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.Analysis.Complex.Basic | {
"line": 403,
"column": 2
} | {
"line": 404,
"column": 10
} | [
{
"pp": "𝕜 : Type u_2\n𝕜' : Type u_3\ninst✝¹ : RCLike 𝕜\ninst✝ : RCLike 𝕜'\nh : RCLike.im RCLike.I = 1\na : 𝕜\n⊢ 0 ≤ (RCLike.map 𝕜 𝕜') a ↔ 0 ≤ a",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
"NonUnitalCommRing.toNonUnitalNon... | rw [RCLike.nonneg_iff, RCLike.nonneg_iff (K := 𝕜)]
simp [h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Basic | {
"line": 403,
"column": 2
} | {
"line": 404,
"column": 10
} | [
{
"pp": "𝕜 : Type u_2\n𝕜' : Type u_3\ninst✝¹ : RCLike 𝕜\ninst✝ : RCLike 𝕜'\nh : RCLike.im RCLike.I = 1\na : 𝕜\n⊢ 0 ≤ (RCLike.map 𝕜 𝕜') a ↔ 0 ≤ a",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.instLE",
"Real",
"NonUnitalCommRing.toNonUnitalNon... | rw [RCLike.nonneg_iff, RCLike.nonneg_iff (K := 𝕜)]
simp [h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.OpenPartialHomeomorph.Basic | {
"line": 116,
"column": 6
} | {
"line": 116,
"column": 46
} | [
{
"pp": "X : Type u_1\nY : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\ne : OpenPartialHomeomorph X Y\nt : Set Y\nhs : t ⊆ e.target\nh : IsOpen[inst✝¹] (↑e.symm '' t)\nhs' : ↑e.symm '' t ⊆ e.source\n⊢ IsOpen[inst✝] t",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Partial... | ← e.image_symm_image_of_subset_target hs | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.RCLike.Basic | {
"line": 642,
"column": 59
} | {
"line": 642,
"column": 96
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nm : ℕ\ns : Bool\ne : ℕ\n⊢ re (ofScientific m s e) = ofScientific m s e",
"usedConstants": [
"Real.instNNRatCast",
"Eq.mpr",
"Real",
"NNRatCast.toOfScientific",
"RCLike.ofReal_re",
"AddMonoid.toAddSemigroup",
"Real.instAddMono... | rw [← ofReal_ofScientific, ofReal_re] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 642,
"column": 59
} | {
"line": 642,
"column": 96
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nm : ℕ\ns : Bool\ne : ℕ\n⊢ re (ofScientific m s e) = ofScientific m s e",
"usedConstants": [
"Real.instNNRatCast",
"Eq.mpr",
"Real",
"NNRatCast.toOfScientific",
"RCLike.ofReal_re",
"AddMonoid.toAddSemigroup",
"Real.instAddMono... | rw [← ofReal_ofScientific, ofReal_re] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.RCLike.Basic | {
"line": 642,
"column": 59
} | {
"line": 642,
"column": 96
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nm : ℕ\ns : Bool\ne : ℕ\n⊢ re (ofScientific m s e) = ofScientific m s e",
"usedConstants": [
"Real.instNNRatCast",
"Eq.mpr",
"Real",
"NNRatCast.toOfScientific",
"RCLike.ofReal_re",
"AddMonoid.toAddSemigroup",
"Real.instAddMono... | rw [← ofReal_ofScientific, ofReal_re] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 861,
"column": 2
} | {
"line": 861,
"column": 38
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx : ℝ\n⊢ 0 ≤ ↑x ↔ 0 ≤ x",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"RCLike.ofReal_le_ofReal",
"Preorder.toLE",
"NormedField.toFie... | rw [← ofReal_zero, ofReal_le_ofReal] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 861,
"column": 2
} | {
"line": 861,
"column": 38
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx : ℝ\n⊢ 0 ≤ ↑x ↔ 0 ≤ x",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"RCLike.ofReal_le_ofReal",
"Preorder.toLE",
"NormedField.toFie... | rw [← ofReal_zero, ofReal_le_ofReal] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.RCLike.Basic | {
"line": 861,
"column": 2
} | {
"line": 861,
"column": 38
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx : ℝ\n⊢ 0 ≤ ↑x ↔ 0 ≤ x",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"RCLike.ofReal_le_ofReal",
"Preorder.toLE",
"NormedField.toFie... | rw [← ofReal_zero, ofReal_le_ofReal] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 865,
"column": 2
} | {
"line": 865,
"column": 38
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx : ℝ\n⊢ ↑x ≤ 0 ↔ x ≤ 0",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"RCLike.ofReal_le_ofReal",
"Preorder.toLE",
"NormedField.toFie... | rw [← ofReal_zero, ofReal_le_ofReal] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 865,
"column": 2
} | {
"line": 865,
"column": 38
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx : ℝ\n⊢ ↑x ≤ 0 ↔ x ≤ 0",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"RCLike.ofReal_le_ofReal",
"Preorder.toLE",
"NormedField.toFie... | rw [← ofReal_zero, ofReal_le_ofReal] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.RCLike.Basic | {
"line": 865,
"column": 2
} | {
"line": 865,
"column": 38
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx : ℝ\n⊢ ↑x ≤ 0 ↔ x ≤ 0",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
"Iff.rfl",
"PartialOrder.toPreorder",
"RCLike.ofReal_le_ofReal",
"Preorder.toLE",
"NormedField.toFie... | rw [← ofReal_zero, ofReal_le_ofReal] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.RCLike.Basic | {
"line": 889,
"column": 2
} | {
"line": 889,
"column": 25
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\nx : K\nhx : 0 ≤ x\n⊢ √(re x * re x + im x * im x) ≤ re x",
"usedConstants": [
"Real.instLE",
"Real",
"HMul.hMul",
"AddMonoid.toAddSemigroup",
"Real.instZero",
"Real.instAddMonoid",
"instReflLe",
"congrArg",
