module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Topology.CompactOpen | {
"line": 123,
"column": 6
} | {
"line": 124,
"column": 69
} | {
"line": 125,
"column": 2
} | [
{
"pp": "case refine_1.inr\nX : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nι : Type u_6\nc : Y\np : ι → Prop\nU : ι → Set Y\nh : (𝓝 c).HasBasis p U\ns : Set C(X, Y)\nhs : s ∈ 𝓝 (const X c)\nS : Set (Set X × Set Y)\nhSf : S.Finite\nhSsub : {g | ∀ (K : Set X) (U : Set Y), (... | [] | refine hf.out.mono (subset_biUnion_of_mem (u := Prod.fst) hKV) (hi.trans ?_)
exact (biInter_subset_of_mem hKV).trans <| iInter_subset _ hKne | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.CompactOpen | {
"line": 123,
"column": 6
} | {
"line": 124,
"column": 69
} | {
"line": 125,
"column": 2
} | [
{
"pp": "case refine_1.inr\nX : Type u_2\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nι : Type u_6\nc : Y\np : ι → Prop\nU : ι → Set Y\nh : (𝓝 c).HasBasis p U\ns : Set C(X, Y)\nhs : s ∈ 𝓝 (const X c)\nS : Set (Set X × Set Y)\nhSf : S.Finite\nhSsub : {g | ∀ (K : Set X) (U : Set Y), (... | [] | refine hf.out.mono (subset_biUnion_of_mem (u := Prod.fst) hKV) (hi.trans ?_)
exact (biInter_subset_of_mem hKV).trans <| iInter_subset _ hKne | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 828,
"column": 4
} | {
"line": 829,
"column": 42
} | {
"line": 830,
"column": 4
} | [
{
"pp": "E : Type u_4\ninst✝⁴ : Ring E\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedRing E\ninst✝¹ : TopologicalSpace E\ninst✝ : OrderClosedTopology E\nl : E\nf : ℕ → E\nhfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)\nhfa : Antitone f\nk n : ℕ\n⊢ ∑ i ∈ Finset.range (2 * n), (-1) ^ i * ... | [
"E : Type u_4\ninst✝⁴ : Ring E\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedRing E\ninst✝¹ : TopologicalSpace E\ninst✝ : OrderClosedTopology E\nl : E\nf : ℕ → E\nhfl : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (𝓝 l)\nhfa : Antitone f\nk n : ℕ\n⊢ ∑ i ∈ Finset.range (2 * n), (-1) ^ i * f i + f (2 *... | simp_rw [_root_.pow_succ', show (-1 : E) ^ (2 * n) = 1 by simp, neg_one_mul, one_mul,
← sub_eq_add_neg, le_sub_iff_add_le] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 865,
"column": 4
} | {
"line": 865,
"column": 46
} | {
"line": 866,
"column": 4
} | [
{
"pp": "E : Type u_5\ninst✝⁶ : Ring E\ninst✝⁵ : LinearOrder E\ninst✝⁴ : IsOrderedRing E\ninst✝³ : UniformSpace E\ninst✝² : IsUniformAddGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : OrderClosedTopology E\nf : ℕ → E\nhfa : Antitone f\nhfs : Summable f\nn✝ : ℕ\nh : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f... | [
"E : Type u_5\ninst✝⁶ : Ring E\ninst✝⁵ : LinearOrder E\ninst✝⁴ : IsOrderedRing E\ninst✝³ : UniformSpace E\ninst✝² : IsUniformAddGroup E\ninst✝¹ : CompleteSpace E\ninst✝ : OrderClosedTopology E\nf : ℕ → E\nhfa : Antitone f\nhfs : Summable f\nn✝ : ℕ\nh : Tendsto (fun n ↦ ∑ i ∈ Finset.range n, (-1) ^ i * f i) atTop (�... | apply le_of_tendsto hfs.tendsto_atTop_zero | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 120,
"column": 76
} | {
"line": 121,
"column": 80
} | {
"line": 123,
"column": 0
} | [
{
"pp": "x : ℝ\nhx : x ≤ -1\n⊢ arcsin x = -(π / 2)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Subtype.coe_mk",
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"instHDiv",
"Real.pi",
"Real.arcsin"... | [] | by
rw [← arcsin_projIcc, projIcc_of_le_left _ hx, Subtype.coe_mk, arcsin_neg_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 126,
"column": 2
} | {
"line": 126,
"column": 75
} | {
"line": 127,
"column": 2
} | [
{
"pp": "case inl\nx : ℝ\nhx₁ : x ≤ -1\n⊢ arcsin (-x) = -arcsin x",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instLE",
"Real",
"instHDiv",
"NonUnitalCommRing.... | [
"case inr\nx : ℝ\nhx₁ : -1 ≤ x\n⊢ arcsin (-x) = -arcsin x"
] | · rw [arcsin_of_le_neg_one hx₁, neg_neg, arcsin_of_one_le (le_neg.2 hx₁)] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 126,
"column": 6
} | {
"line": 126,
"column": 14
} | {
"line": 126,
"column": 15
} | [
{
"pp": "x : ℝ\nhx : 0 < x\n⊢ cosh (log x) = (x + x⁻¹) / 2",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"congrArg",
"Real.instInv",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddOfNat",
"id",
"HDiv.hDi... | [
"x : ℝ\nhx : 0 < x\n⊢ (rexp (log x) + rexp (-log x)) / 2 = (x + x⁻¹) / 2"
] | cosh_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 8
} | {
"line": 112,
"column": 2
} | [
{
"pp": "⊢ ∃ k, -π - π = 2 * π * ↑k",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Int.cast",
"Real",
"Real.pi",
"HMul.hMul",
"Real.instSub",
"Nat.instAtLeastTwoHAddOfNat",
"HSub.hSub",
"Int.instNegInt",
"instOfNatNat",
"Int",
... | [
"case h\n⊢ -π - π = 2 * π * ↑(-1)"
] | use -1 | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 18
} | {
"line": 116,
"column": 19
} | [
{
"pp": "θ : ℝ\n⊢ 2 • ↑(θ / 2) = ↑θ",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"instHDiv",
"Real.Angle",
"Real.Angle.coe",
"Real.instAddMonoid",
"congrArg",
"Real.instDivInvMonoid",
"Nat.instAtL... | [
"θ : ℝ\n⊢ ↑(2 • (θ / 2)) = ↑θ"
] | ← coe_nsmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 587,
"column": 2
} | {
"line": 588,
"column": 33
} | {
"line": 590,
"column": 0
} | [
{
"pp": "e : ℝ\nn : ℕ\nh : NormNum.IsNat e n\n⊢ 0 ≤ Real.log e",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"AddMonoidWithOne.toNatCas... | [] | rw [NormNum.IsNat.to_eq h rfl]
exact Real.log_natCast_nonneg _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 587,
"column": 2
} | {
"line": 588,
"column": 33
} | {
"line": 590,
"column": 0
} | [
{
"pp": "e : ℝ\nn : ℕ\nh : NormNum.IsNat e n\n⊢ 0 ≤ Real.log e",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"AddMonoidWithOne.toNatCas... | [] | rw [NormNum.IsNat.to_eq h rfl]
exact Real.log_natCast_nonneg _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 161,
"column": 16
} | {
"line": 161,
"column": 39
} | {
"line": 161,
"column": 39
} | [
{
"pp": "θ : Angle\n⊢ 2 • θ ≠ 0 ↔ ¬(θ = 0 ∨ θ = ↑π)",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"congrArg",
"Real.Angle.two_nsmul_eq_zero_iff",
"AddMonoid.toNSMul",
"... | [
"θ : Angle\n⊢ 2 • θ ≠ 0 ↔ ¬2 • θ = 0"
] | ← two_nsmul_eq_zero_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Complex | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 83
} | {
"line": 107,
"column": 0
} | [
{
"pp": "case inr\nx : ℂ\nn : ℤ\ny : ℂ\nhx : x ≠ 0\n⊢ x ^ (↑n * y) = (x ^ y) ^ n",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Complex.log",
"HMul.hMul",
"congrArg",
"NonUnitalCommSemiring.toCommSemigroup",
"DivInvMonoid.toZ... | [] | rw [cpow_def_of_ne_zero hx, cpow_def_of_ne_zero hx, mul_left_comm, exp_int_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Complex | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 83
} | {
"line": 107,
