module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Algebra.Polynomial.Div | {
"line": 442,
"column": 6
} | {
"line": 442,
"column": 17
} | [
{
"pp": "R : Type u\ninst✝ : Ring R\np q : R[X]\nhmo : q.Monic\na✝ : Nontrivial R\n⊢ degree 0 < q.degree",
"usedConstants": [
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"Preorder.toLT",
"congrArg",
"id",
"Bot.bot",
"Polynomial.degree",
"Polynomial.degr... | degree_zero | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Eval.SMul | {
"line": 76,
"column": 70
} | {
"line": 76,
"column": 81
} | [
{
"pp": "R : Type u\nS : Type v\ninst✝² : Semiring R\ninst✝¹ : SMulZeroClass S R\ninst✝ : IsScalarTower S R R\ns : S\np q : R[X]\n⊢ (s • 1) • eval₂ C q p = s • eval₂ C q p",
"usedConstants": [
"Eq.mpr",
"Polynomial.C",
"NonAssocSemiring.toAddCommMonoidWithOne",
"MulOne.toOne",
... | smul_assoc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 667,
"column": 35
} | {
"line": 667,
"column": 43
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\np₁ p₂ q : R[X]\nhq : q.Monic\na✝ : Nontrivial R\nf : R[X]\nsub_eq : p₁ - p₂ = q * f\n⊢ p₂ %ₘ q + q * (p₂ /ₘ q + f) = q * f + p₂",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"Semigroup.toMul",
"NonUnitalCommRing.toNonUnitalNonAsso... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 676,
"column": 16
} | {
"line": 676,
"column": 24
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nq p₁ p₂ : R[X]\nhq : q.Monic\nhR : Nontrivial R\n⊢ p₁ %ₘ q + p₂ %ₘ q + q * (p₁ /ₘ q + p₂ /ₘ q) = p₁ + p₂",
"usedConstants": [
"Distrib.leftDistribClass",
"Eq.mpr",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.t... | mul_add, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Polynomial.Div | {
"line": 670,
"column": 2
} | {
"line": 680,
"column": 45
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nq p₁ p₂ : R[X]\n⊢ (p₁ + p₂) %ₘ q = p₁ %ₘ q + p₂ %ₘ q",
"usedConstants": [
"Nontrivial",
"Distrib.leftDistribClass",
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"NonUnitalCom... | by_cases hq : q.Monic
· rcases subsingleton_or_nontrivial R with hR | hR
· simp only [eq_iff_true_of_subsingleton]
· exact
(div_modByMonic_unique (p₁ /ₘ q + p₂ /ₘ q) _ hq
⟨by
rw [mul_add, add_left_comm, add_assoc, modByMonic_add_div, ← add_assoc,
add_comm (q * _), mod... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Polynomial.Div | {
"line": 670,
"column": 2
} | {
"line": 680,
"column": 45
} | [
{
"pp": "R : Type u\ninst✝ : CommRing R\nq p₁ p₂ : R[X]\n⊢ (p₁ + p₂) %ₘ q = p₁ %ₘ q + p₂ %ₘ q",
"usedConstants": [
"Nontrivial",
"Distrib.leftDistribClass",
"WithBot.instPreorder",
"Eq.mpr",
"WithBot",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"NonUnitalCom... | by_cases hq : q.Monic
· rcases subsingleton_or_nontrivial R with hR | hR
· simp only [eq_iff_true_of_subsingleton]
· exact
(div_modByMonic_unique (p₁ /ₘ q + p₂ /ₘ q) _ hq
⟨by
rw [mul_add, add_left_comm, add_assoc, modByMonic_add_div, ← add_assoc,
add_comm (q * _), mod... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Polynomial.Div | {
"line": 779,
"column": 6
} | {
"line": 779,
"column": 56
} | [
{