"AddMo... | simp [nonneg_iff.mp hx] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.RCLike.Basic | {
"line": 967,
"column": 45
} | {
"line": 969,
"column": 61
} | [
{
"pp": "K : Type u_1\ninst✝ : RCLike K\na : K\nha : 0 < a\nr₁ r₂ : ℝ\nhr : r₁ < r₂\n⊢ r₁ • a < r₂ • a",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
"Real",
"instHSMul",
"Preorder.toLT",
"AddMonoidHom.instAddMonoidHomClas... | by
obtain ⟨hare, haim⟩ := RCLike.lt_iff_re_im.1 ha
simp_all [RCLike.lt_iff_re_im (K := K), smul_re, smul_im] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.RCLike.Basic | {
"line": 1318,
"column": 32
} | {
"line": 1318,
"column": 52
} | [
{
"pp": "K : Type u_1\nE : Type u_2\ninst✝ : RCLike K\n𝕜 : Type u_3\nh : RCLike 𝕜\n__spread✝⁻⁰ : NormedField 𝕜 := h.toNormedField\n⊢ ∀ (r : ℝ),\n (⋯ ▸\n let __spread.0 := h.toNormedField;\n im)\n ({ toFun := ⇑(algebraMap ℝ 𝕜), map_one' := ⋯, map_mul' := ⋯, map_zero' := ⋯, map_add... | exact h.ofReal_im_ax | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 293,
"column": 6
} | {
"line": 293,
"column": 63
} | [
{
"pp": "case mp.zero\nα : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝¹ : Norm E\ninst✝ : Norm F\nf : α → E\ng : α → F\nl : Filter α\nh₀ : (∀ (x : α), 0 ≤ ‖f x‖) ∨ ∀ (x : α), 0 ≤ ‖g x‖\nH : f =o[l] g\nx : α\nh₀' : ‖f x‖ ≤ 1 * ‖g x‖\n⊢ 0 ≤ ‖g x‖",
"usedConstants": [
"Norm.norm",
"Real.instLE",
... | refine h₀.elim (fun hf => (hf x).trans ?_) fun hg => hg x | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 493,
"column": 2
} | {
"line": 493,
"column": 32
} | [
{
"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 : f =o[l] g\nhgk : IsBigOWith c l g k\nhc : 0 < c\n⊢ f =o[l] k",
"usedConstants": [
"Eq.mpr",
"Real",
"Asymptotics.Is... | simp only [IsLittleO_def] at * | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 510,
"column": 2
} | {
"line": 510,
"column": 32
} | [
{
"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",
"Asymptotics.Is... | simp only [IsLittleO_def] at * | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 293,
"column": 2
} | {
"line": 293,
"column": 32
} | [
{
"pp": "α : Type u_1\nE' : Type u_6\nF' : Type u_7\nR : Type u_13\n𝕜' : Type u_16\ninst✝⁷ : SeminormedAddCommGroup E'\ninst✝⁶ : SeminormedAddCommGroup F'\ninst✝⁵ : SeminormedRing R\ninst✝⁴ : NormedDivisionRing 𝕜'\nf' : α → E'\ng' : α → F'\nl : Filter α\ninst✝³ : Module R E'\ninst✝² : IsBoundedSMul R E'\ninst... | simp only [IsLittleO_def] at * | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 300,
"column": 2
} | {
"line": 300,
"column": 32
} | [
{
"pp": "α : Type u_1\nE' : Type u_6\nF' : Type u_7\nR : Type u_13\n𝕜' : Type u_16\ninst✝⁷ : SeminormedAddCommGroup E'\ninst✝⁶ : SeminormedAddCommGroup F'\ninst✝⁵ : SeminormedRing R\ninst✝⁴ : NormedDivisionRing 𝕜'\nf' : α → E'\ng' : α → F'\nl : Filter α\ninst✝³ : Module R E'\ninst✝² : IsBoundedSMul R E'\ninst... | simp only [IsLittleO_def] at * | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1326,
"column": 2
} | {
"line": 1326,
"column": 32
} | [
{
"pp": "α : Type u_1\nR : Type u_13\ninst✝² : SeminormedRing R\nS : Type u_17\ninst✝¹ : NormedRing S\ninst✝ : NormMulClass S\nl : Filter α\nf₁ f₂ : α → R\ng₁ g₂ : α → S\nh₁ : f₁ =O[l] g₁\nh₂ : f₂ =o[l] g₂\n⊢ (fun x ↦ f₁ x * f₂ x) =o[l] fun x ↦ g₁ x * g₂ x",
"usedConstants": [
"Eq.mpr",
"Seminor... | simp only [IsLittleO_def] at * | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1329,
"column": 2
} | {
"line": 1329,
"column": 92
} | [
{
"pp": "α : Type u_1\nR : Type u_13\ninst✝² : SeminormedRing R\nS : Type u_17\ninst✝¹ : NormedRing S\ninst✝ : NormMulClass S\nl : Filter α\nf₁ f₂ : α → R\ng₁ g₂ : α → S\nh₁ : f₁ =O[l] g₁\nh₂ : ∀ ⦃c : ℝ⦄, 0 < c → IsBigOWith c l f₂ g₂\nc : ℝ\ncpos : 0 < c\nc' : ℝ\nc'pos : c' > 0\nhc' : IsBigOWith c' l f₁ g₁\n⊢ I... | exact (hc'.mul (h₂ (div_pos cpos c'pos))).congr_const (mul_div_cancel₀ _ (ne_of_gt c'pos)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Asymptotics.Defs | {
"line": 1333,
"column": 2
} | {
"line": 1333,
"column": 32
} | [
{
"pp": "α : Type u_1\nR : Type u_13\ninst✝² : SeminormedRing R\nS : Type u_17\ninst✝¹ : NormedRing S\ninst✝ : NormMulClass S\nl : Filter α\nf₁ f₂ : α → R\ng₁ g₂ : α → S\nh₁ : f₁ =o[l] g₁\nh₂ : f₂ =O[l] g₂\n⊢ (fun x ↦ f₁ x * f₂ x) =o[l] fun x ↦ g₁ x * g₂ x",
"usedConstants": [
"Eq.mpr",
"Seminor... | simp only [IsLittleO_def] at * | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Asymptotics.Lemmas | {
"line": 651,
"column": 6
} | {
"line": 651,
"column": 30
} | [
{
"pp": "E'' : Type u_9\nF'' : Type u_10\ninst✝¹ : NormedAddCommGroup E''\ninst✝ : NormedAddCommGroup F''\nf : ℕ → E''\ng : ℕ → F''\nh : ∀ (x : ℕ), g x = 0 → f x = 0\n⊢ f =O[atTop] g ↔ ∃ C, ∀ (x : ℕ), ‖f x‖ ≤ C * ‖g x‖",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instLE",
"Real"... | ← Nat.cofinite_eq_atTop, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Order.CauSeq.BigOperators | {
"line": 80,
"column": 2
} | {
"line": 84,
"column": 55
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Ring β\nabv : β → α\ninst✝ : IsAbsoluteValue abv\nf g : ℕ → β\nha : IsCauSeq abs fun m ↦ ∑ n ∈ range m, abv (f n)\nhb : IsCauSeq abv fun m ↦ ∑ n ∈ range m, g n\nε : α\nε0 : 0 < ε\nP : α\nhP : ... | have h₃ :
∑ i ∈ range K, f i * ∑ k ∈ range (K - i), g k =
∑ i ∈ range K, f i * (∑ k ∈ range (K - i), g k - ∑ k ∈ range K, g k) +
