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.AbsMax | {
"line": 390,
"column": 44
} | {
"line": 390,
"column": 63
} | [
{
"pp": "case inr\nE : Type u\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedSpace ℂ E\nF : Type v\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℂ F\ninst✝¹ : Nontrivial E\ninst✝ : FiniteDimensional ℂ E\nf : E → F\nU : Set E\nhb : Bornology.IsBounded U\nhne : U.Nonempty\nhd : DiffContOnCl ℂ f U\nhc : IsC... | exact ⟨w, hwU, hle⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.AbsMax | {
"line": 402,
"column": 40
} | {
"line": 419,
"column": 65
} | [
{
"pp": "E : Type u\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℂ E\nF : Type v\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℂ F\ninst✝ : Nontrivial E\nf : E → F\nU : Set E\nhU : Bornology.IsBounded U\nhd : DiffContOnCl ℂ f U\nC : ℝ\nhC : ∀ z ∈ frontier U, ‖f z‖ ≤ C\nz : E\nhz : z ∈ closure[Pse... | by
rw [closure_eq_self_union_frontier, union_comm, mem_union] at hz
rcases hz with hz | hz; · exact hC z hz
/- In case of a finite-dimensional domain, one can just apply
`Complex.exists_mem_frontier_isMaxOn_norm`. To make it work in any Banach space, we restrict
the function to a line first. -/
rcases e... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 153,
"column": 2
} | {
"line": 153,
"column": 66
} | [
{
"pp": "x : ℝ\nh : 0 < x\nh' : x ≤ 1\nhx : |x| = x\n⊢ x - x ^ 3 / 4 < sin x",
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instLE",
"Real",
"instHDiv",
"HMul.hMul",
"Real.lattice",
"abs",
"congrArg",
"Real.instDivInvMonoid",
"Rea... | have := neg_le_of_abs_le (sin_bound <| show |x| ≤ 1 by rwa [hx]) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Bounds | {
"line": 250,
"column": 2
} | {
"line": 253,
"column": 67
} | [
{
"pp": "x : ℝ\n⊢ ‖2 * sin (x / 2)‖ ≤ ‖x‖",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Iff.mpr",
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Mathlib.Tactic.Ring.Common.div_congr",
"Real.partia... | calc
_ = 2 * |Real.sin (x / 2)| := by simp
_ ≤ 2 * |x / 2| := (mul_le_mul_iff_of_pos_left zero_lt_two).mpr Real.abs_sin_le_abs
_ = _ := by rw [abs_div, Nat.abs_ofNat, Real.norm_eq_abs]; ring | Lean.Elab.Tactic._aux_Mathlib_Tactic_Widget_Calc___elabRules_Lean_calcTactic_1 | Lean.calcTactic |
Mathlib.Analysis.Convex.Deriv | {
"line": 515,
"column": 11
} | {
"line": 515,
"column": 62
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nhfc : ConvexOn ℝ S f\nx : ℝ\nhxs : x ∈ interior S\ny : ℝ\nhys : y ∈ interior S\nhxy✝ : x ≤ y\nhxy : x < y\n⊢ x ∈ {y_1 | y_1 ∈ S ∧ y_1 < y} ∧ slope f y x = slope f x y",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
... | simp only [slope_comm, mem_setOf_eq, hxy, and_true] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Convex.Deriv | {
"line": 619,
"column": 38
} | {
"line": 619,
"column": 55
} | [
{
"pp": "S : Set ℝ\nf : ℝ → ℝ\nx y f' : ℝ\nhfc : ConvexOn ℝ S f\nhx : x ∈ S\nhy : y ∈ S\nhxy : x < y\nhf' : HasDerivWithinAt f f' (Iio y) y\n⊢ ∀ᶠ (x_1 : ℝ) in 𝓝 y, x_1 ∈ Iio y → slope f x y ≤ slope f y x_1",
"usedConstants": [
"Eq.mpr",
"NormedCommRing.toSeminormedCommRing",
"Real",
... | slope_comm f x y, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Geometry.Euclidean.Angle.Unoriented.Basic | {
"line": 166,
"column": 6
} | {
"line": 166,
"column": 21
} | [
{
"pp": "V : Type u_1\ninst✝¹ : NormedAddCommGroup V\ninst✝ : InnerProductSpace ℝ V\nx y : V\n⊢ sin (arccos (⟪x, y⟫ / (‖x‖ * ‖y‖))) * (‖x‖ * ‖y‖) = √(⟪x, x⟫ * ⟪y, y⟫ - ⟪x, y⟫ * ⟪x, y⟫)",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"instHDiv",
"HMul.hMul",
"Inner.inn... | Real.sin_arccos | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.BorelCaratheodory | {
"line": 70,
"column": 2
} | {
"line": 70,
"column": 44
} | [
{
"pp": "f : ℂ → ℂ\nM R : ℝ\nz : ℂ\nhM : 0 < M\nhf : DifferentiableOn ℂ f (ball 0 R)\nhf₁ : Set.MapsTo f (ball 0 R) {z | z.re ≤ M}\nhz : z ∈ ball 0 R\nhf₂ : f 0 = 0\n⊢ dist (f z / (2 * ↑M - f z)) 0 ≤ 1 / R * dist z 0",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"GroupWith... | nth_rw 1 [← zero_div (2 * M - f 0), ← hf₂] | Mathlib.Tactic._aux_Mathlib_Tactic_NthRewrite___macroRules_Mathlib_Tactic_tacticNth_rw______1 | Mathlib.Tactic.tacticNth_rw_____ |
Mathlib.Analysis.Complex.CoveringMap | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 28
} | [
{
"pp": "case h.e'_5.h.h.e'_3\n𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : ProperSpace 𝕜\nn : ℕ\nhn : ↑(-↑n) ≠ 0\nx✝ : { x // x ≠ 0 }\n⊢ ↑x✝ ^ (-↑n) = ↑(((fun x ↦ ⟨↑x ^ n, ⋯⟩) ∘ ⇑(Homeomorph.inv₀ 𝕜)) x✝)",
"usedConstants": [
"zpow_natCast",
"NormedCommRing.toNormedRing",
... | simp [Homeomorph.inv₀] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Complex.CoveringMap | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 28
} | [
{
"pp": "case h.e'_5.h.h.e'_3\n𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : ProperSpace 𝕜\nn : ℕ\nhn : ↑(-↑n) ≠ 0\nx✝ : { x // x ≠ 0 }\n⊢ ↑x✝ ^ (-↑n) = ↑(((fun x ↦ ⟨↑x ^ n, ⋯⟩) ∘ ⇑(Homeomorph.inv₀ 𝕜)) x✝)",
"usedConstants": [
"zpow_natCast",
"NormedCommRing.toNormedRing",
... | simp [Homeomorph.inv₀] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.CoveringMap | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 28
} | [
{
"pp": "case h.e'_5.h.h.e'_3\n𝕜 : Type u_1\ninst✝¹ : NontriviallyNormedField 𝕜\ninst✝ : ProperSpace 𝕜\nn : ℕ\nhn : ↑(-↑n) ≠ 0\nx✝ : { x // x ≠ 0 }\n⊢ ↑x✝ ^ (-↑n) = ↑(((fun x ↦ ⟨↑x ^ n, ⋯⟩) ∘ ⇑(Homeomorph.inv₀ 𝕜)) x✝)",
"usedConstants": [
"zpow_natCast",
"NormedCommRing.toNormedRing",
... | simp [Homeomorph.inv₀] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Homotopy.Lifting | {
"line": 98,
"column": 16
} | {
"line": 98,
"column": 48
} | [
{
"pp": "case refine_2.refine_2\nE : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace A\np : E → X\nf : C(↑I × A, X)\ng : ↑I × A → E\ng_lifts : p ∘ g = ⇑f\ncont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)\na✝ : A\ncont_a : Continuous[_, ... | exact ⟨t_mono n.zero_le, le_rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Homotopy.Lifting | {
"line": 105,
"column": 4
} | {
"line": 105,
"column": 60
} | [
{
"pp": "case refine_2.refine_4\nE : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace A\np : E → X\nf : C(↑I × A, X)\ng : ↑I × A → E\ng_lifts : p ∘ g = ⇑f\ncont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)\na : A\ncont_a : Continuous[_, i... | rw [closure_le_eq continuous_fst continuous_const] at ht | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Homotopy.Lifting | {
