module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Data.Real.Sqrt | {
"line": 365,
"column": 43
} | {
"line": 365,
"column": 55
} | [
{
"pp": "x : ℝ\nhx : 0 ≤ x\ny : ℝ\n⊢ √x * (√y)⁻¹ = √x / √y",
"usedConstants": [
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Real.instInv",
"Real.instDivInvMonoid",
"id",
"MulOne.toMul"... | division_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Real.Sqrt | {
"line": 369,
"column": 61
} | {
"line": 369,
"column": 73
} | [
{
"pp": "x y : ℝ\nhy : 0 ≤ y\n⊢ √x * (√y)⁻¹ = √x / √y",
"usedConstants": [
"Eq.mpr",
"Real",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Real.instInv",
"Real.instDivInvMonoid",
"id",
"MulOne.toMul",
... | division_def | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Norm | {
"line": 223,
"column": 24
} | {
"line": 223,
"column": 46
} | [
{
"pp": "z : ℂ\nhz : z = 0\n⊢ |z.re / ‖z‖| ≤ 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"GroupWithZero.toMonoidWithZero",
"Real.instLE",
"Real",
"instHDiv",
"Real.lattice",
"Complex.instNormedAddCommGroup",
"Real.instZero",
... | simp [hz, zero_le_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.Norm | {
"line": 223,
"column": 24
} | {
"line": 223,
"column": 46
} | [
{
"pp": "z : ℂ\nhz : z = 0\n⊢ |z.re / ‖z‖| ≤ 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"GroupWithZero.toMonoidWithZero",
"Real.instLE",
"Real",
"instHDiv",
"Real.lattice",
"Complex.instNormedAddCommGroup",
"Real.instZero",
... | simp [hz, zero_le_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Norm | {
"line": 223,
"column": 24
} | {
"line": 223,
"column": 46
} | [
{
"pp": "z : ℂ\nhz : z = 0\n⊢ |z.re / ‖z‖| ≤ 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"GroupWithZero.toMonoidWithZero",
"Real.instLE",
"Real",
"instHDiv",
"Real.lattice",
"Complex.instNormedAddCommGroup",
"Real.instZero",
... | simp [hz, zero_le_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Norm | {
"line": 228,
"column": 24
} | {
"line": 228,
"column": 46
} | [
{
"pp": "z : ℂ\nhz : z = 0\n⊢ |z.im / ‖z‖| ≤ 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"GroupWithZero.toMonoidWithZero",
"Real.instLE",
"Real",
"instHDiv",
"Real.lattice",
"Complex.instNormedAddCommGroup",
"Real.instZero",
... | simp [hz, zero_le_one] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.Norm | {
"line": 228,
"column": 24
} | {
"line": 228,
"column": 46
} | [
{
"pp": "z : ℂ\nhz : z = 0\n⊢ |z.im / ‖z‖| ≤ 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"GroupWithZero.toMonoidWithZero",
"Real.instLE",
"Real",
"instHDiv",
"Real.lattice",
"Complex.instNormedAddCommGroup",
"Real.instZero",
... | simp [hz, zero_le_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Norm | {
"line": 228,
"column": 24
} | {
"line": 228,
"column": 46
} | [
{
"pp": "z : ℂ\nhz : z = 0\n⊢ |z.im / ‖z‖| ≤ 1",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"GroupWithZero.toMonoidWithZero",
"Real.instLE",
"Real",
"instHDiv",
"Real.lattice",
"Complex.instNormedAddCommGroup",
"Real.instZero",
... | simp [hz, zero_le_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Norm | {
"line": 280,
"column": 35
} | {
"line": 281,
"column": 100
} | [
{
"pp": "f : CauSeq ℂ fun x ↦ ‖x‖\nε : ℝ\nε0 : ε > 0\ni : ℕ\nH : ∀ j ≥ i, ‖↑f j - ↑f i‖ < ε\nj : ℕ\nij : j ≥ i\n⊢ |(fun n ↦ (↑f n).im) j - (fun n ↦ (↑f n).im) i| < ε",
"usedConstants": [
"Norm.norm",
"Real",
"Preorder.toLT",
"Real.lattice",
"AddGroupWithOne.toAddGroup",
"... | by
simpa only [← ofReal_sub, norm_real, sub_re, sub_im] using (abs_im_le_norm _).trans_lt <| H _ ij | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Basic | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 34
} | [
{
"pp": "z w : ℂ\n⊢ dist ((starRingEnd ℂ) z) w = dist z ((starRingEnd ℂ) w)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"congrArg",
"CommSemiring.toSemiring",
"Complex.instNormedField",
"RingHom",
"id",
"RingHom.instFunL... | rw [← dist_conj_conj, conj_conj] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.Basic | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 34
} | [
{
"pp": "z w : ℂ\n⊢ dist ((starRingEnd ℂ) z) w = dist z ((starRingEnd ℂ) w)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"congrArg",
"CommSemiring.toSemiring",
"Complex.instNormedField",
"RingHom",
"id",
"RingHom.instFunL... | rw [← dist_conj_conj, conj_conj] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Basic | {
"line": 230,
"column": 2
} | {
"line": 230,
"column": 34
} | [
{
"pp": "z w : ℂ\n⊢ dist ((starRingEnd ℂ) z) w = dist z ((starRingEnd ℂ) w)",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
"congrArg",
"CommSemiring.toSemiring",
"Complex.instNormedField",
"RingHom",
"id",
"RingHom.instFunL... | rw [← dist_conj_conj, conj_conj] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 265,
"column": 34
} | {
"line": 265,
"column": 65
} | [
{
"pp": "case a.«_@»._internal._hyg.0.«0».«0»\n⊢ (LinearMap.toMatrix basisOneI basisOneI) (↑↑conjAe) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) =\n !![1, 0; 0, -1] ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)",
"usedConstants": [
"Finsupp.instFunLike",
"RingHom.instRingHomClass",
"Complex.i... | simp [LinearMap.toMatrix_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 265,
"column": 34
} | {
"line": 265,
"column": 65
} | [
{