∑ i ∈ range K, f i * ∑ k ∈ range K, g k := by
rw [← sum_add_distrib]; simp [(mul_add _ _ _).symm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.Exponential | {
"line": 40,
"column": 52
} | {
"line": 45,
"column": 39
} | [
{
"pp": "z : ℂ\nn : ℕ\nhn : ‖z‖ < ↑n\nhn0 : 0 < ↑n\nm : ℕ\nhm : n ≤ m\n⊢ |‖z ^ m.succ / ↑m.succ.factorial‖| ≤ ‖z‖ / ↑n * |‖z ^ m / ↑m.factorial‖|",
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"div_le_div₀",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NonAssocSem... | by
rw [abs_norm, abs_norm, Nat.factorial_succ, pow_succ', mul_comm m.succ, Nat.cast_mul,
← div_div, mul_div_assoc, mul_div_right_comm, Complex.norm_mul, Complex.norm_div,
norm_natCast]
gcongr
exact le_trans hm (Nat.le_succ _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Exponential | {
"line": 378,
"column": 2
} | {
"line": 378,
"column": 57
} | [
{
"pp": "x : ℂ\nhx : ‖x‖ ≤ 1\nn : ℕ\nhn : 0 < n\n⊢ (cauSeqNorm (exp' x + -const norm (∑ m ∈ range n, x ^ m / ↑m.factorial))).lim ≤\n ‖x‖ ^ n * (↑n.succ * (↑n.factorial * ↑n)⁻¹)",
"usedConstants": [
"Norm.norm",
"Complex.cauSeqNorm",
"Real",
"instHDiv",
"NonUnitalCommRing.toN... | refine lim_le (CauSeq.le_of_exists ⟨n, fun j hj => ?_⟩) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Complex.Exponential | {
"line": 407,
"column": 2
} | {
"line": 407,
"column": 57
} | [
{
"pp": "x : ℂ\nn : ℕ\nhx : ‖x‖ / ↑n.succ ≤ 1 / 2\n⊢ (cauSeqNorm (exp' x + -const norm (∑ m ∈ range n, x ^ m / ↑m.factorial))).lim ≤ ‖x‖ ^ n / ↑n.factorial * 2",
"usedConstants": [
"Norm.norm",
"Complex.cauSeqNorm",
"Real",
"instHDiv",
"NonUnitalCommRing.toNonUnitalNonAssocComm... | refine lim_le (CauSeq.le_of_exists ⟨n, fun j hj => ?_⟩) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Ring.InfiniteSum | {
"line": 131,
"column": 2
} | {
"line": 132,
"column": 65
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedRing R\nf g : ℕ → R\nhf : Summable fun x ↦ ‖f x‖\nhg : Summable fun x ↦ ‖g x‖\n⊢ Summable fun n ↦ ‖∑ k ∈ range (n + 1), f k * g (n - k)‖",
"usedConstants": [
"Finset.Nat.sum_antidiagonal_eq_sum_range_succ",
"Norm.norm",
"Eq.mpr",
"Real",
"No... | simp_rw [← sum_antidiagonal_eq_sum_range_succ fun k l => f k * g l]
exact summable_norm_sum_mul_antidiagonal_of_summable_norm hf hg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Ring.InfiniteSum | {
"line": 131,
"column": 2
} | {
"line": 132,
"column": 65
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedRing R\nf g : ℕ → R\nhf : Summable fun x ↦ ‖f x‖\nhg : Summable fun x ↦ ‖g x‖\n⊢ Summable fun n ↦ ‖∑ k ∈ range (n + 1), f k * g (n - k)‖",
"usedConstants": [
"Finset.Nat.sum_antidiagonal_eq_sum_range_succ",
"Norm.norm",
"Eq.mpr",
"Real",
"No... | simp_rw [← sum_antidiagonal_eq_sum_range_succ fun k l => f k * g l]
exact summable_norm_sum_mul_antidiagonal_of_summable_norm hf hg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 301,
"column": 4
} | {
"line": 301,
"column": 31
} | [
{
"pp": "x : ℂ\n⊢ I * sin (x * I) = -sinh x",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Eq.mpr",
"Complex.sinh",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"Complex.commRing",
... | rw [mul_comm, ← sinh_mul_I] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Operator.LinearIsometry | {
"line": 954,
"column": 53
} | {
"line": 956,
"column": 5
} | [
{
"pp": "R : Type u_1\nR₂ : Type u_2\nE₂ : Type u_6\nF : Type u_9\ninst✝⁷ : Semiring R\ninst✝⁶ : Semiring R₂\nσ₁₂ : R →+* R₂\nσ₂₁ : R₂ →+* R\ninst✝⁵ : RingHomInvPair σ₁₂ σ₂₁\ninst✝⁴ : RingHomInvPair σ₂₁ σ₁₂\ninst✝³ : SeminormedAddCommGroup E₂\ninst✝² : Module R₂ E₂\ninst✝¹ : NormedAddCommGroup F\ninst✝ : Module... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 533,
"column": 2
} | {
"line": 533,
"column": 33
} | [
{
"pp": "x : ℝ\n⊢ (cexp (↑x * I)).im = Real.sin x",
"usedConstants": [
"Complex.mul_im",
"Real",
"HMul.hMul",
"Complex.cos",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"Complex.im",
"AddMonoid.toAddZeroClass",
"Complex.sin",
"Complex.... | simp [exp_mul_I, sin_ofReal_re] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 533,
"column": 2
} | {
"line": 533,
"column": 33
} | [
{
"pp": "x : ℝ\n⊢ (cexp (↑x * I)).im = Real.sin x",
"usedConstants": [
"Complex.mul_im",
"Real",
"HMul.hMul",
"Complex.cos",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"Complex.im",
"AddMonoid.toAddZeroClass",
"Complex.sin",
"Complex.... | simp [exp_mul_I, sin_ofReal_re] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Trigonometric | {
"line": 533,
"column": 2
} | {
"line": 533,
"column": 33
} | [
{
"pp": "x : ℝ\n⊢ (cexp (↑x * I)).im = Real.sin x",
"usedConstants": [
"Complex.mul_im",
"Real",
"HMul.hMul",
"Complex.cos",
"Real.instZero",
"Real.instAddMonoid",
"congrArg",
"Complex.im",
"AddMonoid.toAddZeroClass",
"Complex.sin",
"Complex.... | simp [exp_mul_I, sin_ofReal_re] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 59,
"column": 2
} | {
"line": 63,
"column": 46
} | [
{
"pp": "r : ℝ\nhr_nonneg : 0 ≤ r\nhr_le : r ≤ 1\nx y : ℂ\nhyx : ‖y - x‖ < r\nhy_eq : y = x + (y - x)\nhyx_sq_le : ‖y - x‖ ^ 2 ≤ r * ‖y - x‖\nh_sq : ∀ (z : ℂ), ‖z‖ ≤ 1 → ‖cexp (x + z) - cexp x‖ ≤ ‖z‖ * ‖cexp x‖ + ‖cexp x‖ * ‖z‖ ^ 2\n⊢ ‖cexp y - cexp x‖ ≤ (1 + r) * ‖cexp x‖ * ‖y - x‖",
"usedConstants": [
... | calc