"line": 116,
"column": 16
} | {
"line": 116,
"column": 37
} | [
{
"pp": "case refine_2.refine_8\nE : Type u_1\nX : Type u_2\nA : Type u_3\ninst✝² : TopologicalSpace E\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace A\np : E → X\nf : C(↑I × A, X)\ng : ↑I × A → E\ng_lifts : p ∘ g = ⇑f\ncont_0 : Continuous[inst✝, inst✝²] fun x ↦ g (0, x)\na : A\ncont_a : Continuous[_, i... | split_ifs with htn hf | Mathlib.Tactic._aux_Mathlib_Tactic_SplitIfs___elabRules_Mathlib_Tactic_splitIfs_1 | Mathlib.Tactic.splitIfs |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 128,
"column": 4
} | {
"line": 132,
"column": 26
} | [
{
"pp": "case neg.hl\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nh : MeromorphicAt f x\nh₂ : ¬meromorphicOrderAt f x = ⊤\ng : 𝕜 → E\nh₁g : AnalyticAt 𝕜 g x\nh₂g : g x ≠ 0\nh₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ ... | filter_upwards [h₃g, self_mem_nhdsWithin] with y h₁y h₂y
rw [zpow_neg, Pi.smul_apply', Pi.inv_apply, Pi.pow_apply, h₁y, ← smul_assoc, smul_eq_mul,
← zpow_neg, ← zpow_add', neg_add_cancel, zpow_zero, one_smul]
left
simp_all [sub_ne_zero] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.TrailingCoefficient | {
"line": 128,
"column": 4
} | {
"line": 132,
"column": 26
} | [
{
"pp": "case neg.hl\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nx : 𝕜\nh : MeromorphicAt f x\nh₂ : ¬meromorphicOrderAt f x = ⊤\ng : 𝕜 → E\nh₁g : AnalyticAt 𝕜 g x\nh₂g : g x ≠ 0\nh₃g : f =ᶠ[𝓝[≠] x] fun z ↦ (z - x) ^ ... | filter_upwards [h₃g, self_mem_nhdsWithin] with y h₁y h₂y
rw [zpow_neg, Pi.smul_apply', Pi.inv_apply, Pi.pow_apply, h₁y, ← smul_assoc, smul_eq_mul,
← zpow_neg, ← zpow_add', neg_add_cancel, zpow_zero, one_smul]
left
simp_all [sub_ne_zero] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.NormalForm | {
"line": 733,
"column": 4
} | {
"line": 733,
"column": 85
} | [
{
"pp": "case pos\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nU : Set 𝕜\nhf : MeromorphicOn f U\nz : 𝕜\nhz : z ∈ U\n⊢ MeromorphicNFAt (toMeromorphicNFOn f U) z",
"usedConstants": [
"Eq.mpr",
"congrArg",... | rw [meromorphicNFAt_congr (toMeromorphicNFOn_eq_toMeromorphicNFAt_on_nhds hf hz)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Meromorphic.FactorizedRational | {
"line": 54,
"column": 2
} | {
"line": 61,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nd : 𝕜 → ℤ\n⊢ (Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u) = support d",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Set.ext",
"Eq.mpr",
"MulOne.toOne",
"False",
"_private.Mathlib.Analysis.Meromorph... | ext u
constructor <;> intro h
· simp_all only [mem_mulSupport, ne_eq, mem_support]
by_contra hCon
simp_all [zpow_zero]
· simp_all only [mem_mulSupport, ne_eq, ne_iff]
use u
simp_all [zero_zpow_eq_one₀] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Meromorphic.FactorizedRational | {
"line": 54,
"column": 2
} | {
"line": 61,
"column": 32
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nd : 𝕜 → ℤ\n⊢ (Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u) = support d",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Set.ext",
"Eq.mpr",
"MulOne.toOne",
"False",
"_private.Mathlib.Analysis.Meromorph... | ext u
constructor <;> intro h
· simp_all only [mem_mulSupport, ne_eq, mem_support]
by_contra hCon
simp_all [zpow_zero]
· simp_all only [mem_mulSupport, ne_eq, ne_iff]
use u
simp_all [zero_zpow_eq_one₀] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.CanonicalDecomposition | {
"line": 84,
"column": 2
} | {
"line": 84,
"column": 32
} | [
{
"pp": "case inl\nw x : ℂ\nhx : x ∈ {w}ᶜ\n⊢ AnalyticAt ℂ (fun z ↦ (↑0 ^ 2 - (starRingEnd ℂ) w * z) / (↑0 * (z - w))) x",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"False",
"Real",
"instHDiv",
"HMul.hMul",
"Complex.instNormedAddCommGroup",
... | · simpa using analyticAt_const | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Basic | {
"line": 45,
"column": 4
} | {
"line": 47,
"column": 91
} | [
{
"pp": "case mp\nR : Type u_1\nn : Type u_2\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing R\ng : GL n R\n⊢ g ∈ Subgroup.center (GL n R) → ↑g ∈ Set.range ⇑(Matrix.scalar n)",
"usedConstants": [
"Units.val",
"HMul.hMul",
"Monoid.toMulOneClass",
"CommSemiring.toSemiring... | intro hg
refine Matrix.mem_range_scalar_of_commute_transvectionStruct fun t ↦ ?_
simpa [Units.ext_iff] using Subgroup.mem_center_iff.mp hg (.mk _ _ t.mul_inv t.inv_mul) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.GeneralLinearGroup.Basic | {
"line": 45,
"column": 4
} | {
"line": 47,
"column": 91
} | [
{
"pp": "case mp\nR : Type u_1\nn : Type u_2\ninst✝² : Fintype n\ninst✝¹ : DecidableEq n\ninst✝ : CommRing R\ng : GL n R\n⊢ g ∈ Subgroup.center (GL n R) → ↑g ∈ Set.range ⇑(Matrix.scalar n)",
"usedConstants": [
"Units.val",
"HMul.hMul",
"Monoid.toMulOneClass",
"CommSemiring.toSemiring... | intro hg
refine Matrix.mem_range_scalar_of_commute_transvectionStruct fun t ↦ ?_
simpa [Units.ext_iff] using Subgroup.mem_center_iff.mp hg (.mk _ _ t.mul_inv t.inv_mul) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.FactorizedRational | {
"line": 212,
"column": 2
} | {
"line": 212,
"column": 22
} | [
{
"pp": "𝕜 : Type u_1\ninst✝ : NontriviallyNormedField 𝕜\nd : 𝕜 → ℤ\nx : 𝕜\nh : HasFiniteSupport d\nthis : (Function.mulSupport fun u ↦ (fun x ↦ x - u) ^ d u) ⊆ ↑(Finite.toFinset h)\ny : 𝕜\nhy : y ∈ Finite.toFinset h\n⊢ meromorphicTrailingCoeffAt (fun x ↦ x - y) x ^ d y = (x - y) ^ update d x 0 y",
"us... | by_cases hxy : x = y | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Complex.Isometry | {
"line": 97,
"column": 2
} | {
"line": 100,
"column": 60
} | [
{
"pp": "f : ℂ →ₗᵢ[ℝ] ℂ\nh₂ : ∀ (z : ℂ), (f z).re = z.re\nz : ℂ\n⊢ (f z).im = z.im ∨ (f z).im = -z.im",
"usedConstants": [
"LinearIsometry",
"NormedCommRing.toNormedRing",
"Norm.norm",
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
... | have h₁ := f.norm_map z
simp only [norm_def] at h₁
rwa [Real.sqrt_inj (normSq_nonneg _) (normSq_nonneg _), normSq_apply (f z), normSq_apply z,
h₂, add_left_cancel_iff, mul_self_eq_mul_self_iff] at h₁ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.Isometry | {
"line": 97,
"column": 2
} | {
"line": 100,
"column": 60
} | [
{
"pp": "f : ℂ →ₗᵢ[ℝ] ℂ\nh₂ : ∀ (z : ℂ), (f z).re = z.re\nz : ℂ\n⊢ (f z).im = z.im ∨ (f z).im = -z.im",
"usedConstants": [
"LinearIsometry",
"NormedCommRing.toNormedRing",
"Norm.norm",
"NegZeroClass.toNeg",
"NormedCommRing.toSeminormedCommRing",
"Real.partialOrder",
... | have h₁ := f.norm_map z
