module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Data.EReal.Basic | {
"line": 367,
"column": 23
} | {
"line": 367,
"column": 28
} | {
"line": 368,
"column": 2
} | [
{
"pp": "x : EReal\nhx : 0 < x\nh'x : x ≠ ⊤\n⊢ x ≠ ⊤ ∧ x ≠ ⊥",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Preorder.toLT",
"eq_false",
"congrArg",
"False.elim",
"PartialOrder.toPreorder",
"EReal",
"not_lt_bot._simp... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.EReal.Basic | {
"line": 367,
"column": 23
} | {
"line": 367,
"column": 28
} | {
"line": 368,
"column": 2
} | [
{
"pp": "x : EReal\nhx : 0 < x\nh'x : x ≠ ⊤\n⊢ x ≠ ⊤ ∧ x ≠ ⊥",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Preorder.toLT",
"eq_false",
"congrArg",
"False.elim",
"PartialOrder.toPreorder",
"EReal",
"not_lt_bot._simp... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.EReal.Basic | {
"line": 367,
"column": 23
} | {
"line": 367,
"column": 28
} | {
"line": 368,
"column": 2
} | [
{
"pp": "x : EReal\nhx : 0 < x\nh'x : x ≠ ⊤\n⊢ x ≠ ⊤ ∧ x ≠ ⊥",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Preorder.toLT",
"eq_false",
"congrArg",
"False.elim",
"PartialOrder.toPreorder",
"EReal",
"not_lt_bot._simp... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.EReal.Basic | {
"line": 371,
"column": 23
} | {
"line": 371,
"column": 28
} | {
"line": 372,
"column": 2
} | [
{
"pp": "x : EReal\nhx : x < 0\nh'x : x ≠ ⊥\n⊢ x ≠ ⊤ ∧ x ≠ ⊥",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Preorder.toLT",
"eq_false",
"and_true",
"congrArg",
"False.elim",
"PartialOrder.toPreorder",
"EReal",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.EReal.Basic | {
"line": 371,
"column": 23
} | {
"line": 371,
"column": 28
} | {
"line": 372,
"column": 2
} | [
{
"pp": "x : EReal\nhx : x < 0\nh'x : x ≠ ⊥\n⊢ x ≠ ⊤ ∧ x ≠ ⊥",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Preorder.toLT",
"eq_false",
"and_true",
"congrArg",
"False.elim",
"PartialOrder.toPreorder",
"EReal",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.EReal.Basic | {
"line": 371,
"column": 23
} | {
"line": 371,
"column": 28
} | {
"line": 372,
"column": 2
} | [
{
"pp": "x : EReal\nhx : x < 0\nh'x : x ≠ ⊥\n⊢ x ≠ ⊤ ∧ x ≠ ⊥",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Preorder.toLT",
"eq_false",
"and_true",
"congrArg",
"False.elim",
"PartialOrder.toPreorder",
"EReal",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.EReal.Basic | {
"line": 378,
"column": 25
} | {
"line": 378,
"column": 30
} | {
"line": 379,
"column": 4
} | [
{
"pp": "y : EReal\nhy0 : 0 < y\nright✝ : y < ⊤\n⊢ y ≠ ⊤ ∧ y ≠ ⊥",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"False",
"Preorder.toLT",
"False.elim",
"PartialOrder.toPreorder",
"EReal",
"lt_self_iff_false._simp_1",
"not_lt_bot._simp_1",
"E... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.EReal.Basic | {
"line": 378,
"column": 25
} | {
"line": 378,
"column": 30
} | {
"line": 379,
"column": 4
} | [
{
"pp": "y : EReal\nhy0 : 0 < y\nright✝ : y < ⊤\n⊢ y ≠ ⊤ ∧ y ≠ ⊥",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"False",
"Preorder.toLT",
"False.elim",
"PartialOrder.toPreorder",
"EReal",
"lt_self_iff_false._simp_1",
"not_lt_bot._simp_1",
"E... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.EReal.Basic | {
"line": 378,
"column": 25
} | {
"line": 378,
"column": 30
} | {
"line": 379,
"column": 4
} | [
{
"pp": "y : EReal\nhy0 : 0 < y\nright✝ : y < ⊤\n⊢ y ≠ ⊤ ∧ y ≠ ⊥",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"False",
"Preorder.toLT",
"False.elim",
"PartialOrder.toPreorder",
"EReal",
"lt_self_iff_false._simp_1",
"not_lt_bot._simp_1",
"E... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.EReal.Basic | {
"line": 388,
"column": 25
} | {
"line": 388,
"column": 30
} | {
"line": 389,
"column": 4
} | [
{
"pp": "y : EReal\nleft✝ : ⊥ < y\nhy0 : y < 0\n⊢ y ≠ ⊤ ∧ y ≠ ⊥",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"False",
"Preorder.toLT",
"False.elim",
"PartialOrder.toPreorder",
"EReal",
"lt_self_iff_false._simp_1",
"CoheytingAlgebra.toOrderTop",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.EReal.Basic | {
"line": 388,
"column": 25
} | {
"line": 388,
"column": 30
} | {
"line": 389,
"column": 4
} | [
{
"pp": "y : EReal\nleft✝ : ⊥ < y\nhy0 : y < 0\n⊢ y ≠ ⊤ ∧ y ≠ ⊥",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"False",
"Preorder.toLT",
"False.elim",
"PartialOrder.toPreorder",
"EReal",
"lt_self_iff_false._simp_1",
"CoheytingAlgebra.toOrderTop",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.EReal.Basic | {
"line": 388,
"column": 25
} | {
"line": 388,
"column": 30
} | {
"line": 389,
"column": 4
} | [
{
"pp": "y : EReal\nleft✝ : ⊥ < y\nhy0 : y < 0\n⊢ y ≠ ⊤ ∧ y ≠ ⊥",
"ppTerm": "?m.56",
"assigned": true,
"usedConstants": [
"False",
"Preorder.toLT",
"False.elim",
"PartialOrder.toPreorder",
"EReal",
"lt_self_iff_false._simp_1",
"CoheytingAlgebra.toOrderTop",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 518,
"column": 4
} | {
"line": 519,
"column": 85
} | {
"line": 520,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nexps : HeightOneSpectrum R →₀ ℤ\n⊢ ∑ i ∈ exps.support, count K v (↑i.asIdeal ^ exps i) = exps v",
"ppTerm": "?m.52",
"assign... | [] | classical simp only [count_zpow, count_maximal, mul_ite, mul_one, mul_zero, Finset.sum_ite_eq',
exps.mem_support_iff, ne_eq, ite_not, ite_eq_right_iff, @eq_comm ℤ 0, imp_self] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 518,
"column": 4
} | {
"line": 519,
"column": 85
} | {
"line": 520,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nexps : HeightOneSpectrum R →₀ ℤ\n⊢ ∑ i ∈ exps.support, count K v (↑i.asIdeal ^ exps i) = exps v",
"ppTerm": "?m.52",
"assign... | [] | classical simp only [count_zpow, count_maximal, mul_ite, mul_one, mul_zero, Finset.sum_ite_eq',
exps.mem_support_iff, ne_eq, ite_not, ite_eq_right_iff, @eq_comm ℤ 0, imp_self] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 518,
"column": 4
} | {
"line": 519,
"column": 85
} | {
"line": 520,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nexps : HeightOneSpectrum R →₀ ℤ\n⊢ ∑ i ∈ exps.support, count K v (↑i.asIdeal ^ exps i) = exps v",
"ppTerm": "?m.52",
"assign... | [] | classical simp only [count_zpow, count_maximal, mul_ite, mul_one, mul_zero, Finset.sum_ite_eq',
exps.mem_support_iff, ne_eq, ite_not, ite_eq_right_iff, @eq_comm ℤ 0, imp_self] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.EReal.Basic | {
"line": 434,
"column": 2
} | {
"line": 434,
"column": 18
} | {
"line": 435,
"column": 2
} | [
{
"pp": "x z : EReal\nh : x < z\na : EReal\nha₁ : x < a\nha₂ : a < z\n⊢ ∃ y, x < ↑y ∧ ↑y < z",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"Real",
"Preorder.toLT",
"False.elim",
"not_top_lt",
"PartialOrder.toPreorder",
"EReal",
"not_lt_bot",
... | [] | induction a with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Data.EReal.Basic | {
"line": 460,
"column": 29
} | {
"line": 460,
"column": 50
} | {
"line": 460,
"column": 50
} | [
{
"pp": "x y : ℝ\n⊢ WithBot.some '' Ioc ↑x ↑y = Ioc ↑x ↑y",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Set.Ioc",
"Real",
"WithBot.some",
"WithBot",
"WithTop.instPreorder",
"congrArg",
"PartialOrder.... | [
"x y : ℝ\n⊢ Ioc ↑↑x ↑↑y = Ioc ↑x ↑y"
] | WithBot.image_coe_Ioc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.EReal.Basic | {
"line": 466,
"column": 29
} | {
"line": 466,
"column": 50
} | {
"line": 466,
"column": 50
} | [
{
"pp": "x y : ℝ\n⊢ WithBot.some '' Ioo ↑x ↑y = Ioo ↑x ↑y",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Real",
"WithBot.some",
"WithBot",
"WithBot.image_coe_Ioo",
"WithTop.instPreorder",
"congrArg",
... | [
"x y : ℝ\n⊢ Ioo ↑↑x ↑↑y = Ioo ↑x ↑y"
] | WithBot.image_coe_Ioo | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.EReal.Basic | {
"line": 472,
"column": 29
} | {
"line": 472,
"column": 50
} | {
"line": 472,
"column": 50
} | [
{
"pp": "x : ℝ\n⊢ WithBot.some '' Ioo ↑x ⊤ = Ioo ↑x ⊤",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Real",
"WithBot.some",
"WithBot",
"WithBot.image_coe_Ioo",
"WithTop.instPreorder",
"congrArg",
"Par... | [
"x : ℝ\n⊢ Ioo ↑↑x ↑⊤ = Ioo ↑x ⊤"
] | WithBot.image_coe_Ioo | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ENNReal.Inv | {
"line": 139,
"column": 2
} | {
"line": 143,
"column": 47
} | {
"line": 145,
"column": 0
} | [
{
"pp": "a b : ℝ≥0∞\nhb₀ : b = 0 → a = 0\nhb : b = ∞ → a = 0\n⊢ a * b * b⁻¹ = a",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"False",
"Semigroup.toMul",
"HMul.hMul",
"eq_false",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"CommSemiring.t... | [] | obtain rfl | hb₀ := eq_or_ne b 0
· simp_all
obtain rfl | hb := eq_or_ne b ⊤
· simp_all
· simp [mul_assoc, ENNReal.mul_inv_cancel, *] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENNReal.Inv | {
"line": 139,
"column": 2
} | {
"line": 143,
"column": 47
} | {
"line": 145,
"column": 0
} | [
{
"pp": "a b : ℝ≥0∞\nhb₀ : b = 0 → a = 0\nhb : b = ∞ → a = 0\n⊢ a * b * b⁻¹ = a",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"False",
"Semigroup.toMul",
"HMul.hMul",
"eq_false",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"CommSemiring.t... | [] | obtain rfl | hb₀ := eq_or_ne b 0
· simp_all
obtain rfl | hb := eq_or_ne b ⊤
· simp_all
· simp [mul_assoc, ENNReal.mul_inv_cancel, *] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENNReal.Inv | {
"line": 654,
"column": 43
} | {
"line": 658,
"column": 28
} | {
"line": 660,
"column": 0
} | [
{
"pp": "a b : ℝ≥0∞\nha : a ≠ ∞\nhb : b ≠ 0\n⊢ ∃ n > 0, (↑n)⁻¹ * a < b",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"ENNReal.exists_nat_pos_mul_gt",
"instHDiv",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
... | [] | by
rcases exists_nat_pos_mul_gt hb ha with ⟨n, npos, hn⟩
use n, npos
rw [← ENNReal.div_eq_inv_mul]
exact div_lt_of_lt_mul' hn | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 627,
"column": 6
} | {
"line": 627,
"column": 46
} | {
"line": 628,
"column": 6
} | [
{
"pp": "case refine_1\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nJ I : Ideal R\nhIJ : J * I ≤ J\nhJ : ¬J = 0\nhI : ¬I = 0\ns : Finset (HeightOneSpectrum R) := ⋯.toFinset\np : HeightOneSpectrum R\nhps : p ∈ s\n⊢ p.asIdeal < 1",
"ppTerm": "?refine_1",
"assigned": true,
"usedConst... | [
"case refine_1\nR : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nJ I : Ideal R\nhIJ : J * I ≤ J\nhJ : ¬J = 0\nhI : ¬I = 0\ns : Finset (HeightOneSpectrum R) := ⋯.toFinset\np : HeightOneSpectrum R\nhps : p ∈ s\n⊢ p.asIdeal ≠ ⊤"
] | rw [Ideal.one_eq_top, lt_top_iff_ne_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.MetricSpace.Defs | {
"line": 143,
"column": 4
} | {
"line": 143,
"column": 32
} | {
"line": 145,
"column": 0
} | [
{
"pp": "case inr\nγ : Type w\ninst✝ : MetricSpace γ\nx : γ\nhr : 0 ≤ 0\n⊢ {x}.Subsingleton",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Set.subsingleton_singleton"
],
"usedFVars": [
"γ",
"x"
],
"usedGoals": []
}
] | [] | exact subsingleton_singleton | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.ENNReal.Inv | {
"line": 939,
"column": 4
} | {
"line": 939,
"column": 23
} | {
"line": 939,
"column": 23
} | [
{
"pp": "x : ℝ\nhx : 0 < x\n⊢ x⁻¹.toNNReal = x.toNNReal⁻¹",
"ppTerm": "?m.19",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"Real.instInv",
"NNReal.instInv",
"id",
"Real.toNNReal_inv",
"NNReal",
"Inv.inv",
"Eq.symm",
... | [
"x : ℝ\nhx : 0 < x\n⊢ x⁻¹.toNNReal = x⁻¹.toNNReal"
] | ← Real.toNNReal_inv | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 631,
"column": 6
} | {
"line": 631,
"column": 23
} | {
"line": 631,
"column": 24
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\n⊢ BoundedSpace α ↔ ∃ C, ∀ (a b : α), dist a b ≤ C",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"PseudoMetricSpace.toBornology",
"congrArg",
"Set.univ",
"Exists",
... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\n⊢ Bornology.IsBounded univ ↔ ∃ C, ∀ (a b : α), dist a b ≤ C"
] | ← isBounded_univ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 651,
"column": 6
} | {
"line": 651,
"column": 23
} | {
"line": 651,
"column": 24
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\n⊢ BoundedSpace α ↔ ∃ C, ∀ (a b : α), nndist a b ≤ C",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NNDist.nndist",
"PseudoMetricSpace.toBornology",
"congrArg",
"Set.univ",
"PartialOrder.toPreor... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\n⊢ Bornology.IsBounded univ ↔ ∃ C, ∀ (a b : α), nndist a b ≤ C"
] | ← isBounded_univ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Real.Pointwise | {
"line": 90,
"column": 2
} | {
"line": 98,
"column": 47
} | {
"line": 100,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Module α ℝ\ninst✝ : IsOrderedModule α ℝ\na : α\nha : a ≤ 0\ns : Set ℝ\n⊢ sSup (a • s) = a • sInf s",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsOrderedModul... | [] | obtain rfl | hs := s.eq_empty_or_nonempty
· rw [smul_set_empty, Real.sSup_empty, Real.sInf_empty, smul_zero]
obtain rfl | ha' := ha.eq_or_lt
· rw [zero_smul_set hs, zero_smul]
exact csSup_singleton 0
by_cases h : BddBelow s
· exact ((OrderIso.smulRightDual ℝ ha').map_csInf' hs h).symm
· rw [Real.sSup_of... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Real.Pointwise | {
"line": 90,
"column": 2
} | {
"line": 98,
"column": 47
} | {
"line": 100,
"column": 0
} | [
{
"pp": "α : Type u_2\ninst✝⁴ : Field α\ninst✝³ : LinearOrder α\ninst✝² : IsStrictOrderedRing α\ninst✝¹ : Module α ℝ\ninst✝ : IsOrderedModule α ℝ\na : α\nha : a ≤ 0\ns : Set ℝ\n⊢ sSup (a • s) = a • sInf s",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"IsOrderedModul... | [] | obtain rfl | hs := s.eq_empty_or_nonempty
· rw [smul_set_empty, Real.sSup_empty, Real.sInf_empty, smul_zero]
obtain rfl | ha' := ha.eq_or_lt
· rw [zero_smul_set hs, zero_smul]
exact csSup_singleton 0
by_cases h : BddBelow s
· exact ((OrderIso.smulRightDual ℝ ha').map_csInf' hs h).symm
· rw [Real.sSup_of... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Pseudo.Basic | {
"line": 42,
"column": 12
} | {
"line": 42,
"column": 57
} | {
"line": 42,
"column": 58
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\nm n✝ n : ℕ\nhle : m ≤ n\nihn : dist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, dist (f i) (f (i + 1))\n⊢ ∑ i ∈ Finset.Ico m n, dist (f i) (f (i + 1)) + dist (f n) (f (n + 1)) =\n ∑ i ∈ Finset.Ico m (n + 1), dist (f i) (f (i + 1))",
"ppTerm": "?m.138",... | [
"α : Type u\ninst✝ : PseudoMetricSpace α\nf : ℕ → α\nm n✝ n : ℕ\nhle : m ≤ n\nihn : dist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, dist (f i) (f (i + 1))\n⊢ ∑ i ∈ Finset.Ico m n, dist (f i) (f (i + 1)) + dist (f n) (f (n + 1)) =\n ∑ i ∈ insert n (Finset.Ico m n), dist (f i) (f (i + 1))"
] | ← Finset.insert_Ico_right_eq_Ico_add_one hle, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 667,
"column": 11
} | {
"line": 667,
"column": 30
} | {
"line": 667,
"column": 31
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nx : R\nhx : x ≠ 0\ny : R\nhxI : x ∈ Ideal.span {x} ⊔ Ideal.span {y}\n⊢ ∃ y_1, Ideal.span {x} ⊔ Ideal.span {y} = Ideal.span {x, y_1}",
"ppTerm": "?m.57",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDedekindDomain R\nx : R\nhx : x ≠ 0\ny : R\nhxI : x ∈ Ideal.span {x} ⊔ Ideal.span {y}\n⊢ ∃ y_1, Ideal.span ({x} ∪ {y}) = Ideal.span {x, y_1}"
] | ← Ideal.span_union, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.EMetricSpace.Basic | {
"line": 45,
"column": 12
} | {
"line": 45,
"column": 57
} | {
"line": 45,
"column": 58
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nm n✝ n : ℕ\nhle : m ≤ n\nihn : edist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, edist (f i) (f (i + 1))\n⊢ ∑ i ∈ Finset.Ico m n, edist (f i) (f (i + 1)) + edist (f n) (f (n + 1)) =\n ∑ i ∈ Finset.Ico m (n + 1), edist (f i) (f (i + 1))",
"ppTerm": "?m... | [
"α : Type u\ninst✝ : PseudoEMetricSpace α\nf : ℕ → α\nm n✝ n : ℕ\nhle : m ≤ n\nihn : edist (f m) (f n) ≤ ∑ i ∈ Finset.Ico m n, edist (f i) (f (i + 1))\n⊢ ∑ i ∈ Finset.Ico m n, edist (f i) (f (i + 1)) + edist (f n) (f (n + 1)) =\n ∑ i ∈ insert n (Finset.Ico m n), edist (f i) (f (i + 1))"
] | ← Finset.insert_Ico_right_eq_Ico_add_one hle, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Pseudo.Basic | {
"line": 207,
"column": 25
} | {
"line": 207,
"column": 37
} | {
"line": 207,
"column": 37
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : PseudoMetricSpace α\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\nu : β → α\n⊢ Tendsto (fun x ↦ dist (u x.1) (Prod.map u u x).2) atTop (𝓝 0) ↔ Tendsto (fun n ↦ dist (u n.1) (u n.2)) atTop (𝓝 0)",
"ppTerm": "?m.44",
"assigned": true,
"usedConstants": [
... | [] | Prod.map_snd | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Order.DenselyOrdered | {
"line": 128,
"column": 22
} | {
"line": 128,
"column": 47
} | {
"line": 128,
"column": 47
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderTopology α\ninst✝² : DenselyOrdered α\ninst✝¹ : NoMinOrder α\ninst✝ : NoMaxOrder α\na b x : α\n⊢ Icc a b ∈ 𝓝 x ↔ x ∈ interior (Icc a b)",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"Filter.i... | [
"α : Type u_1\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderTopology α\ninst✝² : DenselyOrdered α\ninst✝¹ : NoMinOrder α\ninst✝ : NoMaxOrder α\na b x : α\n⊢ Icc a b ∈ 𝓝 x ↔ Icc a b ∈ 𝓝 x"
] | mem_interior_iff_mem_nhds | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.DenselyOrdered | {
"line": 136,
"column": 22
} | {
"line": 136,
"column": 47
} | {
"line": 136,
"column": 47
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\ninst✝ : NoMinOrder α\na b x : α\n⊢ Ico a b ∈ 𝓝 x ↔ x ∈ interior (Ico a b)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"... | [
"α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\ninst✝ : NoMinOrder α\na b x : α\n⊢ Ico a b ∈ 𝓝 x ↔ Ico a b ∈ 𝓝 x"
] | mem_interior_iff_mem_nhds | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.DenselyOrdered | {
"line": 144,
"column": 22
} | {
"line": 144,
"column": 47
} | {
"line": 144,
"column": 47
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\ninst✝ : NoMaxOrder α\na b x : α\n⊢ Ioc a b ∈ 𝓝 x ↔ x ∈ interior (Ioc a b)",
"ppTerm": "?m.26",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
"... | [
"α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\ninst✝ : NoMaxOrder α\na b x : α\n⊢ Ioc a b ∈ 𝓝 x ↔ Ioc a b ∈ 𝓝 x"
] | mem_interior_iff_mem_nhds | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.IsLUB | {
"line": 348,
"column": 2
} | {
"line": 348,
"column": 31
} | {
"line": 349,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderTopology α\ninst✝² : DenselyOrdered α\ninst✝¹ : NoMinOrder α\ninst✝ : FirstCountableTopology α\ns : Set α\nhs : Dense s\nx : α\n⊢ ∃ u, StrictMono u ∧ (∀ (n : ℕ), u n ∈ Iio x ∩ s) ∧ Tendsto u atTop (𝓝 x)",
"ppTerm": "?... | [
"α : Type u_1\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderTopology α\ninst✝² : DenselyOrdered α\ninst✝¹ : NoMinOrder α\ninst✝ : FirstCountableTopology α\ns : Set α\nhs : Dense s\nx y : α\nhy : y < x\n⊢ ∃ u, StrictMono u ∧ (∀ (n : ℕ), u n ∈ Iio x ∩ s) ∧ Tendsto u atTop (𝓝 x)"
] | obtain ⟨y, hy⟩ := exists_lt x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Order.IsLUB | {
"line": 359,
"column": 2
} | {
"line": 359,
"column": 23
} | {
"line": 360,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderTopology α\nβ : Type u_3\ninst✝² : LinearOrder β\ninst✝¹ : DenselyOrdered α\ninst✝ : FirstCountableTopology α\nf : β → α\nx y : α\nhf : DenseRange f\nhmono : Monotone f\nhlt : y < x\nu : ℕ → α\nhu : StrictMono u\nhuyxf : ∀... | [
"α : Type u_1\ninst✝⁵ : TopologicalSpace α\ninst✝⁴ : LinearOrder α\ninst✝³ : OrderTopology α\nβ : Type u_3\ninst✝² : LinearOrder β\ninst✝¹ : DenselyOrdered α\ninst✝ : FirstCountableTopology α\nf : β → α\nx y : α\nhf : DenseRange f\nhmono : Monotone f\nhlt : y < x\nu : ℕ → α\nhu : StrictMono u\nhuyxf : ∀ (n : ℕ), u ... | choose v hv using huf | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.Topology.Order.IsLUB | {
"line": 370,
"column": 2
} | {
"line": 370,
"column": 23
} | {
"line": 371,
"column": 2
} | [
{
"pp": "α : Type u_1\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : OrderTopology α\nβ : Type u_3\ninst✝³ : LinearOrder β\ninst✝² : DenselyOrdered α\ninst✝¹ : NoMinOrder α\ninst✝ : FirstCountableTopology α\nf : β → α\nhf : DenseRange f\nhmono : Monotone f\nx : α\nu : ℕ → α\nhu : StrictMono u\nh... | [
"α : Type u_1\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : LinearOrder α\ninst✝⁴ : OrderTopology α\nβ : Type u_3\ninst✝³ : LinearOrder β\ninst✝² : DenselyOrdered α\ninst✝¹ : NoMinOrder α\ninst✝ : FirstCountableTopology α\nf : β → α\nhf : DenseRange f\nhmono : Monotone f\nx : α\nu : ℕ → α\nhu : StrictMono u\nhuxf : ∀ (n :... | choose v hv using huf | Mathlib.Tactic.Choose._aux_Mathlib_Tactic_Choose___elabRules_Mathlib_Tactic_Choose_choose_1 | Mathlib.Tactic.Choose.choose |
Mathlib.Topology.Order.Monotone | {
"line": 69,
"column": 4
} | {
"line": 71,
"column": 18
} | {
"line": 72,
"column": 2
} | [
{
"pp": "case h'\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : LinearOrder β\ns : Set α\nf : α → β\ninst✝ : SecondCountableTopology α\nhf : MonotoneOn f s\na✝ : Nontrivial α\nt : Set β := ⋯\nx y : β → α\nhxs : ∀ c ∈ t, x c ∈ s\nhys : ∀ c ∈ t... | [] | rintro a ⟨c, hc, rfl⟩
rw [hfx _ hc]
exact hxy _ hc | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.Monotone | {
"line": 69,
"column": 4
} | {
"line": 71,
"column": 18
} | {
"line": 72,
"column": 2
} | [
{
"pp": "case h'\nα : Type u_1\nβ : Type u_2\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : LinearOrder β\ns : Set α\nf : α → β\ninst✝ : SecondCountableTopology α\nhf : MonotoneOn f s\na✝ : Nontrivial α\nt : Set β := ⋯\nx y : β → α\nhxs : ∀ c ∈ t, x c ∈ s\nhys : ∀ c ∈ t... | [] | rintro a ⟨c, hc, rfl⟩
rw [hfx _ hc]
exact hxy _ hc | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.DedekindDomain.Factorization | {
"line": 745,
"column": 6
} | {
"line": 745,
"column": 20
} | {
"line": 745,
"column": 21
} | [
{
"pp": "R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI J I' J' : FractionalIdeal R⁰ K\nH : I * J' = I' * J\nh : J ≤ I\nh' : J' ≤ I'\nhJ' : J' ≠ 0\nhI : I ≠ 0\nthis : (J' ⊓ spanSingleton... | [
"R : Type u_1\ninst✝⁴ : CommRing R\nK : Type u_2\ninst✝³ : Field K\ninst✝² : Algebra R K\ninst✝¹ : IsFractionRing R K\ninst✝ : IsDedekindDomain R\nv : HeightOneSpectrum R\nI J I' J' : FractionalIdeal R⁰ K\nH : I * J' = I' * J\nh : J ≤ I\nh' : J' ≤ I'\nhJ' : J' ≠ 0\nhI : I ≠ 0\nthis : (J' ⊓ spanSingleton R⁰ (I'.divM... | mul_comm I' J, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 919,
"column": 2
} | {
"line": 921,
"column": 75
} | {
"line": 923,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedCommGroup E\na b : E\nr : ℝ\nn : ℕ\nh : a ∈ closedBall b r\n⊢ a ^ n ∈ closedBall (b ^ n) (n • r)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMono... | [] | simp only [mem_closedBall, dist_eq_norm_inv_mul, ← inv_pow, ← mul_pow] at h ⊢
refine norm_pow_le_mul_norm.trans ?_
simpa only [nsmul_eq_mul] using mul_le_mul_of_nonneg_left h n.cast_nonneg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 919,
"column": 2
} | {
"line": 921,
"column": 75
} | {
"line": 923,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : SeminormedCommGroup E\na b : E\nr : ℝ\nn : ℕ\nh : a ∈ closedBall b r\n⊢ a ^ n ∈ closedBall (b ^ n) (n • r)",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Real.instIsOrderedRing",
"Norm.norm",
"Eq.mpr",
"NonAssocSemiring.toAddCommMono... | [] | simp only [mem_closedBall, dist_eq_norm_inv_mul, ← inv_pow, ← mul_pow] at h ⊢
refine norm_pow_le_mul_norm.trans ?_
simpa only [nsmul_eq_mul] using mul_le_mul_of_nonneg_left h n.cast_nonneg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Basic | {
"line": 1003,
"column": 54
} | {
"line": 1003,
"column": 87
} | {
"line": 1005,
"column": 0
} | [
{
"pp": "E : Type u_5\ninst✝ : NormedGroup E\na b : E\n⊢ ‖a / b‖ = 0 ↔ a = b",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Eq.mpr",
"Real",
"instHDiv",
"InvOneClass.toOne",
"DivInvOneMonoid.toInvOneClass",
"Real.instZero",
"co... | [] | by rw [norm_eq_zero', div_eq_one] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.IntermediateValue | {
"line": 342,
"column": 48
} | {
"line": 342,
"column": 53
} | {
"line": 343,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : OrderTopology α\na b : α\ns : Set α\nhs : IsClosed[inst✝²] (s ∩ Icc a b)\nhb : b ∈ s\nhab : a ≤ b\nhgt : ∀ x ∈ s ∩ Ioc a b, (s ∩ Ico a x).Nonempty\nthis : toDual a ∈ ⇑ofDual ⁻¹' s\n⊢ a ∈ s",
"ppTerm": "?m.... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Order.IntermediateValue | {
"line": 342,
"column": 48
} | {
"line": 342,
"column": 53
} | {
"line": 343,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : OrderTopology α\na b : α\ns : Set α\nhs : IsClosed[inst✝²] (s ∩ Icc a b)\nhb : b ∈ s\nhab : a ≤ b\nhgt : ∀ x ∈ s ∩ Ioc a b, (s ∩ Ico a x).Nonempty\nthis : toDual a ∈ ⇑ofDual ⁻¹' s\n⊢ a ∈ s",
"ppTerm": "?m.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.IntermediateValue | {
"line": 342,
"column": 48
} | {
"line": 342,
"column": 53
} | {
"line": 343,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : OrderTopology α\na b : α\ns : Set α\nhs : IsClosed[inst✝²] (s ∩ Icc a b)\nhb : b ∈ s\nhab : a ≤ b\nhgt : ∀ x ∈ s ∩ Ioc a b, (s ∩ Ico a x).Nonempty\nthis : toDual a ∈ ⇑ofDual ⁻¹' s\n⊢ a ∈ s",
"ppTerm": "?m.... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.Diam | {
"line": 80,
"column": 31
} | {
"line": 80,
"column": 77
} | {
"line": 82,
"column": 0
} | [
{
"pp": "X : Type u_2\ns : Set X\nx : X\ninst✝ : PseudoEMetricSpace X\nd : ℝ≥0∞\n⊢ ediam (insert x s) ≤ d ↔ max (⨆ y ∈ s, edist x y) (ediam s) ≤ d",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"PseudoEMetricSpace.edist_comm",
"Pseudo... | [] | by simp +contextual [ediam_le_iff, edist_comm] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.IntermediateValue | {
"line": 347,
"column": 2
} | {
"line": 347,
"column": 7
} | {
"line": 349,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝² : TopologicalSpace α\ninst✝¹ : ConditionallyCompleteLinearOrder α\ninst✝ : OrderTopology α\na b : α\ns : Set α\nhs : IsClosed[inst✝²] (s ∩ Icc a b)\nhb : b ∈ s\nhab : a ≤ b\nhgt : ∀ x ∈ s ∩ Ioc a b, (s ∩ Ico a x).Nonempty\nthis : IsClosed[instTopologicalSpace] (⇑ofDual ⁻¹' s ∩ Icc (t... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 503,
"column": 34
} | {
"line": 503,
"column": 52
} | {
"line": 503,
"column": 52
} | [
{
"pp": "α : Type u\ns : Set α\ninst✝ : PseudoMetricSpace α\nh : ¬Bornology.IsBounded s\n⊢ ∞.toReal = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"id",
"ENNReal.toReal",
"ENNReal",
"Zero.to... | [
"α : Type u\ns : Set α\ninst✝ : PseudoMetricSpace α\nh : ¬Bornology.IsBounded s\n⊢ 0 = 0"
] | ENNReal.toReal_top | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 515,
"column": 7
} | {
"line": 515,
"column": 63
} | {
"line": 515,
"column": 64
} | [
{
"pp": "α : Type u\ns : Set α\nx y : α\ninst✝ : PseudoMetricSpace α\nt : Set α\nxs : x ∈ s\nyt : y ∈ t\n⊢ (ediam (s ∪ t)).toReal ≤ (ediam s).toReal + (edist x y).toReal + (ediam t).toReal",
"ppTerm": "?m.28",
"assigned": true,
"usedConstants": [
"ENNReal.instAdd",
"le_refl",
"Pseu... | [
"α : Type u\ns : Set α\nx y : α\ninst✝ : PseudoMetricSpace α\nt : Set α\nxs : x ∈ s\nyt : y ∈ t\n⊢ (ediam s + edist x y).toReal + (ediam t).toReal ≤ (ediam s).toReal + (edist x y).toReal + (ediam t).toReal",
"case hb\nα : Type u\ns : Set α\nx y : α\ninst✝ : PseudoMetricSpace α\nt : Set α\nxs : x ∈ s\nyt : y ∈ t\n... | ENNReal.toReal_le_add' (ediam_union_le_add_edist xs yt), | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Topology.Order.IntermediateValue | {
"line": 814,
"column": 30
} | {
"line": 814,
"column": 35
} | {
"line": 815,
"column": 2
} | [
{
"pp": "α : Type u\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : ConditionallyCompleteLinearOrder α\ninst✝⁴ : OrderTopology α\ninst✝³ : DenselyOrdered α\nδ : Type u_1\ninst✝² : LinearOrder δ\ninst✝¹ : TopologicalSpace δ\ninst✝ : OrderClosedTopology δ\na b : α\nf : α → δ\nhab : a ≤ b\nhfab : f a ≤ f b\nhf_c : Continuo... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Algebra.Ring.Real | {
"line": 123,
"column": 2
} | {
"line": 123,
"column": 77
} | {
"line": 124,
"column": 2
} | [
{
"pp": "case some.none\nα : Type u\nβ : Type v\nγ : Type w\na : ℝ≥0\n⊢ ContinuousAt (fun p ↦ p.1 + p.2) (some a, none)",
"ppTerm": "?some.none",
"assigned": true,
"usedConstants": [
"ENNReal.instCanonicallyOrderedAdd",
"ENNReal.instAdd",
"le_rfl",
"PartialOrder.toPreorder",
... | [
"case some.some\nα : Type u\nβ : Type v\nγ : Type w\na b : ℝ≥0\n⊢ ContinuousAt (fun p ↦ p.1 + p.2) (some a, some b)"
] | · exact tendsto_nhds_top_mono' continuousAt_snd fun p => le_add_left le_rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Order.Filter.IsBounded | {
"line": 607,
"column": 64
} | {
"line": 612,
"column": 40
} | {
"line": 614,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : LinearOrder β\ninst✝ : OrderBot β\nf : Filter α\nF : ι → α → β\ns : Finset ι\nh : ∀ i ∈ s, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (F i)\n⊢ IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f fun a ↦ s.sup fun i ↦ F i a",
"ppTerm": "?m.30",
"assigned": true,... | [] | by
choose! m hm using h
use sup s m
simp only [eventually_map] at hm ⊢
rw [← eventually_all_finset s] at hm
exact hm.mono fun _ h ↦ sup_mono_fun h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.LiminfLimsup | {
"line": 626,
"column": 2
} | {
"line": 626,
"column": 50
} | {
"line": 627,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\ninst✝ : CompleteLattice γ\ng : sSupHom α γ\n⊢ g (blimsup u f p) ≤ blimsup (⇑g ∘ u) f p",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Filter.instMembership",
... | [
"α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : CompleteLattice α\nf : Filter β\np : β → Prop\nu : β → α\ninst✝ : CompleteLattice γ\ng : sSupHom α γ\n⊢ g (⨅ s ∈ f, ⨆ b, ⨆ (_ : p b ∧ b ∈ s), u b) ≤ ⨅ s ∈ f, ⨆ b, ⨆ (_ : p b ∧ b ∈ s), g (u b)"
] | simp only [blimsup_eq_iInf_biSup, Function.comp] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.EReal.Operations | {
"line": 522,
"column": 2
} | {
"line": 522,
"column": 7
} | {
"line": 524,
"column": 0
} | [
{
"pp": "x y : EReal\nh : ∀ (z : ℝ), x < ↑z → y ≤ ↑z\n⊢ (x < ↑⊤ → y ≤ ↑⊤) ∧ ∀ (x_1 : ℝ), x < ↑↑x_1 → y ≤ ↑↑x_1",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"Real",
"WithBot.some",
"WithBot",
"Preorder.toLT",
"La... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Data.EReal.Operations | {
"line": 527,
"column": 2
} | {
"line": 527,
"column": 7
} | {
"line": 529,
"column": 0
} | [
{
"pp": "x y : EReal\nh : ∀ (z : ℝ), ↑z < y → ↑z ≤ x\n⊢ (↑⊤ < y → ↑⊤ ≤ x) ∧ ∀ (x_1 : ℝ), ↑↑x_1 < y → ↑↑x_1 ≤ x",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"False",
"WithTop.instPartialOrder",
"Real.partialOrder",
"Re... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Order.LiminfLimsup | {
"line": 769,
"column": 69
} | {
"line": 775,
"column": 38
} | {
"line": 777,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\nl : Filter β\nb : ι → Set β\nq : ι → Prop\nhl : l.HasBasis q b\nu : β → Set α\np : β → Prop\nx : α\nhx : x ∈ blimsup u l p\n⊢ ∃ f, ∀ (i : ↑{i | q i}), x ∈ u (f i) ∧ p (f i) ∧ f i ∈ b ↑i",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
... | [] | by
rw [blimsup_eq_iInf_biSup] at hx
simp only [iSup_eq_iUnion, iInf_eq_iInter, mem_iInter, mem_iUnion, exists_prop] at hx
choose g hg hg' using hx
refine ⟨fun i : { i | q i } => g (b i) (hl.mem_of_mem i.2), fun i => ⟨?_, ?_⟩⟩
· exact hg' (b i) (hl.mem_of_mem i.2)
· exact hg (b i) (hl.mem_of_mem i.2) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.EReal.Operations | {
"line": 728,
"column": 19
} | {
"line": 729,
"column": 42
} | {
"line": 731,
"column": 0
} | [
{
"pp": "case pos_bot\nx✝ : ℝ\nh : 0 < x✝\n⊢ -↑x✝ * ⊥ = -(↑x✝ * ⊥)",
"ppTerm": "?pos_bot",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.partialOrder",
"Real",
"HMul.hMul",
"congrArg",
"EReal.coe_neg",
"EReal.instNeg",
... | [] | rw [coe_mul_bot_of_pos h, neg_bot, ← coe_neg,
coe_mul_bot_of_neg (neg_neg_of_pos h)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.EReal.Operations | {
"line": 728,
"column": 19
} | {
"line": 729,
"column": 42
} | {
"line": 731,
"column": 0
} | [
{
"pp": "case pos_bot\nx✝ : ℝ\nh : 0 < x✝\n⊢ -↑x✝ * ⊥ = -(↑x✝ * ⊥)",
"ppTerm": "?pos_bot",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.partialOrder",
"Real",
"HMul.hMul",
"congrArg",
"EReal.coe_neg",
"EReal.instNeg",
... | [] | rw [coe_mul_bot_of_pos h, neg_bot, ← coe_neg,
coe_mul_bot_of_neg (neg_neg_of_pos h)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.EReal.Operations | {
"line": 728,
"column": 19
} | {
"line": 729,
"column": 42
} | {
"line": 731,
"column": 0
} | [
{
"pp": "case pos_bot\nx✝ : ℝ\nh : 0 < x✝\n⊢ -↑x✝ * ⊥ = -(↑x✝ * ⊥)",
"ppTerm": "?pos_bot",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"Real.partialOrder",
"Real",
"HMul.hMul",
"congrArg",
"EReal.coe_neg",
"EReal.instNeg",
... | [] | rw [coe_mul_bot_of_pos h, neg_bot, ← coe_neg,
coe_mul_bot_of_neg (neg_neg_of_pos h)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.EMetricSpace.Lipschitz | {
"line": 270,
"column": 31
} | {
"line": 270,
"column": 73
} | {
"line": 272,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : PseudoEMetricSpace α\nK : ℝ≥0\nf : α → α\nhf : LipschitzWith K f\nn : ℕ\n⊢ LipschitzWith (K ^ n * K) f^[n + 1]",
"ppTerm": "?m.53",
"assigned": true,
"usedConstants": [
"NNReal",
"Nat.iterate",
"NPow.toPow",
"HPow.hPow",
"Nat",
"Semiri... | [] | exact (LipschitzWith.iterate hf n).comp hf | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.InfiniteSum.SummationFilter | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 28
} | {
"line": 208,
"column": 2
} | [
{
"pp": "case mk\nβ : Type u_4\ninst✝² : Finite β\nthis : Fintype β\nF : Filter (Finset β)\ninst✝¹ : { filter := F }.LeAtTop\ninst✝ : { filter := F }.NeBot\nhAtTop : True\nhL : F ≤ pure Finset.univ\nhL' : ∃ s ∈ F, Finset.univ ∉ s\n⊢ ∅ ∈ F",
"ppTerm": "?mk",
"assigned": true,
"usedConstants": [
... | [
"case mk\nβ : Type u_4\ninst✝² : Finite β\nthis : Fintype β\nF : Filter (Finset β)\ninst✝¹ : { filter := F }.LeAtTop\ninst✝ : { filter := F }.NeBot\nhAtTop : True\nhL : F ≤ pure Finset.univ\ns : Set (Finset β)\nhs : s ∈ F\nhs' : Finset.univ ∉ s\n⊢ ∅ ∈ F"
] | obtain ⟨s, hs, hs'⟩ := hL' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.Algebra.InfiniteSum.Defs | {
"line": 187,
"column": 41
} | {
"line": 187,
"column": 62
} | {
"line": 187,
"column": 62
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nL : SummationFilter β\nhL : ¬L.NeBot\nf : β → α\nthis : L.LeAtTop\n⊢ (if L.HasSupport ∧ (mulSupport fun b ↦ f b).Finite then finprod (Set.univ.mulIndicator fun b ↦ f b)\n else if HasProd (fun b ↦ f b) 1 L then 1 else Exis... | [
"α : Type u_1\nβ : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nL : SummationFilter β\nhL : ¬L.NeBot\nf : β → α\nthis : L.LeAtTop\n⊢ (if L.HasSupport ∧ (mulSupport fun b ↦ f b).Finite then ∏ᶠ (b : β), f b\n else if HasProd (fun b ↦ f b) 1 L then 1 else Exists.choose ⋯) =\n ∏ᶠ (b : β), f b"
] | Set.mulIndicator_univ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.InfiniteSum.Defs | {
"line": 319,
"column": 6
} | {
"line": 319,
"column": 15
} | {
"line": 320,
"column": 2
} | [
{
"pp": "case pos.convert_6\nα : Type u_1\nβ : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nL : SummationFilter β\nf : β → α\nha : Multipliable f L\nh : L.HasSupport ∧ ((mulSupport fun b ↦ f b) ∩ L.support).Finite\n⊢ L.HasSupport",
"ppTerm": "?pos.convert_6✝",
"assigned": true,
"used... | [] | exact h.1 | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.InfiniteSum.Defs | {
"line": 319,
"column": 6
} | {
"line": 319,
"column": 15
} | {
"line": 320,
"column": 2
} | [
{
"pp": "case pos.convert_6\nα : Type u_1\nβ : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nL : SummationFilter β\nf : β → α\nha : Multipliable f L\nh : L.HasSupport ∧ ((mulSupport fun b ↦ f b) ∩ L.support).Finite\n⊢ L.HasSupport",
"ppTerm": "?pos.convert_6✝",
"assigned": true,
"used... | [] | exact h.1 | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.Defs | {
"line": 319,
"column": 6
} | {
"line": 319,
"column": 15
} | {
"line": 320,
"column": 2
} | [
{
"pp": "case pos.convert_6\nα : Type u_1\nβ : Type u_2\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nL : SummationFilter β\nf : β → α\nha : Multipliable f L\nh : L.HasSupport ∧ ((mulSupport fun b ↦ f b) ∩ L.support).Finite\n⊢ L.HasSupport",
"ppTerm": "?pos.convert_6✝",
"assigned": true,
"used... | [] | exact h.1 | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 156,
"column": 4
} | {
"line": 158,
"column": 59
} | {
"line": 160,
"column": 0
} | [
{
"pp": "case neg\nι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝³ : CommMonoid α\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf : ι → α\na₂ : α\nha₂ : 1 ≤ a₂\nh : ∀ (s : Finset ι), ∏ i ∈ s, f i ≤ a₂\nhL : ¬L.NeBot\n⊢ ∏'[L] (i : ι), f i ≤ a₂",
"ppTerm": "?neg✝"... | [] | by_cases hf : f.mulSupport.Finite
· simpa [tprod_bot hL, finprod_eq_prod _ hf] using h _
· rwa [tprod_bot hL, finprod_of_infinite_mulSupport hf] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 156,
"column": 4
} | {
"line": 158,
"column": 59
} | {
"line": 160,
"column": 0
} | [
{
"pp": "case neg\nι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝³ : CommMonoid α\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf : ι → α\na₂ : α\nha₂ : 1 ≤ a₂\nh : ∀ (s : Finset ι), ∏ i ∈ s, f i ≤ a₂\nhL : ¬L.NeBot\n⊢ ∏'[L] (i : ι), f i ≤ a₂",
"ppTerm": "?neg✝"... | [] | by_cases hf : f.mulSupport.Finite
· simpa [tprod_bot hL, finprod_eq_prod _ hf] using h _
· rwa [tprod_bot hL, finprod_of_infinite_mulSupport hf] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 453,
"column": 35
} | {
"line": 454,
"column": 93
} | {
"line": 456,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf : β → α\nL : SummationFilter β\ninst✝ : L.LeAtTop\ns : Finset β\nhf : mulSupport f ⊆ ↑s\n⊢ ∏'[L] (b : β), f b = ∏ b ∈ s, f b",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Mul... | [] | by
rw [tprod_eq_finprod (s.finite_toSet.subset hf), finprod_eq_prod_of_mulSupport_subset _ hf] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 410,
"column": 4
} | {
"line": 414,
"column": 45
} | {
"line": 416,
"column": 0
} | [
{
"pp": "case inr\nβ : Type u_2\nG : Type u_4\ninst✝³ : TopologicalSpace G\ninst✝² : CommGroup G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : T2Space G\na : G\nhβ : Infinite β\n⊢ ∏' (x : β), a = a ^ Nat.card β",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
... | [] | simp only [Nat.card_eq_zero_of_infinite, pow_zero]
rcases eq_or_ne a 1 with rfl | ha
· simp
· apply tprod_eq_one_of_not_multipliable
simpa [multipliable_const_iff] using ha | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 410,
"column": 4
} | {
"line": 414,
"column": 45
} | {
"line": 416,
"column": 0
} | [
{
"pp": "case inr\nβ : Type u_2\nG : Type u_4\ninst✝³ : TopologicalSpace G\ninst✝² : CommGroup G\ninst✝¹ : IsTopologicalGroup G\ninst✝ : T2Space G\na : G\nhβ : Infinite β\n⊢ ∏' (x : β), a = a ^ Nat.card β",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
... | [] | simp only [Nat.card_eq_zero_of_infinite, pow_zero]
rcases eq_or_ne a 1 with rfl | ha
· simp
· apply tprod_eq_one_of_not_multipliable
simpa [multipliable_const_iff] using ha | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 666,
"column": 4
} | {
"line": 666,
"column": 9
} | {
"line": 667,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\ns : γ → Set β\nf : β → α\ni : β\nhi : i ∈ ⋃ d, s d\nhs : Pairwise (Disjoint on fun j ↦ s j ∩ mulSupport f)\nj : γ\nhj : i ∈ s j\n⊢ ∏' (x : ↑{j}), (s ↑x).mulIndicator f i = f i",
"ppTerm": "?m.58",
"assi... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 666,
"column": 4
} | {
"line": 666,
"column": 9
} | {
"line": 667,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\ns : γ → Set β\nf : β → α\ni : β\nhi : i ∈ ⋃ d, s d\nhs : Pairwise (Disjoint on fun j ↦ s j ∩ mulSupport f)\nj : γ\nhj : i ∈ s j\n⊢ ∏' (x : ↑{j}), (s ↑x).mulIndicator f i = f i",
"ppTerm": "?m.58",
"assi... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 666,
"column": 4
} | {
"line": 666,
"column": 9
} | {
"line": 667,
"column": 2
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\ns : γ → Set β\nf : β → α\ni : β\nhi : i ∈ ⋃ d, s d\nhs : Pairwise (Disjoint on fun j ↦ s j ∩ mulSupport f)\nj : γ\nhj : i ∈ s j\n⊢ ∏' (x : ↑{j}), (s ↑x).mulIndicator f i = f i",
"ppTerm": "?m.58",
"assi... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 679,
"column": 2
} | {
"line": 679,
"column": 7
} | {
"line": 681,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\ns : γ → Set β\nf : β → α\ni : β\nhi : ∀ (d : γ), i ∉ s d\n⊢ ∏' (d : γ), (s d).mulIndicator f i = 1",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"False",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 679,
"column": 2
} | {
"line": 679,
"column": 7
} | {
"line": 681,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\ns : γ → Set β\nf : β → α\ni : β\nhi : ∀ (d : γ), i ∉ s d\n⊢ ∏' (d : γ), (s d).mulIndicator f i = 1",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"False",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 679,
"column": 2
} | {
"line": 679,
"column": 7
} | {
"line": 681,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\ns : γ → Set β\nf : β → α\ni : β\nhi : ∀ (d : γ), i ∉ s d\n⊢ ∏' (d : γ), (s d).mulIndicator f i = 1",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"MulOne.toOne",
"False",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 687,
"column": 4
} | {
"line": 687,
"column": 9
} | {
"line": 689,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\ns : γ → Set β\nf : β → α\nhs : Pairwise (Disjoint on fun j ↦ s j ∩ mulSupport f)\ni : β\nh₀ : i ∉ ⋃ d, s d\n⊢ (⋃ d, s d).mulIndicator f i = ∏' (d : γ), (s d).mulIndicator f i",
"ppTerm": "?neg✝",
... | [] | aesop | Aesop.evalAesop | Aesop.Frontend.Parser.aesopTactic |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 687,
"column": 4
} | {
"line": 687,
"column": 9
} | {
"line": 689,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\ns : γ → Set β\nf : β → α\nhs : Pairwise (Disjoint on fun j ↦ s j ∩ mulSupport f)\ni : β\nh₀ : i ∉ ⋃ d, s d\n⊢ (⋃ d, s d).mulIndicator f i = ∏' (d : γ), (s d).mulIndicator f i",
"ppTerm": "?neg✝",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 687,
"column": 4
} | {
"line": 687,
"column": 9
} | {
"line": 689,
"column": 0
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\ns : γ → Set β\nf : β → α\nhs : Pairwise (Disjoint on fun j ↦ s j ∩ mulSupport f)\ni : β\nh₀ : i ∉ ⋃ d, s d\n⊢ (⋃ d, s d).mulIndicator f i = ∏' (d : γ), (s d).mulIndicator f i",
"ppTerm": "?neg✝",
... | [] | aesop | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.Ring | {
"line": 145,
"column": 59
} | {
"line": 146,
"column": 69
} | {
"line": 148,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝² : DivisionSemiring α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsTopologicalSemiring α\nf : ι → α\na : α\nh : a ≠ 0\n⊢ Summable (fun i ↦ a / f i) L ↔ Summable (1 / f) L",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"Eq.... | [] | by
simpa only [div_eq_mul_inv, one_mul] using! summable_mul_left_iff h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Instances.NNReal.Lemmas | {
"line": 189,
"column": 2
} | {
"line": 190,
"column": 60
} | {
"line": 192,
"column": 0
} | [
{
"pp": "f : ℕ → ℝ≥0\nk : ℕ\n⊢ (Summable fun i ↦ f (i + k)) ↔ Summable f",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"NNReal.summable_coe",
"Eq.mpr",
"summable_nat_add_iff",
"Real",
"congrArg",
"PseudoMetricSpace.... | [] | rw [← summable_coe, ← summable_coe]
exact @summable_nat_add_iff ℝ _ _ _ (fun i => (f i : ℝ)) k | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.NNReal.Lemmas | {
"line": 189,
"column": 2
} | {
"line": 190,
"column": 60
} | {
"line": 192,
"column": 0
} | [
{
"pp": "f : ℕ → ℝ≥0\nk : ℕ\n⊢ (Summable fun i ↦ f (i + k)) ↔ Summable f",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"NNReal.summable_coe",
"Eq.mpr",
"summable_nat_add_iff",
"Real",
"congrArg",
"PseudoMetricSpace.... | [] | rw [← summable_coe, ← summable_coe]
exact @summable_nat_add_iff ℝ _ _ _ (fun i => (f i : ℝ)) k | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 782,
"column": 86
} | {
"line": 784,
"column": 51
} | {
"line": 786,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝³ : CommMonoidWithZero α\ninst✝² : TopologicalSpace α\nL : SummationFilter β\ninst✝¹ : Nonempty β\ninst✝ : L.LeAtTop\n⊢ HasProd (fun x ↦ 0) 0 L",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"HasProd"... | [] | by
obtain ⟨b⟩ := ‹Nonempty β›
exact hasProd_zero_of_exists_eq_zero ⟨b, by simp⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Sequences | {
"line": 195,
"column": 4
} | {
"line": 195,
"column": 69
} | {
"line": 196,
"column": 4
} | [
{
"pp": "case neg\nX : Type u_1\ninst✝¹ : TopologicalSpace X\nx : ℕ → X\ninst✝ : SequentialSpace X\nhx : ∀ (l : X) (φ : ℕ → ℕ), StrictMono φ → ¬Tendsto (x ∘ φ) atTop (𝓝 l)\ny : ℕ → X\nl : X\nhy : ∀ (n : ℕ), y n ∈ ⋃ i, closure[inst✝¹] {x i}\nhy' : Tendsto y atTop (𝓝 l)\nhm : ∀ (m : ℕ), ∀ᶠ (x_1 : ℕ) in atTop, y... | [
"case neg\nX : Type u_1\ninst✝¹ : TopologicalSpace X\nx : ℕ → X\ninst✝ : SequentialSpace X\nhx : ∀ (l : X) (φ : ℕ → ℕ), StrictMono φ → ¬Tendsto (x ∘ φ) atTop (𝓝 l)\ny : ℕ → X\nl : X\nhy : ∀ (n : ℕ), y n ∈ ⋃ i, closure[inst✝¹] {x i}\nhy' : Tendsto y atTop (𝓝 l)\nhm : ∀ (m : ℕ), ∀ᶠ (x_1 : ℕ) in atTop, y x_1 ∉ closu... | have : Tendsto ψ atTop atTop := tendsto_atTop_mono hψ1 tendsto_id | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Sequences | {
"line": 232,
"column": 75
} | {
"line": 236,
"column": 99
} | {
"line": 238,
"column": 0
} | [
{
"pp": "X : Type u_4\nι : Sort u_3\nt : ι → TopologicalSpace X\nh : ∀ (i : ι), SequentialSpace X\n⊢ SequentialSpace X",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"isClosed_iSup_iff",
"iSup",
"Membership.mem",
"nhds",
"CompleteLattice.toCon... | [] | by
letI : TopologicalSpace X := ⨆ i, t i
refine ⟨fun s hs ↦ isClosed_iSup_iff.2 fun i ↦ ?_⟩
letI := t i
exact IsSeqClosed.isClosed fun u x hus hux ↦ hs hus <| hux.mono_right <| nhds_mono <| le_iSup _ _ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.InfiniteSum.Constructions | {
"line": 86,
"column": 2
} | {
"line": 91,
"column": 47
} | {
"line": 93,
"column": 0
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nM : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nf : α ⊕ β → M\na b : M\nh₁ : HasProd (f ∘ Sum.inl) a\nh₂ : HasProd (f ∘ Sum.inr) b\n⊢ HasProd f (a * b)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Sum... | [] | have : Tendsto ((∏ b ∈ ·, f b) ∘ sumEquiv.symm) (atTop.map sumEquiv) (nhds (a * b)) := by
rw [Finset.sumEquiv.map_atTop, ← prod_atTop_atTop_eq]
convert! (tendsto_mul.comp (nhds_prod_eq (x := a) (y := b) ▸ Tendsto.prodMap h₁ h₂))
ext s
simp
simpa [Tendsto, ← Filter.map_map] using! this | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.Constructions | {
"line": 86,
"column": 2
} | {
"line": 91,
"column": 47
} | {
"line": 93,
"column": 0
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\nM : Type u_6\ninst✝² : CommMonoid M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\nf : α ⊕ β → M\na b : M\nh₁ : HasProd (f ∘ Sum.inl) a\nh₂ : HasProd (f ∘ Sum.inr) b\n⊢ HasProd f (a * b)",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Sum... | [] | have : Tendsto ((∏ b ∈ ·, f b) ∘ sumEquiv.symm) (atTop.map sumEquiv) (nhds (a * b)) := by
rw [Finset.sumEquiv.map_atTop, ← prod_atTop_atTop_eq]
convert! (tendsto_mul.comp (nhds_prod_eq (x := a) (y := b) ▸ Tendsto.prodMap h₁ h₂))
ext s
simp
simpa [Tendsto, ← Filter.map_map] using! this | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 328,
"column": 2
} | {
"line": 329,
"column": 78
} | {
"line": 331,
"column": 0
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedCommGroup E\nx : E\n⊢ Tendsto (fun a ↦ ‖a / x‖) (𝓝 x) (𝓝 0)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"instHDiv",
"Real.instZero",
"congrArg",
"Filter.tendsto_id",
"tendsto_const... | [] | simpa [dist_eq_norm_div] using
tendsto_id.dist (tendsto_const_nhds : Tendsto (fun _a => (x : E)) (𝓝 x) _) | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 328,
"column": 2
} | {
"line": 329,
"column": 78
} | {
"line": 331,
"column": 0
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedCommGroup E\nx : E\n⊢ Tendsto (fun a ↦ ‖a / x‖) (𝓝 x) (𝓝 0)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"instHDiv",
"Real.instZero",
"congrArg",
"Filter.tendsto_id",
"tendsto_const... | [] | simpa [dist_eq_norm_div] using
tendsto_id.dist (tendsto_const_nhds : Tendsto (fun _a => (x : E)) (𝓝 x) _) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Normed.Group.Continuity | {
"line": 328,
"column": 2
} | {
"line": 329,
"column": 78
} | {
"line": 331,
"column": 0
} | [
{
"pp": "E : Type u_4\ninst✝ : SeminormedCommGroup E\nx : E\n⊢ Tendsto (fun a ↦ ‖a / x‖) (𝓝 x) (𝓝 0)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Norm.norm",
"Real",
"instHDiv",
"Real.instZero",
"congrArg",
"Filter.tendsto_id",
"tendsto_const... | [] | simpa [dist_eq_norm_div] using
tendsto_id.dist (tendsto_const_nhds : Tendsto (fun _a => (x : E)) (𝓝 x) _) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 326,
"column": 2
} | {
"line": 326,
"column": 18
} | {
"line": 327,
"column": 2
} | [
{
"pp": "a b : ℝ≥0∞\nha : a ≠ 0 ∨ b ≠ ∞\nhb : b ≠ 0 ∨ a ≠ ∞\nht : ∀ (b : ℝ≥0∞), b ≠ 0 → Tendsto (fun p ↦ p.1 * p.2) (𝓝 (∞, b)) (𝓝 ∞)\n⊢ Tendsto (fun p ↦ p.1 * p.2) (𝓝 (a, b)) (𝓝 (a * b))",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"NNReal.instTopologicalSpace",
"Eq.mpr"... | [] | induction a with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Topology.Instances.ENNReal.Lemmas | {
"line": 665,
"column": 68
} | {
"line": 668,
"column": 77
} | {
"line": 670,
"column": 0
} | [
{
"pp": "α : Type u_4\nβ : Type u_5\ninst✝¹ : PseudoEMetricSpace α\ninst✝ : PseudoEMetricSpace β\nK : ℝ≥0\ns : Set α\n⊢ IsClosed[Pi.topologicalSpace] {f | LipschitzOnWith K f s}",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpa... | [] | by
simp only [LipschitzOnWith, setOf_forall]
refine isClosed_biInter fun x _ => isClosed_biInter fun y _ => isClosed_le ?_ ?_
exacts [.edist (continuous_apply x) (continuous_apply y), continuous_const] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.IsometricSMul | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 46
} | {
"line": 342,
"column": 0
} | [
{
"pp": "M : Type u\ninst✝² : DivInvMonoid M\ninst✝¹ : PseudoMetricSpace M\ninst✝ : IsIsometricSMul Mᵐᵒᵖ M\na b c : M\n⊢ nndist (a / c) (b / c) = nndist a b",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"NNDist.nndist",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul... | [] | simp only [div_eq_mul_inv, nndist_mul_right] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.IsometricSMul | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 46
} | {
"line": 342,
"column": 0
} | [
{
"pp": "M : Type u\ninst✝² : DivInvMonoid M\ninst✝¹ : PseudoMetricSpace M\ninst✝ : IsIsometricSMul Mᵐᵒᵖ M\na b c : M\n⊢ nndist (a / c) (b / c) = nndist a b",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"NNDist.nndist",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul... | [] | simp only [div_eq_mul_inv, nndist_mul_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.IsometricSMul | {
"line": 340,
"column": 2
} | {
"line": 340,
"column": 46
} | {
"line": 342,
"column": 0
} | [
{
"pp": "M : Type u\ninst✝² : DivInvMonoid M\ninst✝¹ : PseudoMetricSpace M\ninst✝ : IsIsometricSMul Mᵐᵒᵖ M\na b c : M\n⊢ nndist (a / c) (b / c) = nndist a b",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"NNDist.nndist",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul... | [] | simp only [div_eq_mul_inv, nndist_mul_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Group.Uniform | {
"line": 447,
"column": 87
} | {
"line": 448,
"column": 66
} | {
"line": 450,
"column": 0
} | [
{
"pp": "E : Type u_2\nF : Type u_3\ninst✝¹ : SeminormedCommGroup E\ninst✝ : SeminormedCommGroup F\nf : E → F\nC : ℝ≥0\ns : Set E\n⊢ LipschitzOnWith C f s ↔ ∀ ⦃x : E⦄, x ∈ s → ∀ ⦃y : E⦄, y ∈ s → ‖f x / f y‖ ≤ ↑C * ‖x / y‖",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Norm.norm",
... | [] | by
simpa [← norm_inv_mul] using lipschitzOnWith_iff_norm_inv_mul_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Normed.Group.Bounded | {
"line": 151,
"column": 2
} | {
"line": 151,
"column": 92
} | {
"line": 153,
"column": 0
} | [
{
"pp": "E : Type u_2\ninst✝³ : SeminormedGroup E\nX : Type u_5\ninst✝² : TopologicalSpace X\ninst✝¹ : DiscreteTopology X\ninst✝ : ProperSpace E\ne : X → E\nhe : Topology.IsClosedEmbedding e\n⊢ Tendsto (norm ∘ e) (cocompact X) atTop",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"No... | [] | apply tendsto_norm_cocompact_atTop'.comp (Topology.IsClosedEmbedding.tendsto_cocompact he) | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
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