‖exp y - exp x‖ = ‖exp (x + (y - x)) - exp x‖ := by nth_rw 1 [hy_eq]
_ ≤ ‖y - x‖ * ‖exp x‖ + ‖exp x‖ * ‖y - x‖ ^ 2 := h_sq (y - x) (hyx.le.trans hr_le)
_ ≤ ‖y - x‖ * ‖exp x‖ + ‖exp x‖ * (r * ‖y - x‖) := by grw [hyx_sq_le]
_ = (1 + r) * ‖exp x‖ * ‖y - x‖ := by ring | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 82,
"column": 4
} | {
"line": 82,
"column": 62
} | [
{
"pp": "case inr\nn : ℕ\nhn : 0 < n\n⊢ ∀ᶠ (x : ℂ) in 𝓝 0, ‖cexp x - ∑ i ∈ Finset.range n, x ^ i / ↑i !‖ ≤ ↑n.succ / (↑n ! * ↑n) * ‖x ^ n‖",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"NormedCommRing.toSeminormedCommR... | rw [NormedAddGroup.nhds_zero_basis_norm_lt.eventually_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Exp | {
"line": 128,
"column": 2
} | {
"line": 129,
"column": 27
} | [
{
"pp": "a : ℝ\nthis : ∀ (a : ℂ), ∀ ε > 0, ∀ᶠ (x : ℂ) in 𝓝 a, dist ((cexp - 1) x) ((cexp - 1) a) < ε\nε : ℝ\nhε : ε > 0\n⊢ ∃ δ > 0, ∀ x ∈ {x | x.re ≤ a}, ∀ y ∈ {x | x.re ≤ a}, dist x y < δ → dist (cexp x) (cexp y) < ε",
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"Re... | simp only [gt_iff_lt, Pi.sub_apply, Pi.one_apply, dist_sub_eq_dist_add_right,
sub_add_cancel] at this | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 432,
"column": 4
} | {
"line": 432,
"column": 89
} | [
{
"pp": "case inr\nx : ℝ\nhx : |x| ≤ π\nh : 0 ≤ x\n⊢ |sin x| = sin |x|",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"Real.pi",
"Real.lattice",
"abs",
"congrArg",
"abs_le",
"PartialOrder.toPreorder",
"Preorder.toLE",
"AddCommGr... | rw [abs_of_nonneg h, abs_of_nonneg (sin_nonneg_of_nonneg_of_le_pi h (abs_le.1 hx).2)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 432,
"column": 4
} | {
"line": 432,
"column": 89
} | [
{
"pp": "case inr\nx : ℝ\nhx : |x| ≤ π\nh : 0 ≤ x\n⊢ |sin x| = sin |x|",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"Real.pi",
"Real.lattice",
"abs",
"congrArg",
"abs_le",
"PartialOrder.toPreorder",
"Preorder.toLE",
"AddCommGr... | rw [abs_of_nonneg h, abs_of_nonneg (sin_nonneg_of_nonneg_of_le_pi h (abs_le.1 hx).2)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 432,
"column": 4
} | {
"line": 432,
"column": 89
} | [
{
"pp": "case inr\nx : ℝ\nhx : |x| ≤ π\nh : 0 ≤ x\n⊢ |sin x| = sin |x|",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real",
"Real.pi",
"Real.lattice",
"abs",
"congrArg",
"abs_le",
"PartialOrder.toPreorder",
"Preorder.toLE",
"AddCommGr... | rw [abs_of_nonneg h, abs_of_nonneg (sin_nonneg_of_nonneg_of_le_pi h (abs_le.1 hx).2)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 831,
"column": 39
} | {
"line": 831,
"column": 55
} | [
{
"pp": "θ : ℝ := π / 5\nhθ : θ = π / 5\nc : ℝ := cos θ\ns : ℝ := sin θ\nhs : s ≠ 0\n⊢ 2 * s * c = sin (2 * θ)",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.cos",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"id",
"instOfNatNat",
"Real.sin_two_... | rw [sin_two_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 831,
"column": 39
} | {
"line": 831,
"column": 55
} | [
{
"pp": "θ : ℝ := π / 5\nhθ : θ = π / 5\nc : ℝ := cos θ\ns : ℝ := sin θ\nhs : s ≠ 0\n⊢ 2 * s * c = sin (2 * θ)",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.cos",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"id",
"instOfNatNat",
"Real.sin_two_... | rw [sin_two_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 831,
"column": 39
} | {
"line": 831,
"column": 55
} | [
{
"pp": "θ : ℝ := π / 5\nhθ : θ = π / 5\nc : ℝ := cos θ\ns : ℝ := sin θ\nhs : s ≠ 0\n⊢ 2 * s * c = sin (2 * θ)",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.cos",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"id",
"instOfNatNat",
"Real.sin_two_... | rw [sin_two_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 835,
"column": 59
} | {
"line": 835,
"column": 75
} | [
{
"pp": "θ : ℝ := π / 5\nhθ : θ = π / 5\nc : ℝ := cos θ\ns : ℝ := sin θ\nhs : s ≠ 0\n⊢ sin (2 * θ) * c + cos (2 * θ) * s = 2 * s * c * c + cos (2 * θ) * s",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.cos",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"id"... | rw [sin_two_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 835,
"column": 59
} | {
"line": 835,
"column": 75
} | [
{
"pp": "θ : ℝ := π / 5\nhθ : θ = π / 5\nc : ℝ := cos θ\ns : ℝ := sin θ\nhs : s ≠ 0\n⊢ sin (2 * θ) * c + cos (2 * θ) * s = 2 * s * c * c + cos (2 * θ) * s",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.cos",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"id"... | rw [sin_two_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 835,
"column": 59
} | {
"line": 835,
"column": 75
} | [
{
"pp": "θ : ℝ := π / 5\nhθ : θ = π / 5\nc : ℝ := cos θ\ns : ℝ := sin θ\nhs : s ≠ 0\n⊢ sin (2 * θ) * c + cos (2 * θ) * s = 2 * s * c * c + cos (2 * θ) * s",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.cos",
"congrArg",
"Nat.instAtLeastTwoHAddOfNat",
"id"... | rw [sin_two_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Basic | {
"line": 853,
"column": 2
} | {
"line": 853,
"column": 66
} | [
{
"pp": "c : ℝ := cos (π / 5)\nthis : 4 * (c * c) + -2 * c + -1 = 0\nhd : discrim 4 (-2) (-1) = 2 * √5 * (2 * √5)\n⊢ c = (1 + √5) / 4",
"usedConstants": [
"NegZeroClass.toNeg",
"False",
"Real",
"instHDiv",
"HMul.hMul",
"CharZero.NeZero.two",
"Real.instRCLike",