simp only [norm_def] at h₁
rwa [Real.sqrt_inj (normSq_nonneg _) (normSq_nonneg _), normSq_apply (f z), normSq_apply z,
h₂, add_left_cancel_iff, mul_self_eq_mul_self_iff] at h₁ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Meromorphic.Order | {
"line": 777,
"column": 4
} | {
"line": 777,
"column": 64
} | [
{
"pp": "case h.inr\n𝕜 : Type u_1\ninst✝² : NontriviallyNormedField 𝕜\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace 𝕜 E\nf : 𝕜 → E\nU : Set 𝕜\nhf : MeromorphicOn f U\nx : 𝕜\nhx : x ∈ U\nh₁f : ∀ᶠ (z : 𝕜) in 𝓝[≠] x, f z = 0\na : 𝕜\nh₁a : ∀ᶠ (x : 𝕜) in 𝓝[{x}ᶜ] a, f x = 0\nhax : a ≠ x... | rw [eventually_nhdsWithin_iff, eventually_nhds_iff] at h₁a ⊢ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.Hadamard | {
"line": 350,
"column": 4
} | {
"line": 353,
"column": 21
} | [
{
"pp": "case h.mp\nE : Type u_1\ninst✝ : NormedAddCommGroup E\nf : ℂ → E\nl u : ℝ\nhul : l < u\ne : E\n⊢ (∃ x, x.re = 0 ∧ f (↑l + x * (↑u - ↑l)) = e) → ∃ x, x.re = l ∧ f x = e",
"usedConstants": [
"Real",
"Complex.mul_re",
"HMul.hMul",
"sub_self",
"Real.instZero",
"Real.... | · intro h
obtain ⟨z, hz₁, hz₂⟩ := h
use ↑l + z * (↑u - ↑l)
simp [hz₁, hz₂] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.Conformal | {
"line": 268,
"column": 6
} | {
"line": 268,
"column": 45
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nf : ℂ → E\nx : ℂ\ns : Set ℂ\nh₁ : DifferentiableWithinAt ℝ f s x\nh₂ : (fderivWithin ℝ f s x) I = I • (fderivWithin ℝ f s x) 1\n⊢ HasDerivWithinAt f (((fderivWithin ℝ f s x).complexOfReal h₂) 1) s x",
"usedConstants": [
"No... | hasDerivWithinAt_iff_hasFDerivWithinAt, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Laplacian | {
"line": 365,
"column": 35
} | {
"line": 367,
"column": 28
} | [
{
"pp": "𝕜 : Type u_1\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAlgebra ℝ 𝕜\nE : Type u_2\ninst✝⁶ : NormedAddCommGroup E\ninst✝⁵ : InnerProductSpace ℝ E\ninst✝⁴ : FiniteDimensional ℝ E\nF : Type u_3\ninst✝³ : NormedAddCommGroup F\ninst✝² : NormedSpace ℝ F\ninst✝¹ : NormedSpace 𝕜 F\ninst✝ : IsScala... | by
filter_upwards [h.eventually (by simp)] with a ha
simp [laplacian_smul v ha] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 127,
"column": 2
} | {
"line": 135,
"column": 75
} | [
{
"pp": "x : ℝ\n⊢ deriv^[2] (fun x ↦ x * log x) x = x⁻¹",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NormedCommRing.toSeminormedCommRing",
"Real",
"DivInvMonoid.toInv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRin... | simp only [Function.iterate_succ, Function.iterate_zero, Function.id_comp, Function.comp_apply]
by_cases hx : x = 0
· rw [hx, inv_zero]
exact deriv_zero_of_not_differentiableAt
(fun h ↦ not_continuousAt_deriv_mul_log_zero h.continuousAt)
· suffices ∀ᶠ y in (𝓝 x), deriv (fun x ↦ x * log x) y = log y + 1... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Log.NegMulLog | {
"line": 127,
"column": 2
} | {
"line": 135,
"column": 75
} | [
{
"pp": "x : ℝ\n⊢ deriv^[2] (fun x ↦ x * log x) x = x⁻¹",
"usedConstants": [
"NormedCommRing.toNormedRing",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NormedCommRing.toSeminormedCommRing",
"Real",
"DivInvMonoid.toInv",
"NonUnitalCommRing.toNonUnitalNonAssocCommRin... | simp only [Function.iterate_succ, Function.iterate_zero, Function.id_comp, Function.comp_apply]
by_cases hx : x = 0
· rw [hx, inv_zero]
exact deriv_zero_of_not_differentiableAt
(fun h ↦ not_continuousAt_deriv_mul_log_zero h.continuousAt)
· suffices ∀ᶠ y in (𝓝 x), deriv (fun x ↦ x * log x) y = log y + 1... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Log.PosLog | {
"line": 163,
"column": 64
} | {
"line": 185,
"column": 65
} | [
{
"pp": "α : Type u_1\ns : Finset α\nf : α → ℝ\n⊢ log⁺ (∑ t ∈ s, f t) ≤ log ↑s.card + ∑ t ∈ s, log⁺ (f t)",
"usedConstants": [
"abs_nonneg._simp_1",
"CharP.cast_eq_zero",
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoid... | by
-- Trivial case: empty sum
by_cases! hs : s = ∅
· simp [hs, posLog]
-- Nontrivial case: Obtain maximal element…
obtain ⟨t_max, ht_max⟩ := s.exists_max_image (fun t ↦ |f t|) hs
-- …then calculate
calc log⁺ (∑ t ∈ s, f t)
_ = log⁺ |∑ t ∈ s, f t| := by
rw [Real.posLog_abs]
_ ≤ log⁺ (∑ t ∈ s, |f t|... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Integrability.LogMeromorphic | {
"line": 47,
"column": 4
} | {
"line": 61,
"column": 14
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nhf : MeromorphicOn f [[a, b]]\nt₀ : ∀ (u : ↑[[a, b]]), meromorphicOrderAt f ↑u ≠ ⊤\n⊢ IntervalIntegrable (fun x ↦ log ‖f x‖) volume a b",
"usedConstants": [
"Filter.codiscreteWithin_mono",
... | obtain ⟨g, h₁g, h₂g, h₃g⟩ := hf.extract_zeros_poles t₀
((MeromorphicOn.divisor f [[a, b]]).finiteSupport isCompact_uIcc)
have h₄g := MeromorphicOn.extract_zeros_poles_log h₂g h₃g
rw [intervalIntegrable_congr_codiscreteWithin
(h₄g.filter_mono (Filter.codiscreteWithin_mono Set.uIoc_subset_uIcc))]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Integrability.LogMeromorphic | {
"line": 47,
"column": 4
} | {
"line": 61,
"column": 14
} | [
{
"pp": "case pos\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nhf : MeromorphicOn f [[a, b]]\nt₀ : ∀ (u : ↑[[a, b]]), meromorphicOrderAt f ↑u ≠ ⊤\n⊢ IntervalIntegrable (fun x ↦ log ‖f x‖) volume a b",
"usedConstants": [
"Filter.codiscreteWithin_mono",
... | obtain ⟨g, h₁g, h₂g, h₃g⟩ := hf.extract_zeros_poles t₀
((MeromorphicOn.divisor f [[a, b]]).finiteSupport isCompact_uIcc)
have h₄g := MeromorphicOn.extract_zeros_poles_log h₂g h₃g
rw [intervalIntegrable_congr_codiscreteWithin
(h₄g.filter_mono (Filter.codiscreteWithin_mono Set.uIoc_subset_uIcc))]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Integrability.LogMeromorphic | {
"line": 72,
"column": 8
} | {
"line": 72,
"column": 54
} | [
{
"pp": "case neg\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nf : ℝ → E\na b : ℝ\nhf : MeromorphicOn f [[a, b]]\nt₀ : ∀ (u : ↑[[a, b]]), meromorphicOrderAt f ↑u = ⊤\nthis : (fun x ↦ log ‖f x‖) =ᶠ[codiscreteWithin (Ι a b)] 0\n⊢ IntervalIntegrable (fun x ↦ log ‖f x‖) volume a b",
"u... | intervalIntegrable_congr_codiscreteWithin this | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 243,
"column": 2
} | {
"line": 248,
"column": 80
} | [
{
"pp": "a b : ℝ\nc : ℂ\nhc : c ≠ 0\n⊢ ∫ (x : ℝ) in a..b, Complex.exp (c * ↑x) = (Complex.exp (c * ↑b) - Complex.exp (c * ↑a)) / c",
"usedConstants": [
"instInnerProductSpaceRealComplex",
"NormedCommRing.toNormedRing",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"NormedCommRin... | have D : ∀ x : ℝ, HasDerivAt (fun y : ℝ => Complex.exp (c * y) / c) (Complex.exp (c * x)) x := by
intro x
conv => congr
rw [← mul_div_cancel_right₀ (Complex.exp (c * x)) hc]
apply ((Complex.hasDerivAt_exp _).comp x _).div_const c
simpa only [mul_one] using ((hasDerivAt_id (x : ℂ)).const_mul _).comp_... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.SpecialFunctions.Integrals.PosLogEqCircleAverage | {
"line": 184,
"column": 17
} | {
"line": 184,
"column": 58
} | [
{
"pp": "case inr.inl\na : ℂ\nh : 1 = ‖a‖\n⊢ log⁺ ‖a‖ = circleAverage (fun x ↦ log ‖x - a‖) 0 1",
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"InnerProductSpace.toNormedSpace",
"Real",
"circleAverage_log_norm_sub_const₁",
"Real.posLog",
"Real.instZero",
"Real.in... | circleAverage_log_norm_sub_const₁ h.symm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.Periodic | {
"line": 171,
"column": 2
} | {
"line": 171,
"column": 43
} | [
{
"pp": "h : ℝ\nf : ℂ → ℂ\nhh : h ≠ 0\nhf : Periodic f ↑h\nz : ℂ\nhol_z : DifferentiableAt ℂ f z\nq : ℂ := 𝕢 h z\nqdiff : HasStrictDerivAt (𝕢 h) (q * (2 * ↑π * I / ↑h)) z\ndiff_ne : q * (2 * ↑π * I / ↑h) ≠ 0\nL : ℂ → ℂ := HasStrictDerivAt.localInverse (𝕢 h) (q * (2 * ↑π * I / ↑h)) z qdiff diff_ne\ndiff_L : D... | have hF := hL.fun_comp (cuspFunction h f) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 168,
"column": 2
} | {
"line": 168,
"column": 49
} | [
{
"pp": "E : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℂ E\nU : Set E\ng : E → ℂ\nhg : AnalyticOnNhd ℂ g U\nhU : IsPreconnected U\n⊢ (∃ w, ∀ z ∈ U, g z = w) ∨ ∀ s ⊆ U, IsOpen[PseudoMetricSpace.toUniformSpace.toTopologicalSpace] s → IsOpen (g '' s)",
"usedConstants": [
"NormedCommRin... | by_cases h : ∃ z₀ ∈ U, ∀ᶠ z in 𝓝 z₀, g z = g z₀ | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Complex.OpenMapping | {
"line": 275,
"column": 4
} | {
"line": 275,
"column": 26
} | [
{
"pp": "case h.e'_5.h.h.e'_3\nn : ℕ\ninst✝ : NeZero (-↑n)\nthis : NeZero n\nx✝ : { z // z ≠ 0 }\n⊢ ↑x✝ ^ (-↑n) = ↑(((fun z ↦ ⟨↑z ^ n, ⋯⟩) ∘ ⇑(Homeomorph.inv₀ ℂ)) x✝)",
"usedConstants": [
"zpow_natCast",
"NormedCommRing.toNormedRing",
"GroupWithZero.toMonoidWithZero",
"NormedCommRing... | simp [Homeomorph.inv₀] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 510,
"column": 6
} | {
"line": 510,
"column": 88
} | [
{
"pp": "a b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nx : ℝ\nx✝ : x ∈ [[a, b]]\n⊢ HasDerivAt (fun y ↦ cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x",
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"N... | simpa only [mul_right_comm, neg_mul, mul_neg] using (hasDerivAt_cos x).pow (n + 1) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 510,
"column": 6
} | {
"line": 510,
"column": 88
} | [
{
"pp": "a b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nx : ℝ\nx✝ : x ∈ [[a, b]]\n⊢ HasDerivAt (fun y ↦ cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x",
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"N... | simpa only [mul_right_comm, neg_mul, mul_neg] using (hasDerivAt_cos x).pow (n + 1) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Integrals.Basic | {
"line": 510,
"column": 6
} | {
"line": 510,
"column": 88
} | [
{
"pp": "a b : ℝ\nn : ℕ\nC : ℝ := cos b ^ (n + 1) * sin b - cos a ^ (n + 1) * sin a\nh : ∀ (α β γ : ℝ), β * α * γ * α = β * (α * α * γ)\nx : ℝ\nx✝ : x ∈ [[a, b]]\n⊢ HasDerivAt (fun y ↦ cos y ^ (n + 1)) (-↑(n + 1) * sin x * cos x ^ n) x",
"usedConstants": [
"IsModuleTopology.toContinuousSMul",
"N... | simpa only [mul_right_comm, neg_mul, mul_neg] using (hasDerivAt_cos x).pow (n + 1) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Complex.Positivity | {
"line": 81,
"column": 2
} | {
"line": 81,
"column": 23
} | [
{
"pp": "case h.e'_3\nf : ℂ → ℂ\nc : ℂ\nhf : Differentiable ℂ f\nh : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f c\nz : ℂ\nhz : z ≤ c\n⊢ f c = f (- -c)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | · simp only [neg_neg] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.Positivity | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 23
} | [
{
"pp": "case h.e'_4\nf : ℂ → ℂ\nc : ℂ\nhf : Differentiable ℂ f\nh : ∀ (n : ℕ), n ≠ 0 → 0 ≤ (-1) ^ n * iteratedDeriv n f c\nz : ℂ\nhz : z ≤ c\n⊢ f z = f (- -z)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"NegZeroClass.toNeg",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
... | · simp only [neg_neg] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Complex.JensenFormula | {
"line": 74,
"column": 43
} | {
"line": 74,
"column": 53
} | [
{
"pp": "r₀ r R : ℝ\nρ : ℂ\nhρ : ‖ρ‖ = R\nhr₀ : 0 < r₀\nhR : 0 < R\nhr₀r : r₀ ≤ r\nhrR : r ≤ R\nθ r₁ : ℝ\n⊢ ↑(normSq (circleMap 0 r₁ θ - ρ)) = ↑(r₁ ^ 2 + R ^ 2 - 2 * r₁ * R * Real.cos (θ - ρ.arg))",
"usedConstants": [
"Eq.mpr",
"Real",
"HMul.hMul",
"Real.cos",
"congrArg",
... | normSq_sub | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.RiemannMapping | {
"line": 92,
"column": 8
} | {
"line": 92,
"column": 17
} | [
{
"pp": "case h\nU : Set ℂ\nhUo : IsOpen U\nhUc : IsSimplyConnected U\nhU : U ≠ univ\nhU₀ : 0 ∉ U\nf : ℂ → ℂ\nhfc : ContinuousOn f U\nhf_inv : LeftInverse (fun x ↦ x ^ 2) f\nhf₀ : ∀ z ∈ U, f z ≠ 0\nhdf : ∀ z ∈ U, HasStrictDerivAt f (2 * f z)⁻¹ z\nx : ℂ\nhx : x ∈ U\nthis : f '' U ∈ 𝓝 (f x)\n⊢ ∀ᶠ (x : ℂ) in 𝓝 (... | nhds_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.TietzeExtension | {
"line": 320,
"column": 2
} | {
"line": 320,
"column": 88
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\ninst✝¹ : NormalSpace Y\ninst✝ : Nonempty X\nf : X →ᵇ ℝ\ne : X → Y\nhe : IsClosedEmbedding e\ninhabited_h : Inhabited X\na : ℝ\nha : IsGLB (range ⇑f) a\n⊢ ∃ g, (∀ (y : Y), ∃ x₁ x₂, g y ∈ Icc (f x₁) (f x₂)) ∧ ⇑g ∘ e = ⇑... | obtain ⟨b, hb⟩ : ∃ b, IsLUB (range f) b := ⟨_, isLUB_ciSup f.isBounded_range.bddAbove⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.TietzeExtension | {
"line": 463,
"column": 2
} | {
"line": 463,
"column": 56
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : NormalSpace Y\nf : C(X, ℝ)\nt : Set ℝ\ne : X → Y\nhs : t.OrdConnected\nhf : ∀ (x : X), f x ∈ t\nhne : t.Nonempty\nhe : IsClosedEmbedding e\n⊢ ∃ g, (∀ (y : Y), g y ∈ t) ∧ ⇑g ∘ e = ⇑f",
"usedConstants": [
... | have h : ℝ ≃o Ioo (-1 : ℝ) 1 := orderIsoIooNegOneOne ℝ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints | {
"line": 164,
"column": 6
} | {
"line": 164,
"column": 41
} | [
{
"pp": "g✝ : GL (Fin 2) ℝ\nz : ℍ\ng : GL (Fin 2) ℝ\nhpos : 0 < (↑g).det\nhell : g.IsElliptic\nhc : 0 < ↑g 1 0\nhd : discrim (↑(↑g 1 0)) (↑(↑g 1 1) - ↑(↑g 0 0)) (-↑(↑g 0 1)) = (Complex.I * ↑√(-(↑g).discr)) ^ 2\n⊢ g • z = z ↔ z = fixedPt g hell",