"pp": "case a.«_@»._internal._hyg.0.«0».«1»\n⊢ (LinearMap.toMatrix basisOneI basisOneI) (↑↑conjAe) ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) =\n !![1, 0; 0, -1] ((fun i ↦ i) ⟨0, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)",
"usedConstants": [
"Finsupp.instFunLike",
"Complex.instAlgebraOfReal",
"instNeZe... | simp [LinearMap.toMatrix_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 265,
"column": 34
} | {
"line": 265,
"column": 65
} | [
{
"pp": "case a.«_@»._internal._hyg.0.«1».«0»\n⊢ (LinearMap.toMatrix basisOneI basisOneI) (↑↑conjAe) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩) =\n !![1, 0; 0, -1] ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨0, ⋯⟩)",
"usedConstants": [
"Finsupp.instFunLike",
"RingHom.instRingHomClass",
"Complex.i... | simp [LinearMap.toMatrix_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.Complex.Module | {
"line": 265,
"column": 34
} | {
"line": 265,
"column": 65
} | [
{
"pp": "case a.«_@»._internal._hyg.0.«1».«1»\n⊢ (LinearMap.toMatrix basisOneI basisOneI) (↑↑conjAe) ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩) =\n !![1, 0; 0, -1] ((fun i ↦ i) ⟨1, ⋯⟩) ((fun i ↦ i) ⟨1, ⋯⟩)",
"usedConstants": [
"Finsupp.instFunLike",
"NegZeroClass.toNeg",
"Complex.instAlg... | simp [LinearMap.toMatrix_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.RCLike.Basic | {
"line": 840,
"column": 62
} | {
"line": 841,
"column": 53
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nz : K\n⊢ 0 < z ↔ ∃ x > 0, ↑x = z",
"usedConstants": [
"Eq.mpr",
"Real",
"RCLike.pos_iff",
"Preorder.toLT",
"RCLike.ofReal_re",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"... | by
simp_rw [pos_iff (K := K), ext_iff (K := K)]; aesop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 509,
"column": 17
} | {
"line": 513,
"column": 75
} | [
{
"pp": "α : Type u_1\nE : Type u_3\nF : Type u_4\nG : Type u_5\ninst✝² : Norm E\ninst✝¹ : Norm F\ninst✝ : Norm G\nc : ℝ\nf : α → E\ng : α → F\nk : α → G\nl : Filter α\nhfg : IsBigOWith c l f g\nhgk : g =o[l] k\nhc : 0 < c\n⊢ f =o[l] k",
"usedConstants": [
"Eq.mpr",
"Real.partialOrder",
"R... | by
simp only [IsLittleO_def] at *
intro c' c'pos
have : 0 < c' / c := div_pos c'pos hc
exact (hfg.trans (hgk this) hc.le).congr_const (mul_div_cancel₀ _ hc.ne') | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Asymptotics.Theta | {
"line": 133,
"column": 52
} | {
"line": 133,
"column": 79
} | [
{
"pp": "α : Type u_1\nE' : Type u_6\ninst✝ : SeminormedAddCommGroup E'\nf' : α → E'\nl : Filter α\ng : α → ℝ\nh : (fun x ↦ ‖f' x‖) =ᶠ[l] g\nx : α\nhx : (fun x ↦ ‖f' x‖) x = g x\n⊢ (fun x ↦ ‖f' x‖) x = (fun x ↦ ‖g x‖) x",
"usedConstants": [
"Norm.norm",
"Real",
"congrArg",
"norm_norm... | simp only [← hx, norm_norm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Asymptotics.Theta | {
"line": 133,
"column": 52
} | {
"line": 133,
"column": 79
} | [
{
"pp": "α : Type u_1\nE' : Type u_6\ninst✝ : SeminormedAddCommGroup E'\nf' : α → E'\nl : Filter α\ng : α → ℝ\nh : (fun x ↦ ‖f' x‖) =ᶠ[l] g\nx : α\nhx : (fun x ↦ ‖f' x‖) x = g x\n⊢ (fun x ↦ ‖f' x‖) x = (fun x ↦ ‖g x‖) x",
"usedConstants": [
"Norm.norm",
"Real",
"congrArg",
"norm_norm... | simp only [← hx, norm_norm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.Theta | {
"line": 133,
"column": 52
} | {
"line": 133,
"column": 79
} | [
{
"pp": "α : Type u_1\nE' : Type u_6\ninst✝ : SeminormedAddCommGroup E'\nf' : α → E'\nl : Filter α\ng : α → ℝ\nh : (fun x ↦ ‖f' x‖) =ᶠ[l] g\nx : α\nhx : (fun x ↦ ‖f' x‖) x = g x\n⊢ (fun x ↦ ‖f' x‖) x = (fun x ↦ ‖g x‖) x",
"usedConstants": [
"Norm.norm",
"Real",
"congrArg",
"norm_norm... | simp only [← hx, norm_norm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 319,
"column": 2
} | {
"line": 319,
"column": 19
} | [
{
"pp": "case cons\nα : Type u_1\nR : Type u_13\n𝕜 : Type u_15\ninst✝¹ : SeminormedRing R\ninst✝ : NormedDivisionRing 𝕜\nl : Filter α\nι : Type u_17\nf : ι → α → R\ng : ι → α → 𝕜\ni : ι\nL : List ι\nihL :\n (∀ i ∈ L, f i =O[l] g i) →\n (fun x ↦ (List.map (fun x_1 ↦ f x_1 x) L).prod) =O[l] fun x ↦ (List.m... | | cons i L ihL => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 339,
"column": 2
} | {
"line": 339,
"column": 19
} | [
{
"pp": "case cons\nα : Type u_1\nR : Type u_13\n𝕜 : Type u_15\ninst✝¹ : SeminormedRing R\ninst✝ : NormedDivisionRing 𝕜\nl : Filter α\nι : Type u_17\nf : ι → α → R\ng : ι → α → 𝕜\ni : ι\nL : List ι\nihL :\n (∀ i ∈ L, f i =O[l] g i) →\n (∃ i ∈ L, f i =o[l] g i) →\n (fun x ↦ (List.map (fun x_1 ↦ f x_1... | | cons i L ihL => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Analysis.Complex.Exponential | {
"line": 538,
"column": 2
} | {
"line": 538,
"column": 34
} | [
{
"pp": "x : ℝ\nh1 : 0 ≤ x\nh2 : x ≤ 1\nn : ℕ\nhn : 0 < n\nh3 : |x| = x\n⊢ rexp x ≤ ∑ m ∈ range n, x ^ m / ↑m.factorial + x ^ n * (↑n + 1) / (↑n.factorial * ↑n)",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.lattice",
"abs",
"congrArg",
"id",
"Rea... | have h4 : |x| ≤ 1 := by rwa [h3] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 267,
"column": 2
} | {
"line": 272,
"column": 6
} | [
{
"pp": "x : ℂ\n⊢ sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.NormNum.isNat_add",
"Complex.sinh",
"Mathlib.Tactic.Ring.Common.mul_congr",
... | have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, sinh_add x (2 * x)]
simp only [cosh_two_mul, sinh_two_mul]
have h2 : cosh x * (2 * sinh x * cosh x) = 2 * sinh x * cosh x ^ 2 := by ring
rw [h2, cosh_sq]
