = (coeff d) 0\nhd : ¬(coeff d) f = 0 ∧ (weight w) d = p\n⊢ (coeff d) f = 0",
"ppTerm": "?m.57",
"assigned": tr... | [
"σ : Type u_1\nR : Type u_2\ninst✝ : Semiring R\nw : σ → ℕ\nf : MvPowerSeries σ R\np : ℕ\nhf : ↑p = weightedOrder w f\nd : σ →₀ ℕ\nhd : ¬(coeff d) f = 0 ∧ (weight w) d = p\nhf' : (weight w) d = p → (coeff d) f = 0\n⊢ (coeff d) f = 0"
] | simp only [coeff_weightedHomogeneousComponent, coeff_zero, ite_eq_right_iff] at hf' | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PowerSeries.Trunc | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 30
} | {
"line": 67,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nm n : ℕ\nφ : R⟦X⟧\n⊢ ((trunc n) φ).coeff m = if m < n then (coeff m) φ else 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"_private.Mathlib.RingTheory.PowerSeries.Trunc.0.PowerSeries.coeff_truncAux",
"con... | [] | simp [trunc, coeff_truncAux] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.PowerSeries.Trunc | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 30
} | {
"line": 67,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nm n : ℕ\nφ : R⟦X⟧\n⊢ ((trunc n) φ).coeff m = if m < n then (coeff m) φ else 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"_private.Mathlib.RingTheory.PowerSeries.Trunc.0.PowerSeries.coeff_truncAux",
"con... | [] | simp [trunc, coeff_truncAux] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Trunc | {
"line": 65,
"column": 2
} | {
"line": 65,
"column": 30
} | {
"line": 67,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nm n : ℕ\nφ : R⟦X⟧\n⊢ ((trunc n) φ).coeff m = if m < n then (coeff m) φ else 0",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"_private.Mathlib.RingTheory.PowerSeries.Trunc.0.PowerSeries.coeff_truncAux",
"con... | [] | simp [trunc, coeff_truncAux] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Evaluation | {
"line": 183,
"column": 4
} | {
"line": 183,
"column": 18
} | {
"line": 184,
"column": 2
} | [
{
"pp": "σ : Type u_1\nR : Type u_2\ninst✝⁶ : CommRing R\ninst✝⁵ : UniformSpace R\nS : Type u_3\ninst✝⁴ : CommRing S\ninst✝³ : UniformSpace S\nφ : R →+* S\na : σ → S\ninst✝² : IsUniformAddGroup R\ninst✝¹ : IsUniformAddGroup S\ninst✝ : IsLinearTopology S S\nhφ : Continuous[inst✝⁵.toTopologicalSpace, inst✝³.toTop... | [] | simpa using hf | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 197,
"column": 6
} | {
"line": 197,
"column": 22
} | {
"line": 197,
"column": 22
} | [
{
"pp": "case neg\nR : Type u_1\ninst✝ : Semiring R\nφ ψ : R⟦X⟧\nn : ℕ\nhn : ↑n < φ.order + ψ.order\ni j : ℕ\nhij : (i, j) ∈ antidiagonal n\nhi : φ.order ≤ ↑i\nhj : ψ.order ≤ ↑j\n⊢ (coeff (i, j).1) φ * (coeff (i, j).2) ψ = 0",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"AddMonoid.... | [
"case neg\nR : Type u_1\ninst✝ : Semiring R\nφ ψ : R⟦X⟧\nn : ℕ\nhn : ↑n < φ.order + ψ.order\ni j : ℕ\nhij : (i, j).1 + (i, j).2 = n\nhi : φ.order ≤ ↑i\nhj : ψ.order ≤ ↑j\n⊢ (coeff (i, j).1) φ * (coeff (i, j).2) ψ = 0"
] | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 239,
"column": 2
} | {
"line": 247,
"column": 23
} | {
"line": 249,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\nn : ℕ\na : R\ninst✝ : Decidable (a = 0)\n⊢ ((monomial n) a).order = if a = 0 then ⊤ else ↑n",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MvPowerSeries.instZero",
"Semiring.toModule",
"instCharZeroENat",
... | [] | split_ifs with h