"usedConstants": [
"UpperHalfPlane.glAction",
"Uni... | gl_smul_eq_self_iff_quadratic hpos, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.UpperHalfPlane.FixedPoints | {
"line": 164,
"column": 42
} | {
"line": 165,
"column": 26
} | [
{
"pp": "g✝ : GL (Fin 2) ℝ\nz : ℍ\ng : GL (Fin 2) ℝ\nhpos : 0 < (↑g).det\nhell : g.IsElliptic\nhc : 0 < ↑g 1 0\nhd : discrim (↑(↑g 1 0)) (↑(↑g 1 1) - ↑(↑g 0 0)) (-↑(↑g 0 1)) = (Complex.I * ↑√(-(↑g).discr)) ^ 2\n⊢ ↑(↑g 1 0) * (↑z * ↑z) + (↑(↑g 1 1) - ↑(↑g 0 0)) * ↑z + -↑(↑g 0 1) = 0 ↔ z = fixedPt g hell",
"u... | quadratic_eq_zero_iff (mod_cast hell.c_ne_zero)
(hd.trans (pow_two _)) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Complex.UpperHalfPlane.FunctionsBoundedAtInfty | {
"line": 92,
"column": 2
} | {
"line": 94,
"column": 63
} | [
{
"pp": "g : GL (Fin 2) ℝ\nhg : ↑g 1 0 = 0\n⊢ Tendsto (fun τ ↦ g • τ) atImInfty atImInfty",
"usedConstants": [
"UpperHalfPlane.glAction",
"Complex.mul_im",
"Real.instIsOrderedRing",
"Units.val",
"Eq.mpr",
"Real",
"instHSMul",
"UpperHalfPlane.im_pos",
"in... | suffices Tendsto (fun τ ↦ |g 0 0 / g 1 1| * τ.im) atImInfty atTop by
simpa [atImInfty, Function.comp_def, im_smul, num, denom, hg, abs_div, abs_mul,
abs_of_pos (UpperHalfPlane.im_pos _), mul_div_right_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.Analysis.Complex.UpperHalfPlane.MoebiusAction | {
"line": 333,
"column": 88
} | {
"line": 344,
"column": 7
} | [
{
"pp": "g : SL(2, ℝ)\nhc : ↑g 1 0 ≠ 0\n⊢ ∃ u v w, (fun x ↦ g • x) = (fun x ↦ w +ᵥ x) ∘ (fun x ↦ ModularGroup.S • x) ∘ (fun x ↦ v +ᵥ x) ∘ fun x ↦ u • x",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Real.partialOrder",
"Real",
"instHSMul",
"Matrix.SpecialLinearGroup",
"... | by
have h_denom (z : ℍ) := denom_ne_zero g z
induction g using Matrix.SpecialLinearGroup.fin_two_induction with | _ a b c d h => ?_
replace hc : c ≠ 0 := by simpa using hc
refine ⟨⟨_, mul_self_pos.mpr hc⟩, c * d, a / c, ?_⟩
ext1 ⟨z, hz⟩; ext1
suffices (↑a * z + b) / (↑c * z + d) = a / c - (c * d + ↑c * ↑c *... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Norm.Transitivity | {
"line": 53,
"column": 39
} | {
"line": 53,
"column": 50
} | [
{
"pp": "S : Type u_2\nm : Type u_5\ninst✝¹ : CommRing S\nM : Matrix m m S\ninst✝ : DecidableEq m\nk i j : m\nlt : i ≠ k ∧ j = k\n⊢ (if i = k then 1 else 0) = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"CommSemiring.toSemiring",
"AddGroupWithOne.toAddMonoidWithOne",
"id",
... | if_neg lt.1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Arsinh | {
"line": 59,
"column": 6
} | {
"line": 59,
"column": 27
} | [
{
"pp": "case hx\nx : ℝ\n⊢ 0 < x + √(1 + x ^ 2)",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.partialOrder",
"Real",
"Real.instZero",
"congrArg",
"instIsLeftCancelAddOfAddLeftReflectLE",
"AddMonoid.toAddZeroCl... | ← neg_lt_iff_pos_add' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.ProperAction.Basic | {
"line": 124,
"column": 4
} | {
"line": 125,
"column": 22
} | [
{
"pp": "case h.e'_3.h\nG : Type u_1\nX : Type u_2\ninst✝⁴ : Group G\ninst✝³ : MulAction G X\ninst✝² : TopologicalSpace G\ninst✝¹ : TopologicalSpace X\ninst✝ : ProperSMul G X\nR : Setoid X := MulAction.orbitRel G X\nπ : X → Quotient R := Quotient.mk'\nthis : IsOpenQuotientMap (Prod.map π π)\nx₁ x₂ : X\n⊢ (x₁, x... | simp only [mem_preimage, map_apply, mem_diagonal_iff, mem_range, Prod.mk.injEq, Prod.exists,
exists_eq_right] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Compactification.OnePoint.Basic | {
"line": 209,
"column": 24
} | {
"line": 216,
"column": 48
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝ : TopologicalSpace X\nS : Set (Set (OnePoint X))\nho : ∀ t ∈ S, (∞ ∈ t → IsCompact (some ⁻¹' t)ᶜ) ∧ IsOpen[inst✝] (some ⁻¹' t)\n⊢ (∞ ∈ ⋃₀ S → IsCompact (some ⁻¹' ⋃₀ S)ᶜ) ∧ IsOpen[inst✝] (some ⁻¹' ⋃₀ S)",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"On... | by
suffices IsOpen ((↑) ⁻¹' ⋃₀ S : Set X) by
refine ⟨?_, this⟩
rintro ⟨s, hsS : s ∈ S, hs : ∞ ∈ s⟩
refine IsCompact.of_isClosed_subset ((ho s hsS).1 hs) this.isClosed_compl ?_
exact compl_subset_compl.mpr (preimage_mono <| subset_sUnion_of_mem hsS)
rw [preimage_sUnion]
exact isOpen_b... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Category.CompHaus.Basic | {
"line": 212,
"column": 4
} | {
"line": 212,
"column": 42
} | [
{
"pp": "case mpr\nX Y : CompHaus\nf : X ⟶ Y\n⊢ Epi (↾⇑(ConcreteCategory.hom f)) → Epi f",
"usedConstants": [
"CategoryTheory.instFaithfulForget",
"ContinuousMap",
"CompHausLike",
"TopCat.str",
"CategoryTheory.Functor.reflectsEpimorphisms_of_faithful",
"TopCat.carrier",
... | apply (forget CompHaus).epi_of_epi_map | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Analysis.Complex.ValueDistribution.Cartan | {
"line": 117,
"column": 8
} | {
"line": 117,
"column": 50
} | [
{
"pp": "case pos\nf : ℂ → ℂ\nh : meromorphicOrderAt f 0 = 0\nhf : MeromorphicAt f 0\n⊢ circleAverage (fun x ↦ log ‖x - meromorphicTrailingCoeffAt f 0‖) 0 1 = log⁺ ‖meromorphicTrailingCoeffAt f 0‖",
"usedConstants": [
"Norm.norm",
"SeminormedAddGroup.toNorm",
"Eq.mpr",
"InnerProductS... | circleAverage_log_norm_sub_const_eq_posLog | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.ConstantSpeed | {
"line": 173,
"column": 2
} | {
"line": 173,
"column": 60
} | [
{
"pp": "E : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : ℝ → E\ns : Set ℝ\nl l' : ℝ≥0\nhl' : l' ≠ 0\nφ : ℝ → ℝ\nφm : MonotoneOn φ s\nhfφ : HasConstantSpeedOnWith (f ∘ φ) s l\nhf : HasConstantSpeedOnWith f (φ '' s) l'\nx : ℝ\nxs : x ∈ s\ny : ℝ\nys : y ∈ s\n⊢ (φ y - φ x) * ↑l' = (y - x) * ↑l",
"usedConstants"... | rw [hasConstantSpeedOnWith_iff_variationOnFromTo_eq] at hf | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 227,
"column": 4
} | {
"line": 229,
"column": 90
} | [
{
"pp": "case pos\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nf : locallyFinsupp E ℤ\nr : ℝ\nh : 0 ≤ f\nhr : 1 ≤ r\nh₃r : 0 < r\na : E\nh₁a : ¬a = 0\nh₂a : a ∈ closedBall 0 |r|\n⊢ 0 ≤ ↑(((toClosedBall r) f) a) * log (r * ‖a‖⁻¹)",
"usedConstants": [
"Iff.mpr",
"AddGroup.t... | refine mul_nonneg ?_ <| log_nonneg ?_
· simpa [h₂a] using h a
· simpa [mul_comm r, one_le_inv_mul₀ (norm_pos_iff.mpr h₁a), abs_of_pos h₃r] using h₂a | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Complex.ValueDistribution.LogCounting.Basic | {
"line": 227,
"column": 4
} | {
"line": 229,
"column": 90
} | [
{
