module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 219,
"column": 4
} | {
"line": 219,
"column": 59
} | {
"line": 220,
"column": 4
} | [
{
"pp": "case mp\nθ ψ : ℝ\nHsin : sin θ = sin ψ\n⊢ ↑θ = ↑ψ ∨ ↑θ + ↑ψ = ↑π",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Real",
"instHDiv",
"Real.pi",
"Real.cos",
"congrArg",
"Real.instDivInvMonoid",
"Real.instSub",
"Nat.instAtLeastTwoHAddOfN... | [
"case mp\nθ ψ : ℝ\nHsin : cos (π / 2 - θ) = cos (π / 2 - ψ)\n⊢ ↑θ = ↑ψ ∨ ↑θ + ↑ψ = ↑π"
] | rw [← cos_pi_div_two_sub, ← cos_pi_div_two_sub] at Hsin | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 241,
"column": 2
} | {
"line": 241,
"column": 60
} | {
"line": 241,
"column": 60
} | [
{
"pp": "case inr\nθ ψ : ℝ\nHcos : cos θ = cos ψ\nHsin : sin θ = sin ψ\nhc : ↑θ = -↑ψ\n⊢ ↑θ = ↑ψ",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Real",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"AddCommGroup.toAddCommMonoid",
"Real.Angle.sin_eq_iff_coe_... | [
"case inr.inl\nθ ψ : ℝ\nHcos : cos θ = cos ψ\nHsin : sin θ = sin ψ\nhc : ↑θ = -↑ψ\nhs : ↑θ = ↑ψ\n⊢ ↑θ = ↑ψ",
"case inr.inr\nθ ψ : ℝ\nHcos : cos θ = cos ψ\nHsin : sin θ = sin ψ\nhc : ↑θ = -↑ψ\nhs : ↑θ + ↑ψ = ↑π\n⊢ ↑θ = ↑ψ"
] | rcases sin_eq_iff_coe_eq_or_add_eq_pi.mp Hsin with hs | hs | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 388,
"column": 4
} | {
"line": 388,
"column": 47
} | {
"line": 389,
"column": 2
} | [
{
"pp": "case inl\nz : ℂ\nhre : z.re < 0\nthis : z ≠ 0\n⊢ (0 ≤ z.re ∨ z.im < 0) ∧ ¬(z.re = 0 ∧ 0 < z.im) ↔ 0 < z.re ∨ z.im < 0 ∨ z = 0",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"False",
"Real.instLE",
"Real",
"Preorder.toLT",
"eq_false",
"and_true",... | [] | simp [hre.ne, hre.not_ge, hre.not_gt, this] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 365,
"column": 2
} | {
"line": 366,
"column": 32
} | {
"line": 368,
"column": 0
} | [
{
"pp": "θ : Angle\n⊢ θ.cos ^ 2 + θ.sin ^ 2 = 1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Real",
"Real.Angle",
"instOfNatNat",
"NPow.toPow",
"Real.instAdd",
"Real.instOne",
"Real.instMonoid",
"Real.Angle.induction_on",
"instHAdd"... | [] | induction θ using Real.Angle.induction_on
exact Real.cos_sq_add_sin_sq _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 365,
"column": 2
} | {
"line": 366,
"column": 32
} | {
"line": 368,
"column": 0
} | [
{
"pp": "θ : Angle\n⊢ θ.cos ^ 2 + θ.sin ^ 2 = 1",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Real",
"Real.Angle",
"instOfNatNat",
"NPow.toPow",
"Real.instAdd",
"Real.instOne",
"Real.instMonoid",
"Real.Angle.induction_on",
"instHAdd"... | [] | induction θ using Real.Angle.induction_on
exact Real.cos_sq_add_sin_sq _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 497,
"column": 66
} | {
"line": 497,
"column": 94
} | {
"line": 499,
"column": 0
} | [
{
"pp": "θ : Angle\n⊢ θ.toReal = π ↔ θ = ↑π",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"congrArg",
"Real.Angle.toReal_pi",
"Iff.rfl",
"Real.Angle.toReal_inj",
"id",
... | [] | rw [← toReal_inj, toReal_pi] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 497,
"column": 66
} | {
"line": 497,
"column": 94
} | {
"line": 499,
"column": 0
} | [
{
"pp": "θ : Angle\n⊢ θ.toReal = π ↔ θ = ↑π",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"congrArg",
"Real.Angle.toReal_pi",
"Iff.rfl",
"Real.Angle.toReal_inj",
"id",
... | [] | rw [← toReal_inj, toReal_pi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 497,
"column": 66
} | {
"line": 497,
"column": 94
} | {
"line": 499,
"column": 0
} | [
{
"pp": "θ : Angle\n⊢ θ.toReal = π ↔ θ = ↑π",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.pi",
"Real.Angle",
"Real.Angle.coe",
"congrArg",
"Real.Angle.toReal_pi",
"Iff.rfl",
"Real.Angle.toReal_inj",
"id",
... | [] | rw [← toReal_inj, toReal_pi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 467,
"column": 2
} | {
"line": 467,
"column": 89
} | {
"line": 469,
"column": 0
} | [
{
"pp": "θ : ℝ\n⊢ (cos ↑θ + sin ↑θ * I).arg = toIocMod Real.two_pi_pos (-π) θ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Real.partialOrder",
"Real",
"Real.instArchimedean",
"Real.pi",
"HMul.hMul",
"Complex.cos"... | [] | rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 467,
"column": 2
} | {
"line": 467,
"column": 89
} | {
"line": 469,
"column": 0
} | [
{
"pp": "θ : ℝ\n⊢ (cos ↑θ + sin ↑θ * I).arg = toIocMod Real.two_pi_pos (-π) θ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Real.partialOrder",
"Real",
"Real.instArchimedean",
"Real.pi",
"HMul.hMul",
"Complex.cos"... | [] | rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 467,
"column": 2
} | {
"line": 467,
"column": 89
} | {
"line": 469,
"column": 0
} | [
{
"pp": "θ : ℝ\n⊢ (cos ↑θ + sin ↑θ * I).arg = toIocMod Real.two_pi_pos (-π) θ",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Real.partialOrder",
"Real",
"Real.instArchimedean",
"Real.pi",
"HMul.hMul",
"Complex.cos"... | [] | rw [← one_mul (_ + _), ← ofReal_one, arg_mul_cos_add_sin_mul_I_eq_toIocMod zero_lt_one] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 652,
"column": 8
} | {
"line": 652,
"column": 10
} | {
"line": 652,
"column": 10
} | [
{
"pp": "case neg\nx : ℂ\nh : x ≠ 0\nhs : x ∉ slitPlane\nha : Function.update ((Real.Angle.coe ∘ arg) ∘ Neg.neg) 0 ↑π = fun z ↦ ↑z.arg + ↑π\n⊢ ContinuousAt (Function.update ((Real.Angle.coe ∘ arg) ∘ Neg.neg) 0 ↑π) (-x)",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"... | [
"case neg\nx : ℂ\nh : x ≠ 0\nhs : x ∉ slitPlane\nha : Function.update ((Real.Angle.coe ∘ arg) ∘ Neg.neg) 0 ↑π = fun z ↦ ↑z.arg + ↑π\n⊢ ContinuousAt (fun z ↦ ↑z.arg + ↑π) (-x)"
] | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 88
} | {
"line": 115,
"column": 0
} | [
{
"pp": "x : ℝ\nhx : x ≤ 0\ny : ℝ\n⊢ x ^ y = if x = 0 then if y = 0 then 1 else 0 else rexp (log x * y) * cos (y * π)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"False",
"Real.partialOrder",
"Real",
"Real.pi",
"HMul.h... | [] | split_ifs with h <;> simp [rpow_def, *]; exact rpow_def_of_neg (lt_of_le_of_ne hx h) _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 88
} | {
"line": 115,
"column": 0
} | [
{
"pp": "x : ℝ\nhx : x ≤ 0\ny : ℝ\n⊢ x ^ y = if x = 0 then if y = 0 then 1 else 0 else rexp (log x * y) * cos (y * π)",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"False",
"Real.partialOrder",
"Real",
"Real.pi",
"HMul.h... | [] | split_ifs with h <;> simp [rpow_def, *]; exact rpow_def_of_neg (lt_of_le_of_ne hx h) _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 175,
"column": 23
} | {
"line": 175,
"column": 42
} | {
"line": 175,
"column": 43
} | [
{
"pp": "case inr\nx y : ℝ\nhx : x < 0\n⊢ |x ^ y| ≤ (-x) ^ y",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instPow",
"Real.instLE",
"Real",
"Real.pi",
"HMul.hMul",
"... | [
"case inr\nx y : ℝ\nhx : x < 0\n⊢ |rexp (log x * y) * cos (y * π)| ≤ (-x) ^ y"
] | rpow_def_of_neg hx, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 152,
"column": 10
} | {
"line": 152,
"column": 12
} | {
"line": 153,
"column": 2
} | [
{
"pp": "case h\ny : ℝ\nhy : 0 < y\nb : ℝ≥0\nc : ℝ\nhc : ∀ (a : ℝ), c ≤ a → ↑b ≤ a ^ y\na : ℝ≥0\n⊢ c.toNNReal ≤ a → b ≤ a ^ y",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
"NNReal",
"LE.le",
"NNReal.instPartialOrder",... | [
"case h\ny : ℝ\nhy : 0 < y\nb : ℝ≥0\nc : ℝ\nhc : ∀ (a : ℝ), c ≤ a → ↑b ≤ a ^ y\na : ℝ≥0\nha : c.toNNReal ≤ a\n⊢ b ≤ a ^ y"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | {
"line": 162,
"column": 30
} | {
"line": 162,
"column": 32
} | {
"line": 163,
"column": 2
} | [
{
"pp": "y : ℝ\nhy : 0 < y\nx c : ℝ≥0\nleft✝ : True\nhc : ∀ x_1 ∈ Set.Ioi c, x_1 ^ y ∈ Set.Ioi x\nhc' : Set.Ioi ↑c ∈ 𝓝 ∞\na : ℝ≥0∞\n⊢ a ∈ Set.Ioi ↑c → ↑x < a ^ y",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"ENNReal.ofNNReal",
"Set.Ioi",
"PartialOrder.toPreorder",
... | [
"y : ℝ\nhy : 0 < y\nx c : ℝ≥0\nleft✝ : True\nhc : ∀ x_1 ∈ Set.Ioi c, x_1 ^ y ∈ Set.Ioi x\nhc' : Set.Ioi ↑c ∈ 𝓝 ∞\na : ℝ≥0∞\nha : a ∈ Set.Ioi ↑c\n⊢ ↑x < a ^ y"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 443,
"column": 86
} | {
"line": 444,
"column": 35
} | {
"line": 446,
"column": 0
} | [
{
"pp": "x y : ℝ\nn : ℕ\nhx : 0 ≤ x\nh : y + ↑n ≠ 0\n⊢ x ^ (y + ↑n) = x ^ y * x ^ n",
"ppTerm": "?m.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real",
"HMul.hMul",
"congrArg",
"id",
"Nat.cast",
"NPow.toPow",
"Real.instAdd... | [] | by
rw [rpow_add' hx h, rpow_natCast] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 354,
"column": 31
} | {
"line": 354,
"column": 38
} | {
"line": 354,
"column": 38
} | [
{
"pp": "case inr.inr\np : ℝ\nx : ℝ≥0\nhx_pos : 0 < x\nrpow_pos_of_nonneg : ∀ {p : ℝ}, 0 < p → 0 < x ^ p\nhp_neg : p < 0\n⊢ 0 < (x ^ (-p))⁻¹",
"ppTerm": "?inr.inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWithZero.toLi... | [
"case inr.inr\np : ℝ\nx : ℝ≥0\nhx_pos : 0 < x\nrpow_pos_of_nonneg : ∀ {p : ℝ}, 0 < p → 0 < x ^ p\nhp_neg : p < 0\n⊢ 0 < x ^ (-p)"
] | inv_pos | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 354,
"column": 4
} | {
"line": 354,
"column": 28
} | {
"line": 355,
"column": 2
} | [
{
"pp": "case inr.inl\ny : ℂ\nh : 0 < y.re ∨ 0 ≠ 0\nA : ContinuousAt (fun p ↦ p.1 ^ p.2) (↑0, y)\nB : ContinuousAt (fun p ↦ (↑p.1, p.2)) (0, y)\n⊢ ContinuousAt (fun p ↦ ↑p.1 ^ p.2) (0, y)",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
... | [] | exact A.comp_of_eq B rfl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 695,
"column": 2
} | {
"line": 697,
"column": 48
} | {
"line": 699,
"column": 0
} | [
{
"pp": "x y : ℝ\nhx : 0 ≤ x\n⊢ x ^ y < 1 ↔ x = 0 ∧ y ≠ 0 ∨ 1 < x ∧ y < 0 ∨ x < 1 ∧ 0 < y",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Real.rpow_lt_one_iff_of_pos",
"LE.le.eq_or_lt",
"Real.instPow",
"False",
"Real.partialOrder",
"Real.instLE",
... | [] | rcases hx.eq_or_lt with (rfl | hx)
· rcases _root_.em (y = 0) with (rfl | hy) <;> simp [*, zero_lt_one]
· simp [rpow_lt_one_iff_of_pos hx, hx.ne.symm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 695,
"column": 2
} | {
"line": 697,
"column": 48
} | {
"line": 699,
"column": 0
} | [
{
"pp": "x y : ℝ\nhx : 0 ≤ x\n⊢ x ^ y < 1 ↔ x = 0 ∧ y ≠ 0 ∨ 1 < x ∧ y < 0 ∨ x < 1 ∧ 0 < y",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Real.rpow_lt_one_iff_of_pos",
"LE.le.eq_or_lt",
"Real.instPow",
"False",
"Real.partialOrder",
"Real.instLE",
... | [] | rcases hx.eq_or_lt with (rfl | hx)
· rcases _root_.em (y = 0) with (rfl | hy) <;> simp [*, zero_lt_one]
· simp [rpow_lt_one_iff_of_pos hx, hx.ne.symm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 795,
"column": 2
} | {
"line": 795,
"column": 62
} | {
"line": 797,
"column": 0
} | [
{
"pp": "x y z : ℝ\nhx : 0 < x\nhy : 0 < y\n⊢ x < y ^ z ↔ log x < z * log y",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.rpow_pos_of_pos",
"Real",
"HMul.hMul",
"congrArg",
"Iff.rfl",
"Real.log_rpow",
"... | [] | rw [← log_lt_log_iff hx (rpow_pos_of_pos hy z), log_rpow hy] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 795,
"column": 2
} | {
"line": 795,
"column": 62
} | {
"line": 797,
"column": 0
} | [
{
"pp": "x y z : ℝ\nhx : 0 < x\nhy : 0 < y\n⊢ x < y ^ z ↔ log x < z * log y",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.rpow_pos_of_pos",
"Real",
"HMul.hMul",
"congrArg",
"Iff.rfl",
"Real.log_rpow",
"... | [] | rw [← log_lt_log_iff hx (rpow_pos_of_pos hy z), log_rpow hy] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 795,
"column": 2
} | {
"line": 795,
"column": 62
} | {
"line": 797,
"column": 0
} | [
{
"pp": "x y z : ℝ\nhx : 0 < x\nhy : 0 < y\n⊢ x < y ^ z ↔ log x < z * log y",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.rpow_pos_of_pos",
"Real",
"HMul.hMul",
"congrArg",
"Iff.rfl",
"Real.log_rpow",
"... | [] | rw [← log_lt_log_iff hx (rpow_pos_of_pos hy z), log_rpow hy] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 732,
"column": 2
} | {
"line": 735,
"column": 77
} | {
"line": 737,
"column": 0
} | [
{
"pp": "ι : Type u_1\ns : Finset ι\nf : ι → ℝ≥0\nr : ℝ\n⊢ ∏ i ∈ s, ↑(f i) ^ r = ↑(∏ i ∈ s, f i) ^ r",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"ENNReal.one_rpow",
"Eq.mpr",
"NNReal.instCommSemiring",
"MulOne.toOne",
"Real",
"ENNReal.ofNNReal",
... | [] | classical
induction s using Finset.induction with
| empty => simp
| insert _ _ hi ih => simp_rw [prod_insert hi, ih, ← coe_mul_rpow, coe_mul] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 732,
"column": 2
} | {
"line": 735,
"column": 77
} | {
"line": 737,
"column": 0
} | [
{
"pp": "ι : Type u_1\ns : Finset ι\nf : ι → ℝ≥0\nr : ℝ\n⊢ ∏ i ∈ s, ↑(f i) ^ r = ↑(∏ i ∈ s, f i) ^ r",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"ENNReal.one_rpow",
"Eq.mpr",
"NNReal.instCommSemiring",
"MulOne.toOne",
"Real",
"ENNReal.ofNNReal",
... | [] | classical
induction s using Finset.induction with
| empty => simp
| insert _ _ hi ih => simp_rw [prod_insert hi, ih, ← coe_mul_rpow, coe_mul] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 732,
"column": 2
} | {
"line": 735,
"column": 77
} | {
"line": 737,
"column": 0
} | [
{
"pp": "ι : Type u_1\ns : Finset ι\nf : ι → ℝ≥0\nr : ℝ\n⊢ ∏ i ∈ s, ↑(f i) ^ r = ↑(∏ i ∈ s, f i) ^ r",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"ENNReal.one_rpow",
"Eq.mpr",
"NNReal.instCommSemiring",
"MulOne.toOne",
"Real",
"ENNReal.ofNNReal",
... | [] | classical
induction s using Finset.induction with
| empty => simp
| insert _ _ hi ih => simp_rw [prod_insert hi, ih, ← coe_mul_rpow, coe_mul] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 843,
"column": 2
} | {
"line": 850,
"column": 48
} | {
"line": 852,
"column": 0
} | [
{
"pp": "x : ℝ≥0∞\ny z : ℝ\nhx : 1 ≤ x\nhyz : y ≤ z\n⊢ x ^ y ≤ x ^ z",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Iff.mpr",
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.... | [] | cases x
· rcases lt_trichotomy y 0 with (Hy | Hy | Hy) <;>
rcases lt_trichotomy z 0 with (Hz | Hz | Hz) <;>
simp [Hy, Hz, top_rpow_of_neg, top_rpow_of_pos] <;>
linarith
· simp only [one_le_coe_iff] at hx
simp [← coe_rpow_of_ne_zero (ne_of_gt (lt_of_lt_of_le zero_lt_one hx)),
NNReal.rpow_le_rpo... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 843,
"column": 2
} | {
"line": 850,
"column": 48
} | {
"line": 852,
"column": 0
} | [
{
"pp": "x : ℝ≥0∞\ny z : ℝ\nhx : 1 ≤ x\nhyz : y ≤ z\n⊢ x ^ y ≤ x ^ z",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"Iff.mpr",
"Real.instIsOrderedRing",
"Mathlib.Tactic.Ring.Common.neg_zero",
"Eq.mpr",
"NegZeroClass.... | [] | cases x
· rcases lt_trichotomy y 0 with (Hy | Hy | Hy) <;>
rcases lt_trichotomy z 0 with (Hz | Hz | Hz) <;>
simp [Hy, Hz, top_rpow_of_neg, top_rpow_of_pos] <;>
linarith
· simp only [one_le_coe_iff] at hx
simp [← coe_rpow_of_ne_zero (ne_of_gt (lt_of_lt_of_le zero_lt_one hx)),
NNReal.rpow_le_rpo... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 920,
"column": 4
} | {
"line": 920,
"column": 73
} | {
"line": 922,
"column": 0
} | [
{
"pp": "case coe\nz : ℝ\nhz : 0 < z\nx✝ : ℝ≥0\nhx : 1 < x✝\n⊢ 1 < ↑x✝ ^ z",
"ppTerm": "?coe",
"assigned": true,
"usedConstants": [
"Real",
"ENNReal.ofNNReal",
"Preorder.toLT",
"Real.instZero",
"congrArg",
"ENNReal.instPowReal",
"PartialOrder.toPreorder",
... | [] | simp [← coe_rpow_of_nonneg _ (le_of_lt hz), NNReal.one_lt_rpow hx hz] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 1080,
"column": 35
} | {
"line": 1081,
"column": 56
} | {
"line": 1083,
"column": 0
} | [
{
"pp": "a b : ℝ\nn d : ℕ\nhb : IsRat b (Int.negOfNat n) d\n⊢ a ^ b = a⁻¹ ^ (↑n / ↑d)",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.instPow",
"Real",
"instHDiv",
"congrArg",
"Real.instInv",
"Real.instDi... | [] | by
rw [← Real.rpow_neg_eq_inv_rpow, hb.neg_to_eq rfl rfl] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Midpoint | {
"line": 219,
"column": 71
} | {
"line": 220,
"column": 26
} | {
"line": 222,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\ninst✝³ : Ring R\ninst✝² : Invertible 2\ninst✝¹ : AddCommGroup V\ninst✝ : Module R V\nx y : V\n⊢ midpoint R (x + y) (x - y) = x",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"AddCommGroup.toAddCommMonoid",
... | [] | by
rw [midpoint_comm]; simp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Midpoint | {
"line": 244,
"column": 40
} | {
"line": 244,
"column": 57
} | {
"line": 245,
"column": 6
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝⁷ : Ring R\ninst✝⁶ : Invertible 2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\ninst✝³ : Ring R'\ninst✝² : Invertible 2\ninst✝¹ : AddCommGroup F\ninst✝ : Module R' F\nf : E → F\nh0 : f 0 = 0\nhm : ∀ (x y : E), f (midpoint R x y) = midpoint ... | [] | rw [h0, zero_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.AffineSpace.Midpoint | {
"line": 244,
"column": 40
} | {
"line": 244,
"column": 57
} | {
"line": 245,
"column": 6
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝⁷ : Ring R\ninst✝⁶ : Invertible 2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\ninst✝³ : Ring R'\ninst✝² : Invertible 2\ninst✝¹ : AddCommGroup F\ninst✝ : Module R' F\nf : E → F\nh0 : f 0 = 0\nhm : ∀ (x y : E), f (midpoint R x y) = midpoint ... | [] | rw [h0, zero_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Midpoint | {
"line": 244,
"column": 40
} | {
"line": 244,
"column": 57
} | {
"line": 245,
"column": 6
} | [
{
"pp": "R : Type u_1\nR' : Type u_2\nE : Type u_3\nF : Type u_4\ninst✝⁷ : Ring R\ninst✝⁶ : Invertible 2\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module R E\ninst✝³ : Ring R'\ninst✝² : Invertible 2\ninst✝¹ : AddCommGroup F\ninst✝ : Module R' F\nf : E → F\nh0 : f 0 = 0\nhm : ∀ (x y : E), f (midpoint R x y) = midpoint ... | [] | rw [h0, zero_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.AffineMap | {
"line": 374,
"column": 4
} | {
"line": 376,
"column": 7
} | {
"line": 378,
"column": 0
} | [
{
"pp": "k : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P... | [] | intro p v
rw [Function.comp_apply, g.map_vadd, f.map_vadd]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.AffineMap | {
"line": 374,
"column": 4
} | {
"line": 376,
"column": 7
} | {
"line": 378,
"column": 0
} | [
{
"pp": "k : Type u_1\nV1 : Type u_2\nP1 : Type u_3\nV2 : Type u_4\nP2 : Type u_5\nV3 : Type u_6\nP3 : Type u_7\nV4 : Type u_8\nP4 : Type u_9\ninst✝¹² : Ring k\ninst✝¹¹ : AddCommGroup V1\ninst✝¹⁰ : Module k V1\ninst✝⁹ : AffineSpace V1 P1\ninst✝⁸ : AddCommGroup V2\ninst✝⁷ : Module k V2\ninst✝⁶ : AffineSpace V2 P... | [] | intro p v
rw [Function.comp_apply, g.map_vadd, f.map_vadd]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Segment | {
"line": 244,
"column": 39
} | {
"line": 244,
"column": 49
} | {
"line": 244,
"column": 50
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : Ring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddRightMono 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nf : E →ᵃ[𝕜] F\na b : E\nx : F\n⊢ x ∈ ⇑f '' ⇑(AffineMap.lineMap a b) '' Icc 0 1 ↔ x ∈ ⇑(AffineMap... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : Ring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddRightMono 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nf : E →ᵃ[𝕜] F\na b : E\nx : F\n⊢ (∃ x_1, (∃ x ∈ Icc 0 1, (AffineMap.lineMap a b) x = x_1) ∧ f x_1 = x) ↔\n ... | mem_image, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Convex.Segment | {
"line": 250,
"column": 43
} | {
"line": 250,
"column": 53
} | {
"line": 250,
"column": 54
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : Ring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddRightMono 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nf : E →ᵃ[𝕜] F\na b : E\nx : F\n⊢ x ∈ ⇑f '' ⇑(AffineMap.lineMap a b) '' Ioo 0 1 ↔ x ∈ ⇑(AffineMap... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : Ring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddRightMono 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nf : E →ᵃ[𝕜] F\na b : E\nx : F\n⊢ (∃ x_1, (∃ x ∈ Ioo 0 1, (AffineMap.lineMap a b) x = x_1) ∧ f x_1 = x) ↔\n ... | mem_image, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Convex.Star | {
"line": 163,
"column": 19
} | {
"line": 163,
"column": 21
} | {
"line": 163,
"column": 22
} | [
{
"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 : E\ns : Set E\nhx : x ∈ s\nh : ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\n⊢ 0 ≤ a → 0 ≤ b → a + b = 1 → a • x ... | [
"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 : E\ns : Set E\nhx : x ∈ s\nh : ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\n⊢ 0 ≤ b → a + b = 1 → a • x + b • y ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Segment | {
"line": 367,
"column": 2
} | {
"line": 373,
"column": 25
} | {
"line": 375,
"column": 0
} | [
{
"pp": "𝕜 : 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\n⊢ x ∈ openSegment 𝕜 x y ↔ x = y",
"ppTerm": "?m.18",
"assigned": true,
"u... | [] | constructor
· rintro ⟨a, b, _, hb, hab, hx⟩
refine smul_right_injective _ hb.ne' ((add_right_inj (a • x)).1 ?_)
rw [hx, ← add_smul, hab, one_smul]
· rintro rfl
rw [openSegment_same]
exact mem_singleton _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Segment | {
"line": 367,
"column": 2
} | {
"line": 373,
"column": 25
} | {
"line": 375,
"column": 0
} | [
{
"pp": "𝕜 : 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\n⊢ x ∈ openSegment 𝕜 x y ↔ x = y",
"ppTerm": "?m.18",
"assigned": true,
"u... | [] | constructor
· rintro ⟨a, b, _, hb, hab, hx⟩
refine smul_right_injective _ hb.ne' ((add_right_inj (a • x)).1 ?_)
rw [hx, ← add_smul, hab, one_smul]
· rintro rfl
rw [openSegment_same]
exact mem_singleton _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Star | {
"line": 176,
"column": 19
} | {
"line": 176,
"column": 21
} | {
"line": 176,
"column": 22
} | [
{
"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 : E\ns : Set E\nhx : x ∈ s\nh : ∀ ⦃y : E⦄, y ∈ s → x ≠ y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\n⊢ 0 ≤ a → 0 ≤ b → a + b = 1 ... | [
"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 : E\ns : Set E\nhx : x ∈ s\nh : ∀ ⦃y : E⦄, y ∈ s → x ≠ y → ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 ≤ a\n⊢ 0 ≤ b → a + b = 1 → a • x ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Star | {
"line": 207,
"column": 17
} | {
"line": 207,
"column": 19
} | {
"line": 207,
"column": 20
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : Semiring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx : E\ns : Set F\nf : E →ₗ[𝕜] F\nhs : StarConvex 𝕜 (f x) s\ny : E\nhy : y ∈ ⇑f ⁻¹' s\na b : 𝕜\n⊢ 0 ≤ a → 0 ≤ b → a... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : Semiring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx : E\ns : Set F\nf : E →ₗ[𝕜] F\nhs : StarConvex 𝕜 (f x) s\ny : E\nhy : y ∈ ⇑f ⁻¹' s\na b : 𝕜\nha : 0 ≤ a\n⊢ 0 ≤ b → a + b = 1... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Star | {
"line": 222,
"column": 17
} | {
"line": 222,
"column": 19
} | {
"line": 222,
"column": 20
} | [
{
"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 : E\ns : Set E\nhs : StarConvex 𝕜 x s\nz y : E\nhy : y ∈ (fun x ↦ z + x) '' s\na b : 𝕜\n⊢ 0 ≤ a → 0 ≤ b → a + b = 1 → a • (z + x) + b • y ∈ (fun x ↦ z + x) '' s",
"ppTerm"... | [
"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 : E\ns : Set E\nhs : StarConvex 𝕜 x s\nz y : E\nhy : y ∈ (fun x ↦ z + x) '' s\na b : 𝕜\nha : 0 ≤ a\n⊢ 0 ≤ b → a + b = 1 → a • (z + x) + b • y ∈ (fun x ↦ z + x) '' s"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Star | {
"line": 229,
"column": 17
} | {
"line": 229,
"column": 19
} | {
"line": 229,
"column": 20
} | [
{
"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 : E\ns : Set E\nhs : StarConvex 𝕜 x s\nz y : E\nhy : y ∈ (fun x ↦ x + z) '' s\na b : 𝕜\n⊢ 0 ≤ a → 0 ≤ b → a + b = 1 → a • (x + z) + b • y ∈ (fun x ↦ x + z) '' s",
"ppTerm"... | [
"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 : E\ns : Set E\nhs : StarConvex 𝕜 x s\nz y : E\nhy : y ∈ (fun x ↦ x + z) '' s\na b : 𝕜\nha : 0 ≤ a\n⊢ 0 ≤ b → a + b = 1 → a • (x + z) + b • y ∈ (fun x ↦ x + z) '' s"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Star | {
"line": 237,
"column": 17
} | {
"line": 237,
"column": 19
} | {
"line": 237,
"column": 20
} | [
{
"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 (z + x) s\ny : E\nhy : y ∈ (fun x ↦ z + x) ⁻¹' s\na b : 𝕜\n⊢ 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ (fun x ↦ z + x) ⁻¹' s",
"ppTer... | [
"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\ny : E\nhy : y ∈ (fun x ↦ z + x) ⁻¹' s\na b : 𝕜\nha : 0 ≤ a\n⊢ 0 ≤ b → a + b = 1 → a • x + b • y ∈ (fun x ↦ z + x) ⁻¹' s"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 78,
"column": 22
} | {
"line": 78,
"column": 33
} | {
"line": 78,
"column": 34
} | [
{
"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\n⊢ Submodule.span k (∅ -ᵥ ∅) = ⊥",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Set.vsub",
"congrArg",
... | [
"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\n⊢ Submodule.span k ∅ = ⊥"
] | vsub_empty, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Segment | {
"line": 486,
"column": 76
} | {
"line": 494,
"column": 38
} | {
"line": 496,
"column": 0
} | [
{
"pp": "𝕜 : 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 : IsOrderedCancelAddMonoid E\ninst✝¹ : Module 𝕜 E\ninst✝ : PosSMulStrictMono 𝕜 E\nx y : E\nh : x < y\n⊢ openSegment 𝕜 x y ⊆ Ioo x y",
"ppTerm": "?m.20",
"ass... | [] | by
rintro z ⟨a, b, ha, hb, hab, rfl⟩
constructor
· calc
x = a • x + b • x := (Convex.combo_self hab _).symm
_ < a • x + b • y := by gcongr
· calc
a • x + b • y < a • y + b • y := by gcongr
_ = y := Convex.combo_self hab _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Basic | {
"line": 148,
"column": 24
} | {
"line": 148,
"column": 26
} | {
"line": 148,
"column": 27
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nh : s.Pairwise fun x y ↦ ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\n⊢ 0 < a → 0 < b → a + b = 1 → a • ... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nh : s.Pairwise fun x y ↦ ∀ ⦃a b : 𝕜⦄, 0 < a → 0 < b → a + b = 1 → a • x + b • y ∈ s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b : 𝕜\nha : 0 < a\n⊢ 0 < b → a + b = 1 → a • x + b • ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Star | {
"line": 339,
"column": 17
} | {
"line": 339,
"column": 19
} | {
"line": 339,
"column": 20
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : Ring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx : E\nf : E →ᵃ[𝕜] F\ns : Set F\nhs : StarConvex 𝕜 (f x) s\ny : E\nhy : y ∈ ⇑f ⁻¹' s\na b : 𝕜\n⊢ 0 ≤ a → 0 ≤ b → a + b =... | [
"𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁵ : Ring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\nx : E\nf : E →ᵃ[𝕜] F\ns : Set F\nhs : StarConvex 𝕜 (f x) s\ny : E\nhy : y ∈ ⇑f ⁻¹' s\na b : 𝕜\nha : 0 ≤ a\n⊢ 0 ≤ b → a + b = 1 → a •... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Basic | {
"line": 234,
"column": 22
} | {
"line": 234,
"column": 24
} | {
"line": 234,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nhs : Convex 𝕜 s\nz x : E\nhx : x ∈ (fun x ↦ z + x) ⁻¹' s\ny : E\nhy : y ∈ (fun x ↦ z + x) ⁻¹' s\na b : 𝕜\n⊢ 0 ≤ a → 0 ≤ b → a + b = 1 → a • x + b • y ∈ (fun x ↦ z + x... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : Module 𝕜 E\ns : Set E\nhs : Convex 𝕜 s\nz x : E\nhx : x ∈ (fun x ↦ z + x) ⁻¹' s\ny : E\nhy : y ∈ (fun x ↦ z + x) ⁻¹' s\na b : 𝕜\nha : 0 ≤ a\n⊢ 0 ≤ b → a + b = 1 → a • x + b • y ∈ (fun x ↦ z + x) ⁻¹' s"... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Basic | {
"line": 275,
"column": 22
} | {
"line": 275,
"column": 24
} | {
"line": 275,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\nβ : Type u_4\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedCancelAddMonoid β\ninst✝¹ : Module 𝕜 β\ninst✝ : PosSMulStrictMono 𝕜 β\nr x : β\nhx : x ∈ Iio r\ny : β\nhy : y ∈ Iio r\na b : 𝕜\n⊢ 0 ≤ a → 0 ≤ b → a + b = ... | [
"𝕜 : Type u_1\nβ : Type u_4\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid β\ninst✝³ : PartialOrder β\ninst✝² : IsOrderedCancelAddMonoid β\ninst✝¹ : Module 𝕜 β\ninst✝ : PosSMulStrictMono 𝕜 β\nr x : β\nhx : x ∈ Iio r\ny : β\nhy : y ∈ Iio r\na b : 𝕜\nha : 0 ≤ a\n⊢ 0 ≤ b → a + b = 1 → a • ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Basic | {
"line": 322,
"column": 22
} | {
"line": 322,
"column": 24
} | {
"line": 322,
"column": 25
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝¹⁰ : Semiring 𝕜\ninst✝⁹ : PartialOrder 𝕜\ninst✝⁸ : AddCommMonoid E\ninst✝⁷ : ZeroLEOneClass 𝕜\ninst✝⁶ : Module 𝕜 E\nR : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : PartialOrder R\ninst✝³ : Module R E\ninst✝² : Module R 𝕜\ninst✝¹ : IsScalarTower R 𝕜 E\ninst✝ : SMulPos... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝¹⁰ : Semiring 𝕜\ninst✝⁹ : PartialOrder 𝕜\ninst✝⁸ : AddCommMonoid E\ninst✝⁷ : ZeroLEOneClass 𝕜\ninst✝⁶ : Module 𝕜 E\nR : Type u_5\ninst✝⁵ : Semiring R\ninst✝⁴ : PartialOrder R\ninst✝³ : Module R E\ninst✝² : Module R 𝕜\ninst✝¹ : IsScalarTower R 𝕜 E\ninst✝ : SMulPosMono R 𝕜\ns... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Star | {
"line": 430,
"column": 17
} | {
"line": 430,
"column": 19
} | {
"line": 430,
"column": 20
} | [
{
"pp": "𝕜 : 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 : y ∈ s... | [
"𝕜 : 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 : y ∈ s\na b : 𝕜\n... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 487,
"column": 2
} | {
"line": 494,
"column": 51
} | {
"line": 496,
"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\ns : Set P\n⊢ (affineSpan k s).direction = vectorSpan k s",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Submod... | [] | apply le_antisymm
· refine Submodule.span_le.2 ?_
rintro v ⟨p₁, ⟨p₂, hp₂, v₁, hv₁, hp₁⟩, p₃, ⟨p₄, hp₄, v₂, hv₂, hp₃⟩, rfl⟩
simp only [SetLike.mem_coe]
rw [hp₁, hp₃, vsub_vadd_eq_vsub_sub, vadd_vsub_assoc]
exact
(vectorSpan k s).sub_mem ((vectorSpan k s).add_mem hv₁ (vsub_mem_vectorSpan k hp₂ hp₄... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 487,
"column": 2
} | {
"line": 494,
"column": 51
} | {
"line": 496,
"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\ns : Set P\n⊢ (affineSpan k s).direction = vectorSpan k s",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Submod... | [] | apply le_antisymm
· refine Submodule.span_le.2 ?_
rintro v ⟨p₁, ⟨p₂, hp₂, v₁, hv₁, hp₁⟩, p₃, ⟨p₄, hp₄, v₂, hv₂, hp₃⟩, rfl⟩
simp only [SetLike.mem_coe]
rw [hp₁, hp₃, vsub_vadd_eq_vsub_sub, vadd_vsub_assoc]
exact
(vectorSpan k s).sub_mem ((vectorSpan k s).add_mem hv₁ (vsub_mem_vectorSpan k hp₂ hp₄... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 744,
"column": 6
} | {
"line": 744,
"column": 30
} | {
"line": 744,
"column": 31
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\n⊢ ⊥.direction = ⊥",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Lattice.toSemilatticeSup",
"vectorSpan",
... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\n⊢ vectorSpan k ↑⊥ = ⊥"
] | direction_eq_vectorSpan, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 744,
"column": 56
} | {
"line": 744,
"column": 67
} | {
"line": 744,
"column": 68
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\n⊢ Submodule.span k (∅ -ᵥ ∅) = ⊥",
"ppTerm": "?m.47",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"Set.vsub",
"congrArg",
... | [
"k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\n⊢ Submodule.span k ∅ = ⊥"
] | vsub_empty, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Basic | {
"line": 647,
"column": 22
} | {
"line": 647,
"column": 24
} | {
"line": 647,
"column": 25
} | [
{
"pp": "R : Type u_5\ninst✝⁹ : CommSemiring R\nA : Type u_6\ninst✝⁸ : Semiring A\ninst✝⁷ : Algebra R A\nM : Type u_7\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module A M\ninst✝⁴ : Module R M\ninst✝³ : IsScalarTower R A M\ninst✝² : PartialOrder R\ninst✝¹ : PartialOrder A\ninst✝ : FaithfulSMul R A\ns : Set M\nhalg : I... | [
"R : Type u_5\ninst✝⁹ : CommSemiring R\nA : Type u_6\ninst✝⁸ : Semiring A\ninst✝⁷ : Algebra R A\nM : Type u_7\ninst✝⁶ : AddCommMonoid M\ninst✝⁵ : Module A M\ninst✝⁴ : Module R M\ninst✝³ : IsScalarTower R A M\ninst✝² : PartialOrder R\ninst✝¹ : PartialOrder A\ninst✝ : FaithfulSMul R A\ns : Set M\nhalg : Ici 0 ⊆ ⇑(alg... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.LocallyConvex.BalancedCoreHull | {
"line": 146,
"column": 10
} | {
"line": 146,
"column": 12
} | {
"line": 147,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\na : 𝕜\n⊢ ‖a‖ ≤ 1 → a • balancedHull 𝕜 s ⊆ balancedHull 𝕜 s",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedRing.toNorm",
"... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\ns : Set E\na : 𝕜\nha : ‖a‖ ≤ 1\n⊢ a • balancedHull 𝕜 s ⊆ balancedHull 𝕜 s"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 241,
"column": 10
} | {
"line": 241,
"column": 12
} | {
"line": 242,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\nA : Set E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nhA : Balanced 𝕜 A\na : 𝕜\n⊢ ‖a‖ ≤ 1 → a • insert 0 (interior A) ⊆ insert 0 (interior A)",
"ppTerm": "?m.20",
"assigned": tr... | [
"𝕜 : Type u_1\nE : Type u_3\ninst✝⁴ : NormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\nA : Set E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nhA : Balanced 𝕜 A\na : 𝕜\nha : ‖a‖ ≤ 1\n⊢ a • insert 0 (interior A) ⊆ insert 0 (interior A)"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 271,
"column": 34
} | {
"line": 271,
"column": 36
} | {
"line": 272,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns : Set E\ninst✝ : PartialOrder 𝕜\nhs : Balanced 𝕜 s\nx : E\nhx : x ∈ {x | ∀ (a : 𝕜), ‖a‖ ≤ 1 → a • x ∈ (convexHull 𝕜) s}\ny : E\nhy : y ∈ {x | ∀ (a : 𝕜), ‖a‖ ≤ 1 → a • x ∈ (convexHull ... | [
"𝕜 : Type u_1\nE : Type u_3\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ns : Set E\ninst✝ : PartialOrder 𝕜\nhs : Balanced 𝕜 s\nx : E\nhx : x ∈ {x | ∀ (a : 𝕜), ‖a‖ ≤ 1 → a • x ∈ (convexHull 𝕜) s}\ny : E\nhy : y ∈ {x | ∀ (a : 𝕜), ‖a‖ ≤ 1 → a • x ∈ (convexHull 𝕜) s}\nu v ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Strict | {
"line": 76,
"column": 26
} | {
"line": 76,
"column": 28
} | {
"line": 76,
"column": 29
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝⁴ : Semiring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : TopologicalSpace E\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\ns t : Set E\nhs : StrictConvex 𝕜 s\nht : StrictConvex 𝕜 t\nx : E\nhx : x ∈ s ∩ t\ny : E\nhy : y ∈ s ∩ t\nhxy : x ≠ y\na b : 𝕜\n⊢ 0 < a → 0 < b → a + ... | [
"𝕜 : Type u_1\nE : Type u_3\ninst✝⁴ : Semiring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : TopologicalSpace E\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\ns t : Set E\nhs : StrictConvex 𝕜 s\nht : StrictConvex 𝕜 t\nx : E\nhx : x ∈ s ∩ t\ny : E\nhy : y ∈ s ∩ t\nhxy : x ≠ y\na b : 𝕜\nha : 0 < a\n⊢ 0 < b → a + b = 1 → ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Strict | {
"line": 131,
"column": 26
} | {
"line": 131,
"column": 28
} | {
"line": 131,
"column": 29
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝⁷ : Semiring 𝕜\ninst✝⁶ : PartialOrder 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns : Set F\nhs : StrictConvex 𝕜 s\nf : E →ₗ[𝕜] F\nhf : Con... | [
"𝕜 : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝⁷ : Semiring 𝕜\ninst✝⁶ : PartialOrder 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommMonoid E\ninst✝² : AddCommMonoid F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns : Set F\nhs : StrictConvex 𝕜 s\nf : E →ₗ[𝕜] F\nhf : Continuous ⇑f\n... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Strict | {
"line": 197,
"column": 26
} | {
"line": 197,
"column": 28
} | {
"line": 197,
"column": 29
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝⁵ : Semiring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : TopologicalSpace E\ninst✝² : AddCancelCommMonoid E\ninst✝¹ : ContinuousAdd E\ninst✝ : Module 𝕜 E\ns : Set E\nhs : StrictConvex 𝕜 s\nz x : E\nhx : x ∈ (fun x ↦ z + x) ⁻¹' s\ny : E\nhy : y ∈ (fun x ↦ z + x) ⁻¹' s\nhxy... | [
"𝕜 : Type u_1\nE : Type u_3\ninst✝⁵ : Semiring 𝕜\ninst✝⁴ : PartialOrder 𝕜\ninst✝³ : TopologicalSpace E\ninst✝² : AddCancelCommMonoid E\ninst✝¹ : ContinuousAdd E\ninst✝ : Module 𝕜 E\ns : Set E\nhs : StrictConvex 𝕜 s\nz x : E\nhx : x ∈ (fun x ↦ z + x) ⁻¹' s\ny : E\nhy : y ∈ (fun x ↦ z + x) ⁻¹' s\nhxy : x ≠ y\na ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Strict | {
"line": 324,
"column": 26
} | {
"line": 324,
"column": 28
} | {
"line": 324,
"column": 29
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝⁷ : Ring 𝕜\ninst✝⁶ : PartialOrder 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns : Set F\nhs : StrictConvex 𝕜 s\nf : E →ᵃ[𝕜] F\nhf : Continuou... | [
"𝕜 : Type u_1\nE : Type u_3\nF : Type u_4\ninst✝⁷ : Ring 𝕜\ninst✝⁶ : PartialOrder 𝕜\ninst✝⁵ : TopologicalSpace E\ninst✝⁴ : TopologicalSpace F\ninst✝³ : AddCommGroup E\ninst✝² : AddCommGroup F\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 F\ns : Set F\nhs : StrictConvex 𝕜 s\nf : E →ᵃ[𝕜] F\nhf : Continuous ⇑f\nhfinj ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 80,
"column": 4
} | {
"line": 81,
"column": 30
} | {
"line": 83,
"column": 0
} | [
{
"pp": "k : 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\ns : AffineSubspace k P\ninst✝ : Nonempty ↥s\na : ↥s.direction\nb : ↥s\n⊢ ⟨↑(a +ᵥ b) -ᵥ ↑b, ⋯⟩ = a",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
... | [] | ext
apply AddTorsor.vadd_vsub' | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 80,
"column": 4
} | {
"line": 81,
"column": 30
} | {
"line": 83,
"column": 0
} | [
{
"pp": "k : 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\ns : AffineSubspace k P\ninst✝ : Nonempty ↥s\na : ↥s.direction\nb : ↥s\n⊢ ⟨↑(a +ᵥ b) -ᵥ ↑b, ⋯⟩ = a",
"ppTerm": "?m.118",
"assigned": true,
"usedConstants": [
... | [] | ext
apply AddTorsor.vadd_vsub' | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 297,
"column": 2
} | {
"line": 297,
"column": 96
} | {
"line": 298,
"column": 2
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni₀ : ι\nv : V\n⊢ v ∈ (fun x ↦ p i₀ -ᵥ x) '' p '' (univ \\ {i₀}) ↔ v ∈ range fun i ↦ p i₀ -ᵥ p ↑i",
"ppTerm": "?m.58",
"assigned": true,
... | [
"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\nv : V\n⊢ (∃ x, (∃ x_1, (x_1 ∈ univ ∧ ¬x_1 = i₀) ∧ p x_1 = x) ∧ p i₀ -ᵥ x = v) ↔ ∃ a, ∃ (_ : a ≠ i₀), p i₀ -ᵥ p a = v"
] | simp only [Set.mem_range, Set.mem_image, Set.mem_sdiff, Set.mem_singleton_iff, Subtype.exists] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 310,
"column": 2
} | {
"line": 310,
"column": 96
} | {
"line": 311,
"column": 2
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nι : Type u_4\np : ι → P\ni₀ : ι\nv : V\n⊢ v ∈ (fun x ↦ x -ᵥ p i₀) '' p '' (univ \\ {i₀}) ↔ v ∈ range fun i ↦ p ↑i -ᵥ p i₀",
"ppTerm": "?m.58",
"assigned": true,
... | [
"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\nv : V\n⊢ (∃ x, (∃ x_1, (x_1 ∈ univ ∧ ¬x_1 = i₀) ∧ p x_1 = x) ∧ x -ᵥ p i₀ = v) ↔ ∃ a, ∃ (_ : a ≠ i₀), p a -ᵥ p i₀ = v"
] | simp only [Set.mem_range, Set.mem_image, Set.mem_sdiff, Set.mem_singleton_iff, Subtype.exists] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 451,
"column": 6
} | {
"line": 451,
"column": 61
} | {
"line": 452,
"column": 6
} | [
{
"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⊢ ∃ y ∈ s₂.direction, ∃ z ∈ k ∙ (p₂ -ᵥ p₁), y + z = p₃ -ᵥ p₂ + (p₂ -ᵥ p₁)"
] | rw [← vsub_add_vsub_cancel p₃ p₂ p₁, Submodule.mem_sup] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Function | {
"line": 1010,
"column": 26
} | {
"line": 1010,
"column": 28
} | {
"line": 1010,
"column": 29
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf : E → β\nh : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃... | [
"𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf : E → β\nh : ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜⦄, 0... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 454,
"column": 8
} | {
"line": 454,
"column": 32
} | {
"line": 454,
"column": 33
} | [
{
"pp": "case refine_2\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\n⊢ k ∙ (p₂ -ᵥ p₁) ≤ (s₁ ⊔ s₂).direction",
"ppTerm": "?refine_2",
"assigned": true,... | [
"case refine_2\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\n⊢ k ∙ (p₂ -ᵥ p₁) ≤ vectorSpan k ↑(s₁ ⊔ s₂)"
] | direction_eq_vectorSpan, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Function | {
"line": 1013,
"column": 26
} | {
"line": 1013,
"column": 28
} | {
"line": 1013,
"column": 29
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf : E → β\nh :\n ∀ ⦃x : E⦄,\n x ∈ s →\n ∀ ⦃y : ... | [
"𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf : E → β\nh :\n ∀ ⦃x : E⦄,\n x ∈ s →\n ∀ ⦃y : E⦄,\n ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Seminorm | {
"line": 350,
"column": 65
} | {
"line": 350,
"column": 80
} | {
"line": 350,
"column": 81
} | [
{
"pp": "case cons\n𝕜 : Type u_3\nE : Type u_7\nι : Type u_11\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : ι → Seminorm 𝕜 E\nx : E\na : ι\ns : Finset ι\nha : a ∉ s\nih : (s.sup p) x = ↑(s.sup fun i ↦ NNReal.mk ((p i) x) ⋯)\n⊢ max ((p a) x) ((s.sup p) x) = ↑(max (NNReal.mk ((p... | [
"case cons\n𝕜 : Type u_3\nE : Type u_7\nι : Type u_11\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : ι → Seminorm 𝕜 E\nx : E\na : ι\ns : Finset ι\nha : a ∉ s\nih : (s.sup p) x = ↑(s.sup fun i ↦ NNReal.mk ((p i) x) ⋯)\n⊢ max ((p a) x) ((s.sup p) x) = max ↑(NNReal.mk ((p a) x) ⋯) ↑(s... | NNReal.coe_max, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Function | {
"line": 1027,
"column": 30
} | {
"line": 1027,
"column": 32
} | {
"line": 1027,
"column": 33
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf : E → β\nh :\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈... | [
"𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf : E → β\nh :\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈ s → x ≠ y →... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Function | {
"line": 1031,
"column": 30
} | {
"line": 1031,
"column": 32
} | {
"line": 1031,
"column": 33
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf : E → β\nh :\n ∀ ⦃x : E⦄,\n x ∈ s →\n ∀ ⦃y : ... | [
"𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁷ : Field 𝕜\ninst✝⁶ : LinearOrder 𝕜\ninst✝⁵ : IsStrictOrderedRing 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : SMul 𝕜 E\ninst✝ : SMul 𝕜 β\ns : Set E\nf : E → β\nh :\n ∀ ⦃x : E⦄,\n x ∈ s →\n ∀ ⦃y : E⦄,\n ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Seminorm | {
"line": 424,
"column": 6
} | {
"line": 426,
"column": 21
} | {
"line": 426,
"column": 21
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np q : Seminorm 𝕜 E\nx : E\n⊢ 0 ∈ lowerBounds (range fun u ↦ p u + q (x - u))",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Seminorm.instSeminormClass",
"NormedCommRi... | [] | by
rintro _ ⟨x, rfl⟩
dsimp; positivity | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 529,
"column": 4
} | {
"line": 531,
"column": 40
} | {
"line": 532,
"column": 4
} | [
{
"pp": "case mp\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_4\nv : V\nx : k\ns : Set ι\np : ι → P\nb : P\n⊢ (∃ fs, ↑fs ⊆ s ∧ ∃ w, ∑ i ∈ fs, w i = x ∧ v = ∑ i ∈ fs, w i • (p i -ᵥ b)) →\n ∃ fs w, ∑ i ∈ fs, w i = x ∧ v... | [
"case mpr\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\nS : AffineSpace V P\nι : Type u_4\nv : V\nx : k\ns : Set ι\np : ι → P\nb : P\n⊢ (∃ fs w, ∑ i ∈ fs, w i = x ∧ v = ∑ i ∈ fs, w i • (p ↑i -ᵥ b)) →\n ∃ fs, ↑fs ⊆ s ∧ ∃ w, ∑ i ∈ fs, w i = x ∧ v = ∑ i ∈ f... | · rintro ⟨fs, hfs, w, rfl, rfl⟩
exact ⟨fs.subtype (· ∈ s), fun i => w i, sum_subtype_of_mem _ hfs,
(sum_subtype_of_mem _ hfs).symm⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.AffineSpace.Centroid | {
"line": 121,
"column": 55
} | {
"line": 122,
"column": 61
} | {
"line": 124,
"column": 0
} | [
{
"pp": "k : 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\nι : Type u_4\nι₂ : Type u_5\ns₂ : Finset ι₂\ne : ι₂ ↪ ι\np : ι → P\n⊢ centroid k (map e s₂) p = centroid k s₂ (p ∘ ⇑e)",
"ppTerm": "?m.25",
"assigned": true,... | [] | by
simp [centroid_def, affineCombination_map, centroidWeights] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Seminorm | {
"line": 852,
"column": 10
} | {
"line": 852,
"column": 12
} | {
"line": 852,
"column": 13
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_7\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nhr₂ : r₂ ≠ 0\na : 𝕜\n⊢ ‖a‖ < r₁ → ∀ (b : E), p b ≤ r₂ → ‖a‖ * p b < r₁ * r₂",
"ppTerm": "?m.52",
"assigned": true,
"usedConstants": [
"Norm.norm",
... | [
"𝕜 : Type u_3\nE : Type u_7\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nr₁ r₂ : ℝ\nhr₂ : r₂ ≠ 0\na : 𝕜\nha : ‖a‖ < r₁\n⊢ ∀ (b : E), p b ≤ r₂ → ‖a‖ * p b < r₁ * r₂"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Seminorm | {
"line": 867,
"column": 10
} | {
"line": 867,
"column": 12
} | {
"line": 867,
"column": 13
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_7\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nr₁ r₂ : ℝ\na : 𝕜\n⊢ ‖a‖ ≤ r₁ → ∀ (b : E), p b ≤ r₂ → ‖a‖ * p b ≤ r₁ * r₂",
"ppTerm": "?m.49",
"assigned": true,
"usedConstants": [
"Norm.norm",
"SeminormedA... | [
"𝕜 : Type u_3\nE : Type u_7\ninst✝² : SeminormedRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nr₁ r₂ : ℝ\na : 𝕜\nha : ‖a‖ ≤ r₁\n⊢ ∀ (b : E), p b ≤ r₂ → ‖a‖ * p b ≤ r₁ * r₂"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Seminorm | {
"line": 921,
"column": 2
} | {
"line": 921,
"column": 41
} | {
"line": 922,
"column": 2
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedDivisionRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nk : 𝕜\nr : ℝ\nx y : E\nhy : y ∈ p.closedBall 0 r\nh : (fun x ↦ k • x) y = x\n⊢ ‖k‖ * p y ≤ ‖k‖ * r",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"... | [
"𝕜 : Type u_3\nE : Type u_7\ninst✝² : NormedDivisionRing 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : Seminorm 𝕜 E\nk : 𝕜\nr : ℝ\nx y : E\nhy : p y ≤ r\nh : (fun x ↦ k • x) y = x\n⊢ ‖k‖ * p y ≤ ‖k‖ * r"
] | rw [Seminorm.mem_closedBall_zero] at hy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 963,
"column": 2
} | {
"line": 963,
"column": 47
} | {
"line": 964,
"column": 2
} | [
{
"pp": "k : Type u_2\nV : Type u_3\nP : Type u_4\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : Nontrivial k\ns : Set P\np : P\nhp : p ∈ s\nw : ↑s → kˣ\n⊢ affineSpan k (Set.range fun q ↦ (AffineMap.lineMap p ↑q) ↑(w q)) = affineSpan k s",
"ppTerm": "?m.33"... | [
"k : Type u_2\nV : Type u_3\nP : Type u_4\ninst✝⁴ : Ring k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : Nontrivial k\ns : Set P\np : P\nhp : p ∈ s\nw : ↑s → kˣ\nthis : s = Set.range Subtype.val\n⊢ affineSpan k (Set.range fun q ↦ (AffineMap.lineMap p ↑q) ↑(w q)) = affineSpan k s"
... | have : s = Set.range ((↑) : s → P) := by simp | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.AffineSpace.Basis | {
"line": 344,
"column": 53
} | {
"line": 344,
"column": 68
} | {
"line": 346,
"column": 0
} | [
{
"pp": "ι : Type u_1\nG : Type u_3\nk : Type u_5\nV : Type u_6\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Ring k\ninst✝³ : Module k V\ninst✝² : Group G\ninst✝¹ : DistribMulAction G V\ninst✝ : SMulCommClass G k V\na : G\nb : AffineBasis ι k V\ni : ι\nj : { j // j ≠ i }\n⊢ ((a • b).basisOf i) j = (a • b.basisOf i) j",
... | [] | simp [smul_sub] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.AffineSpace.Basis | {
"line": 352,
"column": 9
} | {
"line": 352,
"column": 30
} | {
"line": 354,
"column": 0
} | [
{
"pp": "ι : Type u_1\nG : Type u_3\nk : Type u_5\nV : Type u_6\ninst✝⁵ : AddCommGroup V\ninst✝⁴ : Ring k\ninst✝³ : Module k V\ninst✝² : Group G\ninst✝¹ : DistribMulAction G V\ninst✝ : SMulCommClass G k V\na : G\nb : AffineBasis ι k V\ni : ι\nv : V\n⊢ ((a • b).coord i) v = ((b.coord i).comp (↑(DistribMulAction.... | [] | simp [map_sub, coord] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Basic | {
"line": 89,
"column": 2
} | {
"line": 89,
"column": 10
} | {
"line": 90,
"column": 2
} | [
{
"pp": "case mk\nk : Type u_1\nV : Type u_2\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nn : ℕ\ns2 : Simplex k P n\npoints✝ : Fin (n + 1) → P\nindependent✝ : AffineIndependent k points✝\nh : ∀ (i : Fin (n + 1)), { points := points✝, independent := indep... | [
"case mk.mk\nk : Type u_1\nV : Type u_2\nP : Type u_5\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nn : ℕ\npoints✝¹ : Fin (n + 1) → P\nindependent✝¹ : AffineIndependent k points✝¹\npoints✝ : Fin (n + 1) → P\nindependent✝ : AffineIndependent k points✝\nh :\n ∀ (i : Fin (n ... | cases s2 | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 487,
"column": 69
} | {
"line": 487,
"column": 71
} | {
"line": 487,
"column": 71
} | [
{
"pp": "case mp.refine_2\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\ns₁ s₂ : Set ι\nfs₁ : Finset ι\nhfs₁ : ↑fs₁ ⊆ s₁\nw₁ : ι → k\nhw₁ : ∑ i ∈ fs₁, w₁ i = 1\nfs₂ : Finset ι\nhfs... | [
"case mp.refine_2\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\ns₁ s₂ : Set ι\nfs₁ : Finset ι\nhfs₁ : ↑fs₁ ⊆ s₁\nw₁ : ι → k\nhw₁ : ∑ i ∈ fs₁, w₁ i = 1\nfs₂ : Finset ι\nhfs₂ : ↑fs₂ ⊆ s... | ha | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.Affine | {
"line": 35,
"column": 86
} | {
"line": 43,
"column": 77
} | {
"line": 45,
"column": 0
} | [
{
"pp": "R : Type u_1\nV : Type u_2\nP : Type u_3\nW : Type u_4\nQ : Type u_5\ninst✝¹² : AddCommGroup V\ninst✝¹¹ : TopologicalSpace V\ninst✝¹⁰ : AddTorsor V P\ninst✝⁹ : TopologicalSpace P\ninst✝⁸ : IsTopologicalAddTorsor P\ninst✝⁷ : AddCommGroup W\ninst✝⁶ : TopologicalSpace W\ninst✝⁵ : AddTorsor W Q\ninst✝⁴ : T... | [] | by
inhabit P
have :
(f.linear : V → W) =
(Homeomorph.vaddConst <| f default).symm ∘ f ∘ (Homeomorph.vaddConst default) := by
ext v
simp
rw [this]
simp only [Homeomorph.comp_continuous_iff, Homeomorph.comp_continuous_iff'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Combination | {
"line": 131,
"column": 67
} | {
"line": 132,
"column": 76
} | {
"line": 134,
"column": 0
} | [
{
"pp": "R : Type u_1\nE : Type u_3\nι : Type u_5\ninst✝³ : Field R\ninst✝² : AddCommGroup E\ninst✝¹ : Module R E\nt : Finset ι\nw : ι → R\nz : ι → E\ninst✝ : (i : ι) → Decidable (w i ≠ 0)\ni : ι\nhit : i ∈ t\nhit' : i ∉ {i ∈ t | w i ≠ 0}\n⊢ w i = 0",
"ppTerm": "?m.46",
"assigned": true,
"usedConsta... | [] | by
simpa only [hit, mem_filter, true_and, Ne, Classical.not_not] using hit' | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Combination | {
"line": 241,
"column": 24
} | {
"line": 241,
"column": 26
} | {
"line": 241,
"column": 27
} | [
{
"pp": "R : Type u_1\nE : Type u_3\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ns : Set E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nh : ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i ∈ t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x ∈ t, w x • x ∈ s\nx : E\nhx : x ∈ s\ny : E\nhy :... | [
"R : Type u_1\nE : Type u_3\ninst✝⁴ : Field R\ninst✝³ : AddCommGroup E\ninst✝² : Module R E\ns : Set E\ninst✝¹ : LinearOrder R\ninst✝ : IsStrictOrderedRing R\nh : ∀ (t : Finset E) (w : E → R), (∀ i ∈ t, 0 ≤ w i) → ∑ i ∈ t, w i = 1 → (∀ x ∈ t, x ∈ s) → ∑ x ∈ t, w x • x ∈ s\nx : E\nhx : x ∈ s\ny : E\nhy : y ∈ s\na b ... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.AlexandrovDiscrete | {
"line": 48,
"column": 89
} | {
"line": 51,
"column": 57
} | {
"line": 53,
"column": 0
} | [
{
"pp": "α : Type u_3\ninst✝ : TopologicalSpace α\n⊢ AlexandrovDiscrete α ↔ ∀ (S : Set (Set α)), (∀ s ∈ S, IsClosed[inst✝] s) → IsClosed[inst✝] (⋃₀ S)",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"BooleanAlgebra",
"Function.Surjective.forall",
"_private... | [] | by
conv_lhs => tactic =>
simp_rw +singlePass [alexandrovDiscrete_iff, compl_surjective.image_surjective.forall,
forall_mem_image, ← compl_sUnion, isOpen_compl_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.AlexandrovDiscrete | {
"line": 195,
"column": 17
} | {
"line": 195,
"column": 19
} | {
"line": 196,
"column": 4
} | [
{
"pp": "α : Type u_3\ninst✝ : TopologicalSpace α\nhα : ∀ (a : α), 𝓝 a = 𝓟 (nhdsKer {a})\nS : Set (Set α)\nhS : ∀ s ∈ S, ∀ (a : α), (nhdsKer {a} ∩ s).Nonempty → a ∈ s\na : α\n⊢ (nhdsKer {a} ∩ ⋃₀ S).Nonempty → a ∈ ⋃₀ S",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Set.sUnion",
... | [
"α : Type u_3\ninst✝ : TopologicalSpace α\nhα : ∀ (a : α), 𝓝 a = 𝓟 (nhdsKer {a})\nS : Set (Set α)\nhS : ∀ s ∈ S, ∀ (a : α), (nhdsKer {a} ∩ s).Nonempty → a ∈ s\na : α\nha : (nhdsKer {a} ∩ ⋃₀ S).Nonempty\n⊢ a ∈ ⋃₀ S"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Analysis.Convex.Topology | {
"line": 374,
"column": 2
} | {
"line": 381,
"column": 29
} | {
"line": 383,
"column": 0
} | [
{
"pp": "E : Type u_3\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : x ∈ interior s\nt : ℝ\nht : 1 < t\n⊢ closure s ⊆ ⇑(homothety x t) '' interior s",
"ppTerm": "?m.38",
"as... | [] | intro y hy
have hne : t ≠ 0 := (one_pos.trans ht).ne'
refine
⟨homothety x t⁻¹ y, hs.openSegment_interior_closure_subset_interior hx hy ?_,
(AffineEquiv.homothetyUnitsMulHom x (Units.mk0 t hne)).apply_symm_apply y⟩
rw [openSegment_eq_image_lineMap, ← inv_one, ← inv_Ioi₀ (zero_lt_one' ℝ), ← image_inv_eq_i... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Topology | {
"line": 374,
"column": 2
} | {
"line": 381,
"column": 29
} | {
"line": 383,
"column": 0
} | [
{
"pp": "E : Type u_3\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : IsTopologicalAddGroup E\ninst✝ : ContinuousSMul ℝ E\ns : Set E\nhs : Convex ℝ s\nx : E\nhx : x ∈ interior s\nt : ℝ\nht : 1 < t\n⊢ closure s ⊆ ⇑(homothety x t) '' interior s",
"ppTerm": "?m.38",
"as... | [] | intro y hy
have hne : t ≠ 0 := (one_pos.trans ht).ne'
refine
⟨homothety x t⁻¹ y, hs.openSegment_interior_closure_subset_interior hx hy ?_,
(AffineEquiv.homothetyUnitsMulHom x (Units.mk0 t hne)).apply_symm_apply y⟩
rw [openSegment_eq_image_lineMap, ← inv_one, ← inv_Ioi₀ (zero_lt_one' ℝ), ← image_inv_eq_i... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Connected.LocallyPathConnected | {
"line": 283,
"column": 6
} | {
"line": 283,
"column": 83
} | {
"line": 284,
"column": 6
} | [
{
"pp": "case inr.refine_1.hs\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\nx y✝ z : X\nι : Type u_3\nF : Set X\ninst✝¹ : LocallyPathConnectedSpace X\ninst✝ : LocallyPathConnectedSpace Y\nu : Set (X ⊕ Y)\nhu : IsOpen[instTopologicalSpaceSum] u\ny : Y\nhxu : inr y ∈ u\n⊢ ... | [
"case hxs\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\nx y✝ z : X\nι : Type u_3\nF : Set X\ninst✝¹ : LocallyPathConnectedSpace X\ninst✝ : LocallyPathConnectedSpace Y\nu : Set (X ⊕ Y)\nhu : IsOpen[instTopologicalSpaceSum] u\ny : Y\nhxu : inr y ∈ u\n⊢ inr y ∈ inr '' pathCompo... | · exact (isPathConnected_pathComponentIn (by exact hxu)).image continuous_inr | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Connected.LocallyPathConnected | {
"line": 302,
"column": 4
} | {
"line": 302,
"column": 83
} | {
"line": 303,
"column": 2
} | [
{
"pp": "case refine_2\nX✝ : Type u_1\nY : Type u_2\ninst✝⁴ : TopologicalSpace X✝\ninst✝³ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type u_3\nF : Set X✝\ninst✝² : LocallyPathConnectedSpace X✝\nX : ι → Type u_4\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\ninst✝ : ∀ (i : ι), LocallyPathConnectedSpace (X i)\nx : (i : ... | [] | exact isOpenMap_sigmaMk _ <| (hu.preimage continuous_sigmaMk).pathComponentIn _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Connected.LocallyPathConnected | {
"line": 302,
"column": 4
} | {
"line": 302,
"column": 83
} | {
"line": 303,
"column": 2
} | [
{
"pp": "case refine_2\nX✝ : Type u_1\nY : Type u_2\ninst✝⁴ : TopologicalSpace X✝\ninst✝³ : TopologicalSpace Y\nx✝ y z : X✝\nι : Type u_3\nF : Set X✝\ninst✝² : LocallyPathConnectedSpace X✝\nX : ι → Type u_4\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\ninst✝ : ∀ (i : ι), LocallyPathConnectedSpace (X i)\nx : (i : ... | [] | exact isOpenMap_sigmaMk _ <| (hu.preimage continuous_sigmaMk).pathComponentIn _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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