... | rcases (quadratic_eq_zero_iff (by simp) hd c).mp this with h | h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Connected.PathConnected | {
"line": 584,
"column": 33
} | {
"line": 586,
"column": 37
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : PathConnectedSpace X\nf : X → Y\nhf : Continuous[inst✝², inst✝¹] f\n⊢ IsPathConnected (range f)",
"usedConstants": [
"IsPathConnected",
"Eq.mpr",
"Set.image_univ",
"isPathConnected_... | by
rw [← image_univ]
exact isPathConnected_univ.image hf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Path | {
"line": 364,
"column": 59
} | {
"line": 366,
"column": 5
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\n⊢ γ.map ⋯ = γ",
"usedConstants": [
"Path.map",
"Real",
"Path.ext",
"Set.Elem",
"id",
"funext",
"continuous_id",
"Path.instFunLike",
"Path",
"Eq.refl",
"DFunLike.coe",
... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Path | {
"line": 371,
"column": 46
} | {
"line": 373,
"column": 5
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nx y : X\nγ : Path x y\nZ : Type u_4\ninst✝ : TopologicalSpace Z\nf : X → Y\nhf : Continuous[inst✝², inst✝¹] f\ng : Y → Z\nhg : Continuous[inst✝¹, inst✝] g\n⊢ (γ.map hf).map hg = γ.map ⋯",
"usedConstants": [
... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Path | {
"line": 621,
"column": 78
} | {
"line": 623,
"column": 5
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nx y : X\nγ : Path x y\n⊢ γ.reparam id ⋯ ⋯ ⋯ = γ",
"usedConstants": [
"Real.instIsOrderedRing",
"Real.partialOrder",
"Real",
"Set.Icc.instZero",
"PseudoMetricSpace.toUniformSpace",
"Path.ext",
"Membership.mem",
... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.AddCircle | {
"line": 212,
"column": 2
} | {
"line": 213,
"column": 74
} | [
{
"pp": "p : ℝ\nhp : Fact (0 < p)\nu : AddCircle p\nhu' : u ≠ 0\nn : ℕ\nhn : ‖u‖ = p * (↑n / ↑(addOrderOf u))\nhu : ↑(addOrderOf u) ≠ 0\n⊢ p * 1 ≤ addOrderOf u • ‖u‖",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Real.instIsOrderedRing",
"Norm.norm",
... | rw [hn, nsmul_eq_mul, ← mul_assoc, mul_comm _ p, mul_assoc, mul_div_cancel₀ _ hu,
mul_le_mul_iff_right₀ hp.out, Nat.one_le_cast, Nat.one_le_iff_ne_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 80,
"column": 4
} | {
"line": 80,
"column": 41
} | [
{
"pp": "case refine_2\nθ : ℝ\n⊢ ‖cexp (↑θ * I)‖ = 1",
"usedConstants": [
"Complex.norm_exp_ofReal_mul_I"
]
}
] | exact Complex.norm_exp_ofReal_mul_I θ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 173,
"column": 2
} | {
"line": 179,
"column": 73
} | [
{
"pp": "case inr\nz : ℂ\nh₀ : z ≠ 0\n⊢ 0 ≤ z.arg ↔ 0 ≤ z.im",
"usedConstants": [
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"norm_pos_iff",
"Real.partialOrder",
"Real.instLE",
"Real",
"Trans... | calc
0 ≤ arg z ↔ 0 ≤ Real.sin (arg z) :=
⟨fun h => Real.sin_nonneg_of_mem_Icc ⟨h, arg_le_pi z⟩, by
contrapose!
intro h
exact Real.sin_neg_of_neg_of_neg_pi_lt h (neg_pi_lt_arg _)⟩
_ ↔ _ := by rw [sin_arg, le_div_iff₀ (norm_pos_iff.mpr h₀), zero_mul] | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 180,
"column": 56
} | {
"line": 184,
"column": 37
} | [
{
"pp": "x : ℝ\nhx : 0 ≤ x\n⊢ 0 < log x ↔ 1 < x",
"usedConstants": [
"Eq.mpr",
"LE.le.eq_or_lt",
"False",
"Real.partialOrder",
"Real.instLE",
"Real",
"Preorder.toLT",
"Real.instZero",
"Real.instZeroLEOneClass",
"congrArg",
"PartialOrder.toPre... | by
rcases hx.eq_or_lt with (rfl | hx)
· simp [zero_le_one]
rw [← log_one]
exact log_lt_log_iff zero_lt_one hx | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Log.Basic | {
"line": 298,
"column": 53
} | {
"line": 298,
"column": 69
} | [
{
"pp": "case ofNat\nx : ℝ\na✝ : ℕ\n⊢ ↑a✝ * log x = ↑↑a✝ * log x",
"usedConstants": [
"Int.cast",
"Eq.mpr",
"Int.cast_natCast",
"Real",
"HMul.hMul",
"congrArg",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
"AddMonoidWithOne.toNatCast",
"Real.instRin... | Int.cast_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Complex | {
"line": 107,
"column": 71
} | {
"line": 107,
"column": 98
} | [
{
"pp": "x y : ℂ\nn : ℤ\n⊢ x ^ (y * ↑n) = (x ^ y) ^ n",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Int.cast",
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"congrArg",
"Complex.instNormedField",
"... | rw [mul_comm, cpow_int_mul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Complex | {
"line": 107,
"column": 71
} | {
"line": 107,
"column": 98
} | [
{
"pp": "x y : ℂ\nn : ℤ\n⊢ x ^ (y * ↑n) = (x ^ y) ^ n",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Int.cast",
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"congrArg",
"Complex.instNormedField",
"... | rw [mul_comm, cpow_int_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Complex | {
"line": 107,
"column": 71
} | {
"line": 107,
"column": 98
} | [
{
"pp": "x y : ℂ\nn : ℤ\n⊢ x ^ (y * ↑n) = (x ^ y) ^ n",
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"Int.cast",
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"congrArg",
"Complex.instNormedField",
"... | rw [mul_comm, cpow_int_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 183,
"column": 17
} | {
"line": 183,
"column": 33
} | [
{
"pp": "θ : Angle\nh : ↑π = ↑(2 • (π / 2))\n⊢ 2 • θ = 2 • ↑(π / 2) ↔ θ = ↑(π / 2) ∨ θ = ↑(-π / 2)",
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"instHDiv",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"congrArg",