"pp": "case pos\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : ProperSpace E\nf : locallyFinsupp E ℤ\nr : ℝ\nh : 0 ≤ f\nhr : 1 ≤ r\nh₃r : 0 < r\na : E\nh₁a : ¬a = 0\nh₂a : a ∈ closedBall 0 |r|\n⊢ 0 ≤ ↑(((toClosedBall r) f) a) * log (r * ‖a‖⁻¹)",
"usedConstants": [
"Iff.mpr",
"AddGroup.t... | refine mul_nonneg ?_ <| log_nonneg ?_
· simpa [h₂a] using h a
· simpa [mul_comm r, one_le_inv_mul₀ (norm_pos_iff.mpr h₁a), abs_of_pos h₃r] using h₂a | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.BetweenList | {
"line": 110,
"column": 6
} | {
"line": 124,
"column": 13
} | [
{
"pp": "case cons.refine_1\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsOrderedRing R\nhead : P\ntail : List P\nih : List.Wbtw R tail ∧ Pairwise (fun x1 x2 ↦ x1 ≠ x2) tail ↔ Triplewise (Sbtw ... | rcases h with ⟨⟨hp, ht⟩, hpne⟩
refine ⟨⟨?_, ?_⟩, ?_⟩
· clear ih
induction tail with
| nil => simp
| cons head2 tail ih' =>
rw [pairwise_cons] at hp hpne hpne ⊢
refine ⟨fun a ha ↦ ⟨hp.1 a ha, ?_⟩, ?_⟩
· refine ⟨(hpne.1 head2 ?_).symm, hpne.2.1 a ha⟩
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.BetweenList | {
"line": 110,
"column": 6
} | {
"line": 124,
"column": 13
} | [
{
"pp": "case cons.refine_1\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : IsOrderedRing R\nhead : P\ntail : List P\nih : List.Wbtw R tail ∧ Pairwise (fun x1 x2 ↦ x1 ≠ x2) tail ↔ Triplewise (Sbtw ... | rcases h with ⟨⟨hp, ht⟩, hpne⟩
refine ⟨⟨?_, ?_⟩, ?_⟩
· clear ih
induction tail with
| nil => simp
| cons head2 tail ih' =>
rw [pairwise_cons] at hp hpne hpne ⊢
refine ⟨fun a ha ↦ ⟨hp.1 a ha, ?_⟩, ?_⟩
· refine ⟨(hpne.1 head2 ?_).symm, hpne.2.1 a ha⟩
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Hall.Finite | {
"line": 143,
"column": 2
} | {
"line": 144,
"column": 89
} | [
{
"pp": "α : Type v\ninst✝ : DecidableEq α\nι : Type u\nt : ι → Finset α\ns : Finset ι\nhus : #s = #(s.biUnion t)\nht : ∀ (s : Finset ι), #s ≤ #(s.biUnion t)\ns' : Finset ↑(↑s)ᶜ\nthis : DecidableEq ι\ndisj : Disjoint s (image (fun z ↦ ↑z) s')\n⊢ #s' ≤ #(s'.biUnion fun x' ↦ t ↑x' \\ s.biUnion t)",
"usedConst... | have : #s' = #(s ∪ s'.image fun z => z.1) - #s := by
simp [disj, card_image_of_injective _ Subtype.coe_injective, Nat.add_sub_cancel_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.BetweenList | {
"line": 215,
"column": 8
} | {
"line": 216,
"column": 40
} | [
{
"pp": "case pos\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nhead : P\ntail : List P\nih :\n ∀ (hl : tail ≠ []),\n List.Wbtw R tail →\n ∃ l', (∀ a ∈ l', 0 ≤ a) ... | refine ⟨[0], ?_⟩
simp [ht, sortedLE_iff_pairwise] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.BetweenList | {
"line": 215,
"column": 8
} | {
"line": 216,
"column": 40
} | [
{
"pp": "case pos\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nhead : P\ntail : List P\nih :\n ∀ (hl : tail ≠ []),\n List.Wbtw R tail →\n ∃ l', (∀ a ∈ l', 0 ≤ a) ... | refine ⟨[0], ?_⟩
simp [ht, sortedLE_iff_pairwise] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.BetweenList | {
"line": 294,
"column": 8
} | {
"line": 294,
"column": 53
} | [
{
"pp": "case refine_1\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁶ : Field R\ninst✝⁵ : LinearOrder R\ninst✝⁴ : IsStrictOrderedRing R\ninst✝³ : AddCommGroup V\ninst✝² : Module R V\ninst✝¹ : AddTorsor V P\ninst✝ : Nontrivial P\np₁ p₂ : P\nhp₁p₂ : p₁ ≠ p₂\nl' : List R\nhl's : l'.SortedLT\nx✝ : ∃ p₁_1 p₂_1, p... | (lineMap_injective _ hp₁p₂).list_sbtw_map_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.CofilteredSystem | {
"line": 231,
"column": 2
} | {
"line": 241,
"column": 88
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v_1, u} J\nF : J ⥤ Type v\ninst✝ : IsCofilteredOrEmpty J\nh : ∀ (j : J), ∃ i f, (range ⇑(ConcreteCategory.hom (F.map f))).Finite\n⊢ F.IsMittagLeffler",
"usedConstants": [
"PSigma.snd",
"Eq.ge",
"Preorder.toLT",
"CategoryTheory.CategoryStruct.to... | intro j
obtain ⟨i, hi, hf⟩ := h j
obtain ⟨m, ⟨i, f, hm⟩, hmin⟩ := Finset.wellFoundedLT.wf.has_min
{ s : Finset (F.obj j) | ∃ (i : _) (f : i ⟶ j), ↑s = range (F.map f) }
⟨_, i, hi, hf.coe_toFinset⟩
refine ⟨i, f, fun k g =>
(F.ranges_directed j).directedOn_range.is_bot_of_is_min ⟨⟨i, f⟩, rfl⟩ ?_ _ ⟨⟨k, ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.CofilteredSystem | {
"line": 231,
"column": 2
} | {
"line": 241,
"column": 88
} | [
{
"pp": "J : Type u\ninst✝¹ : Category.{v_1, u} J\nF : J ⥤ Type v\ninst✝ : IsCofilteredOrEmpty J\nh : ∀ (j : J), ∃ i f, (range ⇑(ConcreteCategory.hom (F.map f))).Finite\n⊢ F.IsMittagLeffler",
"usedConstants": [
"PSigma.snd",
"Eq.ge",
"Preorder.toLT",
"CategoryTheory.CategoryStruct.to... | intro j
obtain ⟨i, hi, hf⟩ := h j
obtain ⟨m, ⟨i, f, hm⟩, hmin⟩ := Finset.wellFoundedLT.wf.has_min
{ s : Finset (F.obj j) | ∃ (i : _) (f : i ⟶ j), ↑s = range (F.map f) }
⟨_, i, hi, hf.coe_toFinset⟩
refine ⟨i, f, fun k g =>
(F.ranges_directed j).directedOn_range.is_bot_of_is_min ⟨⟨i, f⟩, rfl⟩ ?_ _ ⟨⟨k, ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.PEquiv | {
"line": 228,
"column": 4
} | {
"line": 228,
"column": 31
} | [
{
"pp": "α : Type u\ns : Set α\ninst✝ : DecidablePred fun x ↦ x ∈ s\nh : ofSet s = PEquiv.refl α\n⊢ s = Set.univ",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.univ",
"Set.eq_univ_iff_forall",
"Membership.mem",
"id",
"propext",
"Eq",
"Set.instMembership... | rw [Set.eq_univ_iff_forall] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Approximation | {
"line": 79,
"column": 4
} | {
"line": 79,
"column": 88
} | [
{
"pp": "case refine_1\n𝕜 : Type u_1\nE : Type u_2\ns : Set E\nφ : E → ℝ\ninst✝⁸ : RCLike 𝕜\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : IsScalarTower ℝ 𝕜 E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousSMul 𝕜 E\ninst✝ : LocallyConvexSpac... | have hv (v : 𝕜) : v * L (0, 1) = L (0, v) := by rw [← smul_eq_mul, ← map_smul]; simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.Matrix.Permutation | {
"line": 81,
"column": 2
} | {
"line": 82,
"column": 75
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝² : DecidableEq n\nσ : Perm n\ninst✝¹ : Fintype n\nv : n → R\ninst✝ : CommRing R\n⊢ v ᵥ* Perm.permMatrix R σ = v ∘ ⇑(Equiv.symm σ)",
"usedConstants": [
"dite_congr",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"instSMulOfMul... | ext j
simp [vecMul_eq_sum, Pi.single, Function.update, ← Equiv.symm_apply_eq σ] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.Matrix.Permutation | {
"line": 81,
"column": 2
} | {
"line": 82,
"column": 75
} | [
{