"column": 0
} | [
{
"pp": "case inr\nx : ℂ\nn : ℤ\ny : ℂ\nhx : x ≠ 0\n⊢ x ^ (↑n * y) = (x ^ y) ^ n",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Complex.log",
"HMul.hMul",
"congrArg",
"NonUnitalCommSemiring.toCommSemigroup",
"DivInvMonoid.toZ... | [] | rw [cpow_def_of_ne_zero hx, cpow_def_of_ne_zero hx, mul_left_comm, exp_int_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Complex | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 83
} | {
"line": 107,
"column": 0
} | [
{
"pp": "case inr\nx : ℂ\nn : ℤ\ny : ℂ\nhx : x ≠ 0\n⊢ x ^ (↑n * y) = (x ^ y) ^ n",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Complex.log",
"HMul.hMul",
"congrArg",
"NonUnitalCommSemiring.toCommSemigroup",
"DivInvMonoid.toZ... | [] | rw [cpow_def_of_ne_zero hx, cpow_def_of_ne_zero hx, mul_left_comm, exp_int_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 348,
"column": 35
} | {
"line": 348,
"column": 59
} | {
"line": 348,
"column": 60
} | [
{
"pp": "θ : Angle\n⊢ θ.cos = (↑(π / 2)).cos ↔ θ = ↑(π / 2) ∨ θ = ↑(-π / 2)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"instHDiv",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"congrArg",
"Real... | [
"θ : Angle\n⊢ θ = ↑(π / 2) ∨ θ = -↑(π / 2) ↔ θ = ↑(π / 2) ∨ θ = ↑(-π / 2)"
] | cos_eq_iff_eq_or_eq_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 352,
"column": 2
} | {
"line": 352,
"column": 44
} | {
"line": 353,
"column": 2
} | [
{
"pp": "case h\nθ₂ : Angle\nx✝ : ℝ\n⊢ (↑x✝ + θ₂).sin = (↑x✝).sin * θ₂.cos + (↑x✝).cos * θ₂.sin",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"Real",
"HMul.hMul",
"Real.Angle",
"Real.Angle.coe",
"AddCommGroup.toAddCommMonoid",
"Real.instAdd",
"Real.... | [
"case h.h\nx✝¹ x✝ : ℝ\n⊢ (↑x✝¹ + ↑x✝).sin = (↑x✝¹).sin * (↑x✝).cos + (↑x✝¹).cos * (↑x✝).sin"
] | induction θ₂ using Real.Angle.induction_on | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 356,
"column": 2
} | {
"line": 356,
"column": 44
} | {
"line": 357,
"column": 2
} | [
{
"pp": "θ₁ θ₂ : Angle\n⊢ (θ₁ + θ₂).cos = θ₁.cos * θ₂.cos - θ₁.sin * θ₂.sin",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Real",
"HMul.hMul",
"Real.Angle",
"AddCommGroup.toAddCommMonoid",
"Real.instSub",
"HSub.hSub",
"Real.Angle.induction_on",
... | [
"case h\nθ₁ : Angle\nx✝ : ℝ\n⊢ (θ₁ + ↑x✝).cos = θ₁.cos * (↑x✝).cos - θ₁.sin * (↑x✝).sin"
] | induction θ₂ using Real.Angle.induction_on | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 482,
"column": 48
} | {
"line": 484,
"column": 56
} | {
"line": 486,
"column": 0
} | [
{
"pp": "⊢ toReal 0 = 0",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.partialOrder",
"Real.instLE",
"Real",
"Real.pi",
"Real.Angle",
"Real.Angle.coe... | [] | by
rw [← coe_zero, toReal_coe_eq_self_iff]
exact ⟨Left.neg_neg_iff.2 Real.pi_pos, Real.pi_pos.le⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 564,
"column": 6
} | {
"line": 564,
"column": 18
} | {
"line": 564,
"column": 19
} | [
{
"pp": "n : ℕ\nh : n ≠ 0\nθ : Angle\nh' : 0 < ↑n\n⊢ (n • ↑θ.toReal).toReal = ↑n * θ.toReal ↔ θ.toReal ∈ Set.Ioc (-π / ↑n) (π / ↑n)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Real",
"instHSMul",
"instHDiv",
"Real.pi",
"HM... | [
"n : ℕ\nh : n ≠ 0\nθ : Angle\nh' : 0 < ↑n\n⊢ (↑(n • θ.toReal)).toReal = ↑n * θ.toReal ↔ θ.toReal ∈ Set.Ioc (-π / ↑n) (π / ↑n)"
] | ← coe_nsmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 577,
"column": 2
} | {
"line": 579,
"column": 82
} | {
"line": 581,
"column": 0
} | [
{
"pp": "θ : ℝ\nk : ℤ\n⊢ (↑θ).toReal = θ - 2 * ↑k * π ↔ θ ∈ Set.Ioc ((2 * ↑k - 1) * π) ((2 * ↑k + 1) * π)",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Real.instIsOrd... | [] | rw [← sub_zero (θ : Angle), ← zsmul_zero k, ← coe_two_pi, ← coe_zsmul, ← coe_sub, zsmul_eq_mul, ←
mul_assoc, mul_comm (k : ℝ), toReal_coe_eq_self_iff, Set.mem_Ioc]
exact ⟨fun h => ⟨by linarith, by linarith⟩, fun h => ⟨by linarith, by linarith⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 577,
"column": 2
} | {
"line": 579,
"column": 82
} | {
"line": 581,
"column": 0
} | [
{
"pp": "θ : ℝ\nk : ℤ\n⊢ (↑θ).toReal = θ - 2 * ↑k * π ↔ θ ∈ Set.Ioc ((2 * ↑k - 1) * π) ((2 * ↑k + 1) * π)",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Real.instIsOrd... | [] | rw [← sub_zero (θ : Angle), ← zsmul_zero k, ← coe_two_pi, ← coe_zsmul, ← coe_sub, zsmul_eq_mul, ←
mul_assoc, mul_comm (k : ℝ), toReal_coe_eq_self_iff, Set.mem_Ioc]
exact ⟨fun h => ⟨by linarith, by linarith⟩, fun h => ⟨by linarith, by linarith⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 583,
"column": 2
} | {
"line": 583,
"column": 74
} | {
"line": 585,
"column": 0
} | [
{
"pp": "θ : ℝ\n⊢ (↑θ).toReal = θ - 2 * π ↔ θ ∈ Set.Ioc π (3 * π)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Set.Ioc",
"Mathlib.Meta.NormNum.isNat_add",
"Real",
"Mathlib.Meta.NormNum.instAddMonoidWithOne",
"Real.pi",
... | [] | convert! @toReal_coe_eq_self_sub_two_mul_int_mul_pi_iff θ 1 <;> norm_num | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 583,
"column": 2
} | {
"line": 583,
"column": 74
} | {
"line": 585,
"column": 0
} | [
{
"pp": "θ : ℝ\n⊢ (↑θ).toReal = θ - 2 * π ↔ θ ∈ Set.Ioc π (3 * π)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Set.Ioc",
"Mathlib.Meta.NormNum.isNat_add",
"Real",
"Mathlib.Meta.NormNum.instAddMonoidWithOne",
"Real.pi",
... | [] | convert! @toReal_coe_eq_self_sub_two_mul_int_mul_pi_iff θ 1 <;> norm_num | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 583,
"column": 2
} | {
"line": 583,
"column": 74
} | {
"line": 585,
"column": 0
} | [
{
"pp": "θ : ℝ\n⊢ (↑θ).toReal = θ - 2 * π ↔ θ ∈ Set.Ioc π (3 * π)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Set.Ioc",
"Mathlib.Meta.NormNum.isNat_add",
"Real",
"Mathlib.Meta.NormNum.instAddMonoidWithOne",
"Real.pi",
... | [] | convert! @toReal_coe_eq_self_sub_two_mul_int_mul_pi_iff θ 1 <;> norm_num | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 592,
"column": 6
} | {
"line": 592,
"column": 18
} | {
"line": 592,
"column": 19
} | [
{
"pp": "θ : Angle\n⊢ (2 • ↑θ.toReal).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Real.Angle",
"Real.Angle.coe",
"R... | [
"θ : Angle\n⊢ (↑(2 • θ.toReal)).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal"
] | ← coe_nsmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 433,
"column": 4
} | {
"line": 433,
"column": 46
} | {
"line": 435,
"column": 0
} | [
{
"pp": "case inr.inr\nx : ℂ\nhi : 0 < x.im\n⊢ (-x).arg = x.arg - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"Real.pi",
"Real.instZero",
"congrArg",
"true_or",
"Complex.im",
... | [] | simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 433,
"column": 4
} | {
"line": 433,
"column": 46
} | {
"line": 435,
"column": 0
} | [
{
"pp": "case inr.inr\nx : ℂ\nhi : 0 < x.im\n⊢ (-x).arg = x.arg - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"Real.pi",