"AddCommGroup.toAddCommMonoid",
... | two_nsmul_eq_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 394,
"column": 6
} | {
"line": 394,
"column": 22
} | [
{
"pp": "θ ψ : Angle\nh : 2 • θ = 2 • ψ\n⊢ |θ.sin| = |ψ.sin|",
"usedConstants": [
"instHSMul",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"AddMonoid.toNSMul",
"AddCommGroup.toAddGroup",
"Eq.mp",
"instOfNa... | two_nsmul_eq_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 406,
"column": 6
} | {
"line": 406,
"column": 22
} | [
{
"pp": "θ ψ : Angle\nh : 2 • θ = 2 • ψ\n⊢ |θ.cos| = |ψ.cos|",
"usedConstants": [
"instHSMul",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"AddMonoid.toNSMul",
"AddCommGroup.toAddGroup",
"Eq.mp",
"instOfNa... | two_nsmul_eq_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 650,
"column": 65
} | {
"line": 650,
"column": 81
} | [
{
"pp": "θ ψ : Angle\nh : 2 • θ = 2 • (↑(π / 2) - ψ)\n⊢ |θ.cos| = |ψ.sin|",
"usedConstants": [
"Real",
"instHSMul",
"instHDiv",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"Real.instDivInvMonoid",
"Nat.ins... | two_nsmul_eq_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 693,
"column": 6
} | {
"line": 693,
"column": 22
} | [
{
"pp": "θ ψ : Angle\nh : 2 • θ = 2 • ψ\n⊢ θ.tan = ψ.tan",
"usedConstants": [
"instHSMul",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"AddMonoid.toNSMul",
"AddCommGroup.toAddGroup",
"Eq.mp",
"instOfNatNat... | two_nsmul_eq_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 888,
"column": 12
} | {
"line": 888,
"column": 29
} | [
{
"pp": "case inr.«0»\nθ ψ : Angle\nh✝ : 2 • (θ + ψ) = 0\nhs : θ.sign = ψ.sign\nh0 : θ.sign ≠ 0\nh : ↑(θ.toReal + ψ.toReal) = ↑π\nhk : θ.toReal + ψ.toReal = 2 * π * ↑((Nat.castEmbedding.trans (addLeftEmbedding (-1))) 0) + π\n⊢ θ.toReal + ψ.toReal = π ∨ θ.toReal + ψ.toReal = -π",
"usedConstants": [
"ad... | simp at hk; grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 888,
"column": 12
} | {
"line": 888,
"column": 29
} | [
{
"pp": "case inr.«0»\nθ ψ : Angle\nh✝ : 2 • (θ + ψ) = 0\nhs : θ.sign = ψ.sign\nh0 : θ.sign ≠ 0\nh : ↑(θ.toReal + ψ.toReal) = ↑π\nhk : θ.toReal + ψ.toReal = 2 * π * ↑((Nat.castEmbedding.trans (addLeftEmbedding (-1))) 0) + π\n⊢ θ.toReal + ψ.toReal = π ∨ θ.toReal + ψ.toReal = -π",
"usedConstants": [
"ad... | simp at hk; grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 888,
"column": 12
} | {
"line": 888,
"column": 29
} | [
{
"pp": "case inr.«1»\nθ ψ : Angle\nh✝ : 2 • (θ + ψ) = 0\nhs : θ.sign = ψ.sign\nh0 : θ.sign ≠ 0\nh : ↑(θ.toReal + ψ.toReal) = ↑π\nhk : θ.toReal + ψ.toReal = 2 * π * ↑((Nat.castEmbedding.trans (addLeftEmbedding (-1))) 1) + π\n⊢ θ.toReal + ψ.toReal = π ∨ θ.toReal + ψ.toReal = -π",
"usedConstants": [
"ad... | simp at hk; grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 888,
"column": 12
} | {
"line": 888,
"column": 29
} | [
{
"pp": "case inr.«1»\nθ ψ : Angle\nh✝ : 2 • (θ + ψ) = 0\nhs : θ.sign = ψ.sign\nh0 : θ.sign ≠ 0\nh : ↑(θ.toReal + ψ.toReal) = ↑π\nhk : θ.toReal + ψ.toReal = 2 * π * ↑((Nat.castEmbedding.trans (addLeftEmbedding (-1))) 1) + π\n⊢ θ.toReal + ψ.toReal = π ∨ θ.toReal + ψ.toReal = -π",
"usedConstants": [
"ad... | simp at hk; grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 73,
"column": 2
} | {
"line": 76,
"column": 34
} | [
{
"pp": "case inr.inr\nb : ℝ\nhb₀ : -1 < b\nhb₁ : b < 1\nhb : 0 < b\n⊢ Tendsto (fun x ↦ b ^ x) atTop (𝓝 0)",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Real.instPow",
"Real",
"HMul.hMul",
"Real.rpow_def_of_pos",
"Real.instZero",
"congrArg",
"Filter.tendst... | case inr.inr => -- b > 0
simp_rw [Real.rpow_def_of_pos hb]
refine tendsto_exp_atBot.comp <| (tendsto_const_mul_atBot_of_neg ?_).mpr tendsto_id
exact (log_neg_iff hb).mpr hb₁ | Lean.Elab.Tactic.evalCase | Lean.Parser.Tactic.case |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 57,
"column": 62
} | {
"line": 59,
"column": 22
} | [
{
"pp": "x : ℝ\nn : ℤ\n⊢ x ^ ↑n = x ^ n",
"usedConstants": [
"Complex.cpow_intCast",
"Int.cast",
"Real.instPow",
"Real",
"congrArg",
"Real.instDivInvMonoid",
"DivInvMonoid.toZPow",
"Complex.instPow",
"Complex.instDivInvMonoid",
"Int",
"Comple... | by
simp only [rpow_def, ← Complex.ofReal_zpow, Complex.cpow_intCast, Complex.ofReal_intCast,
Complex.ofReal_re] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 66,
"column": 36
} | {
"line": 66,
"column": 52
} | [
{
"pp": "x : ℝ\nn : ℕ\n⊢ x ^ (-↑n) = x ^ (-↑↑n)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Eq.mpr",
"NegZeroClass.toNeg",
"Int.cast_natCast",
"Real.instPow",
"Real",
"AddGroupWithOne.toAddGroup",
"congrArg",
"AddGroupWithOne.... | Int.cast_natCast | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 325,
"column": 2
} | {
"line": 325,
"column": 40
} | [
{
"pp": "z w : ℂ\n⊢ ‖z ^ w‖ ≤ ‖z‖ ^ w.re / rexp (z.arg * w.im)",
"usedConstants": [
"Norm.norm",
"Real.instPow",
"Real.instLE",
"Real",
"instHDiv",
"HMul.hMul",
"Real.instZero",
"Complex.im",
"Real.instDivInvMonoid",
"Complex.instZero",
"Comp... | by_cases! h : z = 0 → w.re = 0 → w = 0 | Mathlib.Tactic.ByCases._aux_Mathlib_Tactic_ByCases___macroRules_Mathlib_Tactic_ByCases_byCases!_1 | Mathlib.Tactic.ByCases.byCases! |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 593,
"column": 31
} | {
"line": 594,
"column": 78
} | [
{
"pp": "x y z : ℝ\nhx : 0 < x\nhy : 0 < y\nhz : z < 0\n⊢ x ≤ y ^ z⁻¹ ↔ y ≤ x ^ z",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.partialOrder",
"Real.instLE",
"Real.rpow_pos_of_pos",
"Real",
"Real.instZero",
"congrArg",
"Real.instInv",
"Iff.r... | by
rw [← rpow_le_rpow_iff_of_neg _ hx hz, rpow_inv_rpow _ hz.ne] <;> positivity | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 958,
"column": 11
} | {
"line": 958,
"column": 59
} | [
{
"pp": "n : ℕ\nhn : 1 < n\nw z : ℂ\n⊢ ‖↑n ^ w‖ ≤ ‖↑n ^ z‖ ↔ w.re ≤ z.re",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"congrArg",
"Complex.instPow",
"Complex.instNorm",
"Nat.zero_lt_of_lt",
"id",
"instOfN... | norm_natCast_cpow_of_pos (Nat.zero_lt_of_lt hn), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 409,
"column": 2
} | {
"line": 409,
"column": 34
} | [
{
"pp": "x : ℝ≥0\ny z : ℝ\n⊢ x ^ y = x ^ z ↔ y = z ∨ x = 1 ∨ x = 0 ∧ (y = 0 ↔ z = 0)",
"usedConstants": [
"NNReal",
"NNReal.instZero",
"eq_or_ne",
"Zero.toOfNat0",
"OfNat.ofNat"
]
}
] | obtain rfl | hx₀ := eq_or_ne x 0 | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Convex.Segment | {
"line": 297,
"column": 2
} | {
"line": 297,
"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 Hy : y = (y - c) + c := by abel | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.Star | {
"line": 185,
"column": 2
} | {
"line": 185,
"column": 34
} | [
{
"pp": "case inr.inr\n𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\nx : E\ns : Set E\nhx : x ∈ s\nh : ∀ ⦃y : E⦄, y ∈ s → x ≠ y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\nhb... | obtain rfl | hxy := eq_or_ne x y | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 325,
"column": 8
} | {
"line": 325,
"column": 12
} | [
{
"pp": "case h.mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns₁ s₂ : AffineSubspace k P\nhd : s₁.direction = s₂.direction\nhn : (↑s₁ ∩ ↑s₂).Nonempty\np : P\nhq1 : hn.some ∈ ↑s₁\nhq2 : hn.some ∈ ↑s₂\nhp : p ∈ s₁\n⊢ p -ᵥ hn.s... | ← hd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Basic | {
"line": 147,
"column": 2
} | {
"line": 147,
"column": 34
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nh : s.Pairwise fun x y ↦ ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 < a\nhb : 0 < b\nhab : a + ... | obtain rfl | hxy := eq_or_ne x y | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 891,
"column": 2
} | {
"line": 891,
"column": 32
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\ns : AffineSubspace k P\n⊢ affineSpan k ↑s ≤ s",
"usedConstants": []
}
] | rintro p ⟨p₁, hp₁, v, hv, rfl⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.Convex.Basic | {
"line": 471,
"column": 48
} | {
"line": 474,
"column": 66
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns : Set E\ninst✝ : AddRightMono 𝕜\nhs : Convex 𝕜 s\nx y : E\nhx : x ∈ s\nhy : x + y ∈ s\nt : 𝕜\nht : t ∈ Icc 0 1\n⊢ x + t • y ∈ s",
"usedConstants": [
"Eq.mpr",
"Ma... | by
have h : x + t • y = (1 - t) • x + t • (x + y) := by match_scalars <;> noncomm_ring
rw [h]
exact hs hx hy (sub_nonneg_of_le ht.2) ht.1 (sub_add_cancel _ _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Basic | {
"line": 571,
"column": 2
} | {
"line": 571,
"column": 49
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nh_conv : Convex 𝕜 s\np q : 𝕜\nhp : 0 ≤ p\nhq : 0 ≤ q\nv₁ : E\nh₁ : v₁ ∈ s\nv₂ : E\nh₂ : v₂ ∈ s\n⊢ (fun x1 x2 ↦ x1 + x2) ((fun x ↦ p • x) v... | exact h_conv.exists_mem_add_smul_eq h₁ h₂ hp hq | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Basic | {
"line": 606,
"column": 55
} | {
"line": 606,
"column": 83
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\n⊢ (∀ ⦃x : 𝕜⦄, x ∈ s → ∀ ⦃y : 𝕜⦄, y ∈ s → uIcc x y ⊆ s) ↔ s.OrdConnected",
"usedConstants": [
"congrArg",
"PartialOrder.toPreorder",
"Membership.mem",
"SemilatticeInf.toPa... | ordConnected_iff_uIcc_subset | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.LocallyConvex.Bounded | {
"line": 467,
"column": 2
} | {
"line": 467,
"column": 47
} | [
{
"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⊢ Bornology.IsVonNBounded 𝕜 s",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
... | rw [Metric.nhds_basis_ball.isVonNBounded_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.MetricSpace.Equicontinuity | {
"line": 100,
"column": 2
} | {
"line": 100,
"column": 39
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : PseudoMetricSpace α\nι : Type u_4\ninst✝ : PseudoMetricSpace β\nb : ℝ → ℝ\nb_lim : Tendsto b (𝓝 0) (𝓝 0)\nF : ι → β → α\nH : ∀ (x y : β) (i : ι), dist (F i x) (F i y) ≤ b (dist x y)\n⊢ UniformEquicontinuous F",
"usedConstants": [
"Eq.mpr",
"Real",
... | rw [Metric.uniformEquicontinuous_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Function | {
"line": 247,
"column": 8
} | {
"line": 247,
"column": 20
} | [
{
"pp": "case h₂.hb\n𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁸ : Semiring 𝕜\ninst✝⁷ : PartialOrder 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : PosSMulMono 𝕜 β\ns : Set E\nf : E → β\nhf... | · exact hy.2 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Strict | {
"line": 147,
"column": 24
} | {
"line": 152,
"column": 89
} | [
{
"pp": "𝕜 : 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.OrdConnected... | by
refine strictConvex_iff_openSegment_subset.2 fun x hx y hy hxy => ?_
rcases hxy.lt_or_gt with hlt | hlt <;> [skip; rw [openSegment_symm]] <;>
exact
(openSegment_subset_Ioo hlt).trans
(isOpen_Ioo.subset_interior_iff.2 <| Ioo_subset_Icc_self.trans <| hs.out ‹_› ‹_›) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Function | {
"line": 341,
"column": 2
} | {
"line": 341,
"column": 34
} | [
{
"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 : Module 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf : E → β\nh : s.Pairwise fun x y ↦ ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → f (... | obtain rfl | hxy := eq_or_ne x y | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Convex.Strict | {
"line": 197,
"column": 2
} | {
"line": 200,
"column": 83
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝⁵ : Semiring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : TopologicalSpace E\ninst✝² : AddCancelCommMonoid E\ninst✝¹ : ContinuousAdd E\ninst✝ : Module 𝕜 E\ns : Set E\nhs : StrictConvex 𝕜 s\nz : E\n⊢ StrictConvex 𝕜 ((fun x ↦ z + x) ⁻¹' s)",
"usedConstants": [
"No... | intro x hx y hy hxy a b ha hb hab
refine preimage_interior_subset_interior_preimage (continuous_const_add _) ?_
have h := hs hx hy ((add_right_injective _).ne hxy) ha hb hab
rwa [smul_add, smul_add, add_add_add_comm, ← _root_.add_smul, hab, one_smul] at h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Strict | {
"line": 197,
"column": 2
} | {
"line": 200,
"column": 83
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝⁵ : Semiring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : TopologicalSpace E\ninst✝² : AddCancelCommMonoid E\ninst✝¹ : ContinuousAdd E\ninst✝ : Module 𝕜 E\ns : Set E\nhs : StrictConvex 𝕜 s\nz : E\n⊢ StrictConvex 𝕜 ((fun x ↦ z + x) ⁻¹' s)",
"usedConstants": [
"No... | intro x hx y hy hxy a b ha hb hab
refine preimage_interior_subset_interior_preimage (continuous_const_add _) ?_
have h := hs hx hy ((add_right_injective _).ne hxy) ha hb hab
rwa [smul_add, smul_add, add_add_add_comm, ← _root_.add_smul, hab, one_smul] at h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Function | {
"line": 976,
"column": 4
} | {
"line": 977,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 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 : E → β\n⊢ (∀ ⦃x : E⦄,\n x ∈ s →\n ∀ ⦃y : ... | intro h x hx y hy a b ha hb hab
simpa [hab, zero_lt_one] using h hx hy ha hb | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Function | {
"line": 976,
"column": 4
} | {
"line": 977,
"column": 48
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 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 : E → β\n⊢ (∀ ⦃x : E⦄,\n x ∈ s →\n ∀ ⦃y : ... | intro h x hx y hy a b ha hb hab
simpa [hab, zero_lt_one] using h hx hy ha hb | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 215,
"column": 8
} | {
"line": 215,
"column": 13
} | [
{
"pp": "case refine_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np : P\nhp : p ∈ s\nv : V\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhv : p -ᵥ p₂ - (p -ᵥ p₁) = v\n⊢ v ∈ ↑(Submodule.span k ((fun x ↦ p -ᵥ x) '' s... | ← hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 228,
"column": 8
} | {
"line": 228,
"column": 13
} | [
{
"pp": "case refine_1\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np : P\nhp : p ∈ s\nv : V\np₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nhv : p₁ -ᵥ p - (p₂ -ᵥ p) = v\n⊢ v ∈ ↑(Submodule.span k ((fun x ↦ x -ᵥ p) '' s... | ← hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 419,
"column": 68
} | {
"line": 420,
"column": 63
} | [
{
"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 p₁ p₂ : P\n⊢ p ∈ affineSpan k {p₁, p₂} ↔ ∃ r, (AffineMap.lineMap p₂ p₁) r = p",
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"AffineMap.instFunL... | by
rw [Set.pair_comm, mem_affineSpan_pair_iff_exists_lineMap_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Seminorm | {
"line": 78,
"column": 45
} | {
"line": 78,
"column": 50
} | [
{
"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 : AddCommGroup E\ninst✝ : Module 𝕜 E\nf : E → ℝ\nadd_le : ∀ (x y : E), f (x + y) ≤ f x + f y\nsmul : ∀... | smul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Seminorm | {
"line": 81,
"column": 37
} | {
"line": 81,
"column": 42
} | [
{
"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 : AddCommGroup E\ninst✝ : Module 𝕜 E\nf : E → ℝ\nadd_le : ∀ (x y : E), f (x + y) ≤ f x + f y\nsmul : ∀... | smul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 824,
"column": 44
} | {
"line": 824,
"column": 49
} | [
{
"pp": "case right\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\ni0 : ι\nl : ι →₀ k\nleft✝ : l ∈ Finsupp.supported k k Set.univ\nhv : (Finsupp.linearCombination k fun i ↦ p i -ᵥ p i0) l = v\nw :... | ← hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 700,
"column": 77
} | {
"line": 702,
"column": 5
} | [
{
"pp": "k : Type u_1\nV₁ : Type u_2\nP₁ : Type u_3\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V₁\ninst✝³ : Module k V₁\ninst✝² : AffineSpace V₁ P₁\nS₁ S₂ : AffineSubspace k P₁\ninst✝¹ : Nonempty ↥S₁\ninst✝ : Nonempty ↥S₂\nh : S₁ = S₂\n⊢ (ofEq S₁ S₂ h).symm = ofEq S₂ S₁ ⋯",
"usedConstants": [
"Submodule"... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
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