"pp": "n : Type u_1\nR : Type u_2\ninst✝² : DecidableEq n\nσ : Perm n\ninst✝¹ : Fintype n\nv : n → R\ninst✝ : CommRing R\n⊢ v ᵥ* Perm.permMatrix R σ = v ∘ ⇑(Equiv.symm σ)",
"usedConstants": [
"dite_congr",
"instHSMul",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"instSMulOfMul... | ext j
simp [vecMul_eq_sum, Pi.single, Function.update, ← Equiv.symm_apply_eq σ] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Between | {
"line": 542,
"column": 2
} | {
"line": 542,
"column": 22
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁷ : Ring R\ninst✝⁶ : PartialOrder R\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Module R V\ninst✝³ : AddTorsor V P\ninst✝² : IsOrderedRing R\ninst✝¹ : IsDomain R\ninst✝ : IsTorsionFree R V\nx y : P\nr : R\n⊢ Wbtw R x ((lineMap x y) r) y ↔ x = y ∨ r ∈ Set.Icc 0 1",
... | by_cases hxy : x = y | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.LinearAlgebra.Matrix.Stochastic | {
"line": 172,
"column": 57
} | {
"line": 178,
"column": 38
} | [
{
"pp": "R : Type u_1\nn : Type u_2\ninst✝⁴ : Fintype n\ninst✝³ : DecidableEq n\ninst✝² : Semiring R\ninst✝¹ : PartialOrder R\ninst✝ : IsOrderedRing R\nM : Matrix n n R\nx : n → R\nhM : M ∈ colStochastic R n\nhx : ∀ (i : n), 0 ≤ x i\n⊢ ∀ (j : n), 0 ≤ (M *ᵥ x) j",
"usedConstants": [
"Eq.mpr",
"Is... | by
intro j
simp only [Matrix.mulVec, dotProduct]
apply Finset.sum_nonneg
intro k _
refine Left.mul_nonneg ?_ (hx k)
exact nonneg_of_mem_colStochastic hM | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Between | {
"line": 1014,
"column": 4
} | {
"line": 1016,
"column": 46
} | [
{
"pp": "case refine_1.right\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nw x y z : P\nh_ne : x ≠ y\nt₁ : R\nht₁ : t₁ ∈ Set.Icc 0 1\nhx : (lineMap w y) t₁ = x\nt₂ : R\nht₂... | · apply div_le_one_of_le₀
· grind
· nlinarith [ht₁.1, ht₁.2, ht₂.1, ht₂.2] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Between | {
"line": 1054,
"column": 10
} | {
"line": 1054,
"column": 28
} | [
{
"pp": "case inl.inl\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : ∀ (p : P), p = x ∨ p = ty • v +ᵥ x ∨ p = tz • v +ᵥ x → ∃ r, p = r • v +ᵥ x\... | wbtw_comm (z := x) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Between | {
"line": 1063,
"column": 10
} | {
"line": 1063,
"column": 28
} | [
{
"pp": "case inr.inr.inr.inr\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx : P\nv : V\nty tz : R\nh : ∀ (p : P), p = x ∨ p = ty • v +ᵥ x ∨ p = tz • v +ᵥ x → ∃ r, p = r •... | wbtw_comm (z := x) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Between | {
"line": 1090,
"column": 2
} | {
"line": 1093,
"column": 63
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nι : Type u_6\ni j k : ι\nh : Function.Injective ![i, j, k]\nT : ι → P\nhT : AffineIndependent R T\n⊢ ¬Wbtw R (T i) (T ... | replace hT := hT.comp_embedding ⟨_, h⟩
rw [affineIndependent_iff_not_collinear] at hT
contrapose hT
simp [Set.range_comp, Set.image_insert_eq, hT.symm.collinear] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Between | {
"line": 1090,
"column": 2
} | {
"line": 1093,
"column": 63
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nι : Type u_6\ni j k : ι\nh : Function.Injective ![i, j, k]\nT : ι → P\nhT : AffineIndependent R T\n⊢ ¬Wbtw R (T i) (T ... | replace hT := hT.comp_embedding ⟨_, h⟩
rw [affineIndependent_iff_not_collinear] at hT
contrapose hT
simp [Set.range_comp, Set.image_insert_eq, hT.symm.collinear] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Between | {
"line": 1113,
"column": 2
} | {
"line": 1113,
"column": 41
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\nx y : P\nh✝ : x ≠ y\nh : midpoint R x y ≠ x\n⊢ Sbtw R x (midpoint R x y) y",
"usedConstants": [
"Eq.mpr",
... | convert! sbtw_pointReflection_of_ne R h | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.Analysis.Convex.Caratheodory | {
"line": 186,
"column": 6
} | {
"line": 186,
"column": 28
} | [
{
"pp": "case refine_5\n𝕜 : Type u_1\nE : Type u\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\nx : E\nt : Finset E\nht₁ : ↑t ⊆ s\nht₂ : AffineIndependent 𝕜 Subtype.val\nw : E → 𝕜\nhw₁ : ∀ y ∈ t, 0 ≤ w y\nhw₂ : ∑ y ∈ t, w... | rw [contra, zero_smul] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Cone.Dual | {
"line": 127,
"column": 2
} | {
"line": 128,
"column": 79
} | [
{
"pp": "case inr\nE : Type u_1\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : AddCommGroup E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : Module ℝ E\ninst✝¹ : ContinuousSMul ℝ E\ninst✝ : LocallyConvexSpace ℝ E\nK : Set E\nC : ProperCone ℝ E\nhKconv : Convex ℝ K\nhKcomp : IsCompact K\nhKC : Disjoint K ↑C\nx₀ : E\nhx₀✝ :... | simpa [hx₀.ne] using hv ((v * (f x)⁻¹) • x)
(C.smul_mem hx <| le_of_lt <| mul_pos_of_neg_of_neg hv₀ <| inv_neg''.2 hx₀) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Geometry.Convex.Cone.Dual | {
"line": 54,
"column": 64
} | {
"line": 54,
"column": 92
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : CommSemiring R\ninst✝⁵ : PartialOrder R\ninst✝⁴ : IsOrderedRing R\nM : Type u_2\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nN : Type u_3\ninst✝¹ : AddCommMonoid N\ninst✝ : Module R N\np : M →ₗ[R] N →ₗ[R] R\ns✝ t : Set M\ny✝ : N\ns : Set M\nc : failed to pretty print expressio... | exact mul_nonneg c.2 (hy hx) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Convex.Exposed | {
"line": 146,
"column": 4
} | {
"line": 146,
"column": 41
} | [
{
"pp": "case insert\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁷ : TopologicalSpace 𝕜\ninst✝⁶ : Ring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : TopologicalSpace E\ninst✝² : Module 𝕜 E\nA : Set E\ninst✝¹ : IsOrderedRing 𝕜\ninst✝ : ContinuousAdd 𝕜\nC : Set E\nF : Finset (Set E)\na✝ : C ∉ F\nhF'... | rw [Finset.coe_insert, sInter_insert] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Affine.Convex | {
"line": 55,
"column": 43
} | {
"line": 55,
"column": 84
} | [
{
"pp": "E : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nx : E\nhs : s ∈ 𝓝 x\nthis :\n ∀ {E : Type u_1} [inst : NormedAddCommGroup E] [inst_1 : NormedSpace ℝ E] [FiniteDimensional ℝ E] {s : Set E} {x : E},\n s ∈ 𝓝 x → x = 0 → ∃ b, x ∈ interi... | by simpa using vadd_mem_nhds_vadd (-x) hs | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Affine.Convex | {
"line": 64,
"column": 2
} | {
"line": 64,
"column": 83
} | [
{
