module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 573,
"column": 17
} | {
"line": 573,
"column": 29
} | {
"line": 573,
"column": 30
} | [
{
"pp": "θ : Angle\n⊢ (θ + θ).toReal = 2 * θ.toReal ↔ θ.toReal ∈ Set.Ioc (-π / 2) (π / 2)",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioc",
"Real",
"instHSMul",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Real.Angle",
"congr... | [
"θ : Angle\n⊢ (2 • θ).toReal = 2 * θ.toReal ↔ θ.toReal ∈ Set.Ioc (-π / 2) (π / 2)"
] | ← two_nsmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 403,
"column": 17
} | {
"line": 403,
"column": 26
} | {
"line": 403,
"column": 27
} | [
{
"pp": "case inr\nz : ℂ\nhz : z ≠ 0\n⊢ 0 < z.re ∨ 0 ≤ z.im ∧ (z.im < 0 ∨ False) ↔ 0 < z.re ∨ False",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Real.instLE",
"Real",
"Real.instZero",
"congrArg",
"Complex.im",
"Real.inst... | [
"case inr\nz : ℂ\nhz : z ≠ 0\n⊢ 0 < z.re ∨ 0 ≤ z.im ∧ z.im < 0 ↔ 0 < z.re"
] | or_false, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 599,
"column": 17
} | {
"line": 599,
"column": 29
} | {
"line": 599,
"column": 30
} | [
{
"pp": "θ : Angle\n⊢ (θ + θ).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Real.Angle",
"congrArg",
"Real.instDivInv... | [
"θ : Angle\n⊢ (2 • θ).toReal = 2 * θ.toReal - 2 * π ↔ π / 2 < θ.toReal"
] | ← two_nsmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 605,
"column": 2
} | {
"line": 607,
"column": 86
} | {
"line": 609,
"column": 0
} | [
{
"pp": "θ : Angle\n⊢ -3 * π < 2 * θ.toReal ∧ 2 * θ.toReal ≤ -π ↔ θ.toReal ≤ -π / 2",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"NegZeroClass.toNeg... | [] | refine
⟨fun h => by linarith, fun h =>
⟨by linarith [pi_pos, neg_pi_lt_toReal θ], (le_div_iff₀' (zero_lt_two' ℝ)).1 h⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 611,
"column": 17
} | {
"line": 611,
"column": 29
} | {
"line": 611,
"column": 30
} | [
{
"pp": "θ : Angle\n⊢ (θ + θ).toReal = 2 * θ.toReal + 2 * π ↔ θ.toReal ≤ -π / 2",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"instHSMul",
"instHDiv",
"Real.pi",
"HMul.hMul",
"Real.Angle",
"congrArg",
... | [
"θ : Angle\n⊢ (2 • θ).toReal = 2 * θ.toReal + 2 * π ↔ θ.toReal ≤ -π / 2"
] | ← two_nsmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 643,
"column": 22
} | {
"line": 643,
"column": 34
} | {
"line": 643,
"column": 35
} | [
{
"pp": "θ ψ : Angle\nhθ : |θ.toReal| < π / 2\nhψ : |ψ.toReal| < π / 2\n⊢ θ + θ = ψ + ψ ↔ θ = ψ",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instHSMul",
"Real.Angle",
"congrArg",
"AddMonoid.toAddZeroClass",
"AddMonoid.toNSMul",
"AddCo... | [
"θ ψ : Angle\nhθ : |θ.toReal| < π / 2\nhψ : |ψ.toReal| < π / 2\n⊢ 2 • θ = 2 • ψ ↔ θ = ψ"
] | ← two_nsmul, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 831,
"column": 17
} | {
"line": 831,
"column": 29
} | {
"line": 831,
"column": 30
} | [
{
"pp": "θ : Angle\n⊢ (θ + θ).sign = θ.sign ↔ θ = ↑π ∨ |θ.toReal| < π / 2",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"instHDiv",
"Real.pi",
"Real.lattice",
"Real.Angle",
"Real.Angle.coe",
"abs",
... | [
"θ : Angle\n⊢ (2 • θ).sign = θ.sign ↔ θ = ↑π ∨ |θ.toReal| < π / 2"
] | ← two_nsmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 840,
"column": 17
} | {
"line": 840,
"column": 29
} | {
"line": 840,
"column": 30
} | [
{
"pp": "θ : Angle\n⊢ (θ + θ).sign = -θ.sign ↔ θ = 0 ∨ π / 2 < |θ.toReal|",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"instHSMul",
"instHDiv",
"Real.pi",
"Real.lattice",
"Real.Angle",
"abs",
"congrArg",
"Real... | [
"θ : Angle\n⊢ (2 • θ).sign = -θ.sign ↔ θ = 0 ∨ π / 2 < |θ.toReal|"
] | ← two_nsmul, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 309,
"column": 46
} | {
"line": 309,
"column": 61
} | {
"line": 311,
"column": 0
} | [
{
"pp": "x : ℂ\ny z : ℝ\n⊢ (↑(‖x‖ ^ y) * (↑(Real.cos z) + ↑(Real.sin z) * I)).im = ‖x‖ ^ y * Real.sin z",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Complex.mul_im",
"Norm.norm",
"Real.instPow",
"Real",
"Complex.mul_re",
"HMul.hMul",
"Complex.o... | [] | simp [Real.sin] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 353,
"column": 24
} | {
"line": 353,
"column": 71
} | {
"line": 353,
"column": 71
} | [
{
"pp": "n : ℕ\ns : ℂ\nhs : s.re ≠ 0\n⊢ ‖↑↑n ^ s‖ = ↑n ^ s.re",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.instPow",
"Real.partialOrder",
"Real",
... | [
"n : ℕ\ns : ℂ\nhs : s.re ≠ 0\n⊢ ↑n ^ s.re = ↑n ^ s.re"
] | norm_cpow_eq_rpow_re_of_nonneg n.cast_nonneg hs | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 640,
"column": 4
} | {
"line": 640,
"column": 75
} | {
"line": 641,
"column": 2
} | [
{
"pp": "case pos\nx : ℂ\nh : x ≠ 0\nhs : x ∈ slitPlane\n⊢ ContinuousAt (Real.Angle.coe ∘ arg) x",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"Real.Angle",
"Real.Angle.coe",
"Complex.continuousAt_arg",
"C... | [] | exact Real.Angle.continuous_coe.continuousAt.comp (continuousAt_arg hs) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 640,
"column": 4
} | {
"line": 640,
"column": 75
} | {
"line": 641,
"column": 2
} | [
{
"pp": "case pos\nx : ℂ\nh : x ≠ 0\nhs : x ∈ slitPlane\n⊢ ContinuousAt (Real.Angle.coe ∘ arg) x",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"Real.Angle",
"Real.Angle.coe",
"Complex.continuousAt_arg",
"C... | [] | exact Real.Angle.continuous_coe.continuousAt.comp (continuousAt_arg hs) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 640,
"column": 4
} | {
"line": 640,
"column": 75
} | {
"line": 641,
"column": 2
} | [
{
"pp": "case pos\nx : ℂ\nh : x ≠ 0\nhs : x ∈ slitPlane\n⊢ ContinuousAt (Real.Angle.coe ∘ arg) x",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"Real.Angle",
"Real.Angle.coe",
"Complex.continuousAt_arg",
"C... | [] | exact Real.Angle.continuous_coe.continuousAt.comp (continuousAt_arg hs) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 99,
"column": 60
} | {
"line": 112,
"column": 12
} | {
"line": 114,
"column": 0
} | [
{
"pp": "a b c : ℝ\nhb : 0 ≠ b\n⊢ Tendsto (fun x ↦ x ^ (a / (b * x + c))) atTop (𝓝 1)",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.FieldSimp.zpow'_one",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"Mathlib.Tactic.F... | [] | by
refine
Tendsto.congr' ?_
((tendsto_exp_nhds_zero_nhds_one.comp
(by
simpa only [mul_zero, pow_one] using
(tendsto_const_nhds (x := a)).mul
(tendsto_div_pow_mul_exp_add_atTop b c 1 hb))).comp
tendsto_log_atTop)
apply eventuallyEq_of_mem ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 512,
"column": 2
} | {
"line": 513,
"column": 67
} | {
"line": 515,
"column": 0
} | [
{
"pp": "x : ℝ\nn : ℕ\nhx : 0 ≤ x\nhn : n ≠ 0\n⊢ (x ^ (↑n)⁻¹) ^ n = x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"Real.instPow",
"Real.partialOrder",
"Real",
"DivInvM... | [] | have hn0 : (n : ℝ) ≠ 0 := Nat.cast_ne_zero.2 hn
rw [← rpow_natCast, ← rpow_mul hx, inv_mul_cancel₀ hn0, rpow_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 512,
"column": 2
} | {
"line": 513,
"column": 67
} | {
"line": 515,
"column": 0
} | [
{
"pp": "x : ℝ\nn : ℕ\nhx : 0 ≤ x\nhn : n ≠ 0\n⊢ (x ^ (↑n)⁻¹) ^ n = x",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"MulOne.toOne",
"Real.instPow",
"Real.partialOrder",
"Real",
"DivInvM... | [] | have hn0 : (n : ℝ) ≠ 0 := Nat.cast_ne_zero.2 hn
rw [← rpow_natCast, ← rpow_mul hx, inv_mul_cancel₀ hn0, rpow_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 121,
"column": 2
} | {
"line": 122,
"column": 6
} | {
"line": 124,
"column": 0
} | [
{
"pp": "⊢ Tendsto (fun x ↦ x ^ (-1 / x)) atTop (𝓝 1)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.instPow",
"Real.partialOrder",
"Real",
"instHDiv",
"Mathlib.Tactic.Ring.Common.mul_con... | [] | convert! tendsto_rpow_div_mul_add (-(1 : ℝ)) _ (0 : ℝ) zero_ne_one
ring | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 121,
"column": 2
} | {
"line": 122,
"column": 6
} | {
"line": 124,
"column": 0
} | [
{
"pp": "⊢ Tendsto (fun x ↦ x ^ (-1 / x)) atTop (𝓝 1)",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real.instPow",
"Real.partialOrder",
"Real",
"instHDiv",
"Mathlib.Tactic.Ring.Common.mul_con... | [] | convert! tendsto_rpow_div_mul_add (-(1 : ℝ)) _ (0 : ℝ) zero_ne_one
ring | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 137,
"column": 2
} | {
"line": 138,
"column": 26
} | {
"line": 140,
"column": 0
} | [
{
"pp": "s b : ℝ\nhb : 0 < b\nx : ℝ\nhx₀ : 0 ≤ x\n⊢ ((fun x ↦ x ^ b) ∘ fun x ↦ rexp x / x ^ (s / b)) x = rexp (b * x) / x ^ s",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"NonUnitalNonAssocCommRing.toNonUnitalNonAssocCommSemiring",
"GroupWithZero.toMonoidWithZero",
"Re... | [] | simp [Real.div_rpow, (exp_pos x).le, rpow_nonneg, ← Real.rpow_mul, ← exp_mul,
mul_comm x, hb.ne', *] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 549,
"column": 2
} | {
"line": 551,
"column": 41
} | {
"line": 553,
"column": 0
} | [
{
"pp": "x y z : ℝ\nh : 0 ≤ x\nh₁ : x ≤ y\nh₂ : 0 ≤ z\n⊢ x ^ z ≤ y ^ z",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"le_refl",
"Real.instPow",
"Real.partialOrder",
"Real.instLE",
"Real",
"Preorder.toLT",
"Real.instZero",
"instReflLe",
... | [] | rcases eq_or_lt_of_le h₁ with (rfl | h₁'); · rfl
rcases eq_or_lt_of_le h₂ with (rfl | h₂'); · simp
exact le_of_lt (rpow_lt_rpow h h₁' h₂') | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 549,
"column": 2
} | {
"line": 551,
"column": 41
} | {
"line": 553,
"column": 0
} | [
{
"pp": "x y z : ℝ\nh : 0 ≤ x\nh₁ : x ≤ y\nh₂ : 0 ≤ z\n⊢ x ^ z ≤ y ^ z",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"le_refl",
"Real.instPow",
"Real.partialOrder",
"Real.instLE",
"Real",
"Preorder.toLT",
"Real.instZero",
"instReflLe",
... | [] | rcases eq_or_lt_of_le h₁ with (rfl | h₁'); · rfl
rcases eq_or_lt_of_le h₂ with (rfl | h₂'); · simp
exact le_of_lt (rpow_lt_rpow h h₁' h₂') | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 966,
"column": 2
} | {
"line": 967,
"column": 41
} | {
"line": 968,
"column": 2
} | [
{
"pp": "case inr\nn : ℕ\nε : ℝ\nhε : 0 < ε\nh : n > 0\n⊢ ‖log ↑n‖ ≤ ↑n ^ ε / ε",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Complex.log",
"Real.instPow",
"Real.partialOrder",
"Real.instLE",
"Real",
"instHDiv",
... | [
"case inr\nn : ℕ\nε : ℝ\nhε : 0 < ε\nh : n > 0\n⊢ Real.log ↑n ≤ ↑n ^ ε / ε"
] | rw [← natCast_log, norm_real, norm_of_nonneg <| Real.log_nonneg <| by
exact_mod_cast Nat.one_le_of_lt h.lt] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 714,
"column": 31
} | {
"line": 714,
"column": 56
} | {
"line": 715,
"column": 2
} | [
{
"pp": "case inr.inr.top.inl\ny : ℝ≥0∞\nz : ℝ\nhy0 : y ≠ 0\nhxy : ∞ ≤ y\nhx0 : ∞ ≠ 0\nhz : z < 0\n⊢ (∞ * y) ^ z = if (∞ = 0 ∧ y = ∞ ∨ ∞ = ∞ ∧ y = 0) ∧ z < 0 then ∞ else ∞ ^ z * y ^ z",
"ppTerm": "?inr.inr.top.inl",
"assigned": true,
"usedConstants": [
"False",
"Real",
"Preorder.to... | [] | simp [hz, top_unique hxy] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 714,
"column": 31
} | {
"line": 714,
"column": 56
} | {
"line": 715,
"column": 2
} | [
{
"pp": "case inr.inr.top.inr\ny : ℝ≥0∞\nz : ℝ\nhy0 : y ≠ 0\nhxy : ∞ ≤ y\nhx0 : ∞ ≠ 0\nhz : 0 < z\n⊢ (∞ * y) ^ z = if (∞ = 0 ∧ y = ∞ ∨ ∞ = ∞ ∧ y = 0) ∧ z < 0 then ∞ else ∞ ^ z * y ^ z",
"ppTerm": "?inr.inr.top.inr",
"assigned": true,
"usedConstants": [
"False",
"Real",
"HMul.hMul",... | [] | simp [hz, top_unique hxy] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 1201,
"column": 2
} | {
"line": 1206,
"column": 12
} | {
"line": 1208,
"column": 0
} | [
{
"pp": "a : ℝ≥0\nb : ℝ\nm n d r : ℕ\nha : IsNat a m\nhb : IsNNRat b n d\nk : ℕ\nhr : r ^ d = k\nl : ℕ\nhm : m ^ n = l\nhkl : k = l\n⊢ IsNat (a ^ b) r",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"zero_le",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"NonAssocSemi... | [] | rcases ha with ⟨rfl⟩
constructor
have : d ≠ 0 := mod_cast hb.den_nz
rw [hb.to_eq rfl rfl, div_eq_mul_inv, NNReal.rpow_natCast_mul, ← Nat.cast_pow, hm, ← hkl, ← hr,
Nat.cast_pow, NNReal.pow_rpow_inv_natCast]
positivity | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 1201,
"column": 2
} | {
"line": 1206,
"column": 12
} | {
"line": 1208,
"column": 0
} | [
{
"pp": "a : ℝ≥0\nb : ℝ\nm n d r : ℕ\nha : IsNat a m\nhb : IsNNRat b n d\nk : ℕ\nhr : r ^ d = k\nl : ℕ\nhm : m ^ n = l\nhkl : k = l\n⊢ IsNat (a ^ b) r",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"zero_le",
"Eq.mpr",
"Nat.instCanonicallyOrderedAdd",
"NonAssocSemi... | [] | rcases ha with ⟨rfl⟩
constructor
have : d ≠ 0 := mod_cast hb.den_nz
rw [hb.to_eq rfl rfl, div_eq_mul_inv, NNReal.rpow_natCast_mul, ← Nat.cast_pow, hm, ← hkl, ← hr,
Nat.cast_pow, NNReal.pow_rpow_inv_natCast]
positivity | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Segment | {
"line": 314,
"column": 56
} | {
"line": 318,
"column": 87
} | {
"line": 320,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : CommRing 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\nx y z : E\nh : x ∈ [y -[𝕜] z]\n⊢ SameRay 𝕜 (x - y) (z - x)",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Iff.mpr",... | [] | by
rw [segment_eq_image'] at h
rcases h with ⟨θ, ⟨hθ₀, hθ₁⟩, rfl⟩
simpa only [add_sub_cancel_left, ← sub_sub, sub_smul, one_smul] using
(SameRay.sameRay_nonneg_smul_left (z - y) hθ₀).nonneg_smul_right (sub_nonneg.2 hθ₁) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Segment | {
"line": 365,
"column": 4
} | {
"line": 365,
"column": 71
} | {
"line": 366,
"column": 4
} | [
{
"pp": "case mp\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : Ring 𝕜\ninst✝⁵ : LinearOrder 𝕜\ninst✝⁴ : IsStrictOrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\nx y : E\ninst✝¹ : DenselyOrdered 𝕜\ninst✝ : IsTorsionFree 𝕜 E\na b : 𝕜\nleft✝ : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx : a • x + b • y = x\... | [
"case mp\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : Ring 𝕜\ninst✝⁵ : LinearOrder 𝕜\ninst✝⁴ : IsStrictOrderedRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\nx y : E\ninst✝¹ : DenselyOrdered 𝕜\ninst✝ : IsTorsionFree 𝕜 E\na b : 𝕜\nleft✝ : 0 < a\nhb : 0 < b\nhab : a + b = 1\nhx : a • x + b • y = x\n⊢ a • x + (... | refine smul_right_injective _ hb.ne' ((add_right_inj (a • x)).1 ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Convex.Segment | {
"line": 547,
"column": 8
} | {
"line": 547,
"column": 21
} | {
"line": 547,
"column": 22
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nx y : 𝕜\nh : y ≤ x\n⊢ [x -[𝕜] y] = Icc (min x y) (max x y)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"instSMulOfMul",... | [
"case inr\n𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nx y : 𝕜\nh : y ≤ x\n⊢ [y -[𝕜] x] = Icc (min x y) (max x y)"
] | segment_symm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Star | {
"line": 244,
"column": 2
} | {
"line": 244,
"column": 21
} | {
"line": 245,
"column": 2
} | [
{
"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\nx z : E\ns : Set E\nhs : StarConvex 𝕜 (x + z) s\n⊢ StarConvex 𝕜 x ((fun x ↦ x + z) ⁻¹' s)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"congrArg",... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\nx z : E\ns : Set E\nhs : StarConvex 𝕜 (z + x) s\n⊢ StarConvex 𝕜 x ((fun x ↦ x + z) ⁻¹' s)"
] | rw [add_comm] at hs | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Star | {
"line": 433,
"column": 4
} | {
"line": 435,
"column": 38
} | {
"line": 436,
"column": 4
} | [
{
"pp": "case inl.refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedAddMonoid E\ninst✝¹ : Module 𝕜 E\ninst✝ : PosSMulMono 𝕜 E\nx : E\ns : Set E\nhs : s.OrdConnected\nhx : x ∈ s\nh : ∀ y ∈ s, x ≤ y ∨ y ≤ x... | [
"case inl.refine_2\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : PartialOrder E\ninst✝² : IsOrderedAddMonoid E\ninst✝¹ : Module 𝕜 E\ninst✝ : PosSMulMono 𝕜 E\nx : E\ns : Set E\nhs : s.OrdConnected\nhx : x ∈ s\nh : ∀ y ∈ s, x ≤ y ∨ y ≤ x\ny : E\nhy ... | · calc
x = a • x + b • x := (Convex.combo_self hab _).symm
_ ≤ a • x + b • y := by gcongr | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Function | {
"line": 785,
"column": 66
} | {
"line": 794,
"column": 27
} | {
"line": 796,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁷ : Semiring 𝕜\ninst✝⁶ : PartialOrder 𝕜\ninst✝⁵ : AddCommMonoid E\ninst✝⁴ : AddCommGroup β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : SMul 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf : E → β\n⊢ ConvexOn 𝕜 s (-f) ↔ ConcaveOn 𝕜 s f",
... | [] | by
constructor
· rintro ⟨hconv, h⟩
refine ⟨hconv, fun x hx y hy a b ha hb hab => ?_⟩
simpa [add_comm] using h hx hy ha hb hab
· rintro ⟨hconv, h⟩
refine ⟨hconv, fun x hx y hy a b ha hb hab => ?_⟩
rw [← neg_le_neg_iff]
simp_rw [neg_add, Pi.neg_apply, smul_neg, neg_neg]
exact h hx hy ha hb h... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 492,
"column": 6
} | {
"line": 492,
"column": 18
} | {
"line": 492,
"column": 18
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\nw : ι → k\np : ι → P\nhw : (s.weightedVSub p) w = 0\ni : ι\ninst✝ : DecidablePred fun x ↦ x ≠ i\nhis : i ∈ s\nhwi : w i = -1\n⊢ (affineCombination k ... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\nw : ι → k\np : ι → P\nhw : (s.weightedVSub p) w = 0\ni : ι\ninst✝ : DecidablePred fun x ↦ x ≠ i\nhis : i ∈ s\nhwi : w i = -1\n⊢ (affineCombination k ({x ∈ s | x ... | ← filter_ne' | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 590,
"column": 4
} | {
"line": 590,
"column": 26
} | {
"line": 590,
"column": 27
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\ninst✝ : DecidableEq ι\nw : ι → k\np : ι → P\np₁ p₂ : P\ns' : Finset ι\nh : ∑ i ∈ s, w i = 1\nhp₂ : ∀ i ∈ s ∩ s', p i = p₂\nhp₁ : ∀ i ∈ s \\ s', p i =... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\ninst✝ : DecidableEq ι\nw : ι → k\np : ι → P\np₁ p₂ : P\ns' : Finset ι\nh : ∑ i ∈ s, w i = 1\nhp₂ : ∀ i ∈ s ∩ s', p i = p₂\nhp₁ : ∀ i ∈ s \\ s', p i = p₁\n⊢ ∑ x ∈... | vadd_right_cancel_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 295,
"column": 2
} | {
"line": 301,
"column": 75
} | {
"line": 303,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni₀ : ι\n⊢ vectorSpan k (range p) = Submodule.span k (range fun i ↦ p i₀ -ᵥ p ↑i)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants":... | [] | rw [← Set.image_univ, vectorSpan_image_eq_span_vsub_set_left_ne k _ (Set.mem_univ i₀)]
congr with v
simp only [Set.mem_range, Set.mem_image, Set.mem_sdiff, Set.mem_singleton_iff, Subtype.exists]
constructor
· rintro ⟨x, ⟨i₁, ⟨⟨_, hi₁⟩, rfl⟩⟩, hv⟩
exact ⟨i₁, hi₁, hv⟩
· exact fun ⟨i₁, hi₁, hv⟩ => ⟨p i₁, ⟨i₁... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 295,
"column": 2
} | {
"line": 301,
"column": 75
} | {
"line": 303,
"column": 0
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni₀ : ι\n⊢ vectorSpan k (range p) = Submodule.span k (range fun i ↦ p i₀ -ᵥ p ↑i)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants":... | [] | rw [← Set.image_univ, vectorSpan_image_eq_span_vsub_set_left_ne k _ (Set.mem_univ i₀)]
congr with v
simp only [Set.mem_range, Set.mem_image, Set.mem_sdiff, Set.mem_singleton_iff, Subtype.exists]
constructor
· rintro ⟨x, ⟨i₁, ⟨⟨_, hi₁⟩, rfl⟩⟩, hv⟩
exact ⟨i₁, hi₁, hv⟩
· exact fun ⟨i₁, hi₁, hv⟩ => ⟨p i₁, ⟨i₁... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 411,
"column": 8
} | {
"line": 411,
"column": 25
} | {
"line": 411,
"column": 26
} | [
{
"pp": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np p₁ p₂ : P\nh : p ∈ line[k, p₁, p₂]\n⊢ ∃ r, (AffineMap.lineMap p₁ p₂) r = p",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"AddMonoid.to... | [
"case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np p₁ p₂ : P\nh : (p -ᵥ p₁) +ᵥ p₁ ∈ line[k, p₁, p₂]\n⊢ ∃ r, (AffineMap.lineMap p₁ p₂) r = p"
] | ← vsub_vadd p p₁, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 414,
"column": 8
} | {
"line": 414,
"column": 25
} | {
"line": 414,
"column": 26
} | [
{
"pp": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np p₁ p₂ : P\nr : k\nhr : r • (p₂ -ᵥ p₁) = p -ᵥ p₁\n⊢ (AffineMap.lineMap p₁ p₂) r = p",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np p₁ p₂ : P\nr : k\nhr : r • (p₂ -ᵥ p₁) = p -ᵥ p₁\n⊢ (AffineMap.lineMap p₁ p₂) r = (p -ᵥ p₁) +ᵥ p₁"
] | ← vsub_vadd p p₁, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 447,
"column": 10
} | {
"line": 447,
"column": 42
} | {
"line": 447,
"column": 43
} | [
{
"pp": "case right\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns₁ s₂ : AffineSubspace k P\np₁ p₂ : P\nhp₁ : p₁ ∈ ↑s₁\nhp₂ : p₂ ∈ s₂\np₃ : P\nhp₃ : p₃ ∈ ↑s₂\n⊢ p₃ -ᵥ p₁ ∈ s₂.direction ⊔ k ∙ (p₂ -ᵥ p₁)",
"ppTerm": "?right"... | [
"case right\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns₁ s₂ : AffineSubspace k P\np₁ p₂ : P\nhp₁ : p₁ ∈ ↑s₁\nhp₂ : p₂ ∈ s₂\np₃ : P\nhp₃ : p₃ ∈ ↑s₂\n⊢ p₃ -ᵥ p₂ + (p₂ -ᵥ p₁) ∈ s₂.direction ⊔ k ∙ (p₂ -ᵥ p₁)"
] | ← vsub_add_vsub_cancel p₃ p₂ p₁, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 921,
"column": 2
} | {
"line": 922,
"column": 47
} | {
"line": 924,
"column": 0
} | [
{
"pp": "case mpr\nι : Type u_1\nk : Type u_2\nV : Type u_3\nP : Type u_4\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : Nontrivial k\np1 : P\np : ι → P\n⊢ (∃ s w, ∑ i ∈ s, w i = 1 ∧ p1 = (Finset.affineCombination k s p) w) → p1 ∈ affineSpan k (Set.range p)",
... | [] | · rintro ⟨s, w, hw, rfl⟩
exact affineCombination_mem_affineSpan hw p | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 1065,
"column": 39
} | {
"line": 1065,
"column": 71
} | {
"line": 1065,
"column": 72
} | [
{
"pp": "case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np₁ p₂ p₃ p₄ p₅ p₆ : P\nh₂ : p₂ ∉ line[k, p₁, p₃]\nr₁ : kˣ\nhr₁ : ↑r₁ • (p₂ -ᵥ p₁) = p₅ -ᵥ p₄\nr₂ : k\nhr₂ : r₂ • (p₃ -ᵥ p₂) = p₆ -ᵥ p₅\nr₃ : k\nhr₃ : r₃ • ... | [
"case pos\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\np₁ p₂ p₃ p₄ p₅ p₆ : P\nh₂ : p₂ ∉ line[k, p₁, p₃]\nr₁ : kˣ\nhr₁ : ↑r₁ • (p₂ -ᵥ p₁) = p₅ -ᵥ p₄\nr₂ : k\nhr₂ : r₂ • (p₃ -ᵥ p₂) = p₆ -ᵥ p₅\nr₃ : k\nhr₃ : r₃ • (p₁ -ᵥ p₃) =... | ← vsub_add_vsub_cancel p₃ p₂ p₁, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 532,
"column": 2
} | {
"line": 533,
"column": 48
} | {
"line": 535,
"column": 0
} | [
{
"pp": "case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_4\ninst✝⁵ : Ring k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\ninst✝¹ : PartialOrder k\ninst✝ : ZeroLEOneClass k\ns : Simplex k P 0\np : P\n⊢ p = s.points 0 → ∃ w, w 0 = 1 ∧ (∀ (i : Fin 1), 0 ≤ w i ∧ w i ≤ 1) ∧ (affineCombina... | [] | · rintro rfl
exact ⟨1, by simp [affineCombination_apply]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 611,
"column": 6
} | {
"line": 611,
"column": 63
} | {
"line": 613,
"column": 0
} | [
{
"pp": "case mpr.inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_4\ninst✝ : Nontrivial k\np : ι → P\nha : AffineIndependent k p\ns₁ s₂ : Set ι\nh₁ : s₁.Subsingleton\nh₂ : s₂.Subsingleton\n⊢ vectorSpan k (p '' s₁)... | [] | simp [h₁.image p, h₂.image p, vectorSpan_of_subsingleton] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 611,
"column": 6
} | {
"line": 611,
"column": 63
} | {
"line": 613,
"column": 0
} | [
{
"pp": "case mpr.inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_4\ninst✝ : Nontrivial k\np : ι → P\nha : AffineIndependent k p\ns₁ s₂ : Set ι\nh₁ : s₁.Subsingleton\nh₂ : s₂.Subsingleton\n⊢ vectorSpan k (p '' s₁)... | [] | simp [h₁.image p, h₂.image p, vectorSpan_of_subsingleton] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 611,
"column": 6
} | {
"line": 611,
"column": 63
} | {
"line": 613,
"column": 0
} | [
{
"pp": "case mpr.inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_4\ninst✝ : Nontrivial k\np : ι → P\nha : AffineIndependent k p\ns₁ s₂ : Set ι\nh₁ : s₁.Subsingleton\nh₂ : s₂.Subsingleton\n⊢ vectorSpan k (p '' s₁)... | [] | simp [h₁.image p, h₂.image p, vectorSpan_of_subsingleton] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 746,
"column": 4
} | {
"line": 746,
"column": 29
} | {
"line": 747,
"column": 4
} | [
{
"pp": "case inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nh : LinearIndependent k fun (v : ↑((fun p ↦ p -ᵥ p₁) '' (s \\ {p₁}))) ↦ ↑v\nhp₁ : p₁ ∈ s\n⊢ ∃ t, s ⊆ t ∧ (AffineIndependent k fun p ↦ ↑p... | [
"case inr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : Set P\np₁ : P\nh : LinearIndependent k fun (v : ↑((fun p ↦ p -ᵥ p₁) '' (s \\ {p₁}))) ↦ ↑v\nhp₁ : p₁ ∈ s\nbsv : Basis (↑(LinearIndepOn.extend h ⋯)) k V := Basis.ext... | let bsv := Basis.extend h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 258,
"column": 36
} | {
"line": 258,
"column": 57
} | {
"line": 258,
"column": 57
} | [
{
"pp": "case inr\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Subsingleton ι\nh : Nonempty ι\n⊢ Metric.diam {fun x ↦ 1} = 0",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"pseudoMetricSpacePi",
"Real.... | [
"case inr\nι : Type u_1\ninst✝¹ : Fintype ι\ninst✝ : Subsingleton ι\nh : Nonempty ι\n⊢ 0 = 0"
] | Metric.diam_singleton | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.AlexandrovDiscrete | {
"line": 147,
"column": 53
} | {
"line": 148,
"column": 56
} | {
"line": 150,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝¹ : TopologicalSpace α\ninst✝ : AlexandrovDiscrete α\ns : Set α\n⊢ IsOpen[inst✝¹] (nhdsKer s)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"isOpen_sInter",
"setOf",
"id",
"LE.le",
"And",
"Se... | [] | by
rw [nhdsKer_def]; exact isOpen_sInter fun _ ↦ And.left | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Connected.LocallyPathConnected | {
"line": 213,
"column": 2
} | {
"line": 213,
"column": 57
} | {
"line": 214,
"column": 2
} | [
{
"pp": "X : Type u_4\ninst✝ : TopologicalSpace X\n⊢ (∀ (b : Set X), IsOpen[inst✝] b → ∀ (x : X), IsOpen[inst✝] (pathComponentIn b x)) ↔\n ∀ (b : Set X), IsOpen[inst✝] b → ∀ x ∈ b, pathComponentIn b x ∈ 𝓝 x",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
... | [
"X : Type u_4\ninst✝ : TopologicalSpace X\nu : Set X\nx✝ : IsOpen[inst✝] u\n⊢ (∀ (x : X), IsOpen[inst✝] (pathComponentIn u x)) ↔ ∀ x ∈ u, pathComponentIn u x ∈ 𝓝 x"
] | refine forall_congr' fun u ↦ imp_congr_right fun _ ↦ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Module.Convex | {
"line": 190,
"column": 2
} | {
"line": 190,
"column": 38
} | {
"line": 191,
"column": 2
} | [
{
"pp": "E : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx z : E\nu : ℝ\nhu_nonneg : 0 ≤ u\nhu_le_one : u ≤ 1\n⊢ u * ‖z - x‖ ≤ ‖z - x‖",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"MulOne.toOne",
"Real.instLE",
... | [
"E : Type u_1\ninst✝¹ : SeminormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nx z : E\nu : ℝ\nhu_nonneg : 0 ≤ u\nhu_le_one : u ≤ 1\n⊢ u * ‖z - x‖ ≤ 1 * ‖z - x‖"
] | conv_rhs => rw [← one_mul (‖z - x‖)] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convRHS_1 | Mathlib.Tactic.Conv.convRHS |
Mathlib.Analysis.Convex.Combination | {
"line": 604,
"column": 8
} | {
"line": 604,
"column": 27
} | {
"line": 604,
"column": 28
} | [
{
"pp": "𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\ninst✝⁵ : Finite ι\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : (i : ι) → AddCommGroup (E i)\ninst✝ : (i : ι) → Module 𝕜 (E i)\ns : Set ι\nt✝ : (i : ι) → Set (E i)\nx : (i : ι) → E i\nval✝ : Fintype ι\nt : (i : ι) ... | [
"𝕜 : Type u_1\nι : Type u_2\nE : ι → Type u_3\ninst✝⁵ : Finite ι\ninst✝⁴ : Field 𝕜\ninst✝³ : LinearOrder 𝕜\ninst✝² : IsStrictOrderedRing 𝕜\ninst✝¹ : (i : ι) → AddCommGroup (E i)\ninst✝ : (i : ι) → Module 𝕜 (E i)\ns : Set ι\nt✝ : (i : ι) → Set (E i)\nx : (i : ι) → E i\nval✝ : Fintype ι\nt : (i : ι) → Set (E i)\... | Finset.prod_eq_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Operator.Basic | {
"line": 327,
"column": 9
} | {
"line": 327,
"column": 18
} | {
"line": 327,
"column": 19
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_4\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NontrivialTopology E\nx : E\nhx : ‖x‖ ≠ 0\nthis : ‖(ContinuousLinearMap.id 𝕜 E) x‖ / ‖x‖ ≤ ‖ContinuousLinearMap.id 𝕜 E‖\n⊢ 1 ≤ ‖ContinuousLinearMap.id 𝕜 E‖",
"p... | [
"𝕜 : Type u_1\nE : Type u_4\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NontriviallyNormedField 𝕜\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : NontrivialTopology E\nx : E\nhx : ‖x‖ ≠ 0\nthis : ‖x‖ / ‖x‖ ≤ ‖ContinuousLinearMap.id 𝕜 E‖\n⊢ 1 ≤ ‖ContinuousLinearMap.id 𝕜 E‖"
] | id_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 758,
"column": 8
} | {
"line": 758,
"column": 34
} | {
"line": 758,
"column": 34
} | [
{
"pp": "𝕜 : Type u_2\n𝕜₂ : Type u_3\nE : Type u_6\nF : Type u_7\nι' : Type u_10\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : NormedField 𝕜₂\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝³ : RingHomIsometric σ₁₂\nκ : Type u_11\nq : Se... | [
"𝕜 : Type u_2\n𝕜₂ : Type u_3\nE : Type u_6\nF : Type u_7\nι' : Type u_10\ninst✝⁹ : NontriviallyNormedField 𝕜\ninst✝⁸ : AddCommGroup E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : NormedField 𝕜₂\ninst✝⁵ : AddCommGroup F\ninst✝⁴ : Module 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝³ : RingHomIsometric σ₁₂\nκ : Type u_11\nq : SeminormFamily... | ← Seminorm.coe_iSup_eq bdd | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.Germ.Basic | {
"line": 453,
"column": 36
} | {
"line": 453,
"column": 48
} | {
"line": 454,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nl : Filter α\nf g h : α → β\nM : Type u_5\nG : Type u_6\ninst✝ : AddMonoidWithOne M\n⊢ ↑0 = fun x ↦ 0",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddMonoid.toAddZeroClass",
"... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.Germ.Basic | {
"line": 454,
"column": 38
} | {
"line": 454,
"column": 50
} | {
"line": 456,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nl : Filter α\nf g h : α → β\nM : Type u_5\nG : Type u_6\ninst✝ : AddMonoidWithOne M\nx✝ : ℕ\n⊢ ↑(x✝ + 1) = (fun f g x ↦ (fun x1 x2 ↦ x1 + x2) (f x) (g x)) ↑x✝ fun x ↦ 1",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"E... | [] | by simp; rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.AEEqFun | {
"line": 433,
"column": 33
} | {
"line": 433,
"column": 40
} | {
"line": 433,
"column": 40
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace δ\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ng : β → γ → δ\nhg : Continuous[instTopologicalSpaceProd, inst✝²] (uncurry g)\nf₁ : α →ₘ[μ] β\nf₂ : α →ₘ[μ] γ\n⊢ comp (... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace δ\ninst✝¹ : TopologicalSpace β\ninst✝ : TopologicalSpace γ\ng : β → γ → δ\nhg : Continuous[instTopologicalSpaceProd, inst✝²] (uncurry g)\nf₁ : α →ₘ[μ] β\nf₂ : α →ₘ[μ] γ\n⊢ mk (uncurry g ∘ fu... | comp_mk | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Defs | {
"line": 110,
"column": 11
} | {
"line": 110,
"column": 81
} | {
"line": 111,
"column": 4
} | [
{
"pp": "α : Type u_1\nε : Type u_2\nm0 : MeasurableSpace α\ninst✝ : ENorm ε\nμ : Measure α\nf : α → ε\n⊢ eLpNorm f 1 μ = ∫⁻ (x : α), ‖f x‖ₑ ∂μ",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"MeasureTheory.Measure",
"instHDiv",
"congrArg",
... | [
"α : Type u_1\nε : Type u_2\nm0 : MeasurableSpace α\ninst✝ : ENorm ε\nμ : Measure α\nf : α → ε\n⊢ (∫⁻ (x : α), ‖f x‖ₑ ^ ENNReal.toReal 1 ∂μ) ^ (1 / ENNReal.toReal 1) = ∫⁻ (x : α), ‖f x‖ₑ ∂μ"
] | eLpNorm_eq_lintegral_rpow_enorm_toReal one_ne_zero ENNReal.coe_ne_top, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 408,
"column": 27
} | {
"line": 408,
"column": 46
} | {
"line": 408,
"column": 47
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_3\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μ i)\ni : ι\n⊢ (∏ j ∈ Finset.univ.erase i, (μ j) univ) • μ i = μ i",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
... | [
"ι : Type u_1\nα : ι → Type u_3\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝ : ∀ (i : ι), IsProbabilityMeasure (μ i)\ni : ι\n⊢ 1 • μ i = μ i",
"ι : Type u_1\nα : ι → Type u_3\ninst✝² : Fintype ι\ninst✝¹ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (... | Finset.prod_eq_one, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.Indicator | {
"line": 77,
"column": 2
} | {
"line": 77,
"column": 88
} | {
"line": 79,
"column": 0
} | [
{
"pp": "case inr\nα : Type u_1\nm0 : MeasurableSpace α\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\ns : Set α\nc : ε\nhμ0 : μ ≠ 0\n⊢ eLpNormEssSup (s.indicator fun x ↦ c) μ ≤ ‖c‖ₑ",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Set.indic... | [] | · exact (eLpNormEssSup_indicator_le s fun _ => c).trans (eLpNormEssSup_const c hμ0).le | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 130,
"column": 78
} | {
"line": 136,
"column": 76
} | {
"line": 138,
"column": 0
} | [
{
"pp": "α : Type u_1\nε : Type u_2\nm0 : MeasurableSpace α\np : ℝ≥0∞\ninst✝ : ENorm ε\nf : α → ε\n⊢ eLpNorm f p 0 = 0",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"ENNReal.zero_rpow_of_pos",
"Real.partialOrder",
"Real",
"D... | [] | by
by_cases h0 : p = 0
· simp [h0]
by_cases h_top : p = ∞
· simp [h_top]
rw [← Ne] at h0
simp [eLpNorm_eq_eLpNorm' h0 h_top, eLpNorm', ENNReal.toReal_pos h0 h_top] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.LpSeminorm.Indicator | {
"line": 172,
"column": 4
} | {
"line": 173,
"column": 15
} | {
"line": 174,
"column": 4
} | [
{
"pp": "case neg.inr.left\nα : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_7\ninst✝² : TopologicalSpace ε\ninst✝¹ : ESeminormedAddMonoid ε\ns : Set α\nf : α → ε\ninst✝ : DecidablePred fun x ↦ x ∈ s\ng : α → ε\nhs : MeasurableSet s\nhf : MemLp f p (μ.restrict s)\nhg : MemLp g p (μ.rest... | [
"case neg.inr.left\nα : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_7\ninst✝² : TopologicalSpace ε\ninst✝¹ : ESeminormedAddMonoid ε\ns : Set α\nf : α → ε\ninst✝ : DecidablePred fun x ↦ x ∈ s\ng : α → ε\nhs : MeasurableSet s\nhf : MemLp f p (μ.restrict s)\nhg : MemLp g p (μ.restrict sᶜ)\nhp... | have h (x) (hx : x ∈ s) : ‖Set.piecewise s f g x‖ₑ ^ p.toReal = ‖f x‖ₑ ^ p.toReal := by
simp [hx] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 594,
"column": 11
} | {
"line": 594,
"column": 37
} | {
"line": 594,
"column": 37
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝⁵ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝⁴ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝³ : ∀ (i : ι), SigmaFinite (μ i)\ninst✝² : (i : ι) → Group (α i)\ninst✝¹ : ∀ (i : ι), MeasurableMul (α i)\ninst✝ : ∀ (i : ι), ... | [] | measure_preimage_mul_right | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 619,
"column": 2
} | {
"line": 626,
"column": 50
} | {
"line": 628,
"column": 0
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝⁴ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝³ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝² : ∀ (i : ι), SigmaFinite (μ i)\ninst✝¹ : (i : ι) → TopologicalSpace (α i)\ninst✝ : ∀ (i : ι), (μ i).IsOpenPosMeasure\n⊢ (Mea... | [] | constructor
rintro U U_open ⟨a, ha⟩
obtain ⟨s, ⟨hs, hsU⟩⟩ := isOpen_pi_iff'.1 U_open a ha
refine ne_of_gt (lt_of_lt_of_le ?_ (measure_mono hsU))
simp only [pi_pi]
rw [CanonicallyOrderedAdd.prod_pos]
intro i _
apply (hs i).1.measure_pos (μ i) ⟨a i, (hs i).2⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 619,
"column": 2
} | {
"line": 626,
"column": 50
} | {
"line": 628,
"column": 0
} | [
{
"pp": "ι : Type u_1\nι' : Type u_2\nα : ι → Type u_3\ninst✝⁴ : Fintype ι\nm : (i : ι) → OuterMeasure (α i)\ninst✝³ : (i : ι) → MeasurableSpace (α i)\nμ : (i : ι) → Measure (α i)\ninst✝² : ∀ (i : ι), SigmaFinite (μ i)\ninst✝¹ : (i : ι) → TopologicalSpace (α i)\ninst✝ : ∀ (i : ι), (μ i).IsOpenPosMeasure\n⊢ (Mea... | [] | constructor
rintro U U_open ⟨a, ha⟩
obtain ⟨s, ⟨hs, hsU⟩⟩ := isOpen_pi_iff'.1 U_open a ha
refine ne_of_gt (lt_of_lt_of_le ?_ (measure_mono hsU))
simp only [pi_pi]
rw [CanonicallyOrderedAdd.prod_pos]
intro i _
apply (hs i).1.measure_pos (μ i) ⟨a i, (hs i).2⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 170,
"column": 2
} | {
"line": 170,
"column": 21
} | {
"line": 171,
"column": 2
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε' : Type u_6\ninst✝³ : TopologicalSpace ε'\ninst✝² : ContinuousENorm ε'\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\ng : α → ε'\np : ℝ≥0∞\nhg : AEStronglyMeasurable g μ\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ ... | [
"case pos\nα : Type u_1\nm : MeasurableSpace α\nμ : Measure α\nε' : Type u_6\ninst✝³ : TopologicalSpace ε'\ninst✝² : ContinuousENorm ε'\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ESeminormedAddMonoid ε\nf : α → ε\nc : ℝ≥0∞\ng : α → ε'\np : ℝ≥0∞\nhg : AEStronglyMeasurable g μ\nh : ∀ᵐ (x : α) ∂μ, ‖f x‖ₑ ≤ c ... | by_cases h₀ : p = 0 | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Analysis.Convex.Slope | {
"line": 44,
"column": 2
} | {
"line": 44,
"column": 59
} | {
"line": 45,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConcaveOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\n⊢ (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)",
"ppTerm": "?m.48",
"assigned": true,
"us... | [
"𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhf : ConcaveOn 𝕜 s f\nx y z : 𝕜\nhx : x ∈ s\nhz : z ∈ s\nhxy : x < y\nhyz : y < z\nthis : ((-f) y - (-f) x) / (y - x) ≤ ((-f) z - (-f) y) / (z - y)\n⊢ (f z - f y) / (z - y) ≤ (f y - f x) / (y - x)"... | have := ConvexOn.slope_mono_adjacent hf.neg hx hz hxy hyz | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Constructions.Pi | {
"line": 819,
"column": 2
} | {
"line": 819,
"column": 34
} | {
"line": 821,
"column": 0
} | [
{
"pp": "case e_6\nn : ℕ\nα : Fin (n + 1) → Type u\nm : (i : Fin (n + 1)) → MeasurableSpace (α i)\nμ : (i : Fin (n + 1)) → Measure (α i)\ninst✝ : ∀ (i : Fin (n + 1)), SigmaFinite (μ i)\ni : Fin (n + 1)\ne : α i × ((j : Fin n) → α (i.succAbove j)) ≃ᵐ ((j : Fin (n + 1)) → α j) := (MeasurableEquiv.piFinSuccAbove α... | [] | simp [e, i.forall_iff_succAbove] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 51,
"column": 41
} | {
"line": 51,
"column": 72
} | {
"line": 52,
"column": 6
} | [
{
"pp": "case hbc\nx y z : ℝ\nhxy : x < y\nhyz : y < z\nh1 : 0 < y - x\nh2 : x - y < 0\n⊢ 1 - rexp (x - y) < -(x - y)",
"ppTerm": "?hbc",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.toNeg",
... | [] | linarith [add_one_lt_exp h2.ne] | Mathlib.Tactic._aux_Mathlib_Tactic_Linarith_Frontend___elabRules_Mathlib_Tactic_linarith_1 | Mathlib.Tactic.linarith |
Mathlib.Analysis.Convex.Slope | {
"line": 88,
"column": 4
} | {
"line": 88,
"column": 56
} | {
"line": 90,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhs : Convex 𝕜 s\nhf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z → (f y - f x) / (y - x) ≤ (f z - f y) / (z - y)\nx : 𝕜\nhx : x ∈ s\nz : 𝕜\nhz : z ∈ s\nhxz : x < z\na b : 𝕜\nha : 0... | [] | linear_combination key + (- f x * z + x * f z) * hab | Mathlib.Tactic.LinearCombination._aux_Mathlib_Tactic_LinearCombination___elabRules_Mathlib_Tactic_LinearCombination_linearCombination_1 | Mathlib.Tactic.LinearCombination.linearCombination |
Mathlib.Analysis.Convex.Slope | {
"line": 116,
"column": 4
} | {
"line": 116,
"column": 56
} | {
"line": 118,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\ns : Set 𝕜\nf : 𝕜 → 𝕜\nhs : Convex 𝕜 s\nhf : ∀ {x y z : 𝕜}, x ∈ s → z ∈ s → x < y → y < z → (f y - f x) / (y - x) < (f z - f y) / (z - y)\nx : 𝕜\nhx : x ∈ s\nz : 𝕜\nhz : z ∈ s\nhxz : x < z\na b : 𝕜\nha : 0... | [] | linear_combination key + (- f x * z + x * f z) * hab | Mathlib.Tactic.LinearCombination._aux_Mathlib_Tactic_LinearCombination___elabRules_Mathlib_Tactic_LinearCombination_linearCombination_1 | Mathlib.Tactic.LinearCombination.linearCombination |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 480,
"column": 2
} | {
"line": 480,
"column": 69
} | {
"line": 482,
"column": 0
} | [
{
"pp": "case e'_2.e'_5\nα : Type u_1\nF : Type u_5\nm0 : MeasurableSpace α\np : ℝ≥0∞\nq : ℝ\nμ : Measure α\ninst✝ : NormedAddCommGroup F\nf : α → F\nhq_pos : 0 < q\nx✝ : α\n⊢ ENNReal.ofReal ‖f x✝‖ ^ q = ENNReal.ofReal (‖f x✝‖ ^ q)",
"ppTerm": "?e'_2.e'_5",
"assigned": true,
"usedConstants": [
... | [] | exact ENNReal.ofReal_rpow_of_nonneg (by positivity) (by positivity) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 563,
"column": 2
} | {
"line": 563,
"column": 20
} | {
"line": 564,
"column": 2
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\nq : ℝ\nμ ν : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nf : α → ε\nhμν : ν ≤ μ\nhq : 0 ≤ q\n⊢ eLpNorm' f q ν ≤ eLpNorm' f q μ",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"MeasureTheory.eLpNorm'",... | [
"α : Type u_1\nm0 : MeasurableSpace α\nq : ℝ\nμ ν : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nf : α → ε\nhμν : ν ≤ μ\nhq : 0 ≤ q\n⊢ (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂ν) ^ (1 / q) ≤ (∫⁻ (a : α), ‖f a‖ₑ ^ q ∂μ) ^ (1 / q)"
] | simp_rw [eLpNorm'] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 129,
"column": 2
} | {
"line": 129,
"column": 42
} | {
"line": 130,
"column": 2
} | [
{
"pp": "s : ℝ\nhs : -1 ≤ s\np : ℝ\nhp : 1 ≤ p\n⊢ 1 + p * s ≤ (1 + s) ^ p",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real.partialOrder",
"Real.instLE",
"Real",
"Preorder.toLT",
"HMul.hMul",
"PartialOrder.toPreorder",
"LE... | [
"case inl\ns : ℝ\nhs : -1 ≤ s\nhp : 1 ≤ 1\n⊢ 1 + 1 * s ≤ (1 + s) ^ 1",
"case inr\ns : ℝ\nhs : -1 ≤ s\np : ℝ\nhp✝ : 1 ≤ p\nhp : 1 < p\n⊢ 1 + p * s ≤ (1 + s) ^ p"
] | rcases eq_or_lt_of_le hp with (rfl | hp) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 140,
"column": 25
} | {
"line": 140,
"column": 43
} | {
"line": 140,
"column": 44
} | [
{
"pp": "case inl\np : ℝ\nhp1 : 0 < p\nhp2 : p < 1\nhs : -1 ≤ -1\nhs' : -1 ≠ 0\n⊢ 0 ^ p < 1 + p * -1",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"HMul.hMul",
"Real.instZero",
"congrArg",
"AddMonoid.toAddZeroClas... | [
"case inl\np : ℝ\nhp1 : 0 < p\nhp2 : p < 1\nhs : -1 ≤ -1\nhs' : -1 ≠ 0\n⊢ 0 < 1 + p * -1"
] | zero_rpow hp1.ne', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 159,
"column": 4
} | {
"line": 159,
"column": 61
} | {
"line": 160,
"column": 4
} | [
{
"pp": "case inr.inr\ns : ℝ\nhs✝ : -1 ≤ s\nhs'✝ : s ≠ 0\np : ℝ\nhp1 : 0 < p\nhp2 : p < 1\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhs3 : 1 + s ≠ 1\nhs4 : 1 + p * s ≠ 1\nhs' : 0 < s\n⊢ log (1 + s) * p < log (1 + p * s)",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Eq.... | [
"case inr.inr\ns : ℝ\nhs✝ : -1 ≤ s\nhs'✝ : s ≠ 0\np : ℝ\nhp1 : 0 < p\nhp2 : p < 1\nhs : -1 < s\nhs1 : 0 < 1 + s\nhs2 : 0 < 1 + p * s\nhs3 : 1 + s ≠ 1\nhs4 : 1 + p * s ≠ 1\nhs' : 0 < s\n⊢ log (1 + s) / s < log (1 + p * s) / p / s"
] | rw [← lt_div_iff₀ hp1, ← div_lt_div_iff_of_pos_right hs'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 795,
"column": 18
} | {
"line": 795,
"column": 70
} | {
"line": 795,
"column": 70
} | [
{
"pp": "case inr.inr\nα : Type u_1\nF : Type u_5\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝² : NormedAddCommGroup F\nf : α → F\ninst✝¹ : Finite α\ninst✝ : IsFiniteMeasure μ\nhp₀ : p ≠ 0\nhp : p ≠ ∞\n⊢ ∃ c, ∀ (x : α), ↑‖f x‖₊ ^ p.toReal ≤ ↑c",
"ppTerm": "?inr.inr",
"assigned": true,
"us... | [
"case inr.inr\nα : Type u_1\nF : Type u_5\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝² : NormedAddCommGroup F\nf : α → F\ninst✝¹ : Finite α\ninst✝ : IsFiniteMeasure μ\nhp₀ : p ≠ 0\nhp : p ≠ ∞\n⊢ ∃ c, ∀ (x : α), ↑(‖f x‖₊ ^ p.toReal) ≤ ↑c"
] | ← ENNReal.coe_rpow_of_nonneg _ ENNReal.toReal_nonneg | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Real.ConjExponents | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 55
} | {
"line": 239,
"column": 0
} | [
{
"pp": "p q r : ℝ≥0\n⊢ (↑p).HolderTriple ↑q ↑r ↔ p.HolderTriple q r",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"Preorder.toLT",
"Real.instZero",
"congrArg",
"Real.instInv",
"Re... | [] | rw_mod_cast [Real.holderTriple_iff, holderTriple_iff] | Lean.Parser.Tactic._aux_Init_TacticsExtra___macroRules_Lean_Parser_Tactic_tacticRw_mod_cast____1 | Lean.Parser.Tactic.tacticRw_mod_cast___ |
Mathlib.Data.Real.ConjExponents | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 55
} | {
"line": 239,
"column": 0
} | [
{
"pp": "p q r : ℝ≥0\n⊢ (↑p).HolderTriple ↑q ↑r ↔ p.HolderTriple q r",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"Preorder.toLT",
"Real.instZero",
"congrArg",
"Real.instInv",
"Re... | [] | rw_mod_cast [Real.holderTriple_iff, holderTriple_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Real.ConjExponents | {
"line": 237,
"column": 2
} | {
"line": 237,
"column": 55
} | {
"line": 239,
"column": 0
} | [
{
"pp": "p q r : ℝ≥0\n⊢ (↑p).HolderTriple ↑q ↑r ↔ p.HolderTriple q r",
"ppTerm": "?m.1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"Real",
"Preorder.toLT",
"Real.instZero",
"congrArg",
"Real.instInv",
"Re... | [] | rw_mod_cast [Real.holderTriple_iff, holderTriple_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 864,
"column": 2
} | {
"line": 864,
"column": 70
} | {
"line": 866,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nβ : Type u_8\nmβ : MeasurableSpace β\nf : α → β\ng : β → ε\nhg : AEStronglyMeasurable g (Measure.map f μ)\nhf : AEMeasurable f μ\n⊢ MemLp g p (Measure.map f μ) ↔ MemLp (g... | [] | simp [MemLp, eLpNorm_map_measure hg hf, hg.comp_aemeasurable hf, hg] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 864,
"column": 2
} | {
"line": 864,
"column": 70
} | {
"line": 866,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nβ : Type u_8\nmβ : MeasurableSpace β\nf : α → β\ng : β → ε\nhg : AEStronglyMeasurable g (Measure.map f μ)\nhf : AEMeasurable f μ\n⊢ MemLp g p (Measure.map f μ) ↔ MemLp (g... | [] | simp [MemLp, eLpNorm_map_measure hg hf, hg.comp_aemeasurable hf, hg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 864,
"column": 2
} | {
"line": 864,
"column": 70
} | {
"line": 866,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nβ : Type u_8\nmβ : MeasurableSpace β\nf : α → β\ng : β → ε\nhg : AEStronglyMeasurable g (Measure.map f μ)\nhf : AEMeasurable f μ\n⊢ MemLp g p (Measure.map f μ) ↔ MemLp (g... | [] | simp [MemLp, eLpNorm_map_measure hg hf, hg.comp_aemeasurable hf, hg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.SpecificFunctions.Basic | {
"line": 208,
"column": 2
} | {
"line": 208,
"column": 42
} | {
"line": 209,
"column": 2
} | [
{
"pp": "p : ℝ\nhp : 1 ≤ p\n⊢ ConvexOn ℝ (Ici 0) fun x ↦ x ^ p",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real.partialOrder",
"Real.instLE",
"Real",
"Preorder.toLT",
"instSMulOfMul",
"Set.Ici",
"Real.instZero",
"Pa... | [
"case inl\nhp : 1 ≤ 1\n⊢ ConvexOn ℝ (Ici 0) fun x ↦ x ^ 1",
"case inr\np : ℝ\nhp✝ : 1 ≤ p\nhp : 1 < p\n⊢ ConvexOn ℝ (Ici 0) fun x ↦ x ^ p"
] | rcases eq_or_lt_of_le hp with (rfl | hp) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Algebra.Order.Monovary | {
"line": 371,
"column": 2
} | {
"line": 372,
"column": 51
} | {
"line": 374,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁷ : Ring α\ninst✝⁶ : LinearOrder α\ninst✝⁵ : IsStrictOrderedRing α\ninst✝⁴ : AddCommGroup β\ninst✝³ : LinearOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : Module α β\ninst✝ : IsStrictOrderedModule α β\nf : ι → α\ng : ι → β\ns : Set ι\n⊢ MonovaryOn f g s ... | [] | simp_rw [smul_nonneg_iff_pos_imp_nonneg, sub_pos, sub_nonneg, forall_and]
exact (and_iff_right_of_imp MonovaryOn.symm).symm | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Monovary | {
"line": 371,
"column": 2
} | {
"line": 372,
"column": 51
} | {
"line": 374,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\ninst✝⁷ : Ring α\ninst✝⁶ : LinearOrder α\ninst✝⁵ : IsStrictOrderedRing α\ninst✝⁴ : AddCommGroup β\ninst✝³ : LinearOrder β\ninst✝² : IsOrderedAddMonoid β\ninst✝¹ : Module α β\ninst✝ : IsStrictOrderedModule α β\nf : ι → α\ng : ι → β\ns : Set ι\n⊢ MonovaryOn f g s ... | [] | simp_rw [smul_nonneg_iff_pos_imp_nonneg, sub_pos, sub_nonneg, forall_and]
exact (and_iff_right_of_imp MonovaryOn.symm).symm | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.MeanInequalities | {
"line": 135,
"column": 2
} | {
"line": 139,
"column": 25
} | {
"line": 142,
"column": 2
} | [
{
"pp": "case pos\nι : Type u\ns : Finset ι\nw z : ι → ℝ\nhw : ∀ i ∈ s, 0 ≤ w i\nhw' : ∑ i ∈ s, w i = 1\nhz : ∀ i ∈ s, 0 ≤ z i\nA : ∃ i ∈ s, z i = 0 ∧ w i ≠ 0\n⊢ ∏ i ∈ s, z i ^ w i ≤ ∑ i ∈ s, w i * z i",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",... | [
"case neg\nι : Type u\ns : Finset ι\nw z : ι → ℝ\nhw : ∀ i ∈ s, 0 ≤ w i\nhw' : ∑ i ∈ s, w i = 1\nhz : ∀ i ∈ s, 0 ≤ z i\nA : ∀ i ∈ s, z i = 0 → w i = 0\n⊢ ∏ i ∈ s, z i ^ w i ≤ ∑ i ∈ s, w i * z i"
] | · rcases A with ⟨i, his, hzi, hwi⟩
rw [prod_eq_zero his]
· exact sum_nonneg fun j hj => mul_nonneg (hw j hj) (hz j hj)
· rw [hzi]
exact zero_rpow hwi | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Mul | {
"line": 53,
"column": 56
} | {
"line": 53,
"column": 82
} | {
"line": 54,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝¹⁷ : CommRing 𝕜\ninst✝¹⁶ : LinearOrder 𝕜\ninst✝¹⁵ : IsStrictOrderedRing 𝕜\ninst✝¹⁴ : CommRing E\ninst✝¹³ : LinearOrder E\ninst✝¹² : IsStrictOrderedRing E\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : LinearOrder F\ninst✝⁹ : IsOrderedAddMonoid F\ni... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝¹⁷ : CommRing 𝕜\ninst✝¹⁶ : LinearOrder 𝕜\ninst✝¹⁵ : IsStrictOrderedRing 𝕜\ninst✝¹⁴ : CommRing E\ninst✝¹³ : LinearOrder E\ninst✝¹² : IsStrictOrderedRing E\ninst✝¹¹ : AddCommGroup F\ninst✝¹⁰ : LinearOrder F\ninst✝⁹ : IsOrderedAddMonoid F\ninst✝⁸ : AddC... | ← smul_smul_smul_comm a b, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 173,
"column": 16
} | {
"line": 173,
"column": 25
} | {
"line": 173,
"column": 26
} | [
{
"pp": "p q : ℝ\na b : ℝ≥0\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0), a ^ q = (a ^ p) ^ (q / p)\nh_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p\n⊢ ((a ^ p) ^ (q / p) + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)",
"ppTerm": "?m.182",
"assigned... | [
"p q : ℝ\na b : ℝ≥0\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0), a ^ q = (a ^ p) ^ (q / p)\nh_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p\n⊢ ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)"
] | h_rpow b, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 33
} | {
"line": 131,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np : ℝ\nhp0 : 0 ≤ p\nf : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhf_zero : (fun x ↦ f x ^ p) =ᵐ[μ] 0\n⊢ f =ᵐ[μ] 0",
"ppTerm": "?m.46",
"assigned": true,
"usedConstants": [
"MeasureTheory.ae",
"Real",
"MeasureTheory.Measure",
... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np : ℝ\nhp0 : 0 ≤ p\nf : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhf_zero : (fun x ↦ f x ^ p) =ᵐ[μ] 0\nx : α\n⊢ f x ^ p = 0 x → f x = 0 x"
] | filter_upwards [hf_zero] with x | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Analysis.Convex.Mul | {
"line": 73,
"column": 56
} | {
"line": 73,
"column": 82
} | {
"line": 74,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝¹⁸ : CommRing 𝕜\ninst✝¹⁷ : LinearOrder 𝕜\ninst✝¹⁶ : IsStrictOrderedRing 𝕜\ninst✝¹⁵ : CommRing E\ninst✝¹⁴ : LinearOrder E\ninst✝¹³ : IsStrictOrderedRing E\ninst✝¹² : AddCommGroup F\ninst✝¹¹ : LinearOrder F\ninst✝¹⁰ : IsOrderedAddMonoid F\n... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\ninst✝¹⁸ : CommRing 𝕜\ninst✝¹⁷ : LinearOrder 𝕜\ninst✝¹⁶ : IsStrictOrderedRing 𝕜\ninst✝¹⁵ : CommRing E\ninst✝¹⁴ : LinearOrder E\ninst✝¹³ : IsStrictOrderedRing E\ninst✝¹² : AddCommGroup F\ninst✝¹¹ : LinearOrder F\ninst✝¹⁰ : IsOrderedAddMonoid F\ninst✝⁹ : Add... | ← smul_smul_smul_comm a b, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.MeanInequalities | {
"line": 216,
"column": 8
} | {
"line": 216,
"column": 78
} | {
"line": 217,
"column": 6
} | [
{
"pp": "ι : Type u\ns : Finset ι\nw z : ι → ℝ\nhw : ∀ i ∈ s, 0 < w i\nhw' : ∑ i ∈ s, w i = 1\nhz : ∀ i ∈ s, 0 ≤ z i\ni : ι\nhis : i ∈ s\nhzi : z i = 0\nhwi : w i ≠ 0\nh : 0 = ∑ i ∈ s, w i * z i\nj : ι\nhj : j ∈ s\n⊢ ∀ i ∈ s, 0 ≤ w i * z i",
"ppTerm": "?m.144",
"assigned": true,
"usedConstants": [
... | [] | exact fun i hi => (mul_nonneg_iff_of_pos_left (hw i hi)).mpr (hz i hi) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.MeanInequalitiesPow | {
"line": 308,
"column": 16
} | {
"line": 308,
"column": 25
} | {
"line": 308,
"column": 26
} | [
{
"pp": "p q : ℝ\na b : ℝ≥0∞\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0∞), a ^ q = (a ^ p) ^ (q / p)\nh_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p\n⊢ ((a ^ p) ^ (q / p) + b ^ q) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)",
"ppTerm": "?m.165",
"assign... | [
"p q : ℝ\na b : ℝ≥0∞\nhp_pos : 0 < p\nhpq : p ≤ q\nh_rpow : ∀ (a : ℝ≥0∞), a ^ q = (a ^ p) ^ (q / p)\nh_rpow_add_rpow_le_add : ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / (q / p)) ≤ a ^ p + b ^ p\n⊢ ((a ^ p) ^ (q / p) + (b ^ p) ^ (q / p)) ^ (1 / q) ≤ (a ^ p + b ^ p) ^ (1 / p)"
] | h_rpow b, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 123,
"column": 43
} | {
"line": 123,
"column": 53
} | {
"line": 123,
"column": 53
} | [
{
"pp": "α : Type u_1\nε : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nf : α → ε\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\np q : ℝ≥0∞\ninst✝ : IsFiniteMeasure μ\nhpq : ∞ ≤ q\nhfq_m : AEStronglyMeasurable f μ\nhfq_lt_top : eLpNorm f q μ < ∞\nhp0 : p ≠ 0\nhp_top : p = ∞\n⊢ q = ∞",
"ppTerm"... | [
"α : Type u_1\nε : Type u_2\nm : MeasurableSpace α\nμ : Measure α\nf : α → ε\ninst✝² : TopologicalSpace ε\ninst✝¹ : ContinuousENorm ε\np q : ℝ≥0∞\ninst✝ : IsFiniteMeasure μ\nhpq : q = ∞\nhfq_m : AEStronglyMeasurable f μ\nhfq_lt_top : eLpNorm f q μ < ∞\nhp0 : p ≠ 0\nhp_top : p = ∞\n⊢ q = ∞"
] | top_le_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 285,
"column": 2
} | {
"line": 285,
"column": 47
} | {
"line": 286,
"column": 2
} | [
{
"pp": "α : Type u_2\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ∂μ) ... | [
"α : Type u_2\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\n⊢ (∫⁻ (a : α), (f * g) a ^ p ∂μ) ^ (1 / p) ≤ (∫⁻ (a : α), f a ^ q ... | have hq0_ne : q ≠ 0 := (ne_of_lt hq0_lt).symm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 296,
"column": 6
} | {
"line": 298,
"column": 88
} | {
"line": 299,
"column": 4
} | [
{
"pp": "α : Type u_2\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\np2 : ℝ := q / p\n... | [] | gcongr
simp_rw [ENNReal.rpow_mul]
exact ENNReal.lintegral_mul_le_Lp_mul_Lq μ hp2q2 (hf.pow_const _) (hg.pow_const _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 296,
"column": 6
} | {
"line": 298,
"column": 88
} | {
"line": 299,
"column": 4
} | [
{
"pp": "α : Type u_2\ninst✝ : MeasurableSpace α\np q r : ℝ\nhp0_lt : 0 < p\nhpq : p < q\nhpqr : 1 / p = 1 / q + 1 / r\nμ : Measure α\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhg : AEMeasurable g μ\nhp0_ne : p ≠ 0\nhp0 : 0 ≤ p\nhq0_lt : 0 < q\nhq0_ne : q ≠ 0\nh_one_div_r : 1 / r = 1 / p - 1 / q\np2 : ℝ := q / p\n... | [] | gcongr
simp_rw [ENNReal.rpow_mul]
exact ENNReal.lintegral_mul_le_Lp_mul_Lq μ hp2q2 (hf.pow_const _) (hg.pow_const _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 239,
"column": 4
} | {
"line": 240,
"column": 64
} | {
"line": 241,
"column": 2
} | [
{
"pp": "case inr.inr.inr.inl\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nμ : Measure α\nf : α → E\ng : α → F\np r : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\... | [] | have : r = p := by simpa using hpqr
exact this ▸ eLpNorm_le_eLpNorm_mul_eLpNorm_top p hf g b c h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 239,
"column": 4
} | {
"line": 240,
"column": 64
} | {
"line": 241,
"column": 2
} | [
{
"pp": "case inr.inr.inr.inl\nα : Type u_1\nE : Type u_2\nF : Type u_3\nG : Type u_4\nm : MeasurableSpace α\ninst✝² : NormedAddCommGroup E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : NormedAddCommGroup G\nμ : Measure α\nf : α → E\ng : α → F\np r : ℝ≥0∞\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\... | [] | have : r = p := by simpa using hpqr
exact this ▸ eLpNorm_le_eLpNorm_mul_eLpNorm_top p hf g b c h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.MeanInequalities | {
"line": 779,
"column": 40
} | {
"line": 796,
"column": 80
} | {
"line": 798,
"column": 0
} | [
{
"pp": "ι : Type u\ns : Finset ι\nf : ι → ℝ≥0\np q : ℝ\nhpq : p.HolderConjugate q\n⊢ IsGreatest ((fun g ↦ ∑ i ∈ s, f i * g i) '' {g | ∑ i ∈ s, g i ^ q ≤ 1}) ((∑ i ∈ s, f i ^ p) ^ (1 / p))",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"inv_eq_zero._simp_1",
"G... | [] | by
constructor
· use fun i => f i ^ p / f i / (∑ i ∈ s, f i ^ p) ^ (1 / q)
obtain hf | hf := eq_zero_or_pos (∑ i ∈ s, f i ^ p)
· simp [hf, hpq.ne_zero, hpq.symm.ne_zero]
· have A : p + q - q ≠ 0 := by simp [hpq.ne_zero]
have B : ∀ y : ℝ≥0, y * y ^ p / y = y ^ p := by
refine fun y => mul_di... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.MeanInequalities | {
"line": 804,
"column": 2
} | {
"line": 804,
"column": 42
} | {
"line": 805,
"column": 2
} | [
{
"pp": "ι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np : ℝ\nhp : 1 ≤ p\n⊢ (∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i ∈ s, f i ^ p) ^ (1 / p) + (∑ i ∈ s, g i ^ p) ^ (1 / p)",
"ppTerm": "?m.72",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real.instLE",
"Real",
"Pre... | [
"case inl\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\nhp : 1 ≤ 1\n⊢ (∑ i ∈ s, (f i + g i) ^ 1) ^ (1 / 1) ≤ (∑ i ∈ s, f i ^ 1) ^ (1 / 1) + (∑ i ∈ s, g i ^ 1) ^ (1 / 1)",
"case inr\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0\np : ℝ\nhp✝ : 1 ≤ p\nhp : 1 < p\n⊢ (∑ i ∈ s, (f i + g i) ^ p) ^ (1 / p) ≤ (∑ i ∈ s, f i ^ p) ^ (... | rcases eq_or_lt_of_le hp with (rfl | hp) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
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