"Real.instZero",
"congrArg",
"true_or",
"Complex.im",
... | [] | simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 433,
"column": 4
} | {
"line": 433,
"column": 46
} | {
"line": 435,
"column": 0
} | [
{
"pp": "case inr.inr\nx : ℂ\nhi : 0 < x.im\n⊢ (-x).arg = x.arg - π ↔ 0 < x.im ∨ x.im = 0 ∧ x.re < 0",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"Real.pi",
"Real.instZero",
"congrArg",
"true_or",
"Complex.im",
... | [] | simp [hi, arg_neg_eq_arg_sub_pi_of_im_pos] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 604,
"column": 6
} | {
"line": 604,
"column": 18
} | {
"line": 604,
"column": 19
} | [
{
"pp": "θ : Angle\n⊢ (2 • ↑θ.toReal).toReal = 2 * θ.toReal + 2 * π ↔ θ.toReal ≤ -π / 2",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"instHSMul",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Real.Angle",
"Rea... | [
"θ : Angle\n⊢ (↑(2 • θ.toReal)).toReal = 2 * θ.toReal + 2 * π ↔ θ.toReal ≤ -π / 2"
] | ← coe_nsmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 706,
"column": 29
} | {
"line": 706,
"column": 41
} | {
"line": 706,
"column": 42
} | [
{
"pp": "case h.h\nx✝¹ x✝ : ℝ\nh : 2 • ↑(x✝¹ + x✝) = ↑π\n⊢ (↑x✝).tan = (↑x✝¹).tan⁻¹",
"ppTerm": "?h.h",
"assigned": true,
"usedConstants": [
"Real",
"instHSMul",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"Real.instAddMonoid",
"congrArg",
"AddMonoid.to... | [
"case h.h\nx✝¹ x✝ : ℝ\nh : ↑(2 • (x✝¹ + x✝)) = ↑π\n⊢ (↑x✝).tan = (↑x✝¹).tan⁻¹"
] | ← coe_nsmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 199,
"column": 4
} | {
"line": 199,
"column": 24
} | {
"line": 201,
"column": 0
} | [
{
"pp": "case a\nx : ℝ\nhx : 0 < |x|\n⊢ log |x| * (log |x|)⁻¹ ≤ 1",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Real",
"mul_inv_le_one",
"Real.lattice",
"Real.instZeroLEOneClass",
"abs",
"DivisionSemiring.toGroupWithZero",
"Real.instAddGroup",
... | [] | exact mul_inv_le_one | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 338,
"column": 6
} | {
"line": 338,
"column": 55
} | {
"line": 338,
"column": 56
} | [
{
"pp": "x : ℝ\nhx : 0 < x\ny : ℂ\n⊢ ‖↑x ^ y‖ = x ^ y.re",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real",
"instHDiv",
"HMul.hMul",
"Real.instZero",
"congrArg",
"Complex.im",... | [
"x : ℝ\nhx : 0 < x\ny : ℂ\n⊢ ‖↑x‖ ^ y.re / rexp ((↑x).arg * y.im) = x ^ y.re"
] | norm_cpow_of_ne_zero (ofReal_ne_zero.mpr hx.ne'), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 155,
"column": 55
} | {
"line": 155,
"column": 73
} | {
"line": 157,
"column": 0
} | [
{
"pp": "x : ℝ≥0\n⊢ x ^ (-1) = x⁻¹",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Real",
"congrArg",
"NNReal.instInv",
"NNReal",
"Real.instOne",
"Inv.inv",
"HPow.hPow",
"NNReal.instPowReal",
"NNReal.rpow_neg",
"True",
"Rea... | [] | by simp [rpow_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 939,
"column": 2
} | {
"line": 940,
"column": 48
} | {
"line": 941,
"column": 2
} | [
{
"pp": "p : Nat.Primes\ns : ℂ\nhs : 1 < s.re\n⊢ ↑↑p ^ (-s).re ≤ 1 / 2",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemiring.toAddComm... | [
"p : Nat.Primes\ns : ℂ\nhs : 1 < s.re\n⊢ 2 ^ (-s).re ≤ 1 / 2"
] | refine (Real.rpow_le_rpow_of_nonpos zero_lt_two (Nat.cast_le.mpr p.prop.two_le) <|
by rw [neg_re]; linarith only [hs]).trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 387,
"column": 12
} | {
"line": 387,
"column": 31
} | {
"line": 387,
"column": 31
} | [
{
"pp": "case inr\nx : ℝ≥0\nz : ℝ\nhx : x ≤ 1\nh_one_le : 1 ≤ z\nh : 0 < x\n⊢ x ^ z ≤ x",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"id",
"NNReal",
"LE.le",
"R... | [
"case inr\nx : ℝ≥0\nz : ℝ\nhx : x ≤ 1\nh_one_le : 1 ≤ z\nh : 0 < x\n⊢ x ^ z ≤ x ^ 1"
] | ← NNReal.rpow_one x | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 468,
"column": 6
} | {
"line": 468,
"column": 62
} | {
"line": 470,
"column": 0
} | [
{
"pp": "x✝ : ℝ≥0\nw y✝ z y : ℝ\nhy : 0 < y\nx : ℝ≥0\n⊢ (x ^ (1 / y)) ^ y = x",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real",
"instHDiv",
"GroupWithZero.toDivisionMonoid",
"InvOneClass.toOne",
"HM... | [] | rw [← rpow_mul, one_div_mul_cancel hy.ne.symm, rpow_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 580,
"column": 6
} | {
"line": 581,
"column": 65
} | {
"line": 582,
"column": 4
} | [
{
"pp": "case pos\ny : ℝ\nx : ℝ≥0\nh : x = 0\n⊢ ↑x ^ y = 0 ↔ ↑x = 0 ∧ 0 < y ∨ ↑x = ∞ ∧ y < 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"False",
"ENNReal.zero_rpow_of_pos",
"Real",
"ENNReal.ofNNReal",
"Preorder.toLT",
"_private.Mathlib.Analysis.Spec... | [] | rcases lt_trichotomy y 0 with (H | H | H) <;>
simp [h, H, zero_rpow_of_neg, zero_rpow_of_pos, le_of_lt] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 580,
"column": 6
} | {
"line": 581,
"column": 65
} | {
"line": 582,
"column": 4
} | [
{
"pp": "case pos\ny : ℝ\nx : ℝ≥0\nh : x = 0\n⊢ ↑x ^ y = 0 ↔ ↑x = 0 ∧ 0 < y ∨ ↑x = ∞ ∧ y < 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"False",
"ENNReal.zero_rpow_of_pos",
"Real",
"ENNReal.ofNNReal",
"Preorder.toLT",
"_private.Mathlib.Analysis.Spec... | [] | rcases lt_trichotomy y 0 with (H | H | H) <;>
simp [h, H, zero_rpow_of_neg, zero_rpow_of_pos, le_of_lt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 580,
"column": 6
} | {
"line": 581,
"column": 65
} | {
"line": 582,
"column": 4
} | [
{
"pp": "case pos\ny : ℝ\nx : ℝ≥0\nh : x = 0\n⊢ ↑x ^ y = 0 ↔ ↑x = 0 ∧ 0 < y ∨ ↑x = ∞ ∧ y < 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"False",
"ENNReal.zero_rpow_of_pos",
"Real",
"ENNReal.ofNNReal",
"Preorder.toLT",
"_private.Mathlib.Analysis.Spec... | [] | rcases lt_trichotomy y 0 with (H | H | H) <;>
simp [h, H, zero_rpow_of_neg, zero_rpow_of_pos, le_of_lt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 595,
"column": 6
} | {
"line": 596,
"column": 65
} | {
"line": 597,
"column": 4
} | [
{
"pp": "case pos\ny : ℝ\nx : ℝ≥0\nh : x = 0\n⊢ ↑x ^ y = ∞ ↔ ↑x = 0 ∧ y < 0 ∨ ↑x = ∞ ∧ 0 < y",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.SpecialFunctions.Pow.NNReal.0.ENNReal.rpow_eq_top_iff._simp_1_1",
"False",
"ENNReal.zero_rpow_of_pos",
... | [] | rcases lt_trichotomy y 0 with (H | H | H) <;>
simp [h, H, zero_rpow_of_neg, zero_rpow_of_pos, le_of_lt] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 595,
"column": 6
} | {
"line": 596,
"column": 65
} | {
"line": 597,
"column": 4
} | [
{
"pp": "case pos\ny : ℝ\nx : ℝ≥0\nh : x = 0\n⊢ ↑x ^ y = ∞ ↔ ↑x = 0 ∧ y < 0 ∨ ↑x = ∞ ∧ 0 < y",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.SpecialFunctions.Pow.NNReal.0.ENNReal.rpow_eq_top_iff._simp_1_1",
"False",
"ENNReal.zero_rpow_of_pos",
... | [] | rcases lt_trichotomy y 0 with (H | H | H) <;>
simp [h, H, zero_rpow_of_neg, zero_rpow_of_pos, le_of_lt] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 595,
"column": 6
} | {
"line": 596,
"column": 65
} | {
"line": 597,
"column": 4
} | [
{
"pp": "case pos\ny : ℝ\nx : ℝ≥0\nh : x = 0\n⊢ ↑x ^ y = ∞ ↔ ↑x = 0 ∧ y < 0 ∨ ↑x = ∞ ∧ 0 < y",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.SpecialFunctions.Pow.NNReal.0.ENNReal.rpow_eq_top_iff._simp_1_1",
"False",
"ENNReal.zero_rpow_of_pos",
... | [] | rcases lt_trichotomy y 0 with (H | H | H) <;>
simp [h, H, zero_rpow_of_neg, zero_rpow_of_pos, le_of_lt] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 667,
"column": 56
} | {
"line": 667,
"column": 74
} | {
"line": 669,
"column": 0
} | [
{
"pp": "x : ℝ≥0∞\n⊢ x ^ (-1) = x⁻¹",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Real",
"congrArg",
"ENNReal.instPowReal",
"ENNReal.rpow_one",
"Real.instOne",
"Inv.inv",
"HPow.hPow",
"True",
"Real.instNeg",
"eq_self",
"E... | [] | by simp [rpow_neg] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 791,
"column": 4
} | {
"line": 791,
"column": 60
} | {
"line": 793,
"column": 0
} | [
{
"pp": "y : ℝ\nhy : 0 < y\nx : ℝ≥0∞\n⊢ (x ^ (1 / y)) ^ y = x",
"ppTerm": "?m.32",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"ENNReal.rpow_mul",
"Real",
"instHDiv",
"GroupWithZero.toDivisionMonoid",
"InvOneClass.toOne",... | [] | rw [← rpow_mul, one_div_mul_cancel hy.ne.symm, rpow_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 818,
"column": 16
} | {
"line": 818,
"column": 56
} | {
"line": 818,
"column": 56
} | [
{
"pp": "x y : ℝ≥0∞\nz : ℝ\nhz : 0 < z\n⊢ (x ^ z) ^ z⁻¹ ≤ y ^ z⁻¹ ↔ x ^ z ≤ y",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Real.partialOrder",
"Real",
"DivInvMonoid.toInv",
"Preorder.toLT",
"GroupWith... | [
"x y : ℝ≥0∞\nz : ℝ\nhz : 0 < z\n⊢ x ^ z ≤ y ↔ x ^ z ≤ y"
] | @rpow_le_rpow_iff _ _ z⁻¹ (by simp [hz]) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 866,
"column": 8
} | {
"line": 866,
"column": 22
} | {
"line": 866,
"column": 22
} | [
{
"pp": "case neg\ny z : ℝ\nhyz : z ≤ y\nx : ℝ≥0\nhx1 : ↑x ≤ 1\nh : ¬x = 0\n⊢ ↑x ^ y ≤ ↑x ^ z",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"ENNReal.ofNNReal",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Eq.mp",
"NNReal",
"LE.le",
... | [
"case neg\ny z : ℝ\nhyz : z ≤ y\nx : ℝ≥0\nhx1 : x ≤ 1\nh : ¬x = 0\n⊢ ↑x ^ y ≤ ↑x ^ z"
] | coe_le_one_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 905,
"column": 4
} | {
"line": 907,
"column": 48
} | {
"line": 909,
"column": 0
} | [
{
"pp": "case coe\nz : ℝ\nhz : z < 0\nx✝ : ℝ≥0\nhx : 1 < ↑x✝\n⊢ ↑x✝ ^ z < 1",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"ENNReal.ofNNReal",
"Preorder.toLT",
"congrArg",
"ENNReal.instPowReal",
"N... | [] | simp only [one_lt_coe_iff] at hx
simp [← coe_rpow_of_ne_zero (ne_of_gt (lt_trans zero_lt_one hx)),
NNReal.rpow_lt_one_of_one_lt_of_neg hx hz] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 905,
"column": 4
} | {
"line": 907,
"column": 48
} | {
"line": 909,
"column": 0
} | [
{
"pp": "case coe\nz : ℝ\nhz : z < 0\nx✝ : ℝ≥0\nhx : 1 < ↑x✝\n⊢ ↑x✝ ^ z < 1",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"ENNReal.ofNNReal",
"Preorder.toLT",
"congrArg",
"ENNReal.instPowReal",
"N... | [] | simp only [one_lt_coe_iff] at hx
simp [← coe_rpow_of_ne_zero (ne_of_gt (lt_trans zero_lt_one hx)),
NNReal.rpow_lt_one_of_one_lt_of_neg hx hz] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Ray | {
"line": 477,
"column": 42
} | {
"line": 477,
"column": 51
} | {
"line": 477,
"column": 51
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : M\nc₁ c₂ : R\nh : 0 ≤ c₁ * c₂\nhc₁ : c₁ ≠ 0\nhc₂ : c₂ ≠ 0\nhpos : 0 < c₁ * c₂\nh₁ : 0 < c₁\nh₂ : 0 < c₂\n⊢ c₂ • c₁ • v = c₁ • c₂ • v",
"ppTerm": ... | [] | by module | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Ray | {
"line": 478,
"column": 64
} | {
"line": 478,
"column": 73
} | {
"line": 478,
"column": 73
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nv : M\nc₁ c₂ : R\nh : 0 ≤ c₁ * c₂\nhc₁ : c₁ ≠ 0\nhc₂ : c₂ ≠ 0\nhpos : 0 < c₁ * c₂\nh₁ : c₁ < 0\nh₂ : c₂ < 0\n⊢ -c₂ • c₁ • v = -c₁ • c₂ • v",
"ppTerm"... | [] | by module | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Segment | {
"line": 240,
"column": 13
} | {
"line": 240,
"column": 38
} | {
"line": 240,
"column": 39
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : Ring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddRightMono 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nf : E →ᵃ[𝕜] F\na b : E\nx : F\n⊢ x ∈ ⇑f '' [a -[𝕜] b] ↔ x ∈ [f a -[𝕜] f b]",
"ppTerm": "?m... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : Ring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddRightMono 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nf : E →ᵃ[𝕜] F\na b : E\nx : F\n⊢ x ∈ ⇑f '' ⇑(AffineMap.lineMap a b) '' Icc 0 1 ↔ x ∈ ⇑(AffineMap.lineMap (f ... | segment_eq_image_lineMap, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.Ray | {
"line": 535,
"column": 25
} | {
"line": 535,
"column": 96
} | {
"line": 536,
"column": 2
} | [
{
"pp": "case pos\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\ninst✝ : IsTorsionFree R M\nx y : M\nhx : x = 0\n⊢ SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y]",
"ppTerm": "?pos✝",
... | [] | simpa [hx] using fun h : LinearIndependent R ![0, y] => h.ne_zero 0 rfl | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.Ray | {
"line": 535,
"column": 25
} | {
"line": 535,
"column": 96
} | {
"line": 536,
"column": 2
} | [
{
"pp": "case pos\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\ninst✝ : IsTorsionFree R M\nx y : M\nhx : x = 0\n⊢ SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y]",
"ppTerm": "?pos✝",
... | [] | simpa [hx] using fun h : LinearIndependent R ![0, y] => h.ne_zero 0 rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Ray | {
"line": 535,
"column": 25
} | {
"line": 535,
"column": 96
} | {
"line": 536,
"column": 2
} | [
{
"pp": "case pos\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\ninst✝ : IsTorsionFree R M\nx y : M\nhx : x = 0\n⊢ SameRay R x y ∨ SameRay R x (-y) ↔ ¬LinearIndependent R ![x, y]",
"ppTerm": "?pos✝",
... | [] | simpa [hx] using fun h : LinearIndependent R ![0, y] => h.ne_zero 0 rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Ray | {
"line": 660,
"column": 2
} | {
"line": 660,
"column": 46
} | {
"line": 662,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Field R\ninst✝³ : LinearOrder R\ninst✝² : IsStrictOrderedRing R\nM : Type u_2\ninst✝¹ : AddCommGroup M\ninst✝ : Module R M\nx : M\nhx : x ≠ 0\nr : R\nhr : 0 ≤ r\n⊢ SameRay R x (r • x)",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"IsOrderedModule.toP... | [] | exact SameRay.sameRay_nonneg_smul_right x hr | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 706,
"column": 47
} | {
"line": 713,
"column": 34
} | {
"line": 715,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\ns : Set P\nhs : s.Nonempty\n⊢ affineSpan k s = ⊤ ↔ vectorSpan k s = ⊤",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
... | [] | by
refine ⟨vectorSpan_eq_top_of_affineSpan_eq_top k V P, ?_⟩
intro h
suffices Nonempty (affineSpan k s) by
obtain ⟨p, hp : p ∈ affineSpan k s⟩ := this
rw [eq_iff_direction_eq_of_mem hp (mem_top k V p), direction_affineSpan, h, direction_top]
obtain ⟨x, hx⟩ := hs
exact ⟨⟨x, mem_affineSpan k hx⟩⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Star | {
"line": 450,
"column": 80
} | {
"line": 450,
"column": 95
} | {
"line": 450,
"column": 95
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nx : 𝕜\ns : Set 𝕜\nhx : x ∈ s\n⊢ (∀ ⦃y : 𝕜⦄, y ∈ s → [x -[𝕜] y] ⊆ s) ↔ ∀ ⦃y : 𝕜⦄, y ∈ s → uIcc x y ⊆ s",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"instSMulOfMul",
"segme... | [] | segment_eq_uIcc | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 1105,
"column": 2
} | {
"line": 1112,
"column": 60
} | {
"line": 1114,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\ns : Set V\n⊢ affineSpan k s ≤ ↑(Submodule.span k s)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"instHSMul",
"AddMonoid.toAddSemigroup",
... | [] | intro x hx
simp only [Submodule.mem_toAffineSubspace]
induction hx using affineSpan_induction' with
| mem x hx => exact Submodule.subset_span hx
| smul_vsub_vadd c u _ v _ w _ hu hv hw =>
simp only [vsub_eq_sub, vadd_eq_add]
apply Submodule.add_mem _ _ hw
exact Submodule.smul_mem _ _ (Submodule.sub_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 1105,
"column": 2
} | {
"line": 1112,
"column": 60
} | {
"line": 1114,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\ns : Set V\n⊢ affineSpan k s ≤ ↑(Submodule.span k s)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"instHSMul",
"AddMonoid.toAddSemigroup",
... | [] | intro x hx
simp only [Submodule.mem_toAffineSubspace]
induction hx using affineSpan_induction' with
| mem x hx => exact Submodule.subset_span hx
| smul_vsub_vadd c u _ v _ w _ hu hv hw =>
simp only [vsub_eq_sub, vadd_eq_add]
apply Submodule.add_mem _ _ hw
exact Submodule.smul_mem _ _ (Submodule.sub_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.BalancedCoreHull | {
"line": 83,
"column": 2
} | {
"line": 83,
"column": 23
} | {
"line": 84,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝¹ : SeminormedRing 𝕜\ninst✝ : SMul 𝕜 E\ns : Set E\na : 𝕜\nha : ‖a‖ ≤ 1\n⊢ a • balancedCore 𝕜 s ⊆ balancedCore 𝕜 s",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"instHSMul",
"Membership.mem",
"And.casesOn",
"And",
... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝¹ : SeminormedRing 𝕜\ninst✝ : SMul 𝕜 E\ns : Set E\na : 𝕜\nha : ‖a‖ ≤ 1\ny : E\nhy : y ∈ balancedCore 𝕜 s\n⊢ (fun x ↦ a • x) y ∈ balancedCore 𝕜 s"
] | rintro x ⟨y, hy, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 249,
"column": 31
} | {
"line": 251,
"column": 31
} | {
"line": 253,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\nA : Set E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nhA : Balanced 𝕜 A\nh : 0 ∈ interior A\n⊢ Balanced 𝕜 (interior A)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants"... | [] | by
rw [← insert_eq_self.2 h]
exact hA.zero_insert_interior | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 279,
"column": 2
} | {
"line": 280,
"column": 89
} | {
"line": 281,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\nS : Type u_7\ninst✝¹ : SetLike S E\ninst✝ : SMulMemClass S 𝕜 E\nV : S\nhV : Absorbent 𝕜 ↑V\nx : E\n⊢ x ∈ ↑V",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"abs... | [
"𝕜 : Type u_1\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\nS : Type u_7\ninst✝¹ : SetLike S E\ninst✝ : SMulMemClass S 𝕜 E\nV : S\nhV : Absorbent 𝕜 ↑V\nx : E\nc : 𝕜\nhc : c • x ∈ ↑V\nhc' : c ∈ {0}ᶜ\n⊢ x ∈ ↑V"
] | obtain ⟨c, hc, hc'⟩ :=
((absorbent_iff_eventually_nhdsNE_zero.mp hV x).and eventually_mem_nhdsWithin).exists | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.MetricSpace.Equicontinuity | {
"line": 108,
"column": 17
} | {
"line": 108,
"column": 76
} | {
"line": 108,
"column": 76
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoMetricSpace α\nι : Type u_4\ninst✝ : PseudoMetricSpace β\nb : ℝ → ℝ\nb_lim : Tendsto b (𝓝 0) (𝓝 0)\nF : ι → β → α\nH : ∀ (x y : β) (i : ι), dist (F i x) (F i y) ≤ b (dist x y)\nε : ℝ\nε0 : ε > 0\nδ : ℝ\nδ0 : δ > 0\nhδ : ∀ ⦃x : ℝ⦄, dist x 0 < δ → dist (b x) 0... | [] | by simpa only [Real.dist_eq, tsub_zero, abs_dist] using hxy | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Function | {
"line": 405,
"column": 2
} | {
"line": 405,
"column": 80
} | {
"line": 406,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁷ : Semiring 𝕜\ninst✝⁶ : PartialOrder 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid β\ninst✝³ : PartialOrder β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : LinearOrder E\ns : Set E\nf : E → β\nhs : Convex 𝕜 s\nhf :\n ∀ ⦃x : E⦄,\n x ∈ ... | [
"𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁷ : Semiring 𝕜\ninst✝⁶ : PartialOrder 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommMonoid β\ninst✝³ : PartialOrder β\ninst✝² : Module 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : LinearOrder E\ns : Set E\nf : E → β\nhs : Convex 𝕜 s\nhf :\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E... | refine convexOn_iff_pairwise_pos.2 ⟨hs, fun x hx y hy hxy a b ha hb hab => ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 302,
"column": 2
} | {
"line": 302,
"column": 84
} | {
"line": 304,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\nw : ι → k\np₁ : ι → P\np₂ : P\nh : ∑ i ∈ s, w i = 0\n⊢ ∑ i ∈ s, w i • (p₁ i -ᵥ p₂) = (s.weightedVSub p₁) w",
"ppTerm": "?m.34",
"assigned": tr... | [] | rw [sum_smul_vsub_eq_weightedVSub_sub, s.weightedVSub_apply_const _ _ h, sub_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 302,
"column": 2
} | {
"line": 302,
"column": 84
} | {
"line": 304,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\nw : ι → k\np₁ : ι → P\np₂ : P\nh : ∑ i ∈ s, w i = 0\n⊢ ∑ i ∈ s, w i • (p₁ i -ᵥ p₂) = (s.weightedVSub p₁) w",
"ppTerm": "?m.34",
"assigned": tr... | [] | rw [sum_smul_vsub_eq_weightedVSub_sub, s.weightedVSub_apply_const _ _ h, sub_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 302,
"column": 2
} | {
"line": 302,
"column": 84
} | {
"line": 304,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\nw : ι → k\np₁ : ι → P\np₂ : P\nh : ∑ i ∈ s, w i = 0\n⊢ ∑ i ∈ s, w i • (p₁ i -ᵥ p₂) = (s.weightedVSub p₁) w",
"ppTerm": "?m.34",
"assigned": tr... | [] | rw [sum_smul_vsub_eq_weightedVSub_sub, s.weightedVSub_apply_const _ _ h, sub_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 308,
"column": 75
} | {
"line": 308,
"column": 83
} | {
"line": 308,
"column": 83
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\nw : ι → k\np₂ : ι → P\np₁ : P\nh : ∑ i ∈ s, w i = 0\n⊢ 0 - (s.weightedVSub p₂) w = -(s.weightedVSub p₂) w",
"ppTerm": "?m.60",
"assigned": tru... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\nw : ι → k\np₂ : ι → P\np₁ : P\nh : ∑ i ∈ s, w i = 0\n⊢ -(s.weightedVSub p₂) w = -(s.weightedVSub p₂) w"
] | zero_sub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Centroid | {
"line": 218,
"column": 2
} | {
"line": 218,
"column": 96
} | {
"line": 220,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_4\ns : Finset ι\ninst✝ : CharZero k\np : ι → P\np₀ : P\nhs : s.Nonempty\nh : ∑ i ∈ s, centroidWeights k s i = 1\n⊢ (affineCombination k s p) (centroidWei... | [] | grind [sum_smul_vsub_const_eq_affineCombination_vsub, affineCombination_eq_linear_combination] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 102,
"column": 6
} | {
"line": 102,
"column": 66
} | {
"line": 103,
"column": 6
} | [
{
"pp": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i ∈ s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // ... | [
"case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni1 : ι\nh : AffineIndependent k p\ns : Finset { x // x ≠ i1 }\ng : { x // x ≠ i1 } → k\nhg : ∑ i ∈ s, g i • (p ↑i -ᵥ p i1) = 0\ni : { x // x ≠ i1 }\nhi... | let s2 : Finset ι := insert i1 (s.map (Embedding.subtype _)) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 271,
"column": 2
} | {
"line": 274,
"column": 38
} | {
"line": 276,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ns : Set ι\ni : ι\nhi : i ∈ s\n⊢ vectorSpan k (p '' s) = Submodule.span k ((fun x ↦ x -ᵥ p i) '' p '' (s \\ {i}))",
"ppTerm": "?m.33",
"assig... | [] | conv_lhs =>
rw [vectorSpan_eq_span_vsub_set_right k (Set.mem_image_of_mem p hi), ← Set.insert_eq_of_mem hi,
← Set.insert_sdiff_singleton, Set.image_insert_eq, Set.image_insert_eq]
simp [Submodule.span_insert_eq_span] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 271,
"column": 2
} | {
"line": 274,
"column": 38
} | {
"line": 276,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ns : Set ι\ni : ι\nhi : i ∈ s\n⊢ vectorSpan k (p '' s) = Submodule.span k ((fun x ↦ x -ᵥ p i) '' p '' (s \\ {i}))",
"ppTerm": "?m.33",
"assig... | [] | conv_lhs =>
rw [vectorSpan_eq_span_vsub_set_right k (Set.mem_image_of_mem p hi), ← Set.insert_eq_of_mem hi,
← Set.insert_sdiff_singleton, Set.image_insert_eq, Set.image_insert_eq]
simp [Submodule.span_insert_eq_span] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 338,
"column": 4
} | {
"line": 338,
"column": 15
} | {
"line": 338,
"column": 16
} | [
{
"pp": "case convert_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = ⊤\n⊢ Submodule.span k ((fun x ↦ x -ᵥ p) '' ({p} ∪ (fun v ↦ v +ᵥ p) '' s)) = ⊤",
"ppTerm": "?... | [
"case convert_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set V\np : P\nh : Submodule.span k (range Subtype.val) = ⊤\n⊢ ⊤ ≤ Submodule.span k ((fun x ↦ x -ᵥ p) '' ({p} ∪ (fun v ↦ v +ᵥ p) '' s))"
] | eq_top_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 608,
"column": 4
} | {
"line": 608,
"column": 31
} | {
"line": 609,
"column": 2
} | [
{
"pp": "case mp\nk : Type u_1\nV₁ : Type u_2\nP₁ : Type u_3\nV₂ : Type u_4\nP₂ : Type u_5\ninst✝⁶ : Ring k\ninst✝⁵ : AddCommGroup V₁\ninst✝⁴ : Module k V₁\ninst✝³ : AffineSpace V₁ P₁\ninst✝² : AddCommGroup V₂\ninst✝¹ : Module k V₂\ninst✝ : AffineSpace V₂ P₂\nf : P₁ →ᵃ[k] P₂\np : P₁\ndirection : Submodule k V₁\... | [] | exact ⟨r -ᵥ p, hr, by simp⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Seminorm | {
"line": 91,
"column": 6
} | {
"line": 91,
"column": 24
} | {
"line": 92,
"column": 4
} | [
{
"pp": "case pos\nR : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nι : Type u_11\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nf : E → ℝ\nmap_zero : f 0 = 0\nadd_le : ∀ (x y : E), f (x... | [] | simp [h, map_zero] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Seminorm | {
"line": 91,
"column": 6
} | {
"line": 91,
"column": 24
} | {
"line": 92,
"column": 4
} | [
{
"pp": "case pos\nR : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nι : Type u_11\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nf : E → ℝ\nmap_zero : f 0 = 0\nadd_le : ∀ (x y : E), f (x... | [] | simp [h, map_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Seminorm | {
"line": 91,
"column": 6
} | {
"line": 91,
"column": 24
} | {
"line": 92,
"column": 4
} | [
{
"pp": "case pos\nR : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nι : Type u_11\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nf : E → ℝ\nmap_zero : f 0 = 0\nadd_le : ∀ (x y : E), f (x... | [] | simp [h, map_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Seminorm | {
"line": 358,
"column": 2
} | {
"line": 358,
"column": 6
} | {
"line": 359,
"column": 2
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_7\nι : Type u_11\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : ι → Seminorm 𝕜 E\ns : Finset ι\nC : ℝ≥0\nx : E\n⊢ ↑(s.sup fun i ↦ NNReal.mk (((C • p) i) x) ⋯) = C • ↑(s.sup fun i ↦ NNReal.mk ((p i) x) ⋯)",
"ppTerm": "?m.96",
"assigned":... | [
"𝕜 : Type u_3\nE : Type u_7\nι : Type u_11\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : ι → Seminorm 𝕜 E\ns : Finset ι\nC : ℝ≥0\nx : E\n⊢ C • ↑(s.sup fun i ↦ NNReal.mk ((p i) x) ⋯) = ↑(s.sup fun i ↦ NNReal.mk (((C • p) i) x) ⋯)"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.LinearAlgebra.AffineSpace.Basis | {
"line": 220,
"column": 4
} | {
"line": 220,
"column": 38
} | {
"line": 221,
"column": 2
} | [
{
"pp": "ι : Type u_1\nk : Type u_5\nV : Type u_6\nP : Type u_7\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ninst✝ : Fintype ι\nq : P\n⊢ q ∈ ⊤",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"AffineSubspace.mem_top"
... | [] | exact AffineSubspace.mem_top k V q | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.AffineSpace.Basis | {
"line": 230,
"column": 4
} | {
"line": 230,
"column": 38
} | {
"line": 231,
"column": 2
} | [
{
"pp": "ι : Type u_1\nk : Type u_5\nV : Type u_6\nP : Type u_7\ninst✝⁴ : AddCommGroup V\ninst✝³ : AffineSpace V P\ninst✝² : Ring k\ninst✝¹ : Module k V\nb : AffineBasis ι k P\ninst✝ : Fintype ι\nq : P\n⊢ q ∈ ⊤",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"AffineSubspace.mem_top"
... | [] | exact AffineSubspace.mem_top k V q | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Seminorm | {
"line": 493,
"column": 10
} | {
"line": 500,
"column": 58
} | {
"line": 501,
"column": 8
} | [
{
"pp": "case inr\nR : Type u_1\nR' : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\n𝕜₃ : Type u_5\n𝕝 : Type u_6\nE : Type u_7\nE₂ : Type u_8\nE₃ : Type u_9\nF : Type u_10\nι : Type u_11\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np q✝ : Seminorm 𝕜 E\nx✝ : E\ns : Set (Seminorm 𝕜 E)\nx ... | [] | refine ciSup_le fun i =>
((i : Seminorm 𝕜 E).add_le' x y).trans <| add_le_add
-- Porting note: `f` is provided to force `Subtype.val` to appear.
-- A type ascription on `_` would have also worked, but would have been more verbose.
(le_ciSup (f := fun i => (Subtype.... | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.Seminorm | {
"line": 747,
"column": 43
} | {
"line": 747,
"column": 60
} | {
"line": 747,
"column": 61
} | [
{
"pp": "𝕜 : Type u_3\n𝕜₂ : Type u_4\nE : Type u_7\nE₂ : Type u_8\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np : Seminorm 𝕜₂ E₂\nf : E →ₛₗ[σ₁₂] E₂\nx... | [
"𝕜 : Type u_3\n𝕜₂ : Type u_4\nE : Type u_7\nE₂ : Type u_8\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np : Seminorm 𝕜₂ E₂\nf : E →ₛₗ[σ₁₂] E₂\nx : E\nr : ℝ\... | Set.mem_setOf_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Seminorm | {
"line": 752,
"column": 49
} | {
"line": 752,
"column": 66
} | {
"line": 752,
"column": 67
} | [
{
"pp": "𝕜 : Type u_3\n𝕜₂ : Type u_4\nE : Type u_7\nE₂ : Type u_8\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np : Seminorm 𝕜₂ E₂\nf : E →ₛₗ[σ₁₂] E₂\nx... | [
"𝕜 : Type u_3\n𝕜₂ : Type u_4\nE : Type u_7\nE₂ : Type u_8\ninst✝⁶ : SeminormedRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : SeminormedRing 𝕜₂\ninst✝² : AddCommGroup E₂\ninst✝¹ : Module 𝕜₂ E₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝ : RingHomIsometric σ₁₂\np : Seminorm 𝕜₂ E₂\nf : E →ₛₗ[σ₁₂] E₂\nx : E\nr : ℝ\... | Set.mem_setOf_eq, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 402,
"column": 40
} | {
"line": 402,
"column": 76
} | {
"line": 403,
"column": 4
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nI : Set k\nm n : ℕ\ns : Simplex k P n\ne : Fin (n + 1) ≃ Fin (m + 1)\nw : Fin (m + 1) → k\nhw : ∑ i, w i = 1\nhwI : ∀ (i : Fin (m + 1)), w i ∈ I\nx✝ : (affineCombination k u... | [] | rwa [sum_comp_equiv, map_univ_equiv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 410,
"column": 45
} | {
"line": 410,
"column": 81
} | {
"line": 411,
"column": 4
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nI : Set k\nm n : ℕ\ns : Simplex k P n\ne : Fin (n + 1) ≃ Fin (m + 1)\nw : Fin (n + 1) → k\nhw : ∑ i, w i = 1\nhwI : ∀ (i : Fin (n + 1)), w i ∈ I\nx✝ : (affineCombination k u... | [] | rwa [sum_comp_equiv, map_univ_equiv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.Seminorm | {
"line": 899,
"column": 2
} | {
"line": 899,
"column": 37
} | {
"line": 900,
"column": 2
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nk : 𝕜\nr : ℝ\n⊢ p.ball 0 (‖k‖ * r) ⊆ k • p.ball 0 r",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NormedCommRing.toSeminormedCommRing... | [
"case inl\n𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nr : ℝ\n⊢ p.ball 0 (‖0‖ * r) ⊆ 0 • p.ball 0 r",
"case inr\n𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nk : 𝕜\... | rcases eq_or_ne k 0 with (rfl | hk) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Seminorm | {
"line": 1086,
"column": 72
} | {
"line": 1092,
"column": 65
} | {
"line": 1094,
"column": 0
} | [
{
"pp": "𝕝 : Type u_6\nE : Type u_7\ninst✝⁴ : SeminormedRing 𝕝\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕝 E\ninst✝¹ : UniformSpace E\ninst✝ : IsUniformAddGroup E\np : Seminorm 𝕝 E\nhp : ContinuousAt (⇑p) 0\n⊢ UniformContinuous ⇑p",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"... | [] | by
have hp : Filter.Tendsto p (𝓝 0) (𝓝 0) := map_zero p ▸ hp
rw [UniformContinuous, uniformity_eq_comap_nhds_zero_swapped,
Metric.uniformity_eq_comap_nhds_zero, Filter.tendsto_comap_iff]
exact
tendsto_of_tendsto_of_tendsto_of_le_of_le tendsto_const_nhds (hp.comp Filter.tendsto_comap)
(fun xy => di... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.PathConnected | {
"line": 43,
"column": 6
} | {
"line": 43,
"column": 31
} | {
"line": 43,
"column": 32
} | [
{
"pp": "E : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousSMul ℝ E\na b : E\n⊢ range ⇑(Path.segment a b) = [a -[ℝ] b]",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder... | [
"E : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousSMul ℝ E\na b : E\n⊢ range ⇑(Path.segment a b) = ⇑(lineMap a b) '' Icc 0 1"
] | segment_eq_image_lineMap, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Monoid.FunOnFinite | {
"line": 28,
"column": 2
} | {
"line": 28,
"column": 61
} | {
"line": 30,
"column": 0
} | [
{
"pp": "M : Type u_1\ninst✝⁴ : AddCommMonoid M\ninst✝³ : TopologicalSpace M\ninst✝² : ContinuousAdd M\nX : Type u_2\nY : Type u_3\ninst✝¹ : Finite X\ninst✝ : Finite Y\nf : X → Y\nthis : Fintype X\ny : Y\n⊢ Continuous fun a ↦ ∑ x with f x = y, a x",
"ppTerm": "?m.27",
"assigned": true,
"usedConstant... | [] | exact continuous_finsetSum _ (fun _ _ ↦ continuous_apply _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 751,
"column": 4
} | {
"line": 751,
"column": 51
} | {
"line": 752,
"column": 4
} | [
{
"pp": "case inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nh✝ : LinearIndependent k fun (v : ↑((fun p ↦ p -ᵥ p₁) '' (s \\ {p₁}))) ↦ ↑v\nh : LinearIndepOn k id ((fun p ↦ p -ᵥ p₁) '' (s \\ {p₁}))\n... | [
"case inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nh✝ : LinearIndependent k fun (v : ↑((fun p ↦ p -ᵥ p₁) '' (s \\ {p₁}))) ↦ ↑v\nh : LinearIndepOn k id ((fun p ↦ p -ᵥ p₁) '' (s \\ {p₁}))\nhp₁ : p₁ ∈ s... | have hsv := h.subset_extend (Set.subset_univ _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 215,
"column": 43
} | {
"line": 217,
"column": 62
} | {
"line": 217,
"column": 62
} | [
{
"pp": "ι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Nonempty ι\n⊢ (stdSimplex ℝ ι).Nonempty",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.partialOrder",
"Real",
"Real.instZeroLEOneClass",
"single_mem_stdSimplex... | [] | by
classical
exact ⟨_, single_mem_stdSimplex ℝ (Classical.arbitrary ι)⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Combination | {
"line": 126,
"column": 15
} | {
"line": 126,
"column": 18
} | {
"line": 127,
"column": 2
} | [
{
"pp": "R : Type u_1\nE : Type u_3\nι : Type u_5\ninst✝² : Field R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\nt : Finset ι\nw : ι → R\nz : ι → E\nt' : Finset ι\nht : t ⊆ t'\nh : ∀ i ∈ t', i ∉ t → w i = 0\ni : ι\nhit' : i ∈ t'\n⊢ i ∉ t → (∑ x ∈ t', w x)⁻¹ • w i • z i = 0",
"ppTerm": "?m.87",
"assigne... | [
"R : Type u_1\nE : Type u_3\nι : Type u_5\ninst✝² : Field R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\nt : Finset ι\nw : ι → R\nz : ι → E\nt' : Finset ι\nht : t ⊆ t'\nh : ∀ i ∈ t', i ∉ t → w i = 0\ni : ι\nhit' : i ∈ t'\nhit : i ∉ t\n⊢ (∑ x ∈ t', w x)⁻¹ • w i • z i = 0"
] | hit | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.NhdsKer | {
"line": 58,
"column": 55
} | {
"line": 59,
"column": 28
} | {
"line": 61,
"column": 0
} | [
{
"pp": "X : Type u_2\ninst✝ : TopologicalSpace X\nι : Type u_3\ns : Set ι\nt : ι → Set X\n⊢ nhdsKer (⋃ i ∈ s, t i) = ⋃ i ∈ s, nhdsKer (t i)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"congrArg",
"Membership.mem",
"funext",
"True",
"eq_self",
"nhdsK... | [] | by
simp only [nhdsKer_iUnion] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Combination | {
"line": 317,
"column": 27
} | {
"line": 317,
"column": 58
} | {
"line": 318,
"column": 8
} | [
{
"pp": "R : Type u_1\nE : Type u_3\nι : Type u_5\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nv : ι → E\ns : Finset ι\nw : ι → R\nhw₀ : ∀ i ∈ s, 0 ≤ w i\nhw₁ : s.sum w = 1\ns' : Finset ι\nw' : ι → R\nhw₀' : ∀ i ∈ s', 0 ≤ w' i\nhw₁' : s'... | [
"R : Type u_1\nE : Type u_3\nι : Type u_5\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nv : ι → E\ns : Finset ι\nw : ι → R\nhw₀ : ∀ i ∈ s, 0 ≤ w i\nhw₁ : s.sum w = 1\ns' : Finset ι\nw' : ι → R\nhw₀' : ∀ i ∈ s', 0 ≤ w' i\nhw₁' : s'.sum w' = 1\... | ← sum_subset subset_union_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Connected.LocallyPathConnected | {
"line": 309,
"column": 2
} | {
"line": 309,
"column": 6
} | {
"line": 310,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\nx✝ y✝ z : X\nι : Type u_3\nF : Set X\ninst✝¹ : LocallyPathConnectedSpace X\ninst✝ : AlexandrovDiscrete X\nx y : X\nhy : y ⤳ x\n⊢ JoinedIn (nhdsKer {x}) x y",
"ppTerm": "?m.27",
"assigned": true,
"usedConst... | [
"X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\nx✝ y✝ z : X\nι : Type u_3\nF : Set X\ninst✝¹ : LocallyPathConnectedSpace X\ninst✝ : AlexandrovDiscrete X\nx y : X\nhy : y ⤳ x\n⊢ JoinedIn (nhdsKer {x}) y x"
] | symm | Lean.Elab.Tactic.evalSymm | Lean.Parser.Tactic.symm |
Mathlib.Analysis.Convex.Combination | {
"line": 328,
"column": 10
} | {
"line": 328,
"column": 41
} | {
"line": 328,
"column": 42
} | [
{
"pp": "case refine_2.refine_2\nR : Type u_1\nE : Type u_3\nι : Type u_5\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nv : ι → E\ns : Finset ι\nw : ι → R\nhw₀ : ∀ i ∈ s, 0 ≤ w i\nhw₁ : s.sum w = 1\ns' : Finset ι\nw' : ι → R\nhw₀' : ∀ i ∈... | [
"case refine_2.refine_2\nR : Type u_1\nE : Type u_3\nι : Type u_5\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nv : ι → E\ns : Finset ι\nw : ι → R\nhw₀ : ∀ i ∈ s, 0 ≤ w i\nhw₁ : s.sum w = 1\ns' : Finset ι\nw' : ι → R\nhw₀' : ∀ i ∈ s', 0 ≤ w' ... | ← sum_subset subset_union_left, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.Convex | {
"line": 133,
"column": 8
} | {
"line": 133,
"column": 43
} | {
"line": 133,
"column": 43
} | [
{
"pp": "case inl\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : Set E\nhs : Convex ℝ s\nδ : ℝ\nhδ : 0 ≤ δ\n⊢ Convex ℝ (Metric.cthickening δ s)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"Real",
"con... | [
"case inl\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\ns : Set E\nhs : Convex ℝ s\nδ : ℝ\nhδ : 0 ≤ δ\n⊢ Convex ℝ (⋂ ε, ⋂ (_ : δ < ε), Metric.thickening ε s)"
] | cthickening_eq_iInter_thickening hδ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.Convex | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 45
} | {
"line": 177,
"column": 0
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx : E\nhx : ¬x = 0\n⊢ IsConnected {y | ∃ r, 0 ≤ r ∧ r • x = y}",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real",
"instHSMul",
"NormedSpace.toIsBou... | [] | exact isConnected_Ici.image _ (by fun_prop) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 109,
"column": 4
} | {
"line": 109,
"column": 54
} | {
"line": 110,
"column": 4
} | [
{
"pp": "R : Type u_1\nE : Type u_6\nι : Type u_9\ninst✝² : SeminormedRing R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\np : SeminormFamily R E ι\nU V : Set E\nhU✝ : U ∈ p.basisSets\nhV : V ∈ p.basisSets\ns : Finset ι\nr₁ : ℝ\nhr₁ : 0 < r₁\nhU : U = (s.sup p).ball 0 r₁\n⊢ ∃ z ∈ p.basisSets, z ⊆ U ∩ V",
"p... | [
"R : Type u_1\nE : Type u_6\nι : Type u_9\ninst✝² : SeminormedRing R\ninst✝¹ : AddCommGroup E\ninst✝ : Module R E\np : SeminormFamily R E ι\nU V : Set E\nhU✝ : U ∈ p.basisSets\nhV✝ : V ∈ p.basisSets\ns : Finset ι\nr₁ : ℝ\nhr₁ : 0 < r₁\nhU : U = (s.sup p).ball 0 r₁\nt : Finset ι\nr₂ : ℝ\nhr₂ : 0 < r₂\nhV : V = (t.su... | rcases p.basisSets_iff.mp hV with ⟨t, r₂, hr₂, hV⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
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