· rw [h, order_eq_top, map_zero]
· rw [order_eq]
constructor <;> intro i hi
· simp only [Nat.cast_inj] at hi
rwa [hi, coeff_monomial_same]
· simp only [Nat.cast_lt] at hi
rw [coeff_monomial, if_neg]
exact ne_of_lt hi | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 239,
"column": 2
} | {
"line": 247,
"column": 23
} | {
"line": 249,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹ : Semiring R\nn : ℕ\na : R\ninst✝ : Decidable (a = 0)\n⊢ ((monomial n) a).order = if a = 0 then ⊤ else ↑n",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MvPowerSeries.instZero",
"Semiring.toModule",
"instCharZeroENat",
... | [] | split_ifs with h
· rw [h, order_eq_top, map_zero]
· rw [order_eq]
constructor <;> intro i hi
· simp only [Nat.cast_inj] at hi
rwa [hi, coeff_monomial_same]
· simp only [Nat.cast_lt] at hi
rw [coeff_monomial, if_neg]
exact ne_of_lt hi | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 263,
"column": 6
} | {
"line": 263,
"column": 22
} | {
"line": 263,
"column": 22
} | [
{
"pp": "R : Type u_1\ninst✝ : Semiring R\nφ ψ : R⟦X⟧\nn : ℕ\nh : ↑n < ψ.order\nx : ℕ × ℕ\nhx : x ∈ antidiagonal n\n⊢ ↑x.2 ≤ ↑n",
"ppTerm": "?m.85",
"assigned": true,
"usedConstants": [
"AddMonoid.toAddSemigroup",
"congrArg",
"Finset",
"Nat.instAddMonoid",
"Membership.m... | [
"R : Type u_1\ninst✝ : Semiring R\nφ ψ : R⟦X⟧\nn : ℕ\nh : ↑n < ψ.order\nx : ℕ × ℕ\nhx : x.1 + x.2 = n\n⊢ ↑x.2 ≤ ↑n"
] | mem_antidiagonal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 340,
"column": 6
} | {
"line": 350,
"column": 14
} | {
"line": 352,
"column": 0
} | [
{
"pp": "R : Type u_2\ninst✝ : Semiring R\nφ : R⟦X⟧\nhφ : φ ≠ 0\nn : ℕ\nho : φ.order = ↑n\nhn : φ.order.toNat = n\n⊢ emultiplicity X φ ≤ ↑φ.order.toNat",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"not_le",
"PowerSeries.coeff_mul_of_lt_order",
"Eq.mpr",
"NonAssoc... | [] | apply Order.le_of_lt_add_one
rw [← not_le, ← Nat.cast_one, ← Nat.cast_add, ← pow_dvd_iff_le_emultiplicity]
rintro ⟨ψ, H⟩
have := congr_arg (coeff n) H
rw [X_pow_mul, coeff_mul_of_lt_order, ← hn] at this
· exact coeff_order hφ this
· rw [X_pow_eq, order_monomial]
split_ifs
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Order | {
"line": 340,
"column": 6
} | {
"line": 350,
"column": 14
} | {
"line": 352,
"column": 0
} | [
{
"pp": "R : Type u_2\ninst✝ : Semiring R\nφ : R⟦X⟧\nhφ : φ ≠ 0\nn : ℕ\nho : φ.order = ↑n\nhn : φ.order.toNat = n\n⊢ emultiplicity X φ ≤ ↑φ.order.toNat",
"ppTerm": "?m.84",
"assigned": true,
"usedConstants": [
"not_le",
"PowerSeries.coeff_mul_of_lt_order",
"Eq.mpr",
"NonAssoc... | [] | apply Order.le_of_lt_add_one
rw [← not_le, ← Nat.cast_one, ← Nat.cast_add, ← pow_dvd_iff_le_emultiplicity]
rintro ⟨ψ, H⟩
have := congr_arg (coeff n) H
rw [X_pow_mul, coeff_mul_of_lt_order, ← hn] at this
· exact coeff_order hφ this
· rw [X_pow_eq, order_monomial]
split_ifs
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.PowerSeries.Evaluation | {
"line": 88,
"column": 2
} | {
"line": 89,
"column": 21
} | {
"line": 91,
"column": 0
} | [
{
"pp": "S : Type u_2\ninst✝² : CommRing S\ninst✝¹ : TopologicalSpace S\ninst✝ : IsLinearTopology S S\nc x : S\nhx : HasEval x\n⊢ HasEval (c * x)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"CommSemiring.toSemiring",
"Eq.mp",
"id",
... | [] | simp only [hasEval_iff] at hx ⊢
exact hx.mul_left _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.PowerSeries.Evaluation | {