"pp": "E✝ : Type u_1\ninst✝⁴ : NormedAddCommGroup E✝\ninst✝³ : NormedSpace ℝ E✝\ns✝ : Set E✝\nx : E✝\nE : Type u_1\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedSpace ℝ E\ninst✝ : FiniteDimensional ℝ E\ns : Set E\nhs : s ∈ 𝓝 0\nb : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E\n⊢ ∃ b, 0 ∈ interior ((convexHull ℝ)... | set c : AffineBasis (Fin (finrank ℝ E + 1)) ℝ E := -Finset.univ.centroid ℝ b +ᵥ b | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Analysis.Convex.Intrinsic | {
"line": 323,
"column": 89
} | {
"line": 323,
"column": 98
} | [
{
"pp": "V : Type u_2\ninst✝² : NormedAddCommGroup V\ninst✝¹ : NormedSpace ℝ V\ninst✝ : FiniteDimensional ℝ V\ns : Set V\nhscv : Convex ℝ s\nthis : Nonempty ↑s\np : V\nhp : p ∈ s\np' : ↥(_root_.affineSpan ℝ s) := ⟨p, ⋯⟩\n⊢ comap ↑(AffineIsometryEquiv.constVSub ℝ p').symm.toAffineEquiv ⊤ = ⊤",
"usedConstants... | comap_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Integral | {
"line": 332,
"column": 2
} | {
"line": 332,
"column": 85
} | [
{
"pp": "case inr\nα : Type u_1\nE : Type u_2\nm0 : MeasurableSpace α\ninst✝⁴ : NormedAddCommGroup E\ninst✝³ : NormedSpace ℝ E\ninst✝² : CompleteSpace E\nμ : Measure α\nf : α → E\nC : ℝ\ninst✝¹ : StrictConvexSpace ℝ E\ninst✝ : IsFiniteMeasure μ\nh_le : ∀ᵐ (x : α) ∂μ, ‖f x‖ ≤ C\nh₀ : μ ≠ 0\nhμ : 0 < μ.real univ\... | refine (ae_eq_const_or_norm_average_lt_of_norm_le_const h_le).imp_right fun H => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Geometry.Convex.ConvexSpace.AffineSpace | {
"line": 141,
"column": 2
} | {
"line": 141,
"column": 89
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : Ring R\ninst✝⁴ : PartialOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AffineSpace V P\ns t : R\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\nx y : P\n⊢ (Finset.affineCombination R (Finsupp.single x s + Finsupp... | rw [Finset.affineCombination_eq_weightedVSubOfPoint_vadd_of_sum_eq_one _ _ id (b := y)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.MetricSpace | {
"line": 141,
"column": 4
} | {
"line": 142,
"column": 86
} | [
{
"pp": "X : Type u_2\ninst✝² : ConvexSpace ℝ X\ninst✝¹ : MetricSpace X\ninst✝ : IsConvexDist X\nx y : X\ns t s' t' : ℝ\nhs : 0 ≤ s\nht : 0 ≤ t\nh : s + t = 1\nhs' : 0 ≤ s'\nht' : 0 ≤ t'\nh' : s' + t' = 1\nhss' : s' ≤ s\nthis : dist (convexCombPair s t hs ht h x y) (convexCombPair s' t' hs' ht' h' x y) ≤ |s - s... | have : |t - t'| = |s - s'| := by
rw [eq_sub_iff_add_eq.mpr h, eq_sub_iff_add_eq.mpr h']; simp [abs_sub_comm t t'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Convex.SimplicialComplex.Basic | {
"line": 161,
"column": 4
} | {
"line": 161,
"column": 14
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Ring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nK : SimplicialComplex 𝕜 E\ns : Finset E\nx : E\nhx : x ∈ K.vertices\nhs : s ∈ K.faces\nh✝ : x ∈ (convexHull 𝕜) ↑s\nh : x ∈ (convexHull 𝕜) ↑∅\nH : ¬x ∈ s\n⊢ False",
"usedConstant... | coe_empty, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Basic | {
"line": 564,
"column": 94
} | {
"line": 569,
"column": 25
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w : V\n⊢ G.Adj v w ↔ v ≠ w ∧ ∃ e ∈ G.edgeSet, v ∈ e ∧ w ∈ e",
"usedConstants": [
"Eq.mpr",
"Sym2.mem_iff._simp_1",
"and_true",
"Sym2.mk",
"congrArg",
"and_self",
"true_or",
"SimpleGraph.Adj",
"SimpleGraph.ne_of_a... | by
refine ⟨fun _ => ⟨G.ne_of_adj ‹_›, s(v, w), by simpa⟩, ?_⟩
rintro ⟨hne, e, he, hv⟩
rw [Sym2.mem_and_mem_iff hne] at hv
subst e
rwa [mem_edgeSet] at he | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.StoneSeparation | {
"line": 36,
"column": 2
} | {
"line": 77,
"column": 15
} | [
{
"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\np q u v x y z : E\nhz : z ∈ segment 𝕜 x y\nhu : u ∈ segment 𝕜 x p\nhv : v ∈ segment 𝕜 y q\n⊢ ¬Disjoint (segment 𝕜 u v) ((convexHull 𝕜) {p, q, z})"... | rw [not_disjoint_iff]
obtain ⟨az, bz, haz, hbz, habz, rfl⟩ := hz
obtain rfl | haz' := haz.eq_or_lt
· rw [zero_add] at habz
rw [zero_smul, zero_add, habz, one_smul]
refine ⟨v, by apply right_mem_segment, segment_subset_convexHull ?_ ?_ hv⟩ <;> simp
obtain ⟨av, bv, hav, hbv, habv, rfl⟩ := hv
obtain rfl ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.StoneSeparation | {
"line": 36,
"column": 2
} | {
"line": 77,
"column": 15
} | [
{
"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\np q u v x y z : E\nhz : z ∈ segment 𝕜 x y\nhu : u ∈ segment 𝕜 x p\nhv : v ∈ segment 𝕜 y q\n⊢ ¬Disjoint (segment 𝕜 u v) ((convexHull 𝕜) {p, q, z})"... | rw [not_disjoint_iff]
obtain ⟨az, bz, haz, hbz, habz, rfl⟩ := hz
obtain rfl | haz' := haz.eq_or_lt
· rw [zero_add] at habz
rw [zero_smul, zero_add, habz, one_smul]
refine ⟨v, by apply right_mem_segment, segment_subset_convexHull ?_ ?_ hv⟩ <;> simp
obtain ⟨av, bv, hav, hbv, habv, rfl⟩ := hv
obtain rfl ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Side | {
"line": 581,
"column": 2
} | {
"line": 583,
"column": 42
} | [
{
"pp": "case inr.inl\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ns : AffineSubspace R P\nx y p₁ : P\nhp₁ : p₁ ∈ s\np₂ : P\nhp₂ : p₂ ∈ s\nh : p₂ -ᵥ y = 0\n⊢ ∃ p ∈ s, Wbtw... | · rw [vsub_eq_zero_iff_eq] at h
rw [← h]
exact ⟨p₂, hp₂, wbtw_self_right _ _ _⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Side | {
"line": 694,
"column": 8
} | {
"line": 694,
"column": 36
} | [
{
"pp": "case h.mp\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ns : AffineSubspace R P\nx p : P\nhx : x ∉ s\nhp : p ∈ s\ny : P\n⊢ s.WOppSide x y → ∃ a ≤ 0, ∃ b ∈ ↑s, a • (... | wOppSide_iff_exists_left hp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Side | {
"line": 701,
"column": 6
} | {
"line": 701,
"column": 93
} | [
{
"pp": "case h.mp.inr.inr\nR : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Field R\ninst✝⁴ : LinearOrder R\ninst✝³ : IsStrictOrderedRing R\ninst✝² : AddCommGroup V\ninst✝¹ : Module R V\ninst✝ : AddTorsor V P\ns : AffineSubspace R P\nx p : P\nhx : x ∉ s\nhp : p ∈ s\ny p₂ : P\nhp₂ : p₂ ∈ s\nr₁ r₂ : R\nhr₁ : 0... | refine ⟨-r₁ / r₂, (div_neg_of_neg_of_pos (Left.neg_neg_iff.2 hr₁) hr₂).le, p₂, hp₂, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Convex.Visible | {
"line": 172,
"column": 45
} | {
"line": 172,
"column": 72
} | [
{
"pp": "V : Type u_2\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module ℝ V\ns : Set V\ny : V\ninst✝² : TopologicalSpace V\ninst✝¹ : IsTopologicalAddGroup V\ninst✝ : ContinuousSMul ℝ V\nhs : IsClosed[inst✝²] s\nhy : y ∈ s\nx : V\nt : Set ℝ := Set.Ici 0 ∩ ⇑(lineMap x y) ⁻¹' s\nht₁ : 1 ∈ t\nht : BddBelow t\nδ : ℝ := sInf... | rintro hδ₀; simp [hδ₀] at h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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