"pp": "R : Type u\ninst✝¹ : CommRing R\np : R[X]\ninst✝ : IsDomain R\nhi : Irreducible p\nx : R\nhx : p.IsRoot x\ng : R[X]\nhg : p = (X - C x) * g\nthis : IsUnit (X - C x) ∨ IsUnit g\nh : IsUnit (X - C x)\n⊢ p.degree = 1",
"usedConstants": [
"Polynomial.C",
"WithBot",
"Nat.instOne",
... | have h₁ : degree (X - C x) = 1 := degree_X_sub_C x | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 170,
"column": 10
} | {
"line": 170,
"column": 97
} | [
{
"pp": "case a\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : IsCancelMulZero α\neif : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Irreducible b) ∧ f.prod ~ᵤ a\nuif :\n ∀ (f g : Multiset α),\n (∀ x ∈ f, Irreducible x) → (∀ x ∈ g, Irreducible x) → f.prod ~ᵤ g.prod → Multiset.Rel Associated f g\np : α\nthis : D... | · exact fun i hi => (Multiset.mem_cons.1 hi).elim (fun hip => hip.symm ▸ hpi) (hfx.1 _) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.UniqueFactorizationDomain.Basic | {
"line": 348,
"column": 8
} | {
"line": 349,
"column": 38
} | [
{
"pp": "case neg.calc_1\nα : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : IsCancelMulZero α\npf : ∀ (a : α), a ≠ 0 → ∃ f, (∀ b ∈ f, Prime b) ∧ f.prod ~ᵤ a\na b : α\nane0 : a ≠ 0\nc : α\nhc : ¬IsUnit c\nb_eq : b = a * c\nh : ¬b = 0\ncne0 : c ≠ 0\ncon : Classical.choose ⋯ = 0\n⊢ c ~ᵤ 1",
"usedConstants":... | · convert (Classical.choose_spec (pf c cne0)).2.symm
rw [con, Multiset.prod_zero] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Polynomial.Roots | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 94
} | [
{
"pp": "case neg\nR : Type u\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\np : R[X]\nhp0 : ¬p = 0\n⊢ p.roots.card ≤ p.natDegree",
"usedConstants": [
"WithBot.instPreorder",
"WithBot.some",
"WithBot",
"Polynomial.roots",
"Polynomial.degree_eq_natDegree",
"CommSemiring.toSemir... | exact WithBot.coe_le_coe.1 (le_trans (card_roots hp0) (le_of_eq <| degree_eq_natDegree hp0)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Algebra.Polynomial.FieldDivision | {
"line": 380,
"column": 12
} | {
"line": 380,
"column": 21
} | [
{
"pp": "R : Type u\ninst✝ : Field R\np q : R[X]\nhq0 : q ≠ 0\nh : p / q = 0\nthis : q * 0 + p % q = p\n⊢ p.degree < q.degree",
"usedConstants": [
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMul",
"CommRing.toNonUnitalCommRing",
"MulZeroClass.toMul",
"congrArg",
... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet | {
"line": 272,
"column": 21
} | {
"line": 272,
"column": 81
} | [
{
"pp": "α : Type u_1\ninst✝¹ : CommMonoidWithZero α\ninst✝ : UniqueFactorizationMonoid α\ns t : Associates α\nd : Associates α\neq : t = s * d\n⊢ s.factors ≤ t.factors",
"usedConstants": [
"CommMonoidWithZero.toCommMonoid",
"Eq.mpr",
"Associates.factors_mul",
"Semigroup.toMul",
... | by rw [eq, factors_mul]; exact le_add_of_nonneg_right bot_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.Polynomial.UniqueFactorization | {
"line": 66,
"column": 8
} | {
"line": 66,
"column": 61
} | [
{
"pp": "case neg.h\nR : Type u_1\ninst✝² : CommSemiring R\ninst✝¹ : NoZeroDivisors R\ninst✝ : WfDvdMonoid R\nf a : R[X]\nane0 : a ≠ 0\nc : R[X]\nnot_unit_c : ¬IsUnit c\nhac : ¬a * c = 0\ncne0 : c ≠ 0\nhdeg : ¬c.natDegree = 0\n⊢ a.natDegree < a.natDegree + c.natDegree",