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 267,
"column": 2
} | {
"line": 272,
"column": 6
} | [
{
"pp": "x : ℂ\n⊢ sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.NormNum.isNat_add",
"Complex.sinh",
"Mathlib.Tactic.Ring.Common.mul_congr",
... | have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, sinh_add x (2 * x)]
simp only [cosh_two_mul, sinh_two_mul]
have h2 : cosh x * (2 * sinh x * cosh x) = 2 * sinh x * cosh x ^ 2 := by ring
rw [h2, cosh_sq]
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 266,
"column": 71
} | {
"line": 272,
"column": 6
} | [
{
"pp": "x : ℂ\n⊢ sinh (3 * x) = 4 * sinh x ^ 3 + 3 * sinh x",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Meta.NormNum.isNat_add",
"Complex.sinh",
"Mathlib.Tactic.Ring.Common.mul_congr",
... | by
have h1 : x + 2 * x = 3 * x := by ring
rw [← h1, sinh_add x (2 * x)]
simp only [cosh_two_mul, sinh_two_mul]
have h2 : cosh x * (2 * sinh x * cosh x) = 2 * sinh x * cosh x ^ 2 := by ring
rw [h2, cosh_sq]
ring | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 295,
"column": 29
} | {
"line": 295,
"column": 40
} | [
{
"pp": "x : ℂ\n⊢ sinh (x * I) / cosh (x * I) = tan x * I",
"usedConstants": [
"Eq.mpr",
"Complex.sinh",
"instHDiv",
"HMul.hMul",
"Complex.cos",
"congrArg",
"Complex.instDivInvMonoid",
"Complex.instMul",
"id",
"HDiv.hDiv",
"Complex.tan",
... | cosh_mul_I, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 308,
"column": 53
} | {
"line": 308,
"column": 74
} | [
{
"pp": "x : ℂ\n⊢ sinh x / cosh x * I = tanh x * I",
"usedConstants": [
"Complex.tanh",
"Eq.mpr",
"Complex.sinh",
"instHDiv",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
"Monoid.toMulOneClass",
"congrArg",
"Complex.instDivInvMonoid",
"Compl... | tanh_eq_sinh_div_cosh | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 321,
"column": 39
} | {
"line": 321,
"column": 50
} | [
{
"pp": "x y : ℂ\n⊢ cosh (x * I) * cosh (y * I) + sinh (x * I) * sinh (y * I) = cos x * cos y - sin x * sin y",
"usedConstants": [
"Eq.mpr",
"Complex.sinh",
"HMul.hMul",
"Complex.cos",
"congrArg",
"Complex.sin",
"HSub.hSub",
"Complex.instMul",
"id",
... | cosh_mul_I, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 321,
"column": 51
} | {
"line": 321,
"column": 62
} | [
{
"pp": "x y : ℂ\n⊢ cos x * cosh (y * I) + sinh (x * I) * sinh (y * I) = cos x * cos y - sin x * sin y",
"usedConstants": [
"Eq.mpr",
"Complex.sinh",
"HMul.hMul",
"Complex.cos",
"congrArg",
"Complex.sin",
"HSub.hSub",
"Complex.instMul",
"id",
"inst... | cosh_mul_I, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 403,
"column": 53
} | {
"line": 403,
"column": 91
} | [
{
"pp": "x : ℝ\n⊢ (cos ↑x).im = 0",
"usedConstants": [
"Eq.mpr",
"Real",
"Complex.cos",
"Real.instZero",
"congrArg",
"Complex.im",
"Complex.ofReal_cos_ofReal_re",
"id",
"Complex.ofReal",
"Complex.re",
"Zero.toOfNat0",
"Eq.refl",
"... | rw [← ofReal_cos_ofReal_re, ofReal_im] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 403,
"column": 53
} | {
"line": 403,
"column": 91
} | [
{
"pp": "x : ℝ\n⊢ (cos ↑x).im = 0",
"usedConstants": [
"Eq.mpr",
"Real",
"Complex.cos",
"Real.instZero",
"congrArg",
"Complex.im",
"Complex.ofReal_cos_ofReal_re",
"id",
"Complex.ofReal",
"Complex.re",
"Zero.toOfNat0",
"Eq.refl",
"... | rw [← ofReal_cos_ofReal_re, ofReal_im] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 403,
"column": 53
} | {
"line": 403,
"column": 91
} | [
{
"pp": "x : ℝ\n⊢ (cos ↑x).im = 0",
"usedConstants": [
"Eq.mpr",
"Real",
"Complex.cos",
"Real.instZero",
"congrArg",
"Complex.im",
"Complex.ofReal_cos_ofReal_re",
"id",
"Complex.ofReal",
"Complex.re",
"Zero.toOfNat0",
"Eq.refl",
"... | rw [← ofReal_cos_ofReal_re, ofReal_im] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 465,
"column": 19
} | {
"line": 465,
"column": 30
} | [
{
"pp": "x : ℂ\n⊢ sin x ^ 2 + cos x ^ 2 = cosh (x * I) ^ 2 - sinh (x * I) ^ 2",
"usedConstants": [
"Eq.mpr",
"Complex.sinh",
"HMul.hMul",
"Complex.cos",
"congrArg",
"Complex.sin",
"HSub.hSub",
"Complex.instMul",
"id",
"instOfNatNat",
"Monoid.... | cosh_mul_I, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 557,
"column": 78
} | {
"line": 558,
"column": 74
} | [
{
"pp": "x : ℂ\nhx : ‖x‖ ≤ 1\n⊢ ‖cexp (-x * I) - ∑ m ∈ range 4, (-x * I) ^ m / ↑m.factorial‖ / 2 +\n ‖cexp (x * I) - ∑ m ∈ range 4, (x * I) ^ m / ↑m.factorial‖ / 2 ≤\n ‖-x * I‖ ^ 4 * (↑(Nat.succ 4) * (↑(Nat.factorial 4) * ↑4)⁻¹) / 2 +\n ‖x * I‖ ^ 4 * (↑(Nat.succ 4) * (↑(Nat.factorial 4) * ↑4)⁻¹) / ... | by
grw [exp_bound (by simpa) (by simp), exp_bound (by simpa) (by simp)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 573,
"column": 78
} | {
"line": 574,
"column": 74
} | [
{
"pp": "x : ℂ\nhx : ‖x‖ ≤ 1\n⊢ ‖cexp (-x * I) - ∑ m ∈ range 4, (-x * I) ^ m / ↑m.factorial‖ / 2 +\n ‖cexp (x * I) - ∑ m ∈ range 4, (x * I) ^ m / ↑m.factorial‖ / 2 ≤\n ‖-x * I‖ ^ 4 * (↑(Nat.succ 4) * (↑(Nat.factorial 4) * ↑4)⁻¹) / 2 +\n ‖x * I‖ ^ 4 * (↑(Nat.succ 4) * (↑(Nat.factorial 4) * ↑4)⁻¹) / ... | by
grw [exp_bound (by simpa) (by simp), exp_bound (by simpa) (by simp)] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 411,
"column": 2
} | {
"line": 411,
"column": 78
} | [
{
"pp": "α : Type u_1\nl : Filter α\nf : α → ℝ\n⊢ ((fun x ↦ rexp (f x)) =O[l] fun x ↦ 1) ↔ IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l f",
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"Real.lattice",
"abs",
"congrArg",
"Asymptotics.IsBigO",