"line": 88,
"column": 2
} | {
"line": 89,
"column": 21
} | {
"line": 91,
"column": 0
} | [
{
"pp": "S : Type u_2\ninst✝² : CommRing S\ninst✝¹ : TopologicalSpace S\ninst✝ : IsLinearTopology S S\nc x : S\nhx : HasEval x\n⊢ HasEval (c * x)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"CommSemiring.toSemiring",
"Eq.mp",
"id",
... | [] | simp only [hasEval_iff] at hx ⊢
exact hx.mul_left _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 236,
"column": 2
} | {
"line": 236,
"column": 14
} | {
"line": 238,
"column": 0
} | [
{
"pp": "σ : Type u_1\nR : Type u_3\ninst✝ : CommRing R\nthis✝ : UniformSpace R := ⊥\nf : MvPowerSeries σ R\nthis : ∀ (x : MvPowerSeries σ R), (aeval ⋯) x = (AlgHom.id R (MvPowerSeries σ R)) x\n⊢ (aeval ⋯) f = id f",
"ppTerm": "?m.80",
"assigned": true,
"usedConstants": [],
"usedFVars": [
... | [] | exact this f | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 474,
"column": 6
} | {
"line": 474,
"column": 42
} | {
"line": 474,
"column": 43
} | [
{
"pp": "case neg\nσ : Type u_1\nR : Type u_3\ninst✝² : CommRing R\nτ : Type u_4\nS : Type u_5\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\na : σ → MvPowerSeries τ S\nw : τ → ℕ\nha : HasSubst a\nf : MvPowerSeries σ R\nd : τ →₀ ℕ\nhd : ↑((Finsupp.weight w) d) < ⨅ d, ⨅ (_ : (coeff d) f ≠ 0), (Finsupp.weight (weight... | [
"case neg\nσ : Type u_1\nR : Type u_3\ninst✝² : CommRing R\nτ : Type u_4\nS : Type u_5\ninst✝¹ : CommRing S\ninst✝ : Algebra R S\na : σ → MvPowerSeries τ S\nw : τ → ℕ\nha : HasSubst a\nf : MvPowerSeries σ R\nd : τ →₀ ℕ\nhd : ↑((Finsupp.weight w) d) < ⨅ d, ⨅ (_ : (coeff d) f ≠ 0), (Finsupp.weight (weightedOrder w ∘ ... | coeff_eq_zero_of_lt_weightedOrder w, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.NumberTheory.ArithmeticFunction.LFunction | {
"line": 178,
"column": 6
} | {
"line": 178,
"column": 34
} | {
"line": 179,
"column": 2
} | [
{
"pp": "case neg.hb\nR : Type u_1\ninst✝ : CommRing R\nq k : ℕ\nhk : k ≠ 0\nf : PowerSeries R\nhq : 1 < q\ni : ℕ\n⊢ ∃ i_1, q ^ i_1 = (q ^ k) ^ i",
"ppTerm": "?neg.hb✝",
"assigned": true,
"usedConstants": [
"HMul.hMul",
"Nat.instMonoid",
"instMulNat",
"pow_mul",
"Monoid... | [] | exact ⟨k * i, pow_mul q k i⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.MvPowerSeries.Substitution | {
"line": 619,
"column": 4
} | {
"line": 619,
"column": 53
} | {
"line": 620,
"column": 4
} | [
{
"pp": "σ : Type u_1\nA : Type u_2\ninst✝²¹ : CommSemiring A\nR✝ : Type u_3\ninst✝²⁰ : CommRing R✝\ninst✝¹⁹ : Algebra A R✝\nτ : Type u_4\nS : Type u_5\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra A S\ninst✝¹⁶ : Algebra R✝ S\ninst✝¹⁵ : IsScalarTower A R✝ S\na✝¹ a✝ : σ → MvPowerSeries τ S\nT✝ : Type u_6\ninst✝¹⁴ : C... | [
"σ : Type u_1\nA : Type u_2\ninst✝²¹ : CommSemiring A\nR✝ : Type u_3\ninst✝²⁰ : CommRing R✝\ninst✝¹⁹ : Algebra A R✝\nτ : Type u_4\nS : Type u_5\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra A S\ninst✝¹⁶ : Algebra R✝ S\ninst✝¹⁵ : IsScalarTower A R✝ S\na✝¹ a✝ : σ → MvPowerSeries τ S\nT✝ : Type u_6\ninst✝¹⁴ : CommRing T✝\n... | simp only [coeff_one, mul_ite, mul_one, mul_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Calculus.TangentCone.Basic | {
"line": 64,
"column": 2
} | {
"line": 71,
"column": 58
} | {