"usedConstants": [
"lt_add_of_p... | exact lt_add_of_pos_right _ (Nat.pos_of_ne_zero hdeg) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.UniqueFactorizationDomain.FactorSet | {
"line": 593,
"column": 37
} | {
"line": 593,
"column": 46
} | [
{
"pp": "case neg\nα : Type u_1\ninst✝³ : CommMonoidWithZero α\ninst✝² : UniqueFactorizationMonoid α\ninst✝¹ : DecidableEq (Associates α)\ninst✝ : (p : Associates α) → Decidable (Irreducible p)\np a : Associates α\nhp : Irreducible p\nn : ℕ\nh✝ : a ∣ p ^ n\na✝ : Nontrivial α\nhph : p ^ n ≠ 0\nha : a ≠ 0\neq_zer... | mul_zero, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 41
} | [
{
"pp": "R : Type u\nA : Type v\ninst✝² : CommRing R\ninst✝¹ : Ring A\ninst✝ : Algebra R A\na : A\nh : Subsingleton R\n⊢ ¬IsAlgebraic R a",
"usedConstants": [
"is_transcendental_of_subsingleton"
]
}
] | apply is_transcendental_of_subsingleton | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Data.Real.Basic | {
"line": 205,
"column": 18
} | {
"line": 205,
"column": 57
} | [
{
"pp": "x : ℝ\n⊢ { cauchy := ↑0 } = 0",
"usedConstants": [
"Real",
"Real.cauchy",
"CauSeq.Completion.instNatCastCauchy",
"Real.ext_cauchy",
"Real.instZero",
"abs",
"congrArg",
"IsAbsoluteValue.abs_isAbsoluteValue",
"AddMonoid.toAddZeroClass",
"Rat... | by apply ext_cauchy; simp [cauchy_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Real.Basic | {
"line": 288,
"column": 33
} | {
"line": 288,
"column": 58
} | [
{
"pp": "⊢ mk 0 = 0",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"abs",
"Real.ofCauchy_zero",
"congrArg",
"IsAbsoluteValue.abs_isAbsoluteValue",
"Rat",
"CauSeq.Completion.Cauchy",
"Rat.linearOrder",
"Real.ofCauchy",
"id",
... | rw [← ofCauchy_zero]; rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Real.Basic | {
"line": 288,
"column": 33
} | {
"line": 288,
"column": 58
} | [
{
"pp": "⊢ mk 0 = 0",
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"abs",
"Real.ofCauchy_zero",
"congrArg",
"IsAbsoluteValue.abs_isAbsoluteValue",
"Rat",
"CauSeq.Completion.Cauchy",
"Rat.linearOrder",
"Real.ofCauchy",
"id",
... | rw [← ofCauchy_zero]; rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Real.Basic | {
"line": 428,
"column": 4
} | {
"line": 428,
"column": 28
} | [
{
"pp": "case h.h\nx : ℝ\ny✝¹ y✝ : CauSeq ℚ abs\n⊢ y✝¹ ≤ y✝¹ ⊔ y✝",
"usedConstants": [
"Rat",
"Rat.linearOrder",
"CauSeq.le_sup_left",
"Rat.instField",
"Rat.instIsStrictOrderedRing"
]
}
] | exact CauSeq.le_sup_left | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 664,
"column": 4
} | {
"line": 664,
"column": 36
} | [
{
"pp": "case h.e'_3\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\n⊢ -((aeval x) p.divX / (algebraMap K L) (p.coeff 0)) = (aeval x) p.divX / ((aeval x) p - (algebraMap K L) (p.coeff 0))",
"use... | rw [aeval_eq, zero_sub, div_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 664,
"column": 4
} | {
"line": 664,
"column": 36
} | [
{
"pp": "case h.e'_3\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\n⊢ -((aeval x) p.divX / (algebraMap K L) (p.coeff 0)) = (aeval x) p.divX / ((aeval x) p - (algebraMap K L) (p.coeff 0))",