"NormedDivisio... | simp only [isBigO_one_iff, norm_eq_abs, abs_exp, isBoundedUnder_le_exp_comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 411,
"column": 2
} | {
"line": 411,
"column": 78
} | [
{
"pp": "α : Type u_1\nl : Filter α\nf : α → ℝ\n⊢ ((fun x ↦ rexp (f x)) =O[l] fun x ↦ 1) ↔ IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l f",
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"Real.lattice",
"abs",
"congrArg",
"Asymptotics.IsBigO",
"NormedDivisio... | simp only [isBigO_one_iff, norm_eq_abs, abs_exp, isBoundedUnder_le_exp_comp] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 411,
"column": 2
} | {
"line": 411,
"column": 78
} | [
{
"pp": "α : Type u_1\nl : Filter α\nf : α → ℝ\n⊢ ((fun x ↦ rexp (f x)) =O[l] fun x ↦ 1) ↔ IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) l f",
"usedConstants": [
"Norm.norm",
"Real.instLE",
"Real",
"Real.lattice",
"abs",
"congrArg",
"Asymptotics.IsBigO",
"NormedDivisio... | simp only [isBigO_one_iff, norm_eq_abs, abs_exp, isBoundedUnder_le_exp_comp] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Pointwise | {
"line": 66,
"column": 26
} | {
"line": 66,
"column": 53
} | [
{
"pp": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nx : E\ns : Set E\n⊢ infEDist x⁻¹ ((fun x ↦ x⁻¹) '' s) = infEDist x s",
"usedConstants": [
"NormedGroup.to_isIsometricSMul_right",
"Eq.mpr",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
... | infEDist_image isometry_inv | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Group.Pointwise | {
"line": 214,
"column": 2
} | {
"line": 217,
"column": 52
} | [
{
"pp": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nδ : ℝ\ns : Set E\nhs : IsCompact s\nhδ : 0 ≤ δ\n⊢ s * closedBall 1 δ = cthickening δ s",
"usedConstants": [
"Set.ext",
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"InvOneClass.toOne",
"HMul.hM... | rw [hs.cthickening_eq_biUnion_closedBall hδ]
ext x
simp only [mem_mul, dist_eq_norm_div, exists_prop, mem_iUnion, mem_closedBall,
← eq_div_iff_mul_eq'', div_one, exists_eq_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.Pointwise | {
"line": 214,
"column": 2
} | {
"line": 217,
"column": 52
} | [
{
"pp": "E : Type u_1\ninst✝ : SeminormedCommGroup E\nδ : ℝ\ns : Set E\nhs : IsCompact s\nhδ : 0 ≤ δ\n⊢ s * closedBall 1 δ = cthickening δ s",
"usedConstants": [
"Set.ext",
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real",
"instHDiv",
"InvOneClass.toOne",
"HMul.hM... | rw [hs.cthickening_eq_biUnion_closedBall hδ]
ext x
simp only [mem_mul, dist_eq_norm_div, exists_prop, mem_iUnion, mem_closedBall,
← eq_div_iff_mul_eq'', div_one, exists_eq_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Module.Ball.Pointwise | {
"line": 269,
"column": 4
} | {
"line": 269,
"column": 55
} | [
{
"pp": "E : Type u_2\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nδ ε : ℝ\nhε : 0 < ε\nhδ : 0 < δ\ns : Set E\nx z : E\nhz : z ∈ s\nhxz : dist x z < δ + ε\n⊢ ∃ z, (∃ z_1 ∈ s, dist z z_1 < δ) ∧ dist x z < ε",
"usedConstants": [
"exists_dist_lt_lt"
]
}
] | obtain ⟨y, hxy, hyz⟩ := exists_dist_lt_lt hε hδ hxz | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 406,
"column": 6
} | {
"line": 406,
"column": 17
} | [
{
"pp": "R : 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\n⊢ Summable fun n ↦ ‖↑(u n) * r ^ n‖",
"usedConstants": [
"Norm.norm",
"Real",
"congrArg",
"Real.instLT",
"N... | ← norm_norm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecificLimits.Normed | {
"line": 406,
"column": 2
} | {
"line": 406,
"column": 26
} | [
{
"pp": "R : 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\n⊢ Summable fun n ↦ ‖↑(u n) * r ^ n‖",
"usedConstants": [
"Norm.norm",
"Real",
"congrArg",
"Real.instLT",
"N... | rw [← norm_norm] at hrr' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Path | {
"line": 297,
"column": 4
} | {
"line": 297,
"column": 17
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝ : TopologicalSpace X\nx y z : X\nγ : Path x y\nγ' : Path y z\nt : ↑I\nh✝ : ↑(σ t) ≤ 1 / 2\nh : 1 - ↑t ≤ 1 / 2\nh₁ : ↑t ≤ 1 / 2\nht : ↑t = 1 / 2\n⊢ γ ⟨2 * ↑(σ t), ⋯⟩ = γ' (σ ⟨2 * ↑t, ⋯⟩)",
"usedConstants": [
"Iff.mpr",
"Real.instIsOrderedRing",
"NonAss... | norm_num [ht] | Mathlib.Tactic._aux_Mathlib_Tactic_NormNum_Core___elabRules_Mathlib_Tactic_normNum_1 | Mathlib.Tactic.normNum |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 1265,
"column": 66
} | {
"line": 1266,
"column": 49
} | [
{
"pp": "⊢ Function.Antiperiodic sinh (↑π * I)",
"usedConstants": [
"Complex.sinh",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMul.hMul",
"Complex.cos",
"congrArg",
"MulZeroClass.zero_mul",
"AddMonoid.toAddZeroClass",
"Complex.sin",
... | by
simp [Complex.sinh_add, sinh_mul_I, cosh_mul_I] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Path | {
"line": 541,
"column": 6
} | {
"line": 541,
"column": 59
} | [
{
"pp": "case pos\nX✝ : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\nx y z : X✝\nι : Type u_3\nγ✝ : Path x y\nX : Type u_4\ninst✝ : TopologicalSpace X\na b : X\nγ : Path a b\nt₀ t₁ : ℝ\nh₁ : t₀ ≤ ↑0\nh₂ : ¬↑0 ≤ t₁\nh₄✝ : t₀ ≤ t₁\nh₄ : t₁ ≤ 0\n⊢ γ.extend t₁ = γ.extend t₀",
... | simp [γ.extend_of_le_zero h₄, γ.extend_of_le_zero h₁] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Group.AddCircle | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 87
} | [
{