"line": 73,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : AddCommGroup E\ninst✝² : SMul 𝕜 E\ninst✝¹ : TopologicalSpace E\ns t : Set E\nx : E\ninst✝ : ContinuousAdd E\nh : 𝓝[s] x ≤ 𝓝[t] x\n⊢ tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 t x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Filter.in... | [] | simp only [tangentConeAt_def, setOf_subset_setOf]
refine fun y hy ↦ hy.mono ?_
gcongr _ • ?_
rw [nhdsWithin_le_iff]
suffices Tendsto (x + ·) (𝓝[(x + ·) ⁻¹' s] 0) (𝓝[s] x) from
this.mono_right h |> tendsto_nhdsWithin_iff.mp |>.2
refine .inf ?_ (mapsTo_preimage _ _).tendsto
exact (continuous_const_add x... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.TangentCone.Basic | {
"line": 64,
"column": 2
} | {
"line": 71,
"column": 58
} | {
"line": 73,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : AddCommGroup E\ninst✝² : SMul 𝕜 E\ninst✝¹ : TopologicalSpace E\ns t : Set E\nx : E\ninst✝ : ContinuousAdd E\nh : 𝓝[s] x ≤ 𝓝[t] x\n⊢ tangentConeAt 𝕜 s x ⊆ tangentConeAt 𝕜 t x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Filter.in... | [] | simp only [tangentConeAt_def, setOf_subset_setOf]
refine fun y hy ↦ hy.mono ?_
gcongr _ • ?_
rw [nhdsWithin_le_iff]
suffices Tendsto (x + ·) (𝓝[(x + ·) ⁻¹' s] 0) (𝓝[s] x) from
this.mono_right h |> tendsto_nhdsWithin_iff.mp |>.2
refine .inf ?_ (mapsTo_preimage _ _).tendsto
exact (continuous_const_add x... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.TVS | {
"line": 806,
"column": 6
} | {
"line": 811,
"column": 67
} | {
"line": 812,
"column": 2
} | [] | [] | ‖f x‖ₑ ≤ egauge 𝕜 (ball 0 1) (f x) := le_egauge_ball_one ..
_ ≤ egauge 𝕜 (ball 0 r) (g x) := hx
_ ≤ ‖c‖ₑ * ‖g x‖ₑ / ↑r :=
egauge_ball_le_of_one_lt_norm hc <| .inl hr₀.ne'
_ = (‖c‖₊ / r : ℝ≥0) * ‖g x‖ₑ := by
simp [hr₀.ne', ENNReal.mul_div_right_comm, enorm_eq_nnnorm] | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcSteps |
Mathlib.Analysis.Calculus.FDeriv.Basic | {
"line": 940,
"column": 70
} | {
"line": 942,
"column": 78
} | {
"line": 944,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nx₀ : E\ns : Set E\nhs : s ∈ 𝓝 x₀\nC : ℝ≥0\nhlip : LipschitzOnWith C f s\n⊢ ‖fderiv 𝕜 f x₀‖ ≤ ↑C... | [] | by
refine norm_fderiv_le_of_lip' 𝕜 C.coe_nonneg ?_
filter_upwards [hs] with x hx using hlip.norm_sub_le hx (mem_of_mem_nhds hs) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.ConvergenceRadius | {
"line": 345,
"column": 2
} | {
"line": 345,
"column": 42
} | {
"line": 346,
"column": 2
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSerie... | [
"case inr\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\np : FormalMultilinearSeries 𝕜 F G\nu ... | refine le_radius_of_bound _ C fun n ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Analytic.Basic | {
"line": 371,
"column": 2
} | {
"line": 371,
"column": 43
} | {
"line": 372,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nhf : HasFPowerSeriesAt f p x\n⊢ HasFPowerSe... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nhf : HasFPowerSeriesWithinAt f p univ x\n⊢ HasFPowerSer... | rw [← hasFPowerSeriesWithinAt_univ] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Composition | {
"line": 396,
"column": 4
} | {
"line": 396,
"column": 50
} | {
"line": 397,
"column": 4
} | [
{
"pp": "case h₀\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nx : E\nn : ℕ\nb : Composition n\na✝ : b ∈ Finset.univ\nhb : b ≠... | [
"case h₀\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nx : E\nn : ℕ\nb : Composition n\na✝ : b ∈ Finset.univ\nhb : b ≠ Composition... | have A : 1 < b.blocksFun j := by convert! lt_k | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Analytic.Basic | {
"line": 627,
"column": 4
} | {
"line": 627,
"column": 96
} | {
"line": 628,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\ny : E\nhf : HasFPowerSeriesWithin... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\ns : Set E\nx : E\nr : ℝ≥0∞\ny : E\nhf : HasFPowerSeriesWithinOnBall f p s... | have : ContinuousAt (fun z ↦ p.partialSum k z) y := (p.partialSum_continuous k).continuousAt | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Analytic.Composition | {
"line": 491,
"column": 2
} | {
"line": 491,
"column": 26
} | {
"line": 492,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nq : FormalMultilinearSeries 𝕜 F G\n... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nq : FormalMultilinearSeries 𝕜 F G\np : FormalMu... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Analytic.Composition | {
"line": 583,
"column": 2
} | {
"line": 583,
"column": 22
} | {
"line": 584,
"column": 2
} | [
{
"pp": "m M N : ℕ\ni : (n : ℕ) × Composition n\nhi : i ∈ compPartialSumTargetSet m M N\n⊢ ∃ j, ∃ (hj : j ∈ compPartialSumSource m M N), compChangeOfVariables m M N j hj = i",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.mem",
"Exists",
"Forma... | [
"m M N n : ℕ\nc : Composition n\nhi : ⟨n, c⟩ ∈ compPartialSumTargetSet m M N\n⊢ ∃ j, ∃ (hj : j ∈ compPartialSumSource m M N), compChangeOfVariables m M N j hj = ⟨n, c⟩"
] | rcases i with ⟨n, c⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Analytic.CPolynomial | {
"line": 166,
"column": 4
} | {
"line": 166,
"column": 25
} | {
"line": 167,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nF : Type u_3\nG : Type u_4\ninst✝⁷ : NontriviallyNormedField 𝕜\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\ninst✝⁴ : NormedAddCommGroup G\ninst✝³ : NormedSpace 𝕜 G\nι : Type u_5\nEm : ι → Type u_6\ninst✝² : (i : ι) → NormedAddCommGroup (Em i)\ninst✝¹ : (i : ι) → NormedSpa... | [] | exact Nat.ne_of_lt hm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Analytic.Basic | {
"line": 772,
"column": 2
} | {
"line": 772,
"column": 43
} | {
"line": 773,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\nx : E\nhf : HasFPowerSeriesAt f p x\nn : ℕ\n⊢ (fun y ↦ f (x +... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\nx : E\nhf : HasFPowerSeriesWithinAt f p univ x\nn : ℕ\n⊢ (fun y ↦ f (x + ... | rw [← hasFPowerSeriesWithinAt_univ] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Inverse | {
"line": 248,
"column": 48
} | {
"line": 261,
"column": 85
} | {
"line": 263,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nx : E\nh : p 1 = (continuousMultilinearCurryFin1 𝕜 E F)... | [] | by
ext (n v)
match n with
| 0 =>
simp only [comp_coeff_zero', Matrix.zero_empty, id_apply_zero]
congr
ext i
exact i.elim0
| 1 =>
simp only [comp_coeff_one, h, rightInv_coeff_one, ContinuousLinearEquiv.apply_symm_apply,
id_apply_one, ContinuousLinearEquiv.coe_apply, continuousMultilinea... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Analytic.Inverse | {
"line": 278,
"column": 4
} | {
"line": 278,
"column": 78
} | {
"line": 280,
"column": 0
} | [
{
"pp": "case e_f\n𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type u_3\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\ni : E ≃L[𝕜] F\nx : E\nn✝ n : ℕ\nhn : 2 ≤ n + 2\nv : Fin (n + ... | [] | simp [comp_rightInv_aux1 N, this, comp_rightInv_aux2, -Set.toFinset_setOf] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Analytic.Composition | {
"line": 915,
"column": 4
} | {
"line": 915,
"column": 25
} | {
"line": 916,
"column": 2
} | [
{
"pp": "case inl\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nm n : ℕ\ng : F → G\nf : E ... | [] | simp [hg.finite _ hc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Analytic.Composition | {
"line": 915,
"column": 4
} | {
"line": 915,
"column": 25
} | {
"line": 916,
"column": 2
} | [
{
"pp": "case inl\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nm n : ℕ\ng : F → G\nf : E ... | [] | simp [hg.finite _ hc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Composition | {
"line": 915,
"column": 4
} | {
"line": 915,
"column": 25
} | {
"line": 916,
"column": 2
} | [
{
"pp": "case inl\n𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nm n : ℕ\ng : F → G\nf : E ... | [] | simp [hg.finite _ hc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Inverse | {
"line": 384,
"column": 8
} | {
"line": 384,
"column": 19
} | {
"line": 384,
"column": 20
} | [
{
"pp": "case a.a\nn : ℕ\np : ℕ → ℝ\nhp : ∀ (k : ℕ), 0 ≤ p k\nr a : ℝ\nhr : 0 ≤ r\nha : 0 ≤ a\nk : ℕ\na✝¹ : k ∈ Ico 2 (n + 1)\nc : Composition k\na✝ : c ∈ {c | 1 < c.length}.toFinset\n⊢ a ^ k * (r ^ c.length * ∏ j, p (c.blocksFun j)) = (∏ x, r) * (a ^ k * ∏ x, p (c.blocksFun x))",
"ppTerm": "?a.a✝",
"as... | [
"case a.a\nn : ℕ\np : ℕ → ℝ\nhp : ∀ (k : ℕ), 0 ≤ p k\nr a : ℝ\nhr : 0 ≤ r\nha : 0 ≤ a\nk : ℕ\na✝¹ : k ∈ Ico 2 (n + 1)\nc : Composition k\na✝ : c ∈ {c | 1 < c.length}.toFinset\n⊢ a ^ k * (r ^ c.length * ∏ j, p (c.blocksFun j)) = r ^ #univ * (a ^ k * ∏ x, p (c.blocksFun x))"
] | prod_const, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Analytic.Basic | {
"line": 894,
"column": 2
} | {
"line": 894,
"column": 43
} | {
"line": 895,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\nx : E\nhf : HasFPowerSeriesAt f p x\n⊢ (fun y ↦ f y.1 - f y.2... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\np : FormalMultilinearSeries 𝕜 E F\nx : E\nhf : HasFPowerSeriesWithinAt f p univ x\n⊢ (fun y ↦ f y.1 - f y.2 ... | rw [← hasFPowerSeriesWithinAt_univ] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Composition | {
"line": 1166,
"column": 4
} | {
"line": 1166,
"column": 88
} | {
"line": 1168,
"column": 0
} | [
{
"pp": "case succ\nn : ℕ\na : Composition n\nb : Composition a.length\ni : ℕ\nhi : i < b.length\nj : ℕ\nIHj :\n j < b.blocksFun ⟨i, hi⟩ →\n a.sizeUpTo (b.sizeUpTo i + j) = (a.gather b).sizeUpTo i + (a.sigmaCompositionAux b ⟨i, ⋯⟩).sizeUpTo j\nhj : j + 1 < b.blocksFun ⟨i, hi⟩\nA : j < b.blocksFun ⟨i, hi⟩\nB... | [] | rw [getElem_of_eq (getElem_splitWrtComposition _ _ _ _), getElem_drop, getElem_take] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Analytic.Constructions | {
"line": 487,
"column": 25
} | {
"line": 487,
"column": 44
} | {
"line": 487,
"column": 44
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nι : Type u_9\ninst✝² : Fintype ι\nFm : ι → Type u_10\ninst✝¹ : (i : ι) → NormedAddCommGroup (Fm i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (Fm i)\np : (i : ι) → FormalMultilinearSeries ... | [] | simp [radius_pi_le] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Analytic.Constructions | {
"line": 487,
"column": 25
} | {
"line": 487,
"column": 44
} | {
"line": 487,
"column": 44
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nι : Type u_9\ninst✝² : Fintype ι\nFm : ι → Type u_10\ninst✝¹ : (i : ι) → NormedAddCommGroup (Fm i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (Fm i)\np : (i : ι) → FormalMultilinearSeries ... | [] | simp [radius_pi_le] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Constructions | {
"line": 487,
"column": 25
} | {
"line": 487,
"column": 44
} | {
"line": 487,
"column": 44
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁵ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace 𝕜 E\nι : Type u_9\ninst✝² : Fintype ι\nFm : ι → Type u_10\ninst✝¹ : (i : ι) → NormedAddCommGroup (Fm i)\ninst✝ : (i : ι) → NormedSpace 𝕜 (Fm i)\np : (i : ι) → FormalMultilinearSeries ... | [] | simp [radius_pi_le] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Constructions | {
"line": 958,
"column": 2
} | {
"line": 966,
"column": 15
} | {
"line": 968,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\n𝕝 : Type u_8\ninst✝⁴ : NormedDivisionRing 𝕝\ninst✝³ : NormedAlgebra 𝕜 𝕝\ninst✝² : Module 𝕝 F\ninst✝¹ :... | [] | constructor
· exact fun a ↦ h₁f.smul a
· intro hprod
rw [analyticAt_congr (g := (f⁻¹ • f) • g), smul_assoc]
· exact (h₁f.inv h₂f).fun_smul hprod
· filter_upwards [h₁f.continuousAt.preimage_mem_nhds (compl_singleton_mem_nhds_iff.2 h₂f)]
intro y hy
rw [Set.preimage_compl, Set.mem_compl_iff, Se... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Analytic.Constructions | {
"line": 958,
"column": 2
} | {
"line": 966,
"column": 15
} | {
"line": 968,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁹ : NontriviallyNormedField 𝕜\nE : Type u_3\nF : Type u_4\ninst✝⁸ : NormedAddCommGroup E\ninst✝⁷ : NormedSpace 𝕜 E\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NormedSpace 𝕜 F\n𝕝 : Type u_8\ninst✝⁴ : NormedDivisionRing 𝕝\ninst✝³ : NormedAlgebra 𝕜 𝕝\ninst✝² : Module 𝕝 F\ninst✝¹ :... | [] | constructor
· exact fun a ↦ h₁f.smul a
· intro hprod
rw [analyticAt_congr (g := (f⁻¹ • f) • g), smul_assoc]
· exact (h₁f.inv h₂f).fun_smul hprod
· filter_upwards [h₁f.continuousAt.preimage_mem_nhds (compl_singleton_mem_nhds_iff.2 h₂f)]
intro y hy
rw [Set.preimage_compl, Set.mem_compl_iff, Se... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Analytic.Constructions | {
"line": 988,
"column": 2
} | {
"line": 988,
"column": 26
} | {
"line": 988,
"column": 26
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\n𝕝 : Type u_8\ninst✝¹ : NormedDivisionRing 𝕝\ninst✝ : NormedAlgebra 𝕜 𝕝\nf g : E → 𝕝\ns : Set E\nx : E\nfa : AnalyticWithinAt 𝕜 f s x\nga : AnalyticWithinAt 𝕜 g s x\ng0 : g ... | [
"𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\n𝕝 : Type u_8\ninst✝¹ : NormedDivisionRing 𝕝\ninst✝ : NormedAlgebra 𝕜 𝕝\nf g : E → 𝕝\ns : Set E\nx : E\nfa : AnalyticWithinAt 𝕜 f s x\nga : AnalyticWithinAt 𝕜 g s x\ng0 : g x ≠ 0\n⊢ Ana... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Analytic.Constructions | {
"line": 995,
"column": 2
} | {
"line": 995,
"column": 26
} | {
"line": 995,
"column": 26
} | [
{
"pp": "𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\n𝕝 : Type u_8\ninst✝¹ : NormedDivisionRing 𝕝\ninst✝ : NormedAlgebra 𝕜 𝕝\nf g : E → 𝕝\nx : E\nfa : AnalyticAt 𝕜 f x\nga : AnalyticAt 𝕜 g x\ng0 : g x ≠ 0\n⊢ AnalyticAt 𝕜 (f /... | [
"𝕜 : Type u_2\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u_3\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\n𝕝 : Type u_8\ninst✝¹ : NormedDivisionRing 𝕝\ninst✝ : NormedAlgebra 𝕜 𝕝\nf g : E → 𝕝\nx : E\nfa : AnalyticAt 𝕜 f x\nga : AnalyticAt 𝕜 g x\ng0 : g x ≠ 0\n⊢ AnalyticAt 𝕜 (f * g⁻¹) x"
] | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Calculus.FDeriv.Bilinear | {
"line": 96,
"column": 6
} | {
"line": 96,
"column": 62
} | {
"line": 96,
"column": 62
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nb : E × F → G\nu : Set (E × F)\nh : ... | [
"𝕜 : Type u_1\ninst✝⁶ : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace 𝕜 E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace 𝕜 F\nG : Type u_4\ninst✝¹ : NormedAddCommGroup G\ninst✝ : NormedSpace 𝕜 G\nb : E × F → G\nu : Set (E × F)\nh : IsBoundedBil... | DifferentiableAt.fderivWithin (h.differentiableAt p) hxs | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Calculus.FDeriv.Analytic | {
"line": 90,
"column": 2
} | {
"line": 90,
"column": 46
} | {
"line": 91,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nf : E → F\nx : E\ns : Set E\nh : HasFPowerSeriesWithinAt f p s x\n⊢ Tendsto ... | [
"𝕜 : Type u_1\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type u\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type v\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\np : FormalMultilinearSeries 𝕜 E F\nf : E → F\nx : E\ns : Set E\nh : HasFPowerSeriesWithinAt f p s x\n⊢ Tendsto (fun y ↦ ‖y ... | apply Tendsto.mono_left _ nhdsWithin_le_nhds | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 732,
"column": 2
} | {
"line": 734,
"column": 22
} | {
"line": 736,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nn : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUpTo n f p\n⊢ Continuous[PseudoM... | [] | rw [← hasFTaylorSeriesUpToOn_univ_iff] at h
rw [← continuousOn_univ]
exact h.continuousOn | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Calculus.ContDiff.FTaylorSeries | {
"line": 732,
"column": 2
} | {
"line": 734,
"column": 22
} | {
"line": 736,
"column": 0
} | [
{
"pp": "𝕜 : Type u\ninst✝⁴ : NontriviallyNormedField 𝕜\nE : Type uE\ninst✝³ : NormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\nF : Type uF\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedSpace 𝕜 F\nf : E → F\nn : ℕ∞ω\np : E → FormalMultilinearSeries 𝕜 E F\nh : HasFTaylorSeriesUpTo n f p\n⊢ Continuous[PseudoM... | [] | rw [← hasFTaylorSeriesUpToOn_univ_iff] at h
rw [← continuousOn_univ]
exact h.continuousOn | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.Module.Alternating.Topology | {
"line": 315,
"column": 14
} | {
"line": 318,
"column": 72
} | {
"line": 318,
"column": 73
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nι : Type u_5\ninst✝¹⁴ : NormedField 𝕜\ninst✝¹³ : AddCommGroup E\ninst✝¹² : Module 𝕜 E\ninst✝¹¹ : TopologicalSpace E\ninst✝¹⁰ : ContinuousSMul 𝕜 E\ninst✝⁹ : AddCommGroup F\ninst✝⁸ : Module 𝕜 F\ninst✝⁷ : TopologicalSpace F\ninst✝⁶ : IsTopologic... | [] | by
rw [ContinuousAlternatingMap.isEmbedding_toContinuousMultilinearMap.continuous_iff]
exact (map_continuous <| compContinuousMultilinearMapL 𝕜 (fun _ : ι ↦ E) F G g).comp
ContinuousAlternatingMap.continuous_toContinuousMultilinearMap | [anonymous] | Lean.Parser.Term.byTactic |
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