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.Analysis.Complex.Trigonometric | {
"line": 220,
"column": 6
} | {
"line": 220,
"column": 40
} | {
"line": 220,
"column": 41
} | [
{
"pp": "x : ℂ\n⊢ cosh x + sinh x = cexp x",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Complex.sinh",
"HMul.hMul",
"Field.isDomain",
"CharZero.NeZero.two",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.toAddMonoidW... | [
"x : ℂ\n⊢ 2 * (cosh x + sinh x) = 2 * cexp x"
] | ← mul_right_inj' (two_ne_zero' ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 220,
"column": 41
} | {
"line": 220,
"column": 49
} | {
"line": 220,
"column": 50
} | [
{
"pp": "x : ℂ\n⊢ 2 * (cosh x + sinh x) = 2 * cexp x",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Complex.sinh",
"HMul.hMul",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"Complex.instMul",
"id",
... | [
"x : ℂ\n⊢ 2 * cosh x + 2 * sinh x = 2 * cexp x"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 235,
"column": 6
} | {
"line": 235,
"column": 40
} | {
"line": 235,
"column": 41
} | [
{
"pp": "x : ℂ\n⊢ cosh x - sinh x = cexp (-x)",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Complex.sinh",
"HMul.hMul",
"Field.isDomain",
"CharZero.NeZero.two",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.toAddMono... | [
"x : ℂ\n⊢ 2 * (cosh x - sinh x) = 2 * cexp (-x)"
] | ← mul_right_inj' (two_ne_zero' ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Exponential | {
"line": 364,
"column": 82
} | {
"line": 372,
"column": 78
} | {
"line": 373,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝² : Field α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\nn j : ℕ\nhn : 0 < n\n⊢ (↑n.factorial)⁻¹ * ∑ m ∈ range (j - n), (↑n.succ)⁻¹ ^ m =\n (↑n.succ - ↑n.succ * (↑n.succ)⁻¹ ^ (j - n)) / (↑n.factorial * ↑n)",
"ppTerm": "?m.154",
"assigned": true,
"usedConstan... | [] | by
have h₁ : (n.succ : α) ≠ 1 :=
@Nat.cast_one α _ ▸ mt Nat.cast_inj.1 (mt Nat.succ.inj (pos_iff_ne_zero.1 hn))
have h₂ : (n.succ : α) ≠ 0 := by positivity
have h₃ : (n.factorial * n : α) ≠ 0 := by positivity
have h₄ : (n.succ - 1 : α) = n := by simp
rw [geom_sum_inv h₁ h₂, eq_div_... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 288,
"column": 6
} | {
"line": 288,
"column": 40
} | {
"line": 288,
"column": 41
} | [
{
"pp": "x : ℂ\n⊢ sinh (x * I) = sin x * I",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Complex.sinh",
"HMul.hMul",
"Field.isDomain",
"CharZero.NeZero.two",
"congrArg",
"Complex.sin",
"Nat.instAtLeastTwoHAddOfNat",
"AddGro... | [
"x : ℂ\n⊢ 2 * sinh (x * I) = 2 * (sin x * I)"
] | ← mul_right_inj' (two_ne_zero' ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 292,
"column": 6
} | {
"line": 292,
"column": 40
} | {
"line": 292,
"column": 41
} | [
{
"pp": "x : ℂ\n⊢ cosh (x * I) = cos x",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Field.isDomain",
"CharZero.NeZero.two",
"Complex.cos",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne.toAddMonoidWithOn... | [
"x : ℂ\n⊢ 2 * cosh (x * I) = 2 * cos x"
] | ← mul_right_inj' (two_ne_zero' ℂ), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 297,
"column": 51
} | {
"line": 297,
"column": 63
} | {
"line": 297,
"column": 63
} | [
{
"pp": "x : ℂ\n⊢ cos (x * I) = cosh x",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"Complex.cos",
"congrArg",
"Complex.instMul",
"id",
"Complex",
"Eq.symm",
"Eq",
"Complex.cosh_mul_I",
"Complex.cosh... | [
"x : ℂ\n⊢ cosh (x * I * I) = cosh x"
] | ← cosh_mul_I | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 904,
"column": 2
} | {
"line": 904,
"column": 36
} | {
"line": 905,
"column": 2
} | [
{
"pp": "ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\nl : Filter ι\nε : ι → 𝕜\nf : ι → E\nhε : Tendsto ε l (𝓝 0)\nhf : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l (norm ∘ f)\n⊢ Tendsto (ε • f) l (𝓝 0... | [
"ι : Type u_1\n𝕜 : Type u_2\nE : Type u_3\ninst✝³ : NormedDivisionRing 𝕜\ninst✝² : SeminormedAddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : IsBoundedSMul 𝕜 E\nl : Filter ι\nε : ι → 𝕜\nf : ι → E\nhε : ε =o[l] fun _x ↦ 1\nhf : IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l (norm ∘ f)\n⊢ (ε • f) =o[l] fun _x ↦ 1"
] | rw [← isLittleO_one_iff 𝕜] at hε ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Ring.InfiniteSum | {
"line": 37,
"column": 4
} | {
"line": 37,
"column": 69
} | {
"line": 37,
"column": 69
} | [
{
"pp": "ι : Type u_2\nι' : Type u_3\nf : ι → ℝ\ng : ι' → ℝ\nhf : Summable f\nhg : Summable g\nhf' : 0 ≤ f\nhg' : 0 ≤ g\n⊢ Summable fun x ↦ ∑' (y : ι'), f (x, y).1 * g (x, y).2",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"NonUnitalCommRing.toNonUnita... | [] | by simpa only [hg.tsum_mul_left _] using hf.mul_right (∑' x, g x) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 620,
"column": 25
} | {
"line": 620,
"column": 43
} | {
"line": 622,
"column": 0
} | [
{
"pp": "x y : ℝ\n⊢ ↑(cos x + cos y) = ↑(2 * cos ((x + y) / 2) * cos ((x - y) / 2))",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"HMul.hMul",
"Complex.cos",
"Real.cos",
"congrArg",
"Real.instDivInvMonoid",
"Real.instSub... | [] | simp [cos_add_cos] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 620,
"column": 25
} | {
"line": 620,
"column": 43
} | {
"line": 622,
"column": 0
} | [
{
"pp": "x y : ℝ\n⊢ ↑(cos x + cos y) = ↑(2 * cos ((x + y) / 2) * cos ((x - y) / 2))",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"HMul.hMul",
"Complex.cos",
"Real.cos",
"congrArg",
"Real.instDivInvMonoid",
"Real.instSub... | [] | simp [cos_add_cos] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 620,
"column": 25
} | {
"line": 620,
"column": 43
} | {
"line": 622,
"column": 0
} | [
{
"pp": "x y : ℝ\n⊢ ↑(cos x + cos y) = ↑(2 * cos ((x + y) / 2) * cos ((x - y) / 2))",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"HMul.hMul",
"Complex.cos",
"Real.cos",
"congrArg",
"Real.instDivInvMonoid",
"Real.instSub... | [] | simp [cos_add_cos] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 252,
"column": 2
} | {
"line": 255,
"column": 27
} | {
"line": 256,
"column": 2
} | [
{
"pp": "n : ℕ\nC : ℝ\nhC₁ : 1 ≤ C\nhC₀ : 0 < C\nthis : 0 < (rexp 1 * C)⁻¹\n⊢ ∃ ia, True ∧ ∀ x ∈ Set.Ioi ia, rexp x / x ^ n ∈ Set.Ici C",
"ppTerm": "?m.87",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"gt_mem_nhds",
"Real.partialOrder",
"Real",
"Set.Ioi",
"Pre... | [
"n : ℕ\nC : ℝ\nhC₁ : 1 ≤ C\nhC₀ : 0 < C\nthis : 0 < (rexp 1 * C)⁻¹\nN : ℕ\nhN : ∀ k ≥ N, ↑k ^ n / rexp 1 ^ k < (rexp 1 * C)⁻¹\n⊢ ∃ ia, True ∧ ∀ x ∈ Set.Ioi ia, rexp x / x ^ n ∈ Set.Ici C"
] | obtain ⟨N, hN⟩ : ∃ N : ℕ, ∀ k ≥ N, (↑k : ℝ) ^ n / exp 1 ^ k < (exp 1 * C)⁻¹ :=
eventually_atTop.1
((tendsto_pow_const_div_const_pow_of_one_lt n (one_lt_exp_iff.2 zero_lt_one)).eventually
(gt_mem_nhds this)) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 69
} | {
"line": 170,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝³ : Field α\ninst✝² : LinearOrder α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : Archimedean α\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ n ≥ m, |f n| ≤ a\nhnm : ∀ n ≥ m, f n.succ ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ n ≥ m, a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ n ≥ m... | [
"α : Type u_1\ninst✝³ : Field α\ninst✝² : LinearOrder α\ninst✝¹ : IsStrictOrderedRing α\ninst✝ : Archimedean α\nf : ℕ → α\na : α\nm : ℕ\nham : ∀ n ≥ m, |f n| ≤ a\nhnm : ∀ n ≥ m, f n.succ ≤ f n\nε : α\nε0 : ε > 0\nk : ℕ\nhk : a ≤ k • ε\nh : ∃ l, ∀ n ≥ m, a - l • ε < f n\nl : ℕ := Nat.find h\nhl : ∀ n ≥ m, f n > a - ... | rw [abs_of_nonpos (sub_nonpos.2 hfij), neg_sub, sub_lt_iff_lt_add'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 217,
"column": 31
} | {
"line": 219,
"column": 10
} | {
"line": 221,
"column": 0
} | [
{
"pp": "⊢ cos π = -1",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.partialOrder",
"Real",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Mathlib.Meta.NormNum.isInt_eq_true",
"CharZero.NeZero.two",
"Mu... | [] | by
rw [← mul_div_cancel_left₀ π two_ne_zero, mul_div_assoc, cos_two_mul, cos_pi_div_two]
norm_num | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 416,
"column": 16
} | {
"line": 416,
"column": 40
} | {
"line": 417,
"column": 6
} | [
{
"pp": "case h₁\nR : Type u_4\ninst✝ : NormedRing R\nk : ℕ\nr : R\nhr : ‖r‖ < 1\nu : ℕ → ℕ\nhu : (fun n ↦ ↑(u n)) =O[atTop] fun n ↦ ↑(n ^ k)\nr' : ℝ\nhrr' : ‖‖r‖‖ < r'\nh : r' < 1\nn : ℕ\nhn : ‖r ^ n‖ ≤ ‖r‖ ^ n\n⊢ ‖↑(u n)‖ ≤ ↑(u n) * ‖1‖",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
... | [] | exact norm_cast_le (u n) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 519,
"column": 65
} | {
"line": 531,
"column": 45
} | {
"line": 533,
"column": 0
} | [
{
"pp": "R : Type u_4\ninst✝¹ : NormedRing R\ninst✝ : HasSummableGeomSeries R\nx : R\nh : ‖x‖ < 1\n⊢ HasSum (fun n ↦ ↑n * x ^ n) (x * (1 - x)⁻¹ʳ ^ 2)",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"add_mul",
"Eq.mpr",
"MulOne.toOne",
"Semigroup.toMul",
"Trans... | [] | by
have A : HasSum (fun (n : ℕ) ↦ (n + 1) * x ^ n) ((1 - x)⁻¹ʳ ^ 2) := by
convert! hasSum_choose_mul_geometric_of_norm_lt_one' 1 h with n
simp
have B : HasSum (fun (n : ℕ) ↦ x ^ n) ((1 - x)⁻¹ʳ) := hasSum_geom_series_inverse x h
convert! A.sub B using 1
· ext n
simp [add_mul]
· symm
calc (1 - x... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.Ball.Pointwise | {
"line": 85,
"column": 2
} | {
"line": 85,
"column": 84
} | {
"line": 87,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nc : 𝕜\nhc : c ≠ 0\nx : E\nr : ℝ\ny : E\n⊢ c⁻¹ • y ∈ ball (c⁻¹ • c • x) r ↔ y ∈ ball (c • x) (‖c‖ * r)",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Iff.mpr"... | [] | simp [← div_eq_inv_mul, div_lt_iff₀ (norm_pos_iff.2 hc), mul_comm _ r, dist_smul₀] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 660,
"column": 2
} | {
"line": 673,
"column": 9
} | {
"line": 675,
"column": 0
} | [
{
"pp": "α : Type u_4\ninst✝ : SeminormedAddCommGroup α\nf : ℕ → α\nr : ℝ\nhr : 1 < r\nhf : ∃ᶠ (n : ℕ) in atTop, ‖f n‖ ≠ 0\nh : ∀ᶠ (n : ℕ) in atTop, r * ‖f n‖ ≤ ‖f (n + 1)‖\n⊢ ¬Summable f",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real.inst... | [] | rw [eventually_atTop] at h
rcases h with ⟨N₀, hN₀⟩
rw [frequently_atTop] at hf
rcases hf N₀ with ⟨N, hNN₀ : N₀ ≤ N, hN⟩
rw [← @summable_nat_add_iff α _ _ _ _ N]
refine mt Summable.tendsto_atTop_zero
fun h' ↦ not_tendsto_atTop_of_tendsto_nhds (tendsto_norm_zero.comp h') ?_
convert! tendsto_atTop_of_geom_... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 660,
"column": 2
} | {
"line": 673,
"column": 9
} | {
"line": 675,
"column": 0
} | [
{
"pp": "α : Type u_4\ninst✝ : SeminormedAddCommGroup α\nf : ℕ → α\nr : ℝ\nhr : 1 < r\nhf : ∃ᶠ (n : ℕ) in atTop, ‖f n‖ ≠ 0\nh : ∀ᶠ (n : ℕ) in atTop, r * ‖f n‖ ≤ ‖f (n + 1)‖\n⊢ ¬Summable f",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Real.inst... | [] | rw [eventually_atTop] at h
rcases h with ⟨N₀, hN₀⟩
rw [frequently_atTop] at hf
rcases hf N₀ with ⟨N, hNN₀ : N₀ ≤ N, hN⟩
rw [← @summable_nat_add_iff α _ _ _ _ N]
refine mt Summable.tendsto_atTop_zero
fun h' ↦ not_tendsto_atTop_of_tendsto_nhds (tendsto_norm_zero.comp h') ?_
convert! tendsto_atTop_of_geom_... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Connected.PathConnected | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 29
} | {
"line": 254,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y : X\nF : Set X\nf : X → Y\nhf : IsInducing f\nhx : x ∈ F\nhy : y ∈ F\nγ : Path (f x) (f y)\nhγ : ∀ (t : ↑I), γ t ∈ f '' F\n⊢ JoinedIn F x y",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
... | [
"X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nx y : X\nF : Set X\nf : X → Y\nhf : IsInducing f\nhx : x ∈ F\nhy : y ∈ F\nγ : Path (f x) (f y)\nγ' : ↑I → X\nhγ'F : ∀ (t : ↑I), γ' t ∈ F\nhγ' : ∀ (t : ↑I), f (γ' t) = γ t\n⊢ JoinedIn F x y"
] | choose γ' hγ'F hγ' using hγ | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.Topology.Instances.Sign | {
"line": 50,
"column": 2
} | {
"line": 50,
"column": 40
} | {
"line": 51,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝³ : Zero α\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\nh : a ≠ 0\nh_neg : a < 0\n⊢ ContinuousAt (⇑SignType.sign) a",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"SemilatticeInf.toPartialOrder",
"Di... | [
"case inr\nα : Type u_1\ninst✝³ : Zero α\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : OrderTopology α\na : α\nh : a ≠ 0\nh_pos : 0 < a\n⊢ ContinuousAt (⇑SignType.sign) a"
] | · exact continuousAt_sign_of_neg h_neg | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 293,
"column": 55
} | {
"line": 293,
"column": 69
} | {
"line": 293,
"column": 70
} | [
{
"pp": "case succ.inr\nx : ℝ\nn : ℕ\nih : log (x ^ n) = ↑n * log x\nhx : x ≠ 0\n⊢ ↑n * log x + log x = ↑(n + 1) * log x",
"ppTerm": "?succ.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.cast_succ",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"cong... | [
"case succ.inr\nx : ℝ\nn : ℕ\nih : log (x ^ n) = ↑n * log x\nhx : x ≠ 0\n⊢ ↑n * log x + log x = (↑n + 1) * log x"
] | Nat.cast_succ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 298,
"column": 2
} | {
"line": 298,
"column": 70
} | {
"line": 299,
"column": 2
} | [
{
"pp": "case ofNat\nx : ℝ\na✝ : ℕ\n⊢ log (x ^ Int.ofNat a✝) = ↑(Int.ofNat a✝) * log x",
"ppTerm": "?ofNat",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"Real",
"HMul.hMul",
"congrArg",
"Real.instDivIn... | [
"case negSucc\nx : ℝ\na✝ : ℕ\n⊢ log (x ^ Int.negSucc a✝) = ↑(Int.negSucc a✝) * log x"
] | · rw [Int.ofNat_eq_natCast, zpow_natCast, log_pow, Int.cast_natCast] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Complex.Log | {
"line": 43,
"column": 40
} | {
"line": 43,
"column": 48
} | {
"line": 43,
"column": 49
} | [
{
"pp": "x : ℂ\nhx : x ≠ 0\n⊢ ↑‖x‖ * (↑(x.re / ‖x‖) + ↑(x.im / ‖x‖) * I) = x",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Norm.norm",
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"Complex.instNormedAddCommGroup",
... | [
"x : ℂ\nhx : x ≠ 0\n⊢ ↑‖x‖ * ↑(x.re / ‖x‖) + ↑‖x‖ * (↑(x.im / ‖x‖) * I) = x"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Complex | {
"line": 57,
"column": 62
} | {
"line": 57,
"column": 80
} | {
"line": 59,
"column": 0
} | [
{
"pp": "x : ℂ\nh : x ≠ 0\n⊢ 0 ^ x = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Complex.log",
"HMul.hMul",
"eq_false",
"congrArg",
"Complex.instZero",
"Complex.instPow",
"Complex.instMul",
"ite_cond_eq_true",
"HPow.hPow",
... | [] | simp [cpow_def, *] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Pow.Complex | {
"line": 57,
"column": 62
} | {
"line": 57,
"column": 80
} | {
"line": 59,
"column": 0
} | [
{
"pp": "x : ℂ\nh : x ≠ 0\n⊢ 0 ^ x = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Complex.log",
"HMul.hMul",
"eq_false",
"congrArg",
"Complex.instZero",
"Complex.instPow",
"Complex.instMul",
"ite_cond_eq_true",
"HPow.hPow",
... | [] | simp [cpow_def, *] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Complex | {
"line": 57,
"column": 62
} | {
"line": 57,
"column": 80
} | {
"line": 59,
"column": 0
} | [
{
"pp": "x : ℂ\nh : x ≠ 0\n⊢ 0 ^ x = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Complex.log",
"HMul.hMul",
"eq_false",
"congrArg",
"Complex.instZero",
"Complex.instPow",
"Complex.instMul",
"ite_cond_eq_true",
"HPow.hPow",
... | [] | simp [cpow_def, *] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Complex | {
"line": 193,
"column": 21
} | {
"line": 193,
"column": 30
} | {
"line": 193,
"column": 31
} | [
{
"pp": "case inr.inr.inl\na : ℝ\nha : 0 ≤ a\nr : ℂ\nhr : r ≠ 0\nha' : 0 < a\nhb : 0 ≤ 0\n⊢ (↑a * 0) ^ r = ↑a ^ r * 0 ^ r",
"ppTerm": "?inr.inr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"MulZeroClass.toMul",
"congrArg",
"Complex.instZero",
... | [
"case inr.inr.inl\na : ℝ\nha : 0 ≤ a\nr : ℂ\nhr : r ≠ 0\nha' : 0 < a\nhb : 0 ≤ 0\n⊢ 0 ^ r = ↑a ^ r * 0 ^ r"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 274,
"column": 51
} | {
"line": 274,
"column": 56
} | {
"line": 274,
"column": 57
} | [
{
"pp": "case neg.mpr\ny : ℝ\nh₀ : ¬{ re := 0, im := y } = 0\nhy : 0 < y\n⊢ { re := 0, im := y }.arg = { re := y * I.re, im := y * I.im }.arg",
"ppTerm": "?neg.mpr✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.instZero",
"congrArg",
... | [
"case neg.mpr\ny : ℝ\nh₀ : ¬{ re := 0, im := y } = 0\nhy : 0 < y\n⊢ { re := 0, im := y }.arg = { re := y * 0, im := y * I.im }.arg"
] | I_re, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 274,
"column": 63
} | {
"line": 274,
"column": 72
} | {
"line": 274,
"column": 73
} | [
{
"pp": "case neg.mpr\ny : ℝ\nh₀ : ¬{ re := 0, im := y } = 0\nhy : 0 < y\n⊢ { re := 0, im := y }.arg = { re := y * 0, im := y * 1 }.arg",
"ppTerm": "?neg.mpr✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"MulZeroClass.toMul",
"Real.instZero",
... | [
"case neg.mpr\ny : ℝ\nh₀ : ¬{ re := 0, im := y } = 0\nhy : 0 < y\n⊢ { re := 0, im := y }.arg = { re := 0, im := y * 1 }.arg"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 212,
"column": 77
} | {
"line": 212,
"column": 86
} | {
"line": 213,
"column": 6
} | [
{
"pp": "case mpr.inr\nθ ψ : ℝ\nk : ℤ\nH : θ - -ψ = 2 * π * ↑k\n⊢ -2 * 0 * sin ((θ - ψ) / 2) = 0",
"ppTerm": "?mpr.inr",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"MulZeroClass.toMul",
"Re... | [
"case mpr.inr\nθ ψ : ℝ\nk : ℤ\nH : θ - -ψ = 2 * π * ↑k\n⊢ 0 * sin ((θ - ψ) / 2) = 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 231,
"column": 38
} | {
"line": 231,
"column": 47
} | {
"line": 231,
"column": 48
} | [
{
"pp": "case mpr.inl\nθ ψ : ℝ\nk : ℤ\nH : θ - ψ = 2 * π * ↑k\n⊢ 2 * 0 * cos ((θ + ψ) / 2) = 0",
"ppTerm": "?mpr.inl",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"MulZeroClass.toMul",
"Real... | [
"case mpr.inl\nθ ψ : ℝ\nk : ℤ\nH : θ - ψ = 2 * π * ↑k\n⊢ 0 * cos ((θ + ψ) / 2) = 0"
] | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 286,
"column": 2
} | {
"line": 287,
"column": 38
} | {
"line": 289,
"column": 0
} | [
{
"pp": "θ : Angle\nψ : ℝ\n⊢ θ.sin = Real.sin ψ ↔ θ = ↑ψ ∨ θ + ↑ψ = ↑π",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Real",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"AddCommGroup.toAddCommMonoid",
"Real.Angle.sin_eq_iff_coe_eq_or_add_eq_pi",
"R... | [] | induction θ using Real.Angle.induction_on
exact sin_eq_iff_coe_eq_or_add_eq_pi | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 286,
"column": 2
} | {
"line": 287,
"column": 38
} | {
"line": 289,
"column": 0
} | [
{
"pp": "θ : Angle\nψ : ℝ\n⊢ θ.sin = Real.sin ψ ↔ θ = ↑ψ ∨ θ + ↑ψ = ↑π",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Real",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"AddCommGroup.toAddCommMonoid",
"Real.Angle.sin_eq_iff_coe_eq_or_add_eq_pi",
"R... | [] | induction θ using Real.Angle.induction_on
exact sin_eq_iff_coe_eq_or_add_eq_pi | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 102,
"column": 26
} | {
"line": 102,
"column": 34
} | {
"line": 102,
"column": 35
} | [
{
"pp": "x : ℝ\nhx : x < 0\ny : ℝ\nthis : Complex.log ↑x * ↑y = ↑(log (-x) * y) + ↑(y * π) * Complex.I\n⊢ (↑(rexp (log (-x) * y)) * (↑(cos (y * π)) + ↑(sin (y * π)) * Complex.I)).re = rexp (log x * y) * cos (y * π)",
"ppTerm": "?m.76",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribC... | [
"x : ℝ\nhx : x < 0\ny : ℝ\nthis : Complex.log ↑x * ↑y = ↑(log (-x) * y) + ↑(y * π) * Complex.I\n⊢ (↑(rexp (log (-x) * y)) * ↑(cos (y * π)) + ↑(rexp (log (-x) * y)) * (↑(sin (y * π)) * Complex.I)).re =\n rexp (log x * y) * cos (y * π)"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 97,
"column": 2
} | {
"line": 105,
"column": 8
} | {
"line": 106,
"column": 2
} | [
{
"pp": "x : ℝ\nhx : x < 0\ny : ℝ\n⊢ (Complex.exp (Complex.log ↑x * ↑y)).re = rexp (log x * y) * cos (y * π)",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
"Distrib.leftDistribClass",
"Norm.n... | [
"case hnc\nx : ℝ\nhx : x < 0\ny : ℝ\n⊢ ¬↑x = 0"
] | · have : Complex.log x * y = ↑(log (-x) * y) + ↑(y * π) * Complex.I := by
simp only [Complex.log, Complex.norm_real, norm_eq_abs, abs_of_neg hx, log_neg_eq_log,
Complex.arg_ofReal_of_neg hx, Complex.ofReal_mul]
ring
rw [this, Complex.exp_add_mul_I, ← Complex.ofReal_exp, ← Complex.ofReal_cos, ←
... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 458,
"column": 8
} | {
"line": 458,
"column": 28
} | {
"line": 458,
"column": 29
} | [
{
"pp": "case inr.inl.inr.inr\nx : ℂ\nhx : x ≠ 0\nhi : x.im = 0\nhr : 0 < x.re\n⊢ ↑π = ↑0 + ↑π",
"ppTerm": "?inr.inl.inr.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"Real.instZero",
"congrArg",
... | [
"case inr.inl.inr.inr\nx : ℂ\nhx : x ≠ 0\nhi : x.im = 0\nhr : 0 < x.re\n⊢ ↑π = 0 + ↑π"
] | Real.Angle.coe_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 862,
"column": 41
} | {
"line": 873,
"column": 9
} | {
"line": 875,
"column": 0
} | [
{
"pp": "a b : Angle\nha : a.sign ≠ 0\nh : a.sign = b.sign\n⊢ 2 • a = 2 • b ↔ a = b",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"instHSMul",
"Real.pi",
"AddLeftCancelSemigroup.toIsLeftCancelAdd",
"Real.Angle",
"eq_false",
... | [] | by
rw [Real.Angle.two_zsmul_eq_iff]
constructor
· intro h
rcases h with h1 | h2
· exact h1
· have : a.sign = (b + π).sign := by aesop
rw [Real.Angle.sign_add_pi] at this
have := congr_arg (· = b.sign) this
aesop
· intro h
aesop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 581,
"column": 2
} | {
"line": 581,
"column": 48
} | {
"line": 581,
"column": 49
} | [
{
"pp": "x y z : ℝ\nhx : 0 ≤ x\nhy : 0 ≤ y\nhz : 0 < z\n⊢ x ≤ y ^ z⁻¹ ↔ x ^ z ≤ y",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"congrArg",
"Real.instInv",
"Iff.rfl",
"id",
"LE.le",
... | [
"case hx\nx y z : ℝ\nhx : 0 ≤ x\nhy : 0 ≤ y\nhz : 0 < z\n⊢ 0 ≤ y",
"case hy\nx y z : ℝ\nhx : 0 ≤ x\nhy : 0 ≤ y\nhz : 0 < z\n⊢ z ≠ 0",
"x y z : ℝ\nhx : 0 ≤ x\nhy : 0 ≤ y\nhz : 0 < z\n⊢ 0 ≤ y ^ z⁻¹"
] | rw [← rpow_le_rpow_iff hx _ hz, rpow_inv_rpow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 804,
"column": 4
} | {
"line": 804,
"column": 40
} | {
"line": 806,
"column": 0
} | [
{
"pp": "case inr\nx y z : ℝ\nhy : 0 < y\nh : log x < z * log y\nhx : 0 < x\n⊢ x < y ^ z",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instPow",
"Real",
"HMul.hMul",
"Real.instLT",
"Real.log",
"HPow.hPow",
"Real.instMul",
... | [] | exact (lt_rpow_iff_log_lt hx hy).2 h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 804,
"column": 4
} | {
"line": 804,
"column": 40
} | {
"line": 806,
"column": 0
} | [
{
"pp": "case inr\nx y z : ℝ\nhy : 0 < y\nh : log x < z * log y\nhx : 0 < x\n⊢ x < y ^ z",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instPow",
"Real",
"HMul.hMul",
"Real.instLT",
"Real.log",
"HPow.hPow",
"Real.instMul",
... | [] | exact (lt_rpow_iff_log_lt hx hy).2 h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 804,
"column": 4
} | {
"line": 804,
"column": 40
} | {
"line": 806,
"column": 0
} | [
{
"pp": "case inr\nx y z : ℝ\nhy : 0 < y\nh : log x < z * log y\nhx : 0 < x\n⊢ x < y ^ z",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instPow",
"Real",
"HMul.hMul",
"Real.instLT",
"Real.log",
"HPow.hPow",
"Real.instMul",
... | [] | exact (lt_rpow_iff_log_lt hx hy).2 h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 862,
"column": 4
} | {
"line": 862,
"column": 40
} | {
"line": 864,
"column": 0
} | [
{
"pp": "case inr\nx y z : ℝ\nhy : 0 < y\nh : log x < z * log y\nhx : 0 < x\n⊢ x < y ^ z",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instPow",
"Real",
"HMul.hMul",
"Real.instLT",
"Real.log",
"HPow.hPow",
"Real.instMul",
... | [] | exact (lt_rpow_iff_log_lt hx hy).2 h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 862,
"column": 4
} | {
"line": 862,
"column": 40
} | {
"line": 864,
"column": 0
} | [
{
"pp": "case inr\nx y z : ℝ\nhy : 0 < y\nh : log x < z * log y\nhx : 0 < x\n⊢ x < y ^ z",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instPow",
"Real",
"HMul.hMul",
"Real.instLT",
"Real.log",
"HPow.hPow",
"Real.instMul",
... | [] | exact (lt_rpow_iff_log_lt hx hy).2 h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 862,
"column": 4
} | {
"line": 862,
"column": 40
} | {
"line": 864,
"column": 0
} | [
{
"pp": "case inr\nx y z : ℝ\nhy : 0 < y\nh : log x < z * log y\nhx : 0 < x\n⊢ x < y ^ z",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Real.instPow",
"Real",
"HMul.hMul",
"Real.instLT",
"Real.log",
"HPow.hPow",
"Real.instMul",
... | [] | exact (lt_rpow_iff_log_lt hx hy).2 h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 327,
"column": 2
} | {
"line": 331,
"column": 68
} | {
"line": 333,
"column": 0
} | [
{
"pp": "p : ℂ × ℂ\nh₁ : 0 ≤ p.1.re ∨ p.1.im ≠ 0\nh₂ : 0 < p.2.re\n⊢ ContinuousAt (fun x ↦ x.1 ^ x.2) p",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"lt_iff_le_and_ne",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
"Preorder.toLT",
"Re... | [] | obtain ⟨z, w⟩ := p
rw [← not_lt_zero_iff, lt_iff_le_and_ne, not_and_or, Ne, Classical.not_not,
not_le_zero_iff] at h₁
rcases h₁ with (h₁ | (rfl : z = 0))
exacts [continuousAt_cpow h₁, continuousAt_cpow_zero_of_re_pos h₂] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 327,
"column": 2
} | {
"line": 331,
"column": 68
} | {
"line": 333,
"column": 0
} | [
{
"pp": "p : ℂ × ℂ\nh₁ : 0 ≤ p.1.re ∨ p.1.im ≠ 0\nh₂ : 0 < p.2.re\n⊢ ContinuousAt (fun x ↦ x.1 ^ x.2) p",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"lt_iff_le_and_ne",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
"Preorder.toLT",
"Re... | [] | obtain ⟨z, w⟩ := p
rw [← not_lt_zero_iff, lt_iff_le_and_ne, not_and_or, Ne, Classical.not_not,
not_le_zero_iff] at h₁
rcases h₁ with (h₁ | (rfl : z = 0))
exacts [continuousAt_cpow h₁, continuousAt_cpow_zero_of_re_pos h₂] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 253,
"column": 2
} | {
"line": 253,
"column": 47
} | {
"line": 255,
"column": 0
} | [
{
"pp": "case mk\nι : Type u_1\ns : Multiset ι\nf : ι → ℝ\nr : ℝ\nl : List ι\nhs : ∀ i ∈ Quot.mk (⇑(List.isSetoid ι)) l, 0 ≤ f i\n⊢ (Multiset.map (fun x ↦ f x ^ r) (Quot.mk (⇑(List.isSetoid ι)) l)).prod =\n (Multiset.map f (Quot.mk (⇑(List.isSetoid ι)) l)).prod ^ r",
"ppTerm": "?mk",
"assigned": true... | [] | simpa using Real.list_prod_map_rpow' l f hs r | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 389,
"column": 6
} | {
"line": 389,
"column": 79
} | {
"line": 389,
"column": 79
} | [
{
"pp": "s r : ℝ\nhs : s < 0\nx : ℝ\nhx : x ∈ Set.Ioi 0\n⊢ ((fun x ↦ x ^ (-s)) ∘ fun x ↦ x⁻¹) x = (fun x ↦ x ^ s) x",
"ppTerm": "?m.96",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instPow",
"Real.inv_rpow",... | [] | rw [Function.comp_apply, inv_rpow hx.out.le, rpow_neg hx.out.le, inv_inv] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 389,
"column": 6
} | {
"line": 389,
"column": 79
} | {
"line": 389,
"column": 79
} | [
{
"pp": "s r : ℝ\nhs : s < 0\nx : ℝ\nhx : x ∈ Set.Ioi 0\n⊢ ((fun x ↦ x ^ (-s)) ∘ fun x ↦ x⁻¹) x = (fun x ↦ x ^ s) x",
"ppTerm": "?m.96",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instPow",
"Real.inv_rpow",... | [] | rw [Function.comp_apply, inv_rpow hx.out.le, rpow_neg hx.out.le, inv_inv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 389,
"column": 6
} | {
"line": 389,
"column": 79
} | {
"line": 389,
"column": 79
} | [
{
"pp": "s r : ℝ\nhs : s < 0\nx : ℝ\nhx : x ∈ Set.Ioi 0\n⊢ ((fun x ↦ x ^ (-s)) ∘ fun x ↦ x⁻¹) x = (fun x ↦ x ^ s) x",
"ppTerm": "?m.96",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instPow",
"Real.inv_rpow",... | [] | rw [Function.comp_apply, inv_rpow hx.out.le, rpow_neg hx.out.le, inv_inv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 1003,
"column": 64
} | {
"line": 1003,
"column": 72
} | {
"line": 1003,
"column": 73
} | [
{
"pp": "x : ℂ\n⊢ √(‖x‖ * (1 + Real.cos x.arg) / 2) = √((‖x‖ + x.re) / 2)",
"ppTerm": "?m.55",
"assigned": true,
"usedConstants": [
"Distrib.leftDistribClass",
"Norm.norm",
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"Real.cos",
"Monoid.toMulOneClass",
... | [
"x : ℂ\n⊢ √((‖x‖ * 1 + ‖x‖ * Real.cos x.arg) / 2) = √((‖x‖ + x.re) / 2)",
"case hx\nx : ℂ\n⊢ 0 ≤ ‖x‖",
"case hl\nx : ℂ\n⊢ -π ≤ x.arg",
"case hr\nx : ℂ\n⊢ x.arg ≤ π"
] | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 441,
"column": 89
} | {
"line": 443,
"column": 40
} | {
"line": 445,
"column": 0
} | [
{
"pp": "x : ℝ≥0\nn : ℕ\nhn : n ≠ 0\n⊢ (x ^ n) ^ (↑n)⁻¹ = x",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
"Real.instInv",
"NNReal.coe_rpow",
"id",
"NN... | [] | by
rw [← NNReal.coe_inj, coe_rpow, NNReal.coe_pow]
exact Real.pow_rpow_inv_natCast x.2 hn | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 576,
"column": 4
} | {
"line": 576,
"column": 45
} | {
"line": 576,
"column": 46
} | [
{
"pp": "case top\ny : ℝ\n⊢ ∞ ^ y = 0 ↔ ∞ = 0 ∧ 0 < y ∨ ∞ = ∞ ∧ y < 0",
"ppTerm": "?top",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"Real.instZero",
"ENNReal.instPowReal",
"PartialOrder.toPreorder",
"Real.instLT",
"Or.casesOn",
"And... | [
"case top.inl\ny : ℝ\nH : y < 0\n⊢ ∞ ^ y = 0 ↔ ∞ = 0 ∧ 0 < y ∨ ∞ = ∞ ∧ y < 0",
"case top.inr.inl\ny : ℝ\nH : y = 0\n⊢ ∞ ^ y = 0 ↔ ∞ = 0 ∧ 0 < y ∨ ∞ = ∞ ∧ y < 0",
"case top.inr.inr\ny : ℝ\nH : 0 < y\n⊢ ∞ ^ y = 0 ↔ ∞ = 0 ∧ 0 < y ∨ ∞ = ∞ ∧ y < 0"
] | rcases lt_trichotomy y 0 with (H | H | H) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 580,
"column": 6
} | {
"line": 580,
"column": 47
} | {
"line": 580,
"column": 48
} | [
{
"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": [
"Real",
"ENNReal.ofNNReal",
"Preorder.toLT",
"Real.instZero",
"ENNReal.instPowReal",
"PartialOrder.toPreorder",
... | [
"case pos.inl\ny : ℝ\nx : ℝ≥0\nh : x = 0\nH : y < 0\n⊢ ↑x ^ y = 0 ↔ ↑x = 0 ∧ 0 < y ∨ ↑x = ∞ ∧ y < 0",
"case pos.inr.inl\ny : ℝ\nx : ℝ≥0\nh : x = 0\nH : y = 0\n⊢ ↑x ^ y = 0 ↔ ↑x = 0 ∧ 0 < y ∨ ↑x = ∞ ∧ y < 0",
"case pos.inr.inr\ny : ℝ\nx : ℝ≥0\nh : x = 0\nH : 0 < y\n⊢ ↑x ^ y = 0 ↔ ↑x = 0 ∧ 0 < y ∨ ↑x = ∞ ∧ y < 0"... | rcases lt_trichotomy y 0 with (H | H | H) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 591,
"column": 4
} | {
"line": 591,
"column": 45
} | {
"line": 591,
"column": 46
} | [
{
"pp": "case top\ny : ℝ\n⊢ ∞ ^ y = ∞ ↔ ∞ = 0 ∧ y < 0 ∨ ∞ = ∞ ∧ 0 < y",
"ppTerm": "?top",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"Real.instZero",
"ENNReal.instPowReal",
"PartialOrder.toPreorder",
"Real.instLT",
"Or.casesOn",
"And... | [
"case top.inl\ny : ℝ\nH : y < 0\n⊢ ∞ ^ y = ∞ ↔ ∞ = 0 ∧ y < 0 ∨ ∞ = ∞ ∧ 0 < y",
"case top.inr.inl\ny : ℝ\nH : y = 0\n⊢ ∞ ^ y = ∞ ↔ ∞ = 0 ∧ y < 0 ∨ ∞ = ∞ ∧ 0 < y",
"case top.inr.inr\ny : ℝ\nH : 0 < y\n⊢ ∞ ^ y = ∞ ↔ ∞ = 0 ∧ y < 0 ∨ ∞ = ∞ ∧ 0 < y"
] | rcases lt_trichotomy y 0 with (H | H | H) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 595,
"column": 6
} | {
"line": 595,
"column": 47
} | {
"line": 595,
"column": 48
} | [
{
"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": [
"Real",
"ENNReal.ofNNReal",
"Preorder.toLT",
"Real.instZero",
"ENNReal.instPowReal",
"PartialOrder.toPreorder",
... | [
"case pos.inl\ny : ℝ\nx : ℝ≥0\nh : x = 0\nH : y < 0\n⊢ ↑x ^ y = ∞ ↔ ↑x = 0 ∧ y < 0 ∨ ↑x = ∞ ∧ 0 < y",
"case pos.inr.inl\ny : ℝ\nx : ℝ≥0\nh : x = 0\nH : y = 0\n⊢ ↑x ^ y = ∞ ↔ ↑x = 0 ∧ y < 0 ∨ ↑x = ∞ ∧ 0 < y",
"case pos.inr.inr\ny : ℝ\nx : ℝ≥0\nh : x = 0\nH : 0 < y\n⊢ ↑x ^ y = ∞ ↔ ↑x = 0 ∧ y < 0 ∨ ↑x = ∞ ∧ 0 < y"... | rcases lt_trichotomy y 0 with (H | H | H) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 655,
"column": 4
} | {
"line": 655,
"column": 45
} | {
"line": 655,
"column": 46
} | [
{
"pp": "case top\ny : ℝ\n⊢ ∞ ^ (-y) = (∞ ^ y)⁻¹",
"ppTerm": "?top",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"Real.instZero",
"ENNReal.instPowReal",
"PartialOrder.toPreorder",
"Or.casesOn",
"Inv.inv",
"HPow.hPow",
"LT.lt",
... | [
"case top.inl\ny : ℝ\nH : y < 0\n⊢ ∞ ^ (-y) = (∞ ^ y)⁻¹",
"case top.inr.inl\ny : ℝ\nH : y = 0\n⊢ ∞ ^ (-y) = (∞ ^ y)⁻¹",
"case top.inr.inr\ny : ℝ\nH : 0 < y\n⊢ ∞ ^ (-y) = (∞ ^ y)⁻¹"
] | rcases lt_trichotomy y 0 with (H | H | H) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 659,
"column": 6
} | {
"line": 659,
"column": 47
} | {
"line": 659,
"column": 48
} | [
{
"pp": "case pos\ny : ℝ\nx : ℝ≥0\nh : x = 0\n⊢ ↑x ^ (-y) = (↑x ^ y)⁻¹",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Real",
"ENNReal.ofNNReal",
"Preorder.toLT",
"Real.instZero",
"ENNReal.instPowReal",
"PartialOrder.toPreorder",
"Or.casesOn",
... | [
"case pos.inl\ny : ℝ\nx : ℝ≥0\nh : x = 0\nH : y < 0\n⊢ ↑x ^ (-y) = (↑x ^ y)⁻¹",
"case pos.inr.inl\ny : ℝ\nx : ℝ≥0\nh : x = 0\nH : y = 0\n⊢ ↑x ^ (-y) = (↑x ^ y)⁻¹",
"case pos.inr.inr\ny : ℝ\nx : ℝ≥0\nh : x = 0\nH : 0 < y\n⊢ ↑x ^ (-y) = (↑x ^ y)⁻¹"
] | rcases lt_trichotomy y 0 with (H | H | H) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 707,
"column": 2
} | {
"line": 707,
"column": 18
} | {
"line": 708,
"column": 2
} | [
{
"pp": "case inr\nx y : ℝ≥0∞\nz : ℝ\nhz : z < 0 ∨ 0 < z\n⊢ (x * y) ^ z = if (x = 0 ∧ y = ∞ ∨ x = ∞ ∧ y = 0) ∧ z < 0 then ∞ else x ^ z * y ^ z",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"HMul.hMul",
"LinearOrder.toDecidableEq",
"Re... | [
"case inr.inr\nx y : ℝ≥0∞\nz : ℝ\nhz : z < 0 ∨ 0 < z\nthis :\n ∀ (x y : ℝ≥0∞) (z : ℝ),\n z < 0 ∨ 0 < z → x ≤ y → (x * y) ^ z = if (x = 0 ∧ y = ∞ ∨ x = ∞ ∧ y = 0) ∧ z < 0 then ∞ else x ^ z * y ^ z\nhxy : ¬x ≤ y\n⊢ (x * y) ^ z = if (x = 0 ∧ y = ∞ ∨ x = ∞ ∧ y = 0) ∧ z < 0 then ∞ else x ^ z * y ^ z",
"x y : ℝ≥0∞\... | wlog hxy : x ≤ y | Mathlib.Tactic._aux_Mathlib_Tactic_WLOG___elabRules_Mathlib_Tactic_wlog_1 | Mathlib.Tactic.wlog |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 705,
"column": 2
} | {
"line": 721,
"column": 11
} | {
"line": 723,
"column": 0
} | [
{
"pp": "x y : ℝ≥0∞\nz : ℝ\n⊢ (x * y) ^ z = if (x = 0 ∧ y = ∞ ∨ x = ∞ ∧ y = 0) ∧ z < 0 then ∞ else x ^ z * y ^ z",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"ENNReal.coe_ne_top._simp_1",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"ENNReal.rpow_eq_zero_... | [] | rcases eq_or_ne z 0 with (rfl | hz); · simp
replace hz := hz.lt_or_gt
wlog hxy : x ≤ y
· convert! this y x z hz (le_of_not_ge hxy) using 2 <;> simp only [mul_comm, and_comm, or_comm]
rcases eq_or_ne x 0 with (rfl | hx0)
· induction y <;> rcases hz with hz | hz <;> simp [*, hz.not_gt]
rcases eq_or_ne y 0 wit... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 705,
"column": 2
} | {
"line": 721,
"column": 11
} | {
"line": 723,
"column": 0
} | [
{
"pp": "x y : ℝ≥0∞\nz : ℝ\n⊢ (x * y) ^ z = if (x = 0 ∧ y = ∞ ∨ x = ∞ ∧ y = 0) ∧ z < 0 then ∞ else x ^ z * y ^ z",
"ppTerm": "?m.43",
"assigned": true,
"usedConstants": [
"ENNReal.coe_ne_top._simp_1",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"ENNReal.rpow_eq_zero_... | [] | rcases eq_or_ne z 0 with (rfl | hz); · simp
replace hz := hz.lt_or_gt
wlog hxy : x ≤ y
· convert! this y x z hz (le_of_not_ge hxy) using 2 <;> simp only [mul_comm, and_comm, or_comm]
rcases eq_or_ne x 0 with (rfl | hx0)
· induction y <;> rcases hz with hz | hz <;> simp [*, hz.not_gt]
rcases eq_or_ne y 0 wit... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 750,
"column": 40
} | {
"line": 750,
"column": 47
} | {
"line": 750,
"column": 48
} | [
{
"pp": "case insert\nι : Type u_1\nf : ι → ℝ≥0∞\nr : ℝ\ni : ι\ns : Finset ι\nhi : i ∉ s\nih : (∀ i ∈ s, f i ≠ ∞) → ∏ i ∈ s, f i ^ r = (∏ i ∈ s, f i) ^ r\nhf : ∀ i_1 ∈ insert i s, f i_1 ≠ ∞\nh2f : ∀ i ∈ s, f i ≠ ∞\n⊢ f i ^ r * ∏ x ∈ s, f x ^ r = (f i * ∏ x ∈ s, f x) ^ r",
"ppTerm": "?insert",
"assigned"... | [
"case insert\nι : Type u_1\nf : ι → ℝ≥0∞\nr : ℝ\ni : ι\ns : Finset ι\nhi : i ∉ s\nih : (∀ i ∈ s, f i ≠ ∞) → ∏ i ∈ s, f i ^ r = (∏ i ∈ s, f i) ^ r\nhf : ∀ i_1 ∈ insert i s, f i_1 ≠ ∞\nh2f : ∀ i ∈ s, f i ≠ ∞\n⊢ f i ^ r * (∏ i ∈ s, f i) ^ r = (f i * ∏ x ∈ s, f x) ^ r"
] | ih h2f, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 909,
"column": 97
} | {
"line": 914,
"column": 62
} | {
"line": 916,
"column": 0
} | [
{
"pp": "x : ℝ≥0∞\nz : ℝ\nhx : 1 ≤ x\nhz : z < 0\n⊢ x ^ z ≤ 1",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"ENNReal.ofNNReal",
"Real.instZero",
"congrArg",
"... | [] | by
cases x
· simp [top_rpow_of_neg hz]
· simp only [one_le_coe_iff] at hx
simp [← coe_rpow_of_ne_zero (ne_of_gt (lt_of_lt_of_le zero_lt_one hx)),
NNReal.rpow_le_one_of_one_le_of_nonpos hx (le_of_lt hz)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.Ray | {
"line": 530,
"column": 6
} | {
"line": 530,
"column": 47
} | {
"line": 531,
"column": 6
} | [
{
"pp": "case neg.refine_1.inl.inr.inr\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\nhy : ¬y = 0\nr₁ r₂ : R\nhr₁ : 0 < r₁\nleft✝ : 0 < r₂\nh : r₁ • x = r₂ • ... | [
"case neg.refine_1.inl.inr.inr\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\nhy : ¬y = 0\nr₁ r₂ : R\nhr₁ : 0 < r₁\nleft✝ : 0 < r₂\nh : r₁ • x = r₂ • y\n⊢ ![r₁, -... | rw [Fin.sum_univ_two, Fin.exists_fin_two] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.Ray | {
"line": 535,
"column": 6
} | {
"line": 535,
"column": 47
} | {
"line": 536,
"column": 6
} | [
{
"pp": "case neg.refine_1.inr.inr.inr\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\nhy : ¬y = 0\nr₁ r₂ : R\nhr₁ : 0 < r₁\nleft✝ : 0 < r₂\nh : r₁ • x = r₂ • ... | [
"case neg.refine_1.inr.inr.inr\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\nhy : ¬y = 0\nr₁ r₂ : R\nhr₁ : 0 < r₁\nleft✝ : 0 < r₂\nh : r₁ • x = r₂ • -y\n⊢ ![r₁, ... | rw [Fin.sum_univ_two, Fin.exists_fin_two] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Segment | {
"line": 578,
"column": 4
} | {
"line": 578,
"column": 37
} | {
"line": 579,
"column": 2
} | [
{
"pp": "case refine_1.inr\n𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nx y : 𝕜\nh : x < y\na b : 𝕜\nha : 0 ≤ a\nhb : 0 ≤ b\nhab : a + b = 1\nhz : a * x + b * y ∈ Ioc x y\nhb' : 0 < b\n⊢ ∃ a_1 b_1, 0 ≤ a_1 ∧ 0 < b_1 ∧ a_1 + b_1 = 1 ∧ a_1 * x + b_1 * y = a * x + b... | [] | · exact ⟨a, b, ha, hb', hab, rfl⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.AffineSpace.AffineEquiv | {
"line": 150,
"column": 20
} | {
"line": 150,
"column": 75
} | {
"line": 152,
"column": 0
} | [
{
"pp": "k : Type u_1\nP₁ : Type u_2\nP₂ : Type u_3\nP₃ : Type u_4\nP₄ : Type u_5\nV₁ : Type u_6\nV₂ : Type u_7\nV₃ : Type u_8\nV₄ : Type u_9\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V₁\ninst✝¹⁰ : AddCommGroup V₂\ninst✝⁹ : AddCommGroup V₃\ninst✝⁸ : AddCommGroup V₄\ninst✝⁷ : Module k V₁\ninst✝⁶ : Module k V₂\ni... | [] | by simp [h p', h (v +ᵥ p'), vadd_vsub_assoc, vadd_vadd] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Segment | {
"line": 598,
"column": 4
} | {
"line": 599,
"column": 58
} | {
"line": 600,
"column": 4
} | [
{
"pp": "case refine_2.inl\n𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nx y : 𝕜\nh : x < y\na : 𝕜\nha : 0 < a\nhb : 0 ≤ 0\nhab : a + 0 = 1\n⊢ a * x + 0 * y ∈ Ico x y",
"ppTerm": "?refine_2.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [
"case refine_2.inr\n𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nx y : 𝕜\nh : x < y\na b : 𝕜\nha : 0 < a\nhb : 0 ≤ b\nhab : a + b = 1\nhb' : 0 < b\n⊢ a * x + b * y ∈ Ico x y"
] | · rw [add_zero] at hab
rwa [hab, one_mul, zero_mul, add_zero, left_mem_Ico] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 329,
"column": 2
} | {
"line": 330,
"column": 69
} | {
"line": 331,
"column": 2
} | [
{
"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\ns : AffineSubspace k P\np : P\nhp : p ∈ s\nv : V\n⊢ v ∈ ↑s.direction ↔ v ∈ (fun x ↦ p -ᵥ x) '' ↑s",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
... | [
"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\ns : AffineSubspace k P\np : P\nhp : p ∈ s\nv : V\n⊢ (∃ x ∈ ↑s, x -ᵥ p = -v) ↔ ∃ x ∈ ↑s, p -ᵥ x = v"
] | rw [SetLike.mem_coe, ← Submodule.neg_mem_iff, ← SetLike.mem_coe,
coe_direction_eq_vsub_set_right hp, Set.mem_image, Set.mem_image] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 798,
"column": 2
} | {
"line": 800,
"column": 77
} | {
"line": 802,
"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\ns₁ s₂ : AffineSubspace k P\n⊢ Submodule.span k (↑(s₁ ⊓ s₂) -ᵥ ↑(s₁ ⊓ s₂)) ≤ Submodule.span k (↑s₁ -ᵥ ↑s₁) ⊓ Submodule.span k (↑s₂ -ᵥ ↑s₂)",
"ppTerm": "?m.42",
"assigned":... | [] | exact
le_inf (sInf_le_sInf fun p hp => trans (vsub_self_mono inter_subset_left) hp)
(sInf_le_sInf fun p hp => trans (vsub_self_mono inter_subset_right) hp) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Basic | {
"line": 525,
"column": 2
} | {
"line": 525,
"column": 44
} | {
"line": 526,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Semiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsOrderedRing 𝕜\ninst✝ : SMul 𝕜 E\nf : E → 𝕜\nhf1 : ∀ (x y : E), f (x + y) ≤ f x + f y\nhf2 : ∀ ⦃c : 𝕜⦄ (x : E), 0 ≤ c → f (c • x) ≤ c * f x\nB : 𝕜\n⊢ ∀ ⦃x : E⦄, x ∈ {x | f x ≤ B} → ∀ ⦃y ... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Semiring 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : LinearOrder 𝕜\ninst✝¹ : IsOrderedRing 𝕜\ninst✝ : SMul 𝕜 E\nf : E → 𝕜\nhf1 : ∀ (x y : E), f (x + y) ≤ f x + f y\nhf2 : ∀ ⦃c : 𝕜⦄ (x : E), 0 ≤ c → f (c • x) ≤ c * f x\nB : 𝕜\nx : E\nhx : x ∈ {x | f x ≤ B}\ny : E\nhy : y ∈ {x ... | rintro x hx y hy z ⟨a, b, ha, hb, hs, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.Convex.Star | {
"line": 443,
"column": 4
} | {
"line": 445,
"column": 38
} | {
"line": 447,
"column": 0
} | [
{
"pp": "case inr.refine_2\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedAddMonoid E\ninst✝¹ : Module 𝕜 E\ninst✝ : PosSMulMono 𝕜 E\nx : E\ns : Set E\nhs : s.OrdConnected\nhx : x ∈ s\nh : ∀ y ∈ s, x ≤ y ∨ y ≤ x... | [] | calc
a • x + b • y ≤ a • x + b • x := by gcongr
_ = x := Convex.combo_self hab _ | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 1090,
"column": 2
} | {
"line": 1091,
"column": 77
} | {
"line": 1093,
"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\np : P\nps : Set P\nh : p ∈ affineSpan k ps\n⊢ affineSpan k (insert p ps) = affineSpan k ps",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | rw [← mem_coe] at h
rw [← affineSpan_insert_affineSpan, Set.insert_eq_of_mem h, affineSpan_coe] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 1090,
"column": 2
} | {
"line": 1091,
"column": 77
} | {
"line": 1093,
"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\np : P\nps : Set P\nh : p ∈ affineSpan k ps\n⊢ affineSpan k (insert p ps) = affineSpan k ps",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Eq.mp... | [] | rw [← mem_coe] at h
rw [← affineSpan_insert_affineSpan, Set.insert_eq_of_mem h, affineSpan_coe] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 191,
"column": 42
} | {
"line": 191,
"column": 53
} | {
"line": 191,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\ninst✝⁷ : NormedDivisionRing 𝕜\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜 E\ns : Set E\ninst✝⁴ : NormedRing 𝕝\ninst✝³ : Module 𝕜 𝕝\ninst✝² : NormSMulClass 𝕜 𝕝\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\nb : 𝕜\nhs : Balanced 𝕝 s\na : �... | [
"𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\ninst✝⁷ : NormedDivisionRing 𝕜\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜 E\ns : Set E\ninst✝⁴ : NormedRing 𝕝\ninst✝³ : Module 𝕜 𝕝\ninst✝² : NormSMulClass 𝕜 𝕝\ninst✝¹ : SMulWithZero 𝕝 E\ninst✝ : IsScalarTower 𝕜 𝕝 E\nb : 𝕜\nhs : Balanced 𝕝 s\na : 𝕝\nh : ‖a‖ ≤... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 205,
"column": 53
} | {
"line": 205,
"column": 64
} | {
"line": 205,
"column": 65
} | [
{
"pp": "𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\ninst✝⁸ : NormedDivisionRing 𝕜\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜 E\ns : Set E\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : Module 𝕜 𝕝\ninst✝³ : NormSMulClass 𝕜 𝕝\ninst✝² : SMulWithZero 𝕝 E\ninst✝¹ : IsScalarTower 𝕜 𝕝 E\na : 𝕜\nx : E\ninst✝ : SMulCommC... | [
"𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\ninst✝⁸ : NormedDivisionRing 𝕜\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜 E\ns : Set E\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : Module 𝕜 𝕝\ninst✝³ : NormSMulClass 𝕜 𝕝\ninst✝² : SMulWithZero 𝕝 E\ninst✝¹ : IsScalarTower 𝕜 𝕝 E\na : 𝕜\nx : E\ninst✝ : SMulCommClass 𝕝 𝕜 E... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 205,
"column": 38
} | {
"line": 205,
"column": 84
} | {
"line": 207,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\ninst✝⁸ : NormedDivisionRing 𝕜\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜 E\ns : Set E\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : Module 𝕜 𝕝\ninst✝³ : NormSMulClass 𝕜 𝕝\ninst✝² : SMulWithZero 𝕝 E\ninst✝¹ : IsScalarTower 𝕜 𝕝 E\na : 𝕜\nx : E\ninst✝ : SMulCommC... | [] | rw [smul_comm, smul_assoc, smul_inv_smul₀ ha₀] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 205,
"column": 38
} | {
"line": 205,
"column": 84
} | {
"line": 207,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\ninst✝⁸ : NormedDivisionRing 𝕜\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜 E\ns : Set E\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : Module 𝕜 𝕝\ninst✝³ : NormSMulClass 𝕜 𝕝\ninst✝² : SMulWithZero 𝕝 E\ninst✝¹ : IsScalarTower 𝕜 𝕝 E\na : 𝕜\nx : E\ninst✝ : SMulCommC... | [] | rw [smul_comm, smul_assoc, smul_inv_smul₀ ha₀] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 205,
"column": 38
} | {
"line": 205,
"column": 84
} | {
"line": 207,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\n𝕝 : Type u_2\nE : Type u_3\ninst✝⁸ : NormedDivisionRing 𝕜\ninst✝⁷ : AddCommGroup E\ninst✝⁶ : Module 𝕜 E\ns : Set E\ninst✝⁵ : NormedRing 𝕝\ninst✝⁴ : Module 𝕜 𝕝\ninst✝³ : NormSMulClass 𝕜 𝕝\ninst✝² : SMulWithZero 𝕝 E\ninst✝¹ : IsScalarTower 𝕜 𝕝 E\na : 𝕜\nx : E\ninst✝ : SMulCommC... | [] | rw [smul_comm, smul_assoc, smul_inv_smul₀ ha₀] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.BalancedCoreHull | {
"line": 185,
"column": 7
} | {
"line": 185,
"column": 18
} | {
"line": 185,
"column": 19
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NormedDivisionRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nh0 : 0 ∈ balancedCoreAux 𝕜 s\na : 𝕜\nha : ‖a‖ ≤ 1\ny : E\nhy : ∀ (r : 𝕜), 1 ≤ ‖r‖ → y ∈ r • s\nh : a ≠ 0\nr : 𝕜\nhr : 1 ≤ ‖r‖\nh'' : 1 ≤ ‖a⁻¹ • r‖\nh' : y ∈ (a⁻¹ • r) • s\... | [
"case inr\n𝕜 : Type u_1\nE : Type u_2\ninst✝² : NormedDivisionRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nh0 : 0 ∈ balancedCoreAux 𝕜 s\na : 𝕜\nha : ‖a‖ ≤ 1\ny : E\nhy : ∀ (r : 𝕜), 1 ≤ ‖r‖ → y ∈ r • s\nh : a ≠ 0\nr : 𝕜\nhr : 1 ≤ ‖r‖\nh'' : 1 ≤ ‖a⁻¹ • r‖\nh' : y ∈ a⁻¹ • r • s\n⊢ (fun x ↦ a ... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 250,
"column": 2
} | {
"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"... | [] | rw [← insert_eq_self.2 h]
exact hA.zero_insert_interior | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 250,
"column": 2
} | {
"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"... | [] | rw [← insert_eq_self.2 h]
exact hA.zero_insert_interior | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.BalancedCoreHull | {
"line": 231,
"column": 4
} | {
"line": 231,
"column": 46
} | {
"line": 232,
"column": 2
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : IsClosed[inst✝¹] U\nh : 0 ∈ U\na : 𝕜\nha : 1 ≤ ‖a‖\nha' : a ≠ 0\n⊢ IsClosed[inst✝¹] (a • U)",
"ppTerm": "?... | [] | exact isClosedMap_smul_of_ne_zero ha' U hU | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Function | {
"line": 100,
"column": 40
} | {
"line": 101,
"column": 92
} | {
"line": 101,
"column": 92
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConvexOn 𝕜 s f\nhfg : EqOn f g s\nx : E\nhx : x ∈ s\ny : E\nhy : y... | [] | by
simpa only [← hfg hx, ← hfg hy, ← hfg (hf.1 hx hy ha hb hab)] using hf.2 hx hy ha hb hab | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Function | {
"line": 104,
"column": 40
} | {
"line": 105,
"column": 92
} | {
"line": 105,
"column": 92
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf g : E → β\nhf : ConcaveOn 𝕜 s f\nhfg : EqOn f g s\nx : E\nhx : x ∈ s\ny : E\nhy : ... | [] | by
simpa only [← hfg hx, ← hfg hy, ← hfg (hf.1 hx hy ha hb hab)] using hf.2 hx hy ha hb hab | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Function | {
"line": 1032,
"column": 2
} | {
"line": 1032,
"column": 51
} | {
"line": 1033,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_4\nβ : Type u_5\ninst✝⁷ : Semiring 𝕜\ninst✝⁶ : PartialOrder 𝕜\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : PartialOrder α\ninst✝³ : SMul 𝕜 α\ninst✝² : AddCommMonoid β\ninst✝¹ : PartialOrder β\ninst✝ : SMul 𝕜 β\nf : α ≃o β\nhf : ConcaveOn 𝕜 univ ⇑f\nx : β\nx✝¹ : x ∈ univ\ny : β\nx✝... | [
"𝕜 : Type u_1\nα : Type u_4\nβ : Type u_5\ninst✝⁷ : Semiring 𝕜\ninst✝⁶ : PartialOrder 𝕜\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : PartialOrder α\ninst✝³ : SMul 𝕜 α\ninst✝² : AddCommMonoid β\ninst✝¹ : PartialOrder β\ninst✝ : SMul 𝕜 β\nf : α ≃o β\nhf : ConcaveOn 𝕜 univ ⇑f\nx : β\nx✝¹ : x ∈ univ\ny : β\nx✝ : y ∈ univ\... | simp only [hx'', hy'', OrderIso.symm_apply_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Convex.Function | {
"line": 1051,
"column": 2
} | {
"line": 1051,
"column": 51
} | {
"line": 1052,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_4\nβ : Type u_5\ninst✝⁷ : Semiring 𝕜\ninst✝⁶ : PartialOrder 𝕜\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : PartialOrder α\ninst✝³ : SMul 𝕜 α\ninst✝² : AddCommMonoid β\ninst✝¹ : PartialOrder β\ninst✝ : SMul 𝕜 β\nf : α ≃o β\nhf : ConvexOn 𝕜 univ ⇑f\nx : β\nx✝¹ : x ∈ univ\ny : β\nx✝ ... | [
"𝕜 : Type u_1\nα : Type u_4\nβ : Type u_5\ninst✝⁷ : Semiring 𝕜\ninst✝⁶ : PartialOrder 𝕜\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : PartialOrder α\ninst✝³ : SMul 𝕜 α\ninst✝² : AddCommMonoid β\ninst✝¹ : PartialOrder β\ninst✝ : SMul 𝕜 β\nf : α ≃o β\nhf : ConvexOn 𝕜 univ ⇑f\nx : β\nx✝¹ : x ∈ univ\ny : β\nx✝ : y ∈ univ\n... | simp only [hx'', hy'', OrderIso.symm_apply_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 443,
"column": 6
} | {
"line": 448,
"column": 93
} | {
"line": 449,
"column": 2
} | [
{
"pp": "case refine_1.inr\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\ns₁ s₂ : AffineSubspace k P\np₁ p₂ : P\nhp₁ : p₁ ∈ ↑s₁\nhp₂ : p₂ ∈ s₂\np₃ : P\nhp₃ : p₃ ∈ ↑s₂\n⊢ (fun x ↦ x -ᵥ p₁) p₃ ∈ ↑(s₁.direction ⊔ s₂.direction ⊔ k ∙... | [] | rw [sup_assoc, SetLike.mem_coe, Submodule.mem_sup]
use 0, Submodule.zero_mem _, p₃ -ᵥ p₁
rw [and_comm, zero_add]
use rfl
rw [← vsub_add_vsub_cancel p₃ p₂ p₁, Submodule.mem_sup]
use p₃ -ᵥ p₂, vsub_mem_direction hp₃ hp₂, p₂ -ᵥ p₁, Submodule.mem_span_singleton_self _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 443,
"column": 6
} | {
"line": 448,
"column": 93
} | {
"line": 449,
"column": 2
} | [
{
"pp": "case refine_1.inr\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\ns₁ s₂ : AffineSubspace k P\np₁ p₂ : P\nhp₁ : p₁ ∈ ↑s₁\nhp₂ : p₂ ∈ s₂\np₃ : P\nhp₃ : p₃ ∈ ↑s₂\n⊢ (fun x ↦ x -ᵥ p₁) p₃ ∈ ↑(s₁.direction ⊔ s₂.direction ⊔ k ∙... | [] | rw [sup_assoc, SetLike.mem_coe, Submodule.mem_sup]
use 0, Submodule.zero_mem _, p₃ -ᵥ p₁
rw [and_comm, zero_add]
use rfl
rw [← vsub_add_vsub_cancel p₃ p₂ p₁, Submodule.mem_sup]
use p₃ -ᵥ p₂, vsub_mem_direction hp₃ hp₂, p₂ -ᵥ p₁, Submodule.mem_span_singleton_self _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 586,
"column": 4
} | {
"line": 587,
"column": 56
} | {
"line": 589,
"column": 0
} | [
{
"pp": "case inr.hn\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\ns : Set P₁\np : P₁\nhp : p ∈ s\... | [] | exact ⟨f p, mem_image_of_mem f (subset_affineSpan k _ hp),
subset_affineSpan k _ (mem_image_of_mem f hp)⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 586,
"column": 4
} | {
"line": 587,
"column": 56
} | {
"line": 589,
"column": 0
} | [
{
"pp": "case inr.hn\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\ns : Set P₁\np : P₁\nhp : p ∈ s\... | [] | exact ⟨f p, mem_image_of_mem f (subset_affineSpan k _ hp),
subset_affineSpan k _ (mem_image_of_mem f hp)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 586,
"column": 4
} | {
"line": 587,
"column": 56
} | {
"line": 589,
"column": 0
} | [
{
"pp": "case inr.hn\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\ns : Set P₁\np : P₁\nhp : p ∈ s\... | [] | exact ⟨f p, mem_image_of_mem f (subset_affineSpan k _ hp),
subset_affineSpan k _ (mem_image_of_mem f hp)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 598,
"column": 2
} | {
"line": 600,
"column": 7
} | {
"line": 602,
"column": 0
} | [
{
"pp": "k : 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\nhf : Function.Injective ⇑f\ns₁ s₂ : AffineSu... | [] | ext p
simp [mem_inf_iff]
grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 598,
"column": 2
} | {
"line": 600,
"column": 7
} | {
"line": 602,
"column": 0
} | [
{
"pp": "k : 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\nhf : Function.Injective ⇑f\ns₁ s₂ : AffineSu... | [] | ext p
simp [mem_inf_iff]
grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Seminorm | {
"line": 443,
"column": 58
} | {
"line": 443,
"column": 66
} | {
"line": 443,
"column": 67
} | [
{
"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\np q : Seminorm 𝕜 E\na : �... | [
"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\np q : Seminorm 𝕜 E\na : 𝕜\nx : E\nha... | mul_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Seminorm | {
"line": 462,
"column": 8
} | {
"line": 462,
"column": 53
} | {
"line": 463,
"column": 4
} | [
{
"pp": "R : 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\np q : Seminorm 𝕜 E\nx : E\n⊢ p 0 + ... | [] | simp only [map_zero, zero_add, sub_zero]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Seminorm | {
"line": 462,
"column": 8
} | {
"line": 462,
"column": 53
} | {
"line": 463,
"column": 4
} | [
{
"pp": "R : 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\np q : Seminorm 𝕜 E\nx : E\n⊢ p 0 + ... | [] | simp only [map_zero, zero_add, sub_zero]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Seminorm | {
"line": 503,
"column": 10
} | {
"line": 503,
"column": 39
} | {
"line": 504,
"column": 10
} | [
{
"pp": "case inl\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✝ x y : E\nq : E → ℝ\nhq : q ∈ upp... | [
"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 y : E\nq : E... | · simp [Real.iSup_of_isEmpty] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
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