"pp": "p : ℝ\nhp : p ≠ 0\nx✝ : AddCircle p\nε : ℝ\nhε : |p| / 2 ≤ ε\nx : AddCircle p\n⊢ x ∈ closedBall x✝ ε",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
"instHDiv",
"dist_eq_norm",
"QuotientAddGrou... | simpa only [mem_closedBall, dist_eq_norm] using (norm_le_half_period p hp).trans hε | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Group.AddCircle | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 87
} | [
{
"pp": "p : ℝ\nhp : p ≠ 0\nx✝ : AddCircle p\nε : ℝ\nhε : |p| / 2 ≤ ε\nx : AddCircle p\n⊢ x ∈ closedBall x✝ ε",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
"instHDiv",
"dist_eq_norm",
"QuotientAddGrou... | simpa only [mem_closedBall, dist_eq_norm] using (norm_le_half_period p hp).trans hε | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.AddCircle | {
"line": 133,
"column": 4
} | {
"line": 133,
"column": 87
} | [
{
"pp": "p : ℝ\nhp : p ≠ 0\nx✝ : AddCircle p\nε : ℝ\nhε : |p| / 2 ≤ ε\nx : AddCircle p\n⊢ x ∈ closedBall x✝ ε",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
"instHDiv",
"dist_eq_norm",
"QuotientAddGrou... | simpa only [mem_closedBall, dist_eq_norm] using (norm_le_half_period p hp).trans hε | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Inverse | {
"line": 360,
"column": 2
} | {
"line": 360,
"column": 70
} | [
{
"pp": "case pos\nx : ℝ\nhx₁ : -1 ≤ x\nhx₂ : x ≤ 1\n⊢ sin (arccos x) = √(1 - x ^ 2)",
"usedConstants": [
"Eq.mpr",
"Real",
"instHDiv",
"Real.pi",
"Real.arcsin",
"Real.cos",
"Real.cos_arcsin",
"congrArg",
"Real.instDivInvMonoid",
"Real.instSub",
... | rw [arccos_eq_pi_div_two_sub_arcsin, sin_pi_div_two_sub, cos_arcsin] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 249,
"column": 2
} | {
"line": 251,
"column": 13
} | [
{
"pp": "case neg.mp\nz : ℂ\nh₀ : ¬z = 0\n⊢ z.arg = π → z.re < 0 ∧ z.im = 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"False",
"Real.partialOrder",
"Real",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Real.pi",
"HMu... | · intro h
rw [← norm_mul_cos_add_sin_mul_I z, h]
simp [h₀] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 269,
"column": 2
} | {
"line": 271,
"column": 13
} | [
{
"pp": "case neg.mp\nz : ℂ\nh₀ : ¬z = 0\n⊢ z.arg = π / 2 → z.re = 0 ∧ 0 < z.im",
"usedConstants": [
"Complex.mul_im",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"False",
"Real",
"instHDiv",
"Complex.mul_re",
"Real.pi",
"HMul.hMul",
... | · intro h
rw [← norm_mul_cos_add_sin_mul_I z, h]
simp [h₀] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 279,
"column": 2
} | {
"line": 281,
"column": 13
} | [
{
"pp": "case neg.mp\nz : ℂ\nh₀ : ¬z = 0\n⊢ z.arg = -(π / 2) → z.re = 0 ∧ z.im < 0",
"usedConstants": [
"Complex.mul_im",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"False",
"Real.partialOrder",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUni... | · intro h
rw [← norm_mul_cos_add_sin_mul_I z, h]
simp [h₀] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 282,
"column": 4
} | {
"line": 285,
"column": 8
} | [
{
"pp": "case neg.mpr\nz : ℂ\nh₀ : ¬z = 0\n⊢ z.re = 0 ∧ z.im < 0 → z.arg = -(π / 2)",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Complex.mk_eq_add_mul_I",
"Eq.mpr",
"NegZeroClass.toNeg",
"Complex.arg_neg_I",
"Real.partialOrder",
"Real",
... | obtain ⟨x, y⟩ := z
rintro ⟨rfl : x = 0, hy : y < 0⟩
rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I]
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 282,
"column": 4
} | {
"line": 285,
"column": 8
} | [
{
"pp": "case neg.mpr\nz : ℂ\nh₀ : ¬z = 0\n⊢ z.re = 0 ∧ z.im < 0 → z.arg = -(π / 2)",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Complex.mk_eq_add_mul_I",
"Eq.mpr",
"NegZeroClass.toNeg",
"Complex.arg_neg_I",
"Real.partialOrder",
"Real",
... | obtain ⟨x, y⟩ := z
rintro ⟨rfl : x = 0, hy : y < 0⟩
rw [← arg_neg_I, ← arg_real_mul (-I) (neg_pos.2 hy), mk_eq_add_mul_I]
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Complex.Log | {
"line": 64,
"column": 6
} | {
"line": 64,
"column": 33
} | [
{
"pp": "x : ℂ\n⊢ log (cexp x) = x - ↑(toIocDiv Real.two_pi_pos (-π) x.im) * (2 * ↑π * I)",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Complex.log",
"Real",
"Real.instArchimedean",
"Real.pi",
"HMul.hMul",
"congrArg",
"Complex.log_exp_eq_re_add_toIocMod",
... | log_exp_eq_re_add_toIocMod, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 183,
"column": 6
} | {
"line": 183,
"column": 15
} | [
{
"pp": "case inr\nx : ℝ\nhx✝ : 0 ≤ x\nhx : 0 < x\n⊢ 0 < log x ↔ 1 < x",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Real.instLT",
"Real.log_one",
"id",
"Real.log",
"Real.instOne",
"Iff",
"LT.lt",
"One.toOfNat1",
... | ← log_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 196,
"column": 6
} | {
"line": 196,
"column": 15
} | [
{
"pp": "x : ℝ\nh : 0 < x\n⊢ log x < 0 ↔ x < 1",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Real.instLT",
"Real.log_one",
"id",
"Real.log",
"Real.instOne",
"Iff",
"LT.lt",
"One.toOfNat1",
"Zero.toOfNat0",
... | ← log_one | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Log | {
"line": 126,
"column": 90
} | {
"line": 135,
"column": 21
} | [
{
"pp": "x : ℂ\n⊢ log x⁻¹ = if x.arg = π then -(starRingEnd ℂ) (log x) else -log x",
"usedConstants": [
"Complex.log_conj_eq_ite",
"Complex.normSq_pos",
"Norm.norm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg",
"RingHom.instRingHomClass",
... | by
by_cases hx : x = 0
· simp [hx]
rw [inv_def, log_mul_ofReal, Real.log_inv, ofReal_neg, ← sub_eq_neg_add, log_conj_eq_ite]
· simp_rw [log, map_add, map_mul, conj_ofReal, conj_I, normSq_eq_norm_sq, Real.log_pow,
Nat.cast_two, ofReal_mul, neg_add, mul_neg, neg_neg]
norm_num
grind
· rwa [inv_pos,... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 411,
"column": 2
} | {
"line": 412,
"column": 52
} | [
{
"pp": "α : Type u_1\nf : α → ℝ\nh : ∀ (a : α), 0 < f a\n⊢ log (∏ᶠ (a : α), f a) = ∑ᶠ (a : α), log (f a)",
"usedConstants": [
"Real",
"Real.instZero",
"Real.log",
"Real.instOne",
"Function.support",
"Function.mulSupport",
"_private.Mathlib.Analysis.SpecialFunctions... | have H : (fun i ↦ log (f i)).support = f.mulSupport := by
grind [mem_mulSupport, mem_support, log_eq_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 920,
"column": 2
} | {
"line": 920,
"column": 69
} | [
{
"pp": "θ ψ : Angle\nhθs : θ.toReal ∈ Set.Ioo 0 π\nhψs : ψ.toReal ∈ Set.Ioo 0 π\nhsa : (θ + ψ).sign ≠ 1\n⊢ |(θ + ψ).toReal| = 2 * π - (|θ.toReal| + |ψ.toReal|)",
"usedConstants": [
"Real",
"Real.Angle",
"Real.Angle.coe",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"Real.... | have : ((θ + ψ).toReal : Angle) = ↑(θ.toReal + ψ.toReal) := by simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 173,
"column": 2
} | {
"line": 177,
"column": 66
} | [
{
"pp": "x y : ℝ\n⊢ |x ^ y| ≤ |x| ^ y",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.abs_cos_le_one",
"le_refl",
"Real.instPow",
"Real.partialOrder",
"Real.instLE",
... | rcases le_or_gt 0 x with hx | hx
· rw [abs_rpow_of_nonneg hx]
· rw [abs_of_neg hx, rpow_def_of_neg hx, rpow_def_of_pos (neg_pos.2 hx), log_neg_eq_log, abs_mul,
abs_of_pos (exp_pos _)]
exact mul_le_of_le_one_right (exp_pos _).le (abs_cos_le_one _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 173,
"column": 2
} | {
"line": 177,
"column": 66
} | [
{
"pp": "x y : ℝ\n⊢ |x ^ y| ≤ |x| ^ y",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.abs_cos_le_one",
"le_refl",
"Real.instPow",
"Real.partialOrder",
"Real.instLE",
... | rcases le_or_gt 0 x with hx | hx
· rw [abs_rpow_of_nonneg hx]
· rw [abs_of_neg hx, rpow_def_of_neg hx, rpow_def_of_pos (neg_pos.2 hx), log_neg_eq_log, abs_mul,
abs_of_pos (exp_pos _)]
exact mul_le_of_le_one_right (exp_pos _).le (abs_cos_le_one _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 425,
"column": 26
} | {
"line": 425,
"column": 45
} | [
{
"pp": "x : ℝ\nhx : x ≠ 0\ny : ℝ\nn : ℤ\n⊢ (↑x ^ ↑(y + ↑n)).re = (↑x ^ ↑y).re * x ^ n",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Real",
"HMul.hMul",
"congrArg",
"Real.instDivInvMonoid",
"DivInvMonoid.toZPow",
"Complex.instPow",
"Complex.ofReal_add",
... | Complex.ofReal_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 729,
"column": 2
} | {
"line": 729,
"column": 36
} | [
{
"pp": "x y p : ℝ\nhx : 0 ≤ x\nhy : 0 ≤ y\nhp : 0 ≤ p\n⊢ max x y ^ p = max (x ^ p) (y ^ p)",
"usedConstants": [
"Real",
"le_total",
"Real.linearOrder"
]
}
] | rcases le_total x y with hxy | hxy | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 927,
"column": 4
} | {
"line": 927,
"column": 48
} | [
{
"pp": "case inl\nx y z : ℝ\nx1 : 1 ≤ x\nyz : y ≤ z\n⊢ x ^ y ≤ x ^ z",
"usedConstants": [
"Real.rpow_le_rpow_of_exponent_le"
]
}
] | exact Real.rpow_le_rpow_of_exponent_le x1 yz | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 927,
"column": 4
} | {
"line": 927,
"column": 48
} | [
{
"pp": "case inl\nx y z : ℝ\nx1 : 1 ≤ x\nyz : y ≤ z\n⊢ x ^ y ≤ x ^ z",
"usedConstants": [
"Real.rpow_le_rpow_of_exponent_le"
]
}
] | exact Real.rpow_le_rpow_of_exponent_le x1 yz | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 927,
"column": 4
} | {
"line": 927,
"column": 48
} | [
{
"pp": "case inl\nx y z : ℝ\nx1 : 1 ≤ x\nyz : y ≤ z\n⊢ x ^ y ≤ x ^ z",
"usedConstants": [
"Real.rpow_le_rpow_of_exponent_le"
]
}
] | exact Real.rpow_le_rpow_of_exponent_le x1 yz | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 1030,
"column": 2
} | {
"line": 1032,
"column": 12
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\nx : ℝ\n⊢ 0 < (↑n ^ ↑x)⁻¹",
"usedConstants": [
"Iff.mpr",
"zero_le",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"Real.instPow",
"Nat.instMulZeroClass",
"Real.partialOrder",
"Real.rpow_pos_of_pos",
"Real",
"Preorder.toLT",... | refine RCLike.inv_pos_of_pos ?_
rw [show (n : ℂ) ^ (x : ℂ) = (n : ℝ) ^ (x : ℂ) from rfl, ← ofReal_cpow n.cast_nonneg']
positivity | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 1030,
"column": 2
} | {
"line": 1032,
"column": 12
} | [
{
"pp": "n : ℕ\nhn : n ≠ 0\nx : ℝ\n⊢ 0 < (↑n ^ ↑x)⁻¹",
"usedConstants": [
"Iff.mpr",
"zero_le",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"Real.instPow",
"Nat.instMulZeroClass",
"Real.partialOrder",
"Real.rpow_pos_of_pos",
"Real",
"Preorder.toLT",... | refine RCLike.inv_pos_of_pos ?_
rw [show (n : ℂ) ^ (x : ℂ) = (n : ℝ) ^ (x : ℂ) from rfl, ← ofReal_cpow n.cast_nonneg']
positivity | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 811,
"column": 2
} | {
"line": 811,
"column": 36
} | [
{
"pp": "x y : ℝ≥0∞\np : ℝ\nhp : 0 ≤ p\n⊢ max x y ^ p = max (x ^ p) (y ^ p)",
"usedConstants": [
"le_total",
"ENNReal.instLinearOrder",
"ENNReal"
]
}
] | rcases le_total x y with hxy | hxy | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 866,
"column": 2
} | {
"line": 868,
"column": 75
} | [
{
"pp": "case neg\ny z : ℝ\nhyz : z ≤ y\nx : ℝ≥0\nhx1 : ↑x ≤ 1\nh : ¬x = 0\n⊢ ↑x ^ y ≤ ↑x ^ z",
"usedConstants": [
"Iff.mpr",
"Real",
"ENNReal.ofNNReal",
"Preorder.toLT",
"congrArg",
"ENNReal.instPowReal",
"OrderBot.toBot",
"PartialOrder.toPreorder",
"Pr... | · rw [coe_le_one_iff] at hx1
simp [← coe_rpow_of_ne_zero h,
NNReal.rpow_le_rpow_of_exponent_ge (bot_lt_iff_ne_bot.mpr h) hx1 hyz] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 874,
"column": 97
} | {
"line": 876,
"column": 55
} | [
{
"pp": "x : ℝ≥0∞\nz : ℝ\nhx : 1 ≤ x\nh_one_le : 1 ≤ z\n⊢ x ≤ x ^ z",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"ENNReal.instPowReal",
"ENNReal.rpow_one",
"ENNReal.rpow_le_rpow_of_exponent_le",
"id",
"LE.le",
"Real.instOne",
"HPow.hPow",
... | by
nth_rw 1 [← ENNReal.rpow_one x]
exact ENNReal.rpow_le_rpow_of_exponent_le hx h_one_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 889,
"column": 4
} | {
"line": 890,
"column": 78
} | [
{
"pp": "case inr\np : ℝ\nx : ℝ≥0∞\nhx_pos : 0 < x\nhx_ne_top : x ≠ ∞\nhp_nonpos : p ≤ 0\n⊢ 0 < x ^ p",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instLE",
"Real",
"Preorder.toLT",
"NonUnitalCommRing.toN... | rw [← neg_neg p, rpow_neg, ENNReal.inv_pos]
exact rpow_ne_top_of_nonneg (Right.nonneg_neg_iff.mpr hp_nonpos) hx_ne_top | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 889,
"column": 4
} | {
"line": 890,
"column": 78
} | [
{
"pp": "case inr\np : ℝ\nx : ℝ≥0∞\nhx_pos : 0 < x\nhx_ne_top : x ≠ ∞\nhp_nonpos : p ≤ 0\n⊢ 0 < x ^ p",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instLE",
"Real",
"Preorder.toLT",
"NonUnitalCommRing.toN... | rw [← neg_neg p, rpow_neg, ENNReal.inv_pos]
exact rpow_ne_top_of_nonneg (Right.nonneg_neg_iff.mpr hp_nonpos) hx_ne_top | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Segment | {
"line": 296,
"column": 2
} | {
"line": 296,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddRightMono 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nc x y : E\nh : LinearIndependent 𝕜 ![x - c, y - c]\np : 𝕜\np0 : 0 ≤ p\np1 : p ≤ 1\nq : 𝕜\nH : (1 - q) • c + q • y = (1 - p) • c + p • x\nq0 : 0 ≤ q\nq1 : q... | have Hx : x = (x - c) + c := by abel | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.Segment | {
"line": 524,
"column": 4
} | {
"line": 525,
"column": 26
} | [
{
"pp": "case inl\n𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nx z : 𝕜\nhxz : x ≤ z\nhyz : z ≤ x\n⊢ z ∈ [x -[𝕜] x]",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"instSMulOfMul",
... | rw [segment_same]
exact hyz.antisymm hxz | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Segment | {
"line": 524,
"column": 4
} | {
"line": 525,
"column": 26
} | [
{
"pp": "case inl\n𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nx z : 𝕜\nhxz : x ≤ z\nhyz : z ≤ x\n⊢ z ∈ [x -[𝕜] x]",
"usedConstants": [
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"Semiring.toModule",
"instSMulOfMul",
... | rw [segment_same]
exact hyz.antisymm hxz | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 215,
"column": 4
} | {
"line": 215,
"column": 44
} | [
{
"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\nh : (↑s).Nonempty\nc : k\np₁ : P\nhp₁ : p₁ ∈ ↑s\np₂ : P\nhp₂ : p₂ ∈ ↑s\n⊢ c • (p₁ -ᵥ p₂) +ᵥ p₂ ∈ ↑s",
"usedConstants": [
"AffineSubspace.sm... | exact s.smul_vsub_vadd_mem c hp₁ hp₂ hp₂ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 301,
"column": 25
} | {
"line": 301,
"column": 58
} | [
{
"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 ↔ ∃ p₂ ∈ s, v = p -ᵥ p₂",
"usedConstants": [
"Eq.mpr",
"Submodule",
"congrArg",
... | coe_direction_eq_vsub_set_left hp | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Star | {
"line": 332,
"column": 2
} | {
"line": 332,
"column": 37
} | [
{
"pp": "case a\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddRightMono 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx y : E\ns : Set E\nhs : StarConvex 𝕜 x s\nhy : y ∈ s\nt : 𝕜\nht₀ : 0 ≤ t\nht₁ : t ≤ 1\n⊢ x + t • (y - x) ∈ (fun θ ↦ x + θ • (y - x)) '' Icc 0 1",
... | exact mem_image_of_mem _ ⟨ht₀, ht₁⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.LocallyConvex.BalancedCoreHull | {
"line": 226,
"column": 4
} | {
"line": 226,
"column": 33
} | [
{
"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\n⊢ IsClosed[inst✝¹] (balancedCore 𝕜 U)",
"usedConstants": [
"Norm.norm... | rw [balancedCore_eq_iInter h] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.LocallyConvex.BalancedCoreHull | {
"line": 273,
"column": 2
} | {
"line": 273,
"column": 89
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : NormedDivisionRing 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : TopologicalSpace E\ninst✝² : ContinuousSMul 𝕜 E\ninst✝¹ : (𝓝[≠] 0).NeBot\ninst✝ : RegularSpace E\ns : Set E\nhs : s ∈ 𝓝 0 ∧ IsClosed[inst✝³] s\n⊢ ∃ i', (i' ∈ 𝓝 0 ∧ IsClosed[inst✝³] i... | refine ⟨balancedCore 𝕜 s, ⟨balancedCore_mem_nhds_zero hs.1, ?_⟩, balancedCore_subset s⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.LocallyConvex.Bounded | {
"line": 468,
"column": 2
} | {
"line": 468,
"column": 38
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝² : NormedField 𝕜\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\ns : Set E\nh : Bornology.IsBounded s\nr : ℝ\nhr : s ⊆ Metric.ball 0 r\n⊢ ∀ (i : ℝ), 0 < i → Absorbs 𝕜 (Metric.ball 0 i) s",
"usedConstants": [
"NormedCommRing.toSeminormedCommRi... | rw [← ball_normSeminorm 𝕜 E] at hr ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.FilterBasis | {
"line": 148,
"column": 2
} | {
"line": 148,
"column": 91
} | [
{
"pp": "G : Type u\ninst✝ : Group G\nB : GroupFilterBasis G\nx₀ : G\n⊢ 𝓝 x₀ = B.N x₀",
"usedConstants": [
"HMul.hMul",
"instMembershipSetFilterBasis",
"Monoid.toMulOneClass",
"Filter.map",
"FilterBasis.hasBasis",
"Membership.mem",
"id",
"MulOne.toMul",
... | apply TopologicalSpace.nhds_mkOfNhds_of_hasBasis (fun x ↦ (FilterBasis.hasBasis _).map _) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Convex.Strict | {
"line": 149,
"column": 52
} | {
"line": 149,
"column": 68
} | [
{
"pp": "case inr\n𝕜 : Type u_1\nβ : Type u_5\ninst✝⁸ : Semiring 𝕜\ninst✝⁷ : PartialOrder 𝕜\ninst✝⁶ : TopologicalSpace β\ninst✝⁵ : AddCommMonoid β\ninst✝⁴ : LinearOrder β\ninst✝³ : IsOrderedCancelAddMonoid β\ninst✝² : OrderTopology β\ninst✝¹ : Module 𝕜 β\ninst✝ : PosSMulStrictMono 𝕜 β\ns : Set β\nhs : s.Or... | openSegment_symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Strict | {
"line": 319,
"column": 2
} | {
"line": 319,
"column": 37
} | [
{
"pp": "case a\n𝕜 : Type u_1\nE : Type u_3\ninst✝⁵ : Ring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : TopologicalSpace E\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns : Set E\nx y : E\ninst✝ : AddRightMono 𝕜\nh : StrictConvex 𝕜 s\nhx : x ∈ s\nhy : y ∈ s\nhxy : x ≠ y\nt : 𝕜\nht₀ : 0 < t\nht₁ : t < 1\n⊢ x + t... | exact mem_image_of_mem _ ⟨ht₀, ht₁⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 201,
"column": 31
} | {
"line": 201,
"column": 60
} | [
{
"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\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\ninst✝ : DecidableEq ι\ns₂ : Finset ι\nh : s₂ ⊆ s\nw : ι → k\np : ι → P\nb : P\n⊢ ((s \\ s₂).weightedVSubOfPoint p b) w + (s₂.weightedVSubOfPoint p b)... | s.weightedVSubOfPoint_sdiff h | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Centroid | {
"line": 114,
"column": 59
} | {
"line": 116,
"column": 32
} | [
{
"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\ninst✝ : Invertible 2\np : Fin 2 → P\n⊢ centroid k univ p = 2⁻¹ • (p 1 -ᵥ p 0) +ᵥ p 0",
"usedConstants": [
"Eq.mpr",
"instHSMul",
"GroupWithZer... | by
rw [univ_fin2]
convert! centroid_pair k p 0 1 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 436,
"column": 2
} | {
"line": 436,
"column": 23
} | [
{
"pp": "case h.e'_2.h.e'_5\nk : 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\np : ι → P\ni : ι\nhis : i ∈ s\nhwi : w i = 1\nhw0 : ∀ i2 ∈ s, i2 ≠ i → w i2 = 0\nh1 : ∑ i ∈ s, w i = 1\n⊢ ∑ i_1 ∈ s, w ... | refine sum_eq_zero ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 782,
"column": 4
} | {
"line": 782,
"column": 95
} | [
{
"pp": "ι : Type u_1\nk : Type u_2\nV : Type u_3\nP : Type u_4\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : Nontrivial k\ns : Finset ι\nw : ι → k\nh : ∑ i ∈ s, w i = 1\np : ι → P\nhnz : ∑ i ∈ s, w i ≠ 0\ni1 : ι\nhi1 : i1 ∈ s\nw1 : ι → k := Function.update (F... | exact AffineSubspace.vadd_mem_of_mem_direction hv (mem_affineSpan k (Set.mem_range_self _)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Seminorm | {
"line": 118,
"column": 4
} | {
"line": 118,
"column": 23
} | [
{
"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 : SeminormedRing 𝕜\ninst✝¹ : AddGroup E\ninst✝ : SMul 𝕜 E\nf g : Seminorm 𝕜 E\nh : (fun f ↦ f.toFun) f = (fun f ↦ f.toFun) g\n⊢ ... | rcases f with ⟨⟨_⟩⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 831,
"column": 8
} | {
"line": 834,
"column": 12
} | [
{
"pp": "case mp.inl\nι : Type u_1\nk : Type u_2\nV : Type u_3\nP : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nv : V\np : ι → P\nhι : IsEmpty ι\n⊢ v ∈ vectorSpan k (Set.range p) → ∃ s w, ∑ i ∈ s, w i = 0 ∧ v = (s.weightedVSub p) w",
"usedConstants": [
... | rw [Set.range_eq_empty, vectorSpan_empty, Submodule.mem_bot]
rintro rfl
use ∅
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 831,
"column": 8
} | {
"line": 834,
"column": 12
} | [
{
"pp": "case mp.inl\nι : Type u_1\nk : Type u_2\nV : Type u_3\nP : Type u_4\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nv : V\np : ι → P\nhι : IsEmpty ι\n⊢ v ∈ vectorSpan k (Set.range p) → ∃ s w, ∑ i ∈ s, w i = 0 ∧ v = (s.weightedVSub p) w",
"usedConstants": [
... | rw [Set.range_eq_empty, vectorSpan_empty, Submodule.mem_bot]
rintro rfl
use ∅
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Seminorm | {
"line": 381,
"column": 2
} | {
"line": 383,
"column": 23
} | [
{
"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\nx : E\na : ℝ\nha : 0 ≤ a\nh : ∀ i ∈ s, (p i) x ≤ a\n⊢ (s.sup p) x ≤ a",
"usedConstants": [
"Seminorm.instSeminormClass",
"Eq.mpr",
... | lift a to ℝ≥0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Seminorm | {
"line": 381,
"column": 2
} | {
"line": 383,
"column": 23
} | [
{
"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\nx : E\na : ℝ\nha : 0 ≤ a\nh : ∀ i ∈ s, (p i) x ≤ a\n⊢ (s.sup p) x ≤ a",
"usedConstants": [
"Seminorm.instSeminormClass",
"Eq.mpr",
... | lift a to ℝ≥0 using ha
rw [finset_sup_apply, NNReal.coe_le_coe]
exact Finset.sup_le h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 961,
"column": 4
} | {
"line": 961,
"column": 40
} | [
{
"pp": "case h.e'_5\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\np₁ p₂ p₃ p₄ : P\nhp₁₂ : p₁ ≠ p₂\nr₁ : k\nhp₁ : r₁ • (p₄ -ᵥ p₃) = p₁ -ᵥ p₃\nr₂ : k\nhp₂ : r₂ • (p₄ -ᵥ p₃) = p₂ -ᵥ p₃\nhr₀ : r₂ - r₁ ≠ 0\nhr : p₄ -ᵥ p₃ = ... | simp [mul_smul, ← hr, sub_smul, hp₁] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.