"use... | rw [aeval_eq, zero_sub, div_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 664,
"column": 4
} | {
"line": 664,
"column": 36
} | [
{
"pp": "case h.e'_3\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nx : L\np : K[X]\naeval_eq : (aeval x) p = 0\ncoeff_zero_ne : p.coeff 0 ≠ 0\n⊢ -((aeval x) p.divX / (algebraMap K L) (p.coeff 0)) = (aeval x) p.divX / ((aeval x) p - (algebraMap K L) (p.coeff 0))",
"use... | rw [aeval_eq, zero_sub, div_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Algebraic.Basic | {
"line": 690,
"column": 4
} | {
"line": 690,
"column": 35
} | [
{
"pp": "case refine_3\nK : Type u_1\nL : Type u_2\ninst✝² : Field K\ninst✝¹ : Field L\ninst✝ : Algebra K L\nA : Subalgebra K L\nx : ↥A\np : K[X]\n⊢ ∀ (p : K[X]), p ≠ 0 → (p ≠ 0 → (aeval x) p = 0 → (↑x)⁻¹ ∈ A) → p * X ≠ 0 → (aeval x) (p * X) = 0 → (↑x)⁻¹ ∈ A",
"usedConstants": [
"CommSemiring.toSemiri... | intro p hp ih _ne_zero aeval_eq | Lean.Elab.Tactic.evalIntro | Lean.Parser.Tactic.intro |
Mathlib.Data.NNReal.Defs | {
"line": 946,
"column": 2
} | {
"line": 946,
"column": 62
} | [
{
"pp": "Γ₀ : Type u_1\ninst✝ : LinearOrderedCommGroupWithZero Γ₀\nh : Nontrivial Γ₀ˣ\nf : Γ₀ →*₀ ℝ≥0\nhf : StrictMono ⇑f\nr : ℝ≥0\nhr : 0 < r\n⊢ ∃ d, f ↑d < r",
"usedConstants": [
"Nontrivial",
"GroupWithZero.toMonoidWithZero",
"Exists",
"Units",
"Ne",
"nontrivial_iff_ex... | obtain ⟨g, hg1⟩ := (nontrivial_iff_exists_ne (1 : Γ₀ˣ)).mp h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Data.ENNReal.Operations | {
"line": 560,
"column": 68
} | {
"line": 560,
"column": 97
} | [
{
"pp": "a : ℝ≥0∞\ns : Set ℝ≥0∞\n⊢ sInf s + a = ⨅ b ∈ s, b + a",
"usedConstants": [
"iInf",
"congrArg",
"CommSemiring.toSemiring",
"CompletelyDistribLattice.toCompleteLattice",
"Membership.mem",
"Distrib.toAdd",
"ENNReal.iInf_add",
"ENNReal.instCommSemiring",
... | simp [sInf_eq_iInf, iInf_add] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.ENNReal.Operations | {
"line": 560,
"column": 68
} | {
"line": 560,
"column": 97
} | [
{
"pp": "a : ℝ≥0∞\ns : Set ℝ≥0∞\n⊢ sInf s + a = ⨅ b ∈ s, b + a",
"usedConstants": [
"iInf",
"congrArg",
"CommSemiring.toSemiring",
"CompletelyDistribLattice.toCompleteLattice",
"Membership.mem",
"Distrib.toAdd",
"ENNReal.iInf_add",
"ENNReal.instCommSemiring",
... | simp [sInf_eq_iInf, iInf_add] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENNReal.Operations | {
"line": 560,
"column": 68
} | {
"line": 560,
"column": 97
} | [
{
"pp": "a : ℝ≥0∞\ns : Set ℝ≥0∞\n⊢ sInf s + a = ⨅ b ∈ s, b + a",
"usedConstants": [
"iInf",
"congrArg",
"CommSemiring.toSemiring",
"CompletelyDistribLattice.toCompleteLattice",
"Membership.mem",
"Distrib.toAdd",
"ENNReal.iInf_add",
"ENNReal.instCommSemiring",
... | simp [sInf_eq_iInf, iInf_add] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENNReal.Operations | {
"line": 575,
"column": 67
} | {
"line": 575,
"column": 92
} | [
{
"pp": "ι : Type u_2\ninst✝¹ : Preorder ι\ninst✝ : IsCodirectedOrder ι\nf g : ι → ℝ≥0∞\nhf : Monotone f\nhg : Monotone g\ni j _k : ι\nx✝ : _k ≤ i ∧ _k ≤ j\nhi : _k ≤ i\nhj : _k ≤ j\n⊢ f _k + g _k ≤ f i + g j",
"usedConstants": [
"ENNReal.instAddCommMonoid",
"CommSemiring.toSemiring",
"cov... | by gcongr <;> apply_rules | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.ENNReal.Operations | {
"line": 623,
"column": 4
} | {
"line": 623,
"column": 73
} | [
{
"pp": "case neg\nι : Sort u_1\nf : ι → ℝ≥0\nh : ¬BddAbove (range f)\n⊢ (⨆ i, ↑(f i)).toNNReal = ⨆ i, f i",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"ENNReal.toNNReal_top",
"NNReal.iSup_of_not_bddAbove",
"ENNReal.ofNNReal",
"congrArg",
"iSup",
"PartialOrder.to... | rw [NNReal.iSup_of_not_bddAbove h, iSup_coe_eq_top.2 h, toNNReal_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.ENNReal.Operations | {
"line": 623,
"column": 4
} | {
"line": 623,
"column": 73
} | [
{
"pp": "case neg\nι : Sort u_1\nf : ι → ℝ≥0\nh : ¬BddAbove (range f)\n⊢ (⨆ i, ↑(f i)).toNNReal = ⨆ i, f i",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"ENNReal.toNNReal_top",
"NNReal.iSup_of_not_bddAbove",
"ENNReal.ofNNReal",
"congrArg",
"iSup",
"PartialOrder.to... | rw [NNReal.iSup_of_not_bddAbove h, iSup_coe_eq_top.2 h, toNNReal_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.ENNReal.Operations | {
"line": 623,
"column": 4
} | {
"line": 623,
"column": 73
} | [
{
"pp": "case neg\nι : Sort u_1\nf : ι → ℝ≥0\nh : ¬BddAbove (range f)\n⊢ (⨆ i, ↑(f i)).toNNReal = ⨆ i, f i",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"ENNReal.toNNReal_top",
"NNReal.iSup_of_not_bddAbove",
"ENNReal.ofNNReal",
"congrArg",
"iSup",
"PartialOrder.to... | rw [NNReal.iSup_of_not_bddAbove h, iSup_coe_eq_top.2 h, toNNReal_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENNReal.Operations | {
"line": 701,
"column": 67
} | {
"line": 701,
"column": 92
} | [
{
"pp": "ι : Type u_3\ninst✝¹ : Preorder ι\ninst✝ : IsDirectedOrder ι\nf g : ι → ℝ≥0∞\nhf : Monotone f\nhg : Monotone g\ni j _k : ι\nx✝ : i ≤ _k ∧ j ≤ _k\nhi : i ≤ _k\nhj : j ≤ _k\n⊢ f i + g j ≤ f _k + g _k",
"usedConstants": [
"ENNReal.instAddCommMonoid",
"CommSemiring.toSemiring",
"covar... | by gcongr <;> apply_rules | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Sign.Defs | {
"line": 293,
"column": 2
} | {
"line": 293,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝² : Zero α\ninst✝¹ : Preorder α\ninst✝ : DecidableLT α\na : α\nh : sign a = -1\n⊢ a < 0",
"usedConstants": [
"Preorder.toLT",
"SignType.instOne",
"congrArg",
"PartialOrder.toPreorder",
"SignType.instLinearOrder",
"SemilatticeInf.toPartialOrder"... | rw [sign_apply] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Sign.Defs | {
"line": 312,
"column": 2
} | {
"line": 312,
"column": 22
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Zero α\ninst✝ : LinearOrder α\na : α\nh : sign a = 0\n⊢ a = 0",
"usedConstants": [
"Preorder.toLT",
"SignType.instOne",
"congrArg",
"PartialOrder.toPreorder",
"SignType.instLinearOrder",
"SemilatticeInf.toPartialOrder",
"Eq.mp",
... | rw [sign_apply] at h | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Sign.Basic | {
"line": 193,
"column": 15
} | {
"line": 193,
"column": 27
} | [
{
"pp": "α : Type u\ninst✝¹ : Nonempty α\ninst✝ : DecidableEq α\ns : Finset α\nf : α → ℤ\nn : ℕ\nh : ∑ i ∈ s, (f i).natAbs ≤ n\nβ : Type u\nw✝ : Fintype β\nsgn : β → SignType\ng : β → α\nhg : ∀ (b : β), g b ∈ s\nhβ : Fintype.card β = ∑ a ∈ s, (f a).natAbs\nhf : ∀ a ∈ s, