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 371
values | kind stringclasses 375
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 148,
"column": 46
} | {
"line": 148,
"column": 64
} | {
"line": 150,
"column": 0
} | [
{
"pp": "x : ℝ\n⊢ x ^ 1 = x",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real",
"congrArg",
"Complex.instPow",
"Complex.ofReal",
"Complex.re",
"Real.instOne",
"Complex.cpow_one",
"HPow.hPow",
"True",
"eq_s... | [] | by simp [rpow_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 154,
"column": 46
} | {
"line": 154,
"column": 64
} | {
"line": 156,
"column": 0
} | [
{
"pp": "x : ℝ\n⊢ 1 ^ x = 1",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Real.instPow",
"Real",
"congrArg",
"Complex.instPow",
"Complex.ofReal",
"Complex.re",
"Real.instOne",
"HPow.hPow",
"True",
"Complex.one_cpow",
"eq_... | [] | by simp [rpow_def] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 536,
"column": 2
} | {
"line": 536,
"column": 47
} | {
"line": 538,
"column": 0
} | [
{
"pp": "⊢ π / 2 ≠ 0",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"GroupWithZero.toMonoidWithZero",
"Real.partialOrder",
"Real",
"Real.pi",
"CharZero.NeZero.two",
"FloorRing.toFloorSemiring",
"Nat.instAtLeastTwoHAddOfNat",
"AddGroupWithOne... | [] | exact div_ne_zero Real.pi_ne_zero two_ne_zero | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 303,
"column": 9
} | {
"line": 303,
"column": 21
} | {
"line": 303,
"column": 21
} | [
{
"pp": "case inr\nx : ℂ\ny : ℝ\nhx : x ≠ 0\n⊢ 0 < ‖x‖",
"ppTerm": "?inr✝",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"Eq.mpr",
"norm_pos_iff",
"Real",
"Complex.instNormedAddCommGroup",
"Real.instZero",
"congrArg... | [
"case inr\nx : ℂ\ny : ℝ\nhx : x ≠ 0\n⊢ x ≠ 0"
] | norm_pos_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 365,
"column": 4
} | {
"line": 365,
"column": 81
} | {
"line": 366,
"column": 2
} | [
{
"pp": "case h₁\nx : ℝ\nhx : 0 ≤ x\ny : ℝ\nz : ℂ\n⊢ -π < (log ↑x * ↑y).im",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Complex.log",
"Real.partialOrder",
"Real",
"Real.pi",
"H... | [] | rw [← ofReal_log hx, ← ofReal_mul, ofReal_im, neg_lt_zero]; exact Real.pi_pos | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 365,
"column": 4
} | {
"line": 365,
"column": 81
} | {
"line": 366,
"column": 2
} | [
{
"pp": "case h₁\nx : ℝ\nhx : 0 ≤ x\ny : ℝ\nz : ℂ\n⊢ -π < (log ↑x * ↑y).im",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"Complex.log",
"Real.partialOrder",
"Real",
"Real.pi",
"H... | [] | rw [← ofReal_log hx, ← ofReal_mul, ofReal_im, neg_lt_zero]; exact Real.pi_pos | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 798,
"column": 2
} | {
"line": 798,
"column": 33
} | {
"line": 800,
"column": 0
} | [
{
"pp": "θ ψ : Angle\n⊢ θ = ψ ↔ θ.sign = ψ.sign ∧ |θ.toReal| = |ψ.toReal|",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle.0.Real.Angle.eq_iff_sign_eq_and_abs_toReal_eq._proof_1_1"
],
"usedFVars": [
"θ",
... | [] | grind [toReal_neg_iff_sign_neg] | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 798,
"column": 2
} | {
"line": 798,
"column": 33
} | {
"line": 800,
"column": 0
} | [
{
"pp": "θ ψ : Angle\n⊢ θ = ψ ↔ θ.sign = ψ.sign ∧ |θ.toReal| = |ψ.toReal|",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle.0.Real.Angle.eq_iff_sign_eq_and_abs_toReal_eq._proof_1_1"
],
"usedFVars": [
"θ",
... | [] | grind [toReal_neg_iff_sign_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle | {
"line": 798,
"column": 2
} | {
"line": 798,
"column": 33
} | {
"line": 800,
"column": 0
} | [
{
"pp": "θ ψ : Angle\n⊢ θ = ψ ↔ θ.sign = ψ.sign ∧ |θ.toReal| = |ψ.toReal|",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.SpecialFunctions.Trigonometric.Angle.0.Real.Angle.eq_iff_sign_eq_and_abs_toReal_eq._proof_1_1"
],
"usedFVars": [
"θ",
... | [] | grind [toReal_neg_iff_sign_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 479,
"column": 2
} | {
"line": 479,
"column": 22
} | {
"line": 481,
"column": 0
} | [
{
"pp": "case neg.hx\nx y z : ℝ\nhx : 0 ≤ x\nhy : 0 ≤ y\nh_ifs : ¬x = 0 ∧ ¬y = 0\n⊢ 0 ≤ x",
"ppTerm": "?neg.hx✝",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hx"
],
"usedGoals": []
},
{
"pp": "case hx\nx y z : ℝ\nhx : 0 ≤ x\nhy : 0 ≤ y\n⊢ 0 ≤ x * y",
"ppTerm"... | [] | all_goals positivity | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 559,
"column": 2
} | {
"line": 563,
"column": 98
} | {
"line": 565,
"column": 0
} | [
{
"pp": "x : ℂ\nhx_re : x.re < 0\nhx_im : 0 < x.im\n⊢ arg =ᶠ[𝓝 x] fun x ↦ Real.arcsin ((-x).im / ‖x‖) + π",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHDiv",
"Real.pi",
"Real.lattice",
... | [] | suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im from
h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le
refine IsOpen.eventually_mem ?_ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im)
exact
IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_i... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.Arg | {
"line": 559,
"column": 2
} | {
"line": 563,
"column": 98
} | {
"line": 565,
"column": 0
} | [
{
"pp": "x : ℂ\nhx_re : x.re < 0\nhx_im : 0 < x.im\n⊢ arg =ᶠ[𝓝 x] fun x ↦ Real.arcsin ((-x).im / ‖x‖) + π",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Norm.norm",
"NormedCommRing.toSeminormedCommRing",
"Real",
"instHDiv",
"Real.pi",
"Real.lattice",
... | [] | suffices h_forall_nhds : ∀ᶠ y : ℂ in 𝓝 x, y.re < 0 ∧ 0 < y.im from
h_forall_nhds.mono fun y hy => arg_of_re_neg_of_im_nonneg hy.1 hy.2.le
refine IsOpen.eventually_mem ?_ (⟨hx_re, hx_im⟩ : x.re < 0 ∧ 0 < x.im)
exact
IsOpen.and (isOpen_lt continuous_re continuous_zero) (isOpen_lt continuous_zero continuous_i... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 559,
"column": 2
} | {
"line": 559,
"column": 22
} | {
"line": 561,
"column": 0
} | [
{
"pp": "x y z : ℝ\nhx : 0 < x\nhxy : x < y\nhz : z < 0\nthis : 0 < y\n⊢ 0 ≤ x",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real",
"Real.instZero",
"PartialOrder.toPreorder",
"le_of_lt",
"Zero.toOfNat0",
"OfNat.ofNat"
]... | [] | all_goals positivity | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 565,
"column": 2
} | {
"line": 565,
"column": 22
} | {
"line": 567,
"column": 0
} | [
{
"pp": "x y z : ℝ\nhx : 0 < x\nhxy : x ≤ y\nhz : z ≤ 0\nthis : 0 < y\n⊢ 0 ≤ x",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Real.partialOrder",
"Real",
"Real.instZero",
"PartialOrder.toPreorder",
"le_of_lt",
"Zero.toOfNat0",
"OfNat.ofNat"
]... | [] | all_goals positivity | Lean.Elab.Tactic.evalAllGoals | Lean.Parser.Tactic.allGoals |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 699,
"column": 2
} | {
"line": 699,
"column": 63
} | {
"line": 701,
"column": 0
} | [
{
"pp": "x y : ℝ\nhx : 0 ≤ x\nhy : 0 < y\n⊢ x ^ y < 1 ↔ x < 1",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"Real.instZero",
"Real.instZeroLEOneClass",
"congrArg",
"Iff.rfl",
"Real.inst... | [] | rw [← Real.rpow_lt_rpow_iff hx zero_le_one hy, Real.one_rpow] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 699,
"column": 2
} | {
"line": 699,
"column": 63
} | {
"line": 701,
"column": 0
} | [
{
"pp": "x y : ℝ\nhx : 0 ≤ x\nhy : 0 < y\n⊢ x ^ y < 1 ↔ x < 1",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"Real.instZero",
"Real.instZeroLEOneClass",
"congrArg",
"Iff.rfl",
"Real.inst... | [] | rw [← Real.rpow_lt_rpow_iff hx zero_le_one hy, Real.one_rpow] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.Real | {
"line": 699,
"column": 2
} | {
"line": 699,
"column": 63
} | {
"line": 701,
"column": 0
} | [
{
"pp": "x y : ℝ\nhx : 0 ≤ x\nhy : 0 < y\n⊢ x ^ y < 1 ↔ x < 1",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Real.instLE",
"Real",
"Real.instZero",
"Real.instZeroLEOneClass",
"congrArg",
"Iff.rfl",
"Real.inst... | [] | rw [← Real.rpow_lt_rpow_iff hx zero_le_one hy, Real.one_rpow] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.Continuity | {
"line": 343,
"column": 2
} | {
"line": 343,
"column": 53
} | {
"line": 344,
"column": 2
} | [
{
"pp": "x : ℝ\ny : ℂ\nh : 0 < y.re ∨ x ≠ 0\n⊢ ContinuousAt (fun p ↦ ↑p.1 ^ p.2) (x, y)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"NormedCommRing.toSeminormedCommRing",
"Real",
"Preorder.toLT",
"Real.instZero",
"ContinuousAt",
"PartialOrder.toPreor... | [
"case inl\nx : ℝ\ny : ℂ\nh : 0 < y.re ∨ x ≠ 0\nhx : 0 < x\n⊢ ContinuousAt (fun p ↦ ↑p.1 ^ p.2) (x, y)",
"case inr.inl\ny : ℂ\nh : 0 < y.re ∨ 0 ≠ 0\n⊢ ContinuousAt (fun p ↦ ↑p.1 ^ p.2) (0, y)",
"case inr.inr\nx : ℝ\ny : ℂ\nh : 0 < y.re ∨ x ≠ 0\nhx : x < 0\n⊢ ContinuousAt (fun p ↦ ↑p.1 ^ p.2) (x, y)"
] | rcases lt_trichotomy (0 : ℝ) x with (hx | rfl | hx) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 546,
"column": 2
} | {
"line": 550,
"column": 34
} | {
"line": 552,
"column": 0
} | [
{
"pp": "x : ℝ≥0\ny : ℝ\nh : 0 ≤ y\n⊢ ↑(x ^ y) = ↑x ^ y",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"le_iff_eq_or_lt",
"NNReal.zero_rpow",
"ENNReal.zero_rpow_of_pos",
"Real.partialOrder",
"Real",
"ENNReal.ofNNReal",
"Preorder.toLT",
"Line... | [] | by_cases hx : x = 0
· rcases le_iff_eq_or_lt.1 h with (H | H)
· simp [hx, H.symm]
· simp [hx, zero_rpow_of_pos H, NNReal.zero_rpow (ne_of_gt H)]
· exact coe_rpow_of_ne_zero hx _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 546,
"column": 2
} | {
"line": 550,
"column": 34
} | {
"line": 552,
"column": 0
} | [
{
"pp": "x : ℝ≥0\ny : ℝ\nh : 0 ≤ y\n⊢ ↑(x ^ y) = ↑x ^ y",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"le_iff_eq_or_lt",
"NNReal.zero_rpow",
"ENNReal.zero_rpow_of_pos",
"Real.partialOrder",
"Real",
"ENNReal.ofNNReal",
"Preorder.toLT",
"Line... | [] | by_cases hx : x = 0
· rcases le_iff_eq_or_lt.1 h with (H | H)
· simp [hx, H.symm]
· simp [hx, zero_rpow_of_pos H, NNReal.zero_rpow (ne_of_gt H)]
· exact coe_rpow_of_ne_zero hx _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 710,
"column": 47
} | {
"line": 710,
"column": 66
} | {
"line": 711,
"column": 2
} | [
{
"pp": "case inl.top.inl\nz : ℝ\nhxy : 0 ≤ ∞\nhz : z < 0\n⊢ (0 * ∞) ^ z = if (0 = 0 ∧ ∞ = ∞ ∨ 0 = ∞ ∧ ∞ = 0) ∧ z < 0 then ∞ else 0 ^ z * ∞ ^ z",
"ppTerm": "?inl.top.inl",
"assigned": true,
"usedConstants": [
"False",
"Real",
"Preorder.toLT",
"HMul.hMul",
"ENNReal.top_n... | [] | simp [*, hz.not_gt] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 710,
"column": 47
} | {
"line": 710,
"column": 66
} | {
"line": 711,
"column": 2
} | [
{
"pp": "case inl.top.inr\nz : ℝ\nhxy : 0 ≤ ∞\nhz : 0 < z\n⊢ (0 * ∞) ^ z = if (0 = 0 ∧ ∞ = ∞ ∨ 0 = ∞ ∧ ∞ = 0) ∧ z < 0 then ∞ else 0 ^ z * ∞ ^ z",
"ppTerm": "?inl.top.inr",
"assigned": true,
"usedConstants": [
"False",
"ENNReal.zero_rpow_of_pos",
"Real",
"Preorder.toLT",
... | [] | simp [*, hz.not_gt] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 710,
"column": 47
} | {
"line": 710,
"column": 66
} | {
"line": 711,
"column": 2
} | [
{
"pp": "case inl.coe.inl\nz : ℝ\nx✝ : ℝ≥0\nhxy : 0 ≤ ↑x✝\nhz : z < 0\n⊢ (0 * ↑x✝) ^ z = if (0 = 0 ∧ ↑x✝ = ∞ ∨ 0 = ∞ ∧ ↑x✝ = 0) ∧ z < 0 then ∞ else 0 ^ z * ↑x✝ ^ z",
"ppTerm": "?inl.coe.inl",
"assigned": true,
"usedConstants": [
"ENNReal.coe_ne_top._simp_1",
"ENNReal.rpow_eq_zero_iff._si... | [] | simp [*, hz.not_gt] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 710,
"column": 47
} | {
"line": 710,
"column": 66
} | {
"line": 711,
"column": 2
} | [
{
"pp": "case inl.coe.inr\nz : ℝ\nx✝ : ℝ≥0\nhxy : 0 ≤ ↑x✝\nhz : 0 < z\n⊢ (0 * ↑x✝) ^ z = if (0 = 0 ∧ ↑x✝ = ∞ ∨ 0 = ∞ ∧ ↑x✝ = 0) ∧ z < 0 then ∞ else 0 ^ z * ↑x✝ ^ z",
"ppTerm": "?inl.coe.inr",
"assigned": true,
"usedConstants": [
"ENNReal.coe_ne_top._simp_1",
"False",
"ENNReal.zero_... | [] | simp [*, hz.not_gt] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.AffineSpace.Midpoint | {
"line": 215,
"column": 22
} | {
"line": 215,
"column": 36
} | {
"line": 215,
"column": 37
} | [
{
"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.33",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instVAddOfAdd",
"congrArg",
"AddCommGroup.... | [
"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 +ᵥ -y) (x + y) = x"
] | ← vadd_eq_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Midpoint | {
"line": 215,
"column": 37
} | {
"line": 215,
"column": 51
} | {
"line": 215,
"column": 52
} | [
{
"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 +ᵥ -y) (x + y) = x",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instVAddOfAdd",
"congrArg",
"AddCommGroup... | [
"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 +ᵥ -y) (x +ᵥ y) = x"
] | ← vadd_eq_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 936,
"column": 2
} | {
"line": 939,
"column": 72
} | {
"line": 941,
"column": 0
} | [
{
"pp": "x : ℝ≥0∞\nz : ℝ\nhx1 : 0 < x\nhx2 : x ≤ 1\nhz : z < 0\n⊢ 1 ≤ x ^ z",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.SpecialFunctions.Pow.NNReal.0.ENNReal.one_le_rpow_of_pos_of_le_one_of_neg._simp_1_1",
"ENNReal.one_le_coe_iff._simp_1",
"... | [] | lift x to ℝ≥0 using ne_of_lt (lt_of_le_of_lt hx2 coe_lt_top)
simp only [coe_le_one_iff, coe_pos] at hx1 hx2 ⊢
simp [← coe_rpow_of_ne_zero (ne_of_gt hx1),
NNReal.one_le_rpow_of_pos_of_le_one_of_nonpos hx1 hx2 (le_of_lt hz)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Pow.NNReal | {
"line": 936,
"column": 2
} | {
"line": 939,
"column": 72
} | {
"line": 941,
"column": 0
} | [
{
"pp": "x : ℝ≥0∞\nz : ℝ\nhx1 : 0 < x\nhx2 : x ≤ 1\nhz : z < 0\n⊢ 1 ≤ x ^ z",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.SpecialFunctions.Pow.NNReal.0.ENNReal.one_le_rpow_of_pos_of_le_one_of_neg._simp_1_1",
"ENNReal.one_le_coe_iff._simp_1",
"... | [] | lift x to ℝ≥0 using ne_of_lt (lt_of_le_of_lt hx2 coe_lt_top)
simp only [coe_le_one_iff, coe_pos] at hx1 hx2 ⊢
simp [← coe_rpow_of_ne_zero (ne_of_gt hx1),
NNReal.one_le_rpow_of_pos_of_le_one_of_nonpos hx1 hx2 (le_of_lt hz)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Segment | {
"line": 267,
"column": 11
} | {
"line": 267,
"column": 25
} | {
"line": 267,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddRightMono 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\na x b c : E\n⊢ a + x ∈ [a + b -[𝕜] a + c] ↔ x ∈ [b -[𝕜] c]",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"DistribMulAction.toD... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddRightMono 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\na x b c : E\n⊢ a +ᵥ x ∈ [a +ᵥ b -[𝕜] a +ᵥ c] ↔ x ∈ [b -[𝕜] c]"
] | ← vadd_eq_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Convex.Segment | {
"line": 272,
"column": 11
} | {
"line": 272,
"column": 25
} | {
"line": 272,
"column": 26
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddRightMono 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\na x b c : E\n⊢ a + x ∈ openSegment 𝕜 (a + b) (a + c) ↔ x ∈ openSegment 𝕜 b c",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Di... | [
"𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : Ring 𝕜\ninst✝³ : PartialOrder 𝕜\ninst✝² : AddRightMono 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\na x b c : E\n⊢ a +ᵥ x ∈ openSegment 𝕜 (a +ᵥ b) (a +ᵥ c) ↔ x ∈ openSegment 𝕜 b c"
] | ← vadd_eq_add, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Analysis.Convex.Segment | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 23
} | {
"line": 349,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁵ : Ring 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\ninst✝ : Invertible 2\nx y : E\n⊢ x = midpoint 𝕜 (x - y) (x + y)",
"ppTerm": "?m.98",
"assigned": true,
"usedConstants": [
"Eq.mpr",
... | [] | rw [midpoint_sub_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Segment | {
"line": 552,
"column": 2
} | {
"line": 552,
"column": 84
} | {
"line": 554,
"column": 0
} | [
{
"pp": "case inr\n𝕜 : Type u_1\ninst✝² : Field 𝕜\ninst✝¹ : LinearOrder 𝕜\ninst✝ : IsStrictOrderedRing 𝕜\nx y : 𝕜\nhxy : x ≠ y\nh : y < x\n⊢ openSegment 𝕜 x y = Ioo (min x y) (max x y)",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
... | [] | · rw [openSegment_symm, openSegment_eq_Ioo h, max_eq_left h.le, min_eq_right h.le] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 334,
"column": 4
} | {
"line": 334,
"column": 40
} | {
"line": 336,
"column": 0
} | [
{
"pp": "case p\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p ∈ s\nv : V\nx✝ : P\n| x✝ ∈ ↑s ∧ x✝ -ᵥ p = -v",
"ppTerm": "?p",
"assigned": true,
"usedConstants": [
"NegZeroC... | [
"case p\nk : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : Ring k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\ns : AffineSubspace k P\np : P\nhp : p ∈ s\nv : V\nx✝ : P\n| x✝ ∈ ↑s ∧ p -ᵥ x✝ = v"
] | rw [← neg_vsub_eq_vsub_rev, neg_inj] | Lean.Parser.Tactic.Conv._aux_Init_Conv___macroRules_Lean_Parser_Tactic_Conv_convRw___1 | Lean.Parser.Tactic.Conv.convRw__ |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 677,
"column": 2
} | {
"line": 678,
"column": 54
} | {
"line": 679,
"column": 2
} | [
{
"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\np : P\nv : V\n_hv : v ∈ ⊤\n⊢ v ∈ ⊤.direction",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"Submodule",
"Lattice.toSemilatticeSup",
"Add... | [
"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\np : P\nv : V\n_hv : v ∈ ⊤\nhpv : (v +ᵥ p) -ᵥ p ∈ ⊤.direction\n⊢ v ∈ ⊤.direction"
] | have hpv : ((v +ᵥ p) -ᵥ p : V) ∈ (⊤ : AffineSubspace k P).direction :=
vsub_mem_direction (mem_top k V _) (mem_top k V _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Defs | {
"line": 1124,
"column": 6
} | {
"line": 1124,
"column": 28
} | {
"line": 1124,
"column": 29
} | [
{
"pp": "k : Type u_1\nV : Type u_2\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\ns : Set V\n⊢ ↑(Submodule.span k (insert 0 s)) ⊆ ↑(affineSpan k (insert 0 s))",
"ppTerm": "?m.39",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Submodule",
"vectorSpan",
"cong... | [
"k : Type u_1\nV : Type u_2\ninst✝² : Ring k\ninst✝¹ : AddCommGroup V\ninst✝ : Module k V\ns : Set V\n⊢ ↑(Submodule.span k (insert 0 s)) ⊆ ↑(vectorSpan k (insert 0 s)) + insert 0 s"
] | ← vectorSpan_add_self, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Basic | {
"line": 648,
"column": 8
} | {
"line": 648,
"column": 14
} | {
"line": 648,
"column": 15
} | [
{
"pp": "case e'_6\nR : 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 ... | [
"case e'_6\nR : 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 : Ic... | ← hd2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Basic | {
"line": 649,
"column": 13
} | {
"line": 649,
"column": 19
} | {
"line": 649,
"column": 20
} | [
{
"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... | ← hd2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.BalancedCoreHull | {
"line": 230,
"column": 8
} | {
"line": 230,
"column": 20
} | {
"line": 230,
"column": 20
} | [
{
"pp": "case pos\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : IsClosed[inst✝¹] U\nh : 0 ∈ U\na : 𝕜\nha : 1 ≤ ‖a‖\nha' : 0 < ‖a‖\n⊢ IsClosed[inst✝¹] (a • U)",
"ppTerm": ... | [
"case pos\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁴ : NormedDivisionRing 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\ninst✝¹ : TopologicalSpace E\ninst✝ : ContinuousSMul 𝕜 E\nU : Set E\nhU : IsClosed[inst✝¹] U\nh : 0 ∈ U\na : 𝕜\nha : 1 ≤ ‖a‖\nha' : a ≠ 0\n⊢ IsClosed[inst✝¹] (a • U)"
] | norm_pos_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.Basic | {
"line": 281,
"column": 2
} | {
"line": 281,
"column": 29
} | {
"line": 282,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\nS : Type u_7\ninst✝¹ : SetLike S E\ninst✝ : SMulMemClass S 𝕜 E\nV : S\nhV : Absorbent 𝕜 ↑V\nx : E\nc : 𝕜\nhc : c • x ∈ ↑V\nhc' : c ∈ {0}ᶜ\n⊢ x ∈ ↑V",
"ppTerm": "?m.51",
"assigned"... | [
"𝕜 : Type u_1\nE : Type u_3\ninst✝⁴ : NontriviallyNormedField 𝕜\ninst✝³ : AddCommGroup E\ninst✝² : Module 𝕜 E\nS : Type u_7\ninst✝¹ : SetLike S E\ninst✝ : SMulMemClass S 𝕜 E\nV : S\nhV : Absorbent 𝕜 ↑V\nx : E\nc : 𝕜\nhc : c • x ∈ ↑V\nhc' : c ∈ {0}ᶜ\n⊢ c⁻¹ • c • x ∈ ↑V"
] | rw [← inv_smul_smul₀ hc' x] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Function | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 54
} | {
"line": 257,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁸ : Semiring 𝕜\ninst✝⁷ : PartialOrder 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : PosSMulMono 𝕜 β\ns : Set E\nf : E → β\nhf : ConvexOn ... | [
"𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁸ : Semiring 𝕜\ninst✝⁷ : PartialOrder 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid β\ninst✝⁴ : PartialOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : PosSMulMono 𝕜 β\ns : Set E\nf : E → β\nhf : ConvexOn 𝕜 s f\nx : ... | rintro ⟨x, r⟩ ⟨hx, hr⟩ ⟨y, t⟩ ⟨hy, ht⟩ a b ha hb hab | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Analysis.Convex.Function | {
"line": 337,
"column": 2
} | {
"line": 337,
"column": 30
} | {
"line": 338,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf : E → β\n⊢ ConvexOn 𝕜 s f ↔\n Convex 𝕜 s ∧ s.Pairwise fun x y ↦ ∀ ⦃a b : �... | [
"𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁶ : Semiring 𝕜\ninst✝⁵ : PartialOrder 𝕜\ninst✝⁴ : AddCommMonoid E\ninst✝³ : AddCommMonoid β\ninst✝² : PartialOrder β\ninst✝¹ : Module 𝕜 E\ninst✝ : Module 𝕜 β\ns : Set E\nf : E → β\n⊢ (Convex 𝕜 s ∧\n ∀ ⦃x : E⦄,\n x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ∀ ⦃a b : 𝕜... | rw [convexOn_iff_forall_pos] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Convex.Function | {
"line": 611,
"column": 53
} | {
"line": 614,
"column": 26
} | {
"line": 615,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁸ : Semiring 𝕜\ninst✝⁷ : PartialOrder 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid β\ninst✝⁴ : LinearOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : PosSMulStrictMono 𝕜 β\ns : Set E\nf : E → β\nhf : Conv... | [] | by
gcongr
· apply le_max_left
· apply le_max_right | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Function | {
"line": 640,
"column": 53
} | {
"line": 643,
"column": 26
} | {
"line": 644,
"column": 4
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nβ : Type u_5\ninst✝⁸ : Semiring 𝕜\ninst✝⁷ : PartialOrder 𝕜\ninst✝⁶ : AddCommMonoid E\ninst✝⁵ : AddCommMonoid β\ninst✝⁴ : LinearOrder β\ninst✝³ : IsOrderedAddMonoid β\ninst✝² : SMul 𝕜 E\ninst✝¹ : Module 𝕜 β\ninst✝ : PosSMulStrictMono 𝕜 β\ns : Set E\nf : E → β\nhf : Stri... | [] | by
gcongr
· apply le_max_left
· apply le_max_right | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Function | {
"line": 1029,
"column": 2
} | {
"line": 1034,
"column": 66
} | {
"line": 1036,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_4\nβ : Type u_5\ninst✝⁷ : Semiring 𝕜\ninst✝⁶ : PartialOrder 𝕜\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : PartialOrder α\ninst✝³ : SMul 𝕜 α\ninst✝² : AddCommMonoid β\ninst✝¹ : PartialOrder β\ninst✝ : SMul 𝕜 β\nf : α ≃o β\nhf : ConcaveOn 𝕜 univ ⇑f\n⊢ ConvexOn 𝕜 univ ⇑f.symm",
... | [] | refine ⟨convex_univ, fun x _ y _ a b ha hb hab => ?_⟩
obtain ⟨x', hx''⟩ := f.surjective.exists.mp ⟨x, rfl⟩
obtain ⟨y', hy''⟩ := f.surjective.exists.mp ⟨y, rfl⟩
simp only [hx'', hy'', OrderIso.symm_apply_apply]
rw [← f.le_iff_le, OrderIso.apply_symm_apply]
exact hf.2 (by simp : x' ∈ univ) (by simp : y' ∈ univ)... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Function | {
"line": 1029,
"column": 2
} | {
"line": 1034,
"column": 66
} | {
"line": 1036,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_4\nβ : Type u_5\ninst✝⁷ : Semiring 𝕜\ninst✝⁶ : PartialOrder 𝕜\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : PartialOrder α\ninst✝³ : SMul 𝕜 α\ninst✝² : AddCommMonoid β\ninst✝¹ : PartialOrder β\ninst✝ : SMul 𝕜 β\nf : α ≃o β\nhf : ConcaveOn 𝕜 univ ⇑f\n⊢ ConvexOn 𝕜 univ ⇑f.symm",
... | [] | refine ⟨convex_univ, fun x _ y _ a b ha hb hab => ?_⟩
obtain ⟨x', hx''⟩ := f.surjective.exists.mp ⟨x, rfl⟩
obtain ⟨y', hy''⟩ := f.surjective.exists.mp ⟨y, rfl⟩
simp only [hx'', hy'', OrderIso.symm_apply_apply]
rw [← f.le_iff_le, OrderIso.apply_symm_apply]
exact hf.2 (by simp : x' ∈ univ) (by simp : y' ∈ univ)... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Convex.Function | {
"line": 1048,
"column": 2
} | {
"line": 1053,
"column": 66
} | {
"line": 1055,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_4\nβ : Type u_5\ninst✝⁷ : Semiring 𝕜\ninst✝⁶ : PartialOrder 𝕜\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : PartialOrder α\ninst✝³ : SMul 𝕜 α\ninst✝² : AddCommMonoid β\ninst✝¹ : PartialOrder β\ninst✝ : SMul 𝕜 β\nf : α ≃o β\nhf : ConvexOn 𝕜 univ ⇑f\n⊢ ConcaveOn 𝕜 univ ⇑f.symm",
... | [] | refine ⟨convex_univ, fun x _ y _ a b ha hb hab => ?_⟩
obtain ⟨x', hx''⟩ := f.surjective.exists.mp ⟨x, rfl⟩
obtain ⟨y', hy''⟩ := f.surjective.exists.mp ⟨y, rfl⟩
simp only [hx'', hy'', OrderIso.symm_apply_apply]
rw [← f.le_iff_le, OrderIso.apply_symm_apply]
exact hf.2 (by simp : x' ∈ univ) (by simp : y' ∈ univ)... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Function | {
"line": 1048,
"column": 2
} | {
"line": 1053,
"column": 66
} | {
"line": 1055,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nα : Type u_4\nβ : Type u_5\ninst✝⁷ : Semiring 𝕜\ninst✝⁶ : PartialOrder 𝕜\ninst✝⁵ : AddCommMonoid α\ninst✝⁴ : PartialOrder α\ninst✝³ : SMul 𝕜 α\ninst✝² : AddCommMonoid β\ninst✝¹ : PartialOrder β\ninst✝ : SMul 𝕜 β\nf : α ≃o β\nhf : ConvexOn 𝕜 univ ⇑f\n⊢ ConcaveOn 𝕜 univ ⇑f.symm",
... | [] | refine ⟨convex_univ, fun x _ y _ a b ha hb hab => ?_⟩
obtain ⟨x', hx''⟩ := f.surjective.exists.mp ⟨x, rfl⟩
obtain ⟨y', hy''⟩ := f.surjective.exists.mp ⟨y, rfl⟩
simp only [hx'', hy'', OrderIso.symm_apply_apply]
rw [← f.le_iff_le, OrderIso.apply_symm_apply]
exact hf.2 (by simp : x' ∈ univ) (by simp : y' ∈ univ)... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 219,
"column": 2
} | {
"line": 219,
"column": 18
} | {
"line": 221,
"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\nS : AffineSpace V P\nι : Type u_4\ns : Finset ι\nw : ι → k\np : ι → P\nb : P\npred : ι → Prop\ninst✝ : DecidablePred pred\nh : ∀ i ∈ s, w i ≠ 0 → pred i\ni : ι\nhi : i ∈ s\nhne : w i • (p i -ᵥ b) ≠ ... | [] | simp [hw] at hne | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.AffineSpace.AffineSubspace.Basic | {
"line": 449,
"column": 4
} | {
"line": 449,
"column": 43
} | {
"line": 450,
"column": 4
} | [
{
"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⊢ s₁.direction ⊔ s₂.direction ⊔ k ∙ (p₂ -ᵥ p₁) ≤ (s₁ ⊔ s₂).direction",
"ppTerm": "?re... | [
"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"
] | refine sup_le (sup_direction_le _ _) ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.LinearAlgebra.AffineSpace.Centroid | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 52
} | {
"line": 158,
"column": 0
} | [
{
"pp": "k : Type u_1\ninst✝² : DivisionRing k\nι : Type u_4\ns : Finset ι\ninst✝¹ : CharZero k\ninst✝ : Fintype ι\nh : s.Nonempty\n⊢ ∑ i ∈ s, centroidWeights k s i = 1",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Finset.sum_centroidWeights_eq_one_of_nonempty"
],
"usedFVa... | [] | exact s.sum_centroidWeights_eq_one_of_nonempty k h | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 396,
"column": 6
} | {
"line": 396,
"column": 20
} | {
"line": 396,
"column": 21
} | [
{
"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\nι : Type u_4\ns : Finset ι\nw₁ w₂ : ι → k\np : ι → P\n⊢ (s.weightedVSub p) w₁ +ᵥ (affineCombination k s p) w₂ = (affineCombination k s p) (w₁ + w₂)",
"ppTerm": "?m.38",
"... | [
"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\nι : Type u_4\ns : Finset ι\nw₁ w₂ : ι → k\np : ι → P\n⊢ (s.weightedVSub p) w₁ +ᵥ (affineCombination k s p) w₂ = (affineCombination k s p) (w₁ +ᵥ w₂)"
] | ← vadd_eq_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 727,
"column": 2
} | {
"line": 728,
"column": 47
} | {
"line": 730,
"column": 0
} | [
{
"pp": "k : Type u_6\nV : Type u_7\nP : Type u_8\ninst✝⁴ : CommRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_9\ninst✝ : DecidableEq ι\ns : Finset ι\np : ι → P\nw : ι → k\ni : ι\nhi : i ∈ s\nr : k\n⊢ (AffineMap.homothety (p i) r) ((affineCombination k s p) w) =\n ... | [] | rw [AffineMap.homothety_eq_lineMap, ← Finset.lineMap_affineCombination,
Finset.affineCombination_piSingle _ _ _ hi] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 727,
"column": 2
} | {
"line": 728,
"column": 47
} | {
"line": 730,
"column": 0
} | [
{
"pp": "k : Type u_6\nV : Type u_7\nP : Type u_8\ninst✝⁴ : CommRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_9\ninst✝ : DecidableEq ι\ns : Finset ι\np : ι → P\nw : ι → k\ni : ι\nhi : i ∈ s\nr : k\n⊢ (AffineMap.homothety (p i) r) ((affineCombination k s p) w) =\n ... | [] | rw [AffineMap.homothety_eq_lineMap, ← Finset.lineMap_affineCombination,
Finset.affineCombination_piSingle _ _ _ hi] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Combination | {
"line": 727,
"column": 2
} | {
"line": 728,
"column": 47
} | {
"line": 730,
"column": 0
} | [
{
"pp": "k : Type u_6\nV : Type u_7\nP : Type u_8\ninst✝⁴ : CommRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\nι : Type u_9\ninst✝ : DecidableEq ι\ns : Finset ι\np : ι → P\nw : ι → k\ni : ι\nhi : i ∈ s\nr : k\n⊢ (AffineMap.homothety (p i) r) ((affineCombination k s p) w) =\n ... | [] | rw [AffineMap.homothety_eq_lineMap, ← Finset.lineMap_affineCombination,
Finset.affineCombination_piSingle _ _ _ hi] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Seminorm | {
"line": 1020,
"column": 21
} | {
"line": 1020,
"column": 30
} | {
"line": 1020,
"column": 31
} | [
{
"pp": "𝕜 : Type u_3\nE : Type u_7\ninst✝⁵ : NormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : NormedSpace ℝ 𝕜\ninst✝² : Module 𝕜 E\ninst✝¹ : Module ℝ E\ninst✝ : IsScalarTower ℝ 𝕜 E\np : Seminorm 𝕜 E\nx : E\nr : ℝ\ny : E\n⊢ y ∈ p.ball x r ↔ y ∈ {x_1 | x_1 ∈ univ ∧ (⇑p ∘ fun z ↦ z + -x) x_1 < r}",
"ppT... | [
"𝕜 : Type u_3\nE : Type u_7\ninst✝⁵ : NormedField 𝕜\ninst✝⁴ : AddCommGroup E\ninst✝³ : NormedSpace ℝ 𝕜\ninst✝² : Module 𝕜 E\ninst✝¹ : Module ℝ E\ninst✝ : IsScalarTower ℝ 𝕜 E\np : Seminorm 𝕜 E\nx : E\nr : ℝ\ny : E\n⊢ y ∈ p.ball x r ↔ y ∈ {x_1 | (⇑p ∘ fun z ↦ z + -x) x_1 < r}"
] | sep_univ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.AffineSpace.Independent | {
"line": 525,
"column": 29
} | {
"line": 525,
"column": 57
} | {
"line": 525,
"column": 57
} | [
{
"pp": "case mp\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\ni : ι\ns : Set ι\nhs : p i ∈ affineSpan k (p '' s)\nh : (s ∩ {i}).Nonempty\n⊢ i ∈ s",
... | [
"case mp\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\ni : ι\ns : Set ι\nhs : p i ∈ affineSpan k (p '' s)\nh : i ∈ s\n⊢ i ∈ s"
] | Set.inter_singleton_nonempty | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.PathConnected | {
"line": 76,
"column": 7
} | {
"line": 76,
"column": 26
} | {
"line": 76,
"column": 27
} | [
{
"pp": "case h\nE : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousSMul ℝ E\nx y : E\ns : Set E\nh : [x -[ℝ] y] ⊆ s\n⊢ ∀ (t : ↑I), (Path.segment x y) t ∈ s",
"ppTerm": "?h",
"assigned": true,
"usedConstants": [
... | [
"case h\nE : Type u_1\ninst✝⁴ : AddCommGroup E\ninst✝³ : Module ℝ E\ninst✝² : TopologicalSpace E\ninst✝¹ : ContinuousAdd E\ninst✝ : ContinuousSMul ℝ E\nx y : E\ns : Set E\nh : [x -[ℝ] y] ⊆ s\n⊢ range ⇑(Path.segment x y) ⊆ s"
] | ← range_subset_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.StdSimplex | {
"line": 375,
"column": 4
} | {
"line": 378,
"column": 45
} | {
"line": 379,
"column": 4
} | [
{
"pp": "S : Type u_1\ninst✝⁶ : Semiring S\ninst✝⁵ : PartialOrder S\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : Fintype X\ninst✝³ : Fintype Y\ninst✝² : Fintype Z\ninst✝¹ : IsOrderedRing S\ninst✝ : Subsingleton X\ns t : ↑(stdSimplex S X)\ni : X\n⊢ s i = t i",
"ppTerm": "?m.24",
"assigned": true,
... | [
"S : Type u_1\ninst✝⁶ : Semiring S\ninst✝⁵ : PartialOrder S\nX : Type u_2\nY : Type u_3\nZ : Type u_4\ninst✝⁴ : Fintype X\ninst✝³ : Fintype Y\ninst✝² : Fintype Z\ninst✝¹ : IsOrderedRing S\ninst✝ : Subsingleton X\ns t : ↑(stdSimplex S X)\ni : X\nthis : ∀ (u : ↑(stdSimplex S X)), u i = 1\n⊢ s i = t i"
] | have (u : stdSimplex S X) : u i = 1 := by
rw [← sum_eq_one u, Finset.sum_eq_single i _ (by simp)]
intro j _ hj
exact (hj (Subsingleton.elim j i)).elim | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.NhdsKer | {
"line": 137,
"column": 20
} | {
"line": 137,
"column": 55
} | {
"line": 137,
"column": 56
} | [
{
"pp": "ι : Type u_3\nX : ι → Type u_4\ninst✝ : (i : ι) → TopologicalSpace (X i)\ns : (i : ι) → Set (X i)\n| nhdsKer (univ.pi s)",
"ppTerm": "?m.108",
"assigned": true,
"usedConstants": [
"Pi.topologicalSpace",
"congrArg",
"Set.univ",
"Membership.mem",
"Set.biUnion_of_... | [
"ι : Type u_3\nX : ι → Type u_4\ninst✝ : (i : ι) → TopologicalSpace (X i)\ns : (i : ι) → Set (X i)\n| nhdsKer (⋃ x ∈ univ.pi s, {x})"
] | ← biUnion_of_singleton (univ.pi s), | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Analysis.Convex.Topology | {
"line": 268,
"column": 73
} | {
"line": 277,
"column": 57
} | {
"line": 279,
"column": 0
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_3\ninst✝⁹ : Field 𝕜\ninst✝⁸ : LinearOrder 𝕜\ninst✝⁷ : IsStrictOrderedRing 𝕜\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : TopologicalSpace E\ninst✝³ : IsTopologicalAddGroup E\ninst✝² : TopologicalSpace 𝕜\ninst✝¹ : OrderTopology 𝕜\ninst✝ : ContinuousSMul 𝕜 E\ns... | [] | by
refine subset_antisymm ?_ (interior_mono subset_closure)
intro y hy
rcases hs' with ⟨x, hx⟩
have h := AffineMap.lineMap_apply_one (k := 𝕜) x y
obtain ⟨t, ht1, ht⟩ := AffineMap.lineMap_continuous.tendsto' _ _ h |>.eventually_mem
(mem_interior_iff_mem_nhds.1 hy) |>.exists_gt
apply hs.openSegment_inter... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Module.LocallyConvex | {
"line": 141,
"column": 91
} | {
"line": 151,
"column": 10
} | {
"line": 153,
"column": 0
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : Field 𝕜\ninst✝⁷ : PartialOrder 𝕜\ninst✝⁶ : ZeroLEOneClass 𝕜\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : Module 𝕜 E\ninst✝³ : TopologicalSpace E\ninst✝² : IsTopologicalAddGroup E\ninst✝¹ : ContinuousConstSMul 𝕜 E\ninst✝ : LocallyConvexSpace 𝕜 E\ns t : Set E\ndisj : Dis... | [] | by
letI : UniformSpace E := IsTopologicalAddGroup.rightUniformSpace E
haveI : IsUniformAddGroup E := isUniformAddGroup_of_addCommGroup
have := (LocallyConvexSpace.convex_open_basis_zero 𝕜 E).comap fun x : E × E => x.2 - x.1
rw [← uniformity_eq_comap_nhds_zero] at this
rcases disj.exists_uniform_thickening_of... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Module.Convex | {
"line": 97,
"column": 22
} | {
"line": 97,
"column": 36
} | {
"line": 97,
"column": 37
} | [
{
"pp": "F : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : Nontrivial F\nx : F\nr : ℝ\nhr : 0 ≤ r\nthis : (convexHull ℝ) (sphere 0 r) = closedBall 0 r\n⊢ (convexHull ℝ) (sphere (x + 0) r) = closedBall (x + 0) r",
"ppTerm": "?m.41",
"assigned": true,
"usedConstants": [
... | [
"F : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : Nontrivial F\nx : F\nr : ℝ\nhr : 0 ≤ r\nthis : (convexHull ℝ) (sphere 0 r) = closedBall 0 r\n⊢ (convexHull ℝ) (sphere (x +ᵥ 0) r) = closedBall (x +ᵥ 0) r"
] | ← vadd_eq_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Module.Convex | {
"line": 96,
"column": 2
} | {
"line": 119,
"column": 82
} | {
"line": 121,
"column": 0
} | [
{
"pp": "F : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : Nontrivial F\nx : F\nr : ℝ\nhr : 0 ≤ r\n⊢ (convexHull ℝ) (sphere x r) = closedBall x r",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.Normed.Module.Convex.0.convexHull_... | [] | suffices convexHull ℝ (sphere (0 : F) r) = closedBall 0 r by
rw [← add_zero x, ← vadd_eq_add, ← vadd_sphere, convexHull_vadd,
this, vadd_closedBall_zero, vadd_eq_add, add_zero]
refine subset_antisymm (convexHull_min sphere_subset_closedBall (convex_closedBall 0 r))
(fun x h ↦ mem_convexHull_iff.mpr fun ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Module.Convex | {
"line": 96,
"column": 2
} | {
"line": 119,
"column": 82
} | {
"line": 121,
"column": 0
} | [
{
"pp": "F : Type u_2\ninst✝² : NormedAddCommGroup F\ninst✝¹ : NormedSpace ℝ F\ninst✝ : Nontrivial F\nx : F\nr : ℝ\nhr : 0 ≤ r\n⊢ (convexHull ℝ) (sphere x r) = closedBall x r",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Analysis.Normed.Module.Convex.0.convexHull_... | [] | suffices convexHull ℝ (sphere (0 : F) r) = closedBall 0 r by
rw [← add_zero x, ← vadd_eq_add, ← vadd_sphere, convexHull_vadd,
this, vadd_closedBall_zero, vadd_eq_add, add_zero]
refine subset_antisymm (convexHull_min sphere_subset_closedBall (convex_closedBall 0 r))
(fun x h ↦ mem_convexHull_iff.mpr fun ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 256,
"column": 2
} | {
"line": 256,
"column": 37
} | {
"line": 258,
"column": 0
} | [
{
"pp": "case h\n𝕜 : Type u_2\n𝕜₂ : Type u_3\nE : Type u_6\nF : Type u_7\nι' : Type u_10\ninst✝⁷ : SeminormedRing 𝕜\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module 𝕜 E\ninst✝⁴ : SeminormedRing 𝕜₂\ninst✝³ : AddCommGroup F\ninst✝² : Module 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹ : RingHomIsometric σ₁₂\nι : Type u_11\ninst... | [] | simp only [h, Finset.sup_singleton] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 547,
"column": 4
} | {
"line": 547,
"column": 12
} | {
"line": 547,
"column": 13
} | [
{
"pp": "case mp\n𝕜 : Type u_2\nE : Type u_6\nι : Type u_9\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\n⊢ (∀ (I : Finset ι), ∃ r > 0, ∀ x ∈ s, (I.sup p) x < r) → ∀ (i : ι), ∃ r > 0, ∀... | [
"case mp\n𝕜 : Type u_2\nE : Type u_6\nι : Type u_9\ninst✝³ : NontriviallyNormedField 𝕜\ninst✝² : AddCommGroup E\ninst✝¹ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\ninst✝ : TopologicalSpace E\ns : Set E\nhp : WithSeminorms p\nhI : ∀ (I : Finset ι), ∃ r > 0, ∀ x ∈ s, (I.sup p) x < r\n⊢ ∀ (i : ι), ∃ r > 0, ∀ x ∈ s, (p... | intro hI | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Analysis.Normed.Operator.Basic | {
"line": 69,
"column": 50
} | {
"line": 69,
"column": 69
} | {
"line": 69,
"column": 69
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\n𝓕 : Type u_8\ninst✝⁸ : SeminormedAddCommGroup E\ninst✝⁷ : SeminormedAddCommGroup F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝² : Fu... | [
"𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\n𝓕 : Type u_8\ninst✝⁸ : SeminormedAddCommGroup E\ninst✝⁷ : SeminormedAddCommGroup F\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\ninst✝⁴ : NormedSpace 𝕜 E\ninst✝³ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝² : FunLike 𝓕 E F... | ← LinearMap.coe_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 812,
"column": 18
} | {
"line": 812,
"column": 31
} | {
"line": 812,
"column": 31
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_6\nι : Type u_9\nι' : Type u_10\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nq : SeminormFamily 𝕜 E ι'\nhpq : Seminorm.IsBounded p q LinearMap.id\nhqp : Seminorm.IsBounded q p LinearMap.id\n⊢ Seminorm.IsBounded p q LinearM... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.LocallyConvex.WithSeminorms | {
"line": 812,
"column": 18
} | {
"line": 812,
"column": 31
} | {
"line": 812,
"column": 31
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_6\nι : Type u_9\nι' : Type u_10\ninst✝² : NormedField 𝕜\ninst✝¹ : AddCommGroup E\ninst✝ : Module 𝕜 E\np : SeminormFamily 𝕜 E ι\nq : SeminormFamily 𝕜 E ι'\nhpq : Seminorm.IsBounded p q LinearMap.id\nhqp : Seminorm.IsBounded q p LinearMap.id\n⊢ Seminorm.IsBounded q p LinearM... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Operator.NNNorm | {
"line": 205,
"column": 18
} | {
"line": 205,
"column": 59
} | {
"line": 205,
"column": 59
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nE : Type u_4\nF : Type u_5\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : SeminormedAddCommGroup F\ninst✝⁵ : DenselyNormedField 𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜₂\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹ : RingHomIsometric σ₁₂\ninst... | [] | by simpa using! congrArg NNReal.toReal hx | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.AEEqFun | {
"line": 618,
"column": 2
} | {
"line": 621,
"column": 22
} | {
"line": 623,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf' f g : α →ₘ[μ] β\nhf : f' ≤ f\nhg : f' ≤ g\n⊢ f' ≤ f ⊓ g",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"MeasureTheory.a... | [] | rw [← coeFn_le] at hf hg ⊢
filter_upwards [hf, hg, coeFn_inf f g] with _ haf hag ha_inf
rw [ha_inf]
exact le_inf haf hag | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.AEEqFun | {
"line": 618,
"column": 2
} | {
"line": 621,
"column": 22
} | {
"line": 623,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : MeasurableSpace α\nμ : Measure α\ninst✝² : TopologicalSpace β\ninst✝¹ : SemilatticeInf β\ninst✝ : ContinuousInf β\nf' f g : α →ₘ[μ] β\nhf : f' ≤ f\nhg : f' ≤ g\n⊢ f' ≤ f ⊓ g",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"MeasureTheory.a... | [] | rw [← coeFn_le] at hf hg ⊢
filter_upwards [hf, hg, coeFn_inf f g] with _ haf hag ha_inf
rw [ha_inf]
exact le_inf haf hag | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Probability.ConditionalProbability | {
"line": 231,
"column": 2
} | {
"line": 231,
"column": 15
} | {
"line": 232,
"column": 2
} | [
{
"pp": "Ω : Type u_1\nm : MeasurableSpace Ω\nμ : Measure Ω\ns t : Set Ω\nhms : MeasurableSet s\nhcst : μ[t | s] ≠ 0\n⊢ (μ s)⁻¹ * μ (s ∩ t) ≠ 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MeasureTheory.Measure",
"HMul.hMul",
"MulZeroClass.toMul",
... | [
"Ω : Type u_1\nm : MeasurableSpace Ω\nμ : Measure Ω\ns t : Set Ω\nhms : MeasurableSet s\nhcst : μ[t | s] ≠ 0\n⊢ (μ s)⁻¹ * μ (s ∩ t) = μ[t | s]"
] | convert! hcst | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___macroRules_Mathlib_Tactic_convert!_1 | Mathlib.Tactic.convert! |
Mathlib.MeasureTheory.MeasurableSpace.Pi | {
"line": 86,
"column": 14
} | {
"line": 86,
"column": 30
} | {
"line": 86,
"column": 30
} | [
{
"pp": "case intro.a\nι : Type u_1\nα : ι → Type u_2\ninst✝ : Finite ι\nC : (i : ι) → Set (Set (α i))\nval✝ : Encodable ι\ni : ι\ns : Set (α i)\nhs : s ∈ C i\nt : (i : ι) → ℕ → Set (α i)\nh1t : ∀ (i : ι) (n : ℕ), t i n ∈ C i\nh2t : ∀ (i : ι), ⋃ n, t i n = univ\nthis : univ.pi (update (fun i' ↦ iUnion (t i')) i... | [
"case intro.a\nι : Type u_1\nα : ι → Type u_2\ninst✝ : Finite ι\nC : (i : ι) → Set (Set (α i))\nval✝ : Encodable ι\ni : ι\ns : Set (α i)\nhs : s ∈ C i\nt : (i : ι) → ℕ → Set (α i)\nh1t : ∀ (i : ι) (n : ℕ), t i n ∈ C i\nh2t : ∀ (i : ι), ⋃ n, t i n = univ\nthis : univ.pi (update (fun i' ↦ iUnion (t i')) i (⋃ x, s)) =... | ← iUnion_univ_pi | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ENNReal.Holder | {
"line": 166,
"column": 4
} | {
"line": 166,
"column": 39
} | {
"line": 167,
"column": 4
} | [
{
"pp": "case mpr\np : ℝ≥0∞\ninst✝ : p.HolderConjugate 1\n⊢ p = ∞",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"ENNReal.HolderConjugate.one_sub_inv",
"ENNReal.HolderConjugate.symm",
"congrArg",
"InvolutiveInv.toInv",
"HSub.hSub",
"id",
... | [
"case mpr\np : ℝ≥0∞\ninst✝ : p.HolderConjugate 1\n⊢ (1 - 1⁻¹)⁻¹ = ∞"
] | rw [← inv_inv p, ← one_sub_inv 1 p] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 143,
"column": 76
} | {
"line": 144,
"column": 14
} | {
"line": 146,
"column": 0
} | [
{
"pp": "α : Type u_1\nm0 : MeasurableSpace α\np : ℝ≥0∞\nε : Type u_7\ninst✝¹ : TopologicalSpace ε\ninst✝ : ContinuousENorm ε\nf : α → ε\n⊢ MemLp f p 0",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"MeasureTheory.Measure",
"Preorder.toLT",
"congrArg",
"and_self",
... | [] | by
simp [MemLp] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.LpSeminorm.Monotonicity | {
"line": 129,
"column": 86
} | {
"line": 136,
"column": 17
} | {
"line": 138,
"column": 0
} | [
{
"pp": "α : Type u_7\ninst✝² : Semiring α\ninst✝¹ : LinearOrder α\ninst✝ : IsStrictOrderedRing α\na b c : α\nha : 0 ≤ a\nhb : b < 0\nhc : 0 ≤ c\n⊢ a ≤ b * c ↔ a = 0 ∧ c = 0",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"le_refl",
"mul_nonpos_of_nonpos_of_nonn... | [] | by
constructor
· intro h
exact
⟨(h.trans (mul_nonpos_of_nonpos_of_nonneg hb.le hc)).antisymm ha,
(nonpos_of_mul_nonneg_right (ha.trans h) hb).antisymm hc⟩
· rintro ⟨rfl, rfl⟩
rw [mul_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 639,
"column": 2
} | {
"line": 639,
"column": 65
} | {
"line": 641,
"column": 0
} | [
{
"pp": "α : Type u_1\nε : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : ENorm ε\nc : ℝ≥0∞\nhc : c ≠ 0\nf : α → ε\n⊢ eLpNormEssSup f (c • μ) = eLpNormEssSup f μ",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"instHSMul",
"MeasureTheory.Measure",
"instSMulOfMul... | [] | simp_rw [eLpNormEssSup]; exact essSup_ennreal_smul_measure hc _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.Basic | {
"line": 639,
"column": 2
} | {
"line": 639,
"column": 65
} | {
"line": 641,
"column": 0
} | [
{
"pp": "α : Type u_1\nε : Type u_2\nm0 : MeasurableSpace α\nμ : Measure α\ninst✝ : ENorm ε\nc : ℝ≥0∞\nhc : c ≠ 0\nf : α → ε\n⊢ eLpNormEssSup f (c • μ) = eLpNormEssSup f μ",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"instHSMul",
"MeasureTheory.Measure",
"instSMulOfMul... | [] | simp_rw [eLpNormEssSup]; exact essSup_ennreal_smul_measure hc _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Real.ConjExponents | {
"line": 529,
"column": 2
} | {
"line": 529,
"column": 33
} | {
"line": 530,
"column": 2
} | [
{
"pp": "p : ℝ≥0∞\nhp : 1 ≤ p\nthis : p ≠ 0\n⊢ 1 + (p - 1)⁻¹ = (1⁻¹ - p⁻¹)⁻¹",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
"LE.le.eq_or_lt",
"Preorder.toLT",
"InvolutiveInv.toInv",
"PartialOrder.toPreorder",
"HSub.hSub",
"Ne",
... | [
"case inl\nhp : 1 ≤ 1\nthis : 1 ≠ 0\n⊢ 1 + (1 - 1)⁻¹ = (1⁻¹ - 1⁻¹)⁻¹",
"case inr\np : ℝ≥0∞\nhp : 1 ≤ p\nthis : p ≠ 0\nhp₁ : 1 < p\n⊢ 1 + (p - 1)⁻¹ = (1⁻¹ - p⁻¹)⁻¹"
] | obtain rfl | hp₁ := hp.eq_or_lt | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.MeasureTheory.Integral.MeanInequalities | {
"line": 116,
"column": 6
} | {
"line": 117,
"column": 79
} | {
"line": 118,
"column": 6
} | [
{
"pp": "α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.HolderConjugate q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhf_nontop : ∫⁻ (a : α), f a ^ p ∂μ ≠ ∞\nhg_nontop : ∫⁻ (a : α), g a ^ q ∂μ ≠ ∞\nhf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0\nhg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0\nnpf : ℝ≥0∞... | [
"α : Type u_1\ninst✝ : MeasurableSpace α\nμ : Measure α\np q : ℝ\nhpq : p.HolderConjugate q\nf g : α → ℝ≥0∞\nhf : AEMeasurable f μ\nhf_nontop : ∫⁻ (a : α), f a ^ p ∂μ ≠ ∞\nhg_nontop : ∫⁻ (a : α), g a ^ q ∂μ ≠ ∞\nhf_nonzero : ∫⁻ (a : α), f a ^ p ∂μ ≠ 0\nhg_nonzero : ∫⁻ (a : α), g a ^ q ∂μ ≠ 0\nnpf : ℝ≥0∞ := (∫⁻ (c :... | rw [Pi.mul_apply, fun_eq_funMulInvSnorm_mul_eLpNorm f hf_nonzero hf_nontop,
fun_eq_funMulInvSnorm_mul_eLpNorm g hg_nonzero hg_nontop, Pi.mul_apply] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 76
} | {
"line": 107,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nμ : Measure α\nf g : α → E\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\np : ℝ≥0∞\n⊢ eLpNorm (f - g) p μ ≤ p.LpAddConst * (eLpNorm f p μ + eLpNorm g p μ)",
"ppTerm": "?m.40",
"assigned": true,
... | [] | simpa only [sub_eq_add_neg, eLpNorm_neg] using eLpNorm_add_le' hf hg.neg p | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 76
} | {
"line": 107,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nμ : Measure α\nf g : α → E\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\np : ℝ≥0∞\n⊢ eLpNorm (f - g) p μ ≤ p.LpAddConst * (eLpNorm f p μ + eLpNorm g p μ)",
"ppTerm": "?m.40",
"assigned": true,
... | [] | simpa only [sub_eq_add_neg, eLpNorm_neg] using eLpNorm_add_le' hf hg.neg p | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSeminorm.TriangleInequality | {
"line": 105,
"column": 2
} | {
"line": 105,
"column": 76
} | {
"line": 107,
"column": 0
} | [
{
"pp": "α : Type u_1\nE : Type u_2\nm : MeasurableSpace α\ninst✝ : NormedAddCommGroup E\nμ : Measure α\nf g : α → E\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\np : ℝ≥0∞\n⊢ eLpNorm (f - g) p μ ≤ p.LpAddConst * (eLpNorm f p μ + eLpNorm g p μ)",
"ppTerm": "?m.40",
"assigned": true,
... | [] | simpa only [sub_eq_add_neg, eLpNorm_neg] using eLpNorm_add_le' hf hg.neg p | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.MeanInequalities | {
"line": 394,
"column": 24
} | {
"line": 394,
"column": 37
} | {
"line": 396,
"column": 0
} | [
{
"pp": "w₁ w₂ p₁ p₂ : ℝ\nhw₁ : 0 ≤ w₁\nhw₂ : 0 ≤ w₂\nhp₁ : 0 ≤ p₁\nhp₂ : 0 ≤ p₂\nhw : w₁ + w₂ = 1\n⊢ ↑(⟨w₁, hw₁⟩ + ⟨w₂, hw₂⟩) = ↑1",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"hw"
],
"usedGoals": []
}
] | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.MeanInequalities | {
"line": 409,
"column": 24
} | {
"line": 409,
"column": 37
} | {
"line": 411,
"column": 0
} | [
{
"pp": "w₁ w₂ w₃ w₄ p₁ p₂ p₃ p₄ : ℝ\nhw₁ : 0 ≤ w₁\nhw₂ : 0 ≤ w₂\nhw₃ : 0 ≤ w₃\nhw₄ : 0 ≤ w₄\nhp₁ : 0 ≤ p₁\nhp₂ : 0 ≤ p₂\nhp₃ : 0 ≤ p₃\nhp₄ : 0 ≤ p₄\nhw : w₁ + w₂ + w₃ + w₄ = 1\n⊢ ↑(⟨w₁, hw₁⟩ + ⟨w₂, hw₂⟩ + ⟨w₃, hw₃⟩ + ⟨w₄, hw₄⟩) = ↑1",
"ppTerm": "?m.126",
"assigned": true,
"usedConstants": [],
"... | [] | by assumption | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Function.LpSeminorm.CompareExp | {
"line": 215,
"column": 6
} | {
"line": 215,
"column": 19
} | {
"line": 216,
"column": 6
} | [
{
"pp": "α : 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 q r : ℝ\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nc : ℝ≥0... | [
"case h₁\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 q r : ℝ\nhf : AEStronglyMeasurable f μ\nhg : AEStronglyMeasurable g μ\nb : E → F → G\nc : ℝ≥0\nh... | gcongr ?_ ^ _ | Mathlib.Tactic.GCongr._aux_Mathlib_Tactic_GCongr_Core___elabRules_Mathlib_Tactic_GCongr_gcongr_1 | Mathlib.Tactic.GCongr.gcongr |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 334,
"column": 4
} | {
"line": 335,
"column": 69
} | {
"line": 336,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\nc : ℝ\nf : ↥(Lp E p μ)\ng : ↥(Lp F p μ)\nh : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ c * ‖↑↑g x‖\nhc : 0 ≤ c\n⊢ ‖f‖ ≤ c * ‖g‖",
"ppTerm": "?inl",
"as... | [] | lift c to ℝ≥0 using hc
exact NNReal.coe_le_coe.mpr (nnnorm_le_mul_nnnorm_of_ae_le_mul h) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Function.LpSpace.Basic | {
"line": 334,
"column": 4
} | {
"line": 335,
"column": 69
} | {
"line": 336,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\nE : Type u_4\nF : Type u_5\nm : MeasurableSpace α\np : ℝ≥0∞\nμ : Measure α\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedAddCommGroup F\nc : ℝ\nf : ↥(Lp E p μ)\ng : ↥(Lp F p μ)\nh : ∀ᵐ (x : α) ∂μ, ‖↑↑f x‖ ≤ c * ‖↑↑g x‖\nhc : 0 ≤ c\n⊢ ‖f‖ ≤ c * ‖g‖",
"ppTerm": "?inl",
"as... | [] | lift c to ℝ≥0 using hc
exact NNReal.coe_le_coe.mpr (nnnorm_le_mul_nnnorm_of_ae_le_mul h) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.MeanInequalities | {
"line": 1038,
"column": 2
} | {
"line": 1038,
"column": 17
} | {
"line": 1039,
"column": 2
} | [
{
"pp": "ι : Type u\np : ℝ\nhp : 1 ≤ p\nf : ι → ℝ≥0\nhf_sum : Summable fun i ↦ ↑(f i) ^ p\ng : ι → ℝ≥0\nhg_sum : Summable fun i ↦ ↑(g i) ^ p\n⊢ (Summable fun i ↦ (↑(f i) + ↑(g i)) ^ p) ∧\n (∑' (i : ι), (↑(f i) + ↑(g i)) ^ p) ^ (1 / p) ≤\n (∑' (i : ι), ↑(f i) ^ p) ^ (1 / p) + (∑' (i : ι), ↑(g i) ^ p) ^ (... | [
"ι : Type u\np : ℝ\nf g : ι → ℝ≥0\nhp : 1 ≤ p\nhf_sum : Summable fun a ↦ f a ^ p\nhg_sum : Summable fun a ↦ g a ^ p\n⊢ (Summable fun a ↦ (f a + g a) ^ p) ∧\n (∑' (a : ι), (f a + g a) ^ p) ^ (1 / p) ≤ (∑' (a : ι), f a ^ p) ^ (1 / p) + (∑' (a : ι), g a ^ p) ^ (1 / p)"
] | norm_cast0 at * | Lean.Elab.Tactic.NormCast.evalNormCast0 | Lean.Parser.Tactic.normCast0 |
Mathlib.Analysis.MeanInequalities | {
"line": 1100,
"column": 2
} | {
"line": 1102,
"column": 52
} | {
"line": 1104,
"column": 0
} | [
{
"pp": "case neg\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0∞\np q : ℝ\nhpq : p.HolderConjugate q\nH : (∑ i ∈ s, f i ^ p) ^ (1 / p) ≠ 0 ∧ (∑ i ∈ s, g i ^ q) ^ (1 / q) ≠ 0\nH' : (∀ i ∈ s, f i ≠ ∞) ∧ ∀ i ∈ s, g i ≠ ∞\nthis :\n ∑ x ∈ s, ↑(f x).toNNReal * ↑(g x).toNNReal ≤\n (∑ x ∈ s, ↑(f x).toNNReal ^ p) ^ p⁻¹ *... | [] | convert! this using 1 <;> [skip; congr 2] <;> [skip; skip; simp; skip; simp] <;>
· refine Finset.sum_congr rfl fun i hi => ?_
simp [H'.1 i hi, H'.2 i hi, -WithZero.coe_mul] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Analysis.MeanInequalities | {
"line": 1114,
"column": 4
} | {
"line": 1114,
"column": 85
} | {
"line": 1115,
"column": 4
} | [
{
"pp": "case pos\nι : Type u\ns : Finset ι\np : ℝ\nhp✝ : 1 ≤ p\nw f : ι → ℝ≥0∞\nhp : 1 < p\nhp₀ : 0 < p\nhp₁ : p⁻¹ < 1\nH : (∀ i ∈ s, w i = 0) ∨ ∀ i ∈ s, w i = 0 ∨ f i = 0\n⊢ ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹",
"ppTerm": "?pos✝",
"assigned": true,
"use... | [
"case pos\nι : Type u\ns : Finset ι\np : ℝ\nhp✝ : 1 ≤ p\nw f : ι → ℝ≥0∞\nhp : 1 < p\nhp₀ : 0 < p\nhp₁ : p⁻¹ < 1\nH : (∀ i ∈ s, w i = 0) ∨ ∀ i ∈ s, w i = 0 ∨ f i = 0\nthis : ∀ i ∈ s, w i * f i = 0\n⊢ ∑ i ∈ s, w i * f i ≤ (∑ i ∈ s, w i) ^ (1 - p⁻¹) * (∑ i ∈ s, w i * f i ^ p) ^ p⁻¹"
] | have (i) (hi : i ∈ s) : w i * f i = 0 := by rcases H with H | H <;> simp [H i hi] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.MeanInequalities | {
"line": 1159,
"column": 31
} | {
"line": 1159,
"column": 50
} | {
"line": 1160,
"column": 2
} | [
{
"pp": "case pos.inl\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\nH' : (∑ i ∈ s, f i ^ p) ^ (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": "?pos.inl✝",
"assigned": true,
"usedConstants": [
"ENNReal... | [] | simp [H', -one_div] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.MeanInequalities | {
"line": 1159,
"column": 31
} | {
"line": 1159,
"column": 50
} | {
"line": 1160,
"column": 2
} | [
{
"pp": "case pos.inr\nι : Type u\ns : Finset ι\nf g : ι → ℝ≥0∞\np : ℝ\nhp : 1 ≤ p\nH' : (∑ i ∈ s, g i ^ p) ^ (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": "?pos.inr✝",
"assigned": true,
"usedConstants": [
"ENNReal... | [] | simp [H', -one_div] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.Normed.Operator.Mul | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 94
} | {
"line": 279,
"column": 2
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nR : Type u_3\ninst✝⁵ : NormedDivisionRing R\ninst✝⁴ : NormedAlgebra 𝕜 R\ninst✝³ : Module R E\ninst✝² : NormSMulClass R E\ninst✝¹ : IsScalarTower 𝕜 R E\ninst✝ : Nontrivial E\na :... | [
"case refine_1\n𝕜 : Type u_1\nE : Type u_2\ninst✝⁸ : NontriviallyNormedField 𝕜\ninst✝⁷ : NormedAddCommGroup E\ninst✝⁶ : NormedSpace 𝕜 E\nR : Type u_3\ninst✝⁵ : NormedDivisionRing R\ninst✝⁴ : NormedAlgebra 𝕜 R\ninst✝³ : Module R E\ninst✝² : NormSMulClass R E\ninst✝¹ : IsScalarTower 𝕜 R E\ninst✝ : Nontrivial E\n... | refine ContinuousLinearMap.opNorm_eq_of_bounds (norm_nonneg _) (fun x => ?_) fun N _ h => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Operator.NormedSpace | {
"line": 369,
"column": 6
} | {
"line": 371,
"column": 49
} | {
"line": 372,
"column": 4
} | [
{
"pp": "case mp\n𝕜 : Type u_1\n𝕜₂ : Type u_3\nE : Type u_5\nF : Type u_6\nι : Type u_9\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁴ : RingHomIsometric σ₁₂\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\n... | [] | intro ⟨p, hp, hpf⟩
rcases p.bound_of_continuous_normedSpace hp with ⟨C, -, hC⟩
exact ⟨C, fun i x ↦ (hpf i x).trans (hC x)⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Operator.NormedSpace | {
"line": 369,
"column": 6
} | {
"line": 371,
"column": 49
} | {
"line": 372,
"column": 4
} | [
{
"pp": "case mp\n𝕜 : Type u_1\n𝕜₂ : Type u_3\nE : Type u_5\nF : Type u_6\nι : Type u_9\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁴ : RingHomIsometric σ₁₂\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\n... | [] | intro ⟨p, hp, hpf⟩
rcases p.bound_of_continuous_normedSpace hp with ⟨C, -, hC⟩
exact ⟨C, fun i x ↦ (hpf i x).trans (hC x)⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Operator.NormedSpace | {
"line": 372,
"column": 6
} | {
"line": 372,
"column": 19
} | {
"line": 373,
"column": 6
} | [
{
"pp": "case mpr\n𝕜 : Type u_1\n𝕜₂ : Type u_3\nE : Type u_5\nF : Type u_6\nι : Type u_9\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁴ : RingHomIsometric σ₁₂\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\... | [
"case mpr\n𝕜 : Type u_1\n𝕜₂ : Type u_3\nE : Type u_5\nF : Type u_6\nι : Type u_9\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁴ : RingHomIsometric σ₁₂\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : Nor... | intro ⟨C, hC⟩ | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Analysis.Normed.Operator.NormedSpace | {
"line": 372,
"column": 6
} | {
"line": 372,
"column": 19
} | {
"line": 373,
"column": 6
} | [
{
"pp": "case mpr\n𝕜 : Type u_1\n𝕜₂ : Type u_3\nE : Type u_5\nF : Type u_6\nι : Type u_9\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁴ : RingHomIsometric σ₁₂\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\... | [
"case mpr\n𝕜 : Type u_1\n𝕜₂ : Type u_3\nE : Type u_5\nF : Type u_6\nι : Type u_9\ninst✝⁶ : NontriviallyNormedField 𝕜\ninst✝⁵ : NontriviallyNormedField 𝕜₂\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝⁴ : RingHomIsometric σ₁₂\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : SeminormedAddCommGroup F\ninst✝¹ : NormedSpace 𝕜 E\ninst✝ : Nor... | intro ⟨C, hC⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Topology.Algebra.Module.Complement | {
"line": 319,
"column": 47
} | {
"line": 325,
"column": 16
} | {
"line": 327,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : Ring R\nM : Type u_2\ninst✝³ : TopologicalSpace M\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\np q : Submodule R M\ninst✝ : IsTopologicalAddGroup M\nh : IsTopCompl p q\nhq : IsClosed[inst✝³] ↑q\n⊢ T3Space ↥p",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
... | [] | by
have : IsClosed ({0} : Set p) := by
rw [← (isQuotientMap_projectionOntoL h).isClosed_preimage]
rwa [← ker_projectionOntoL h] at hq
have : T1Space p := IsTopologicalAddGroup.t1Space _ this
rw [RegularSpace.t3Space_iff_t0Space]
infer_instance | [anonymous] | Lean.Parser.Term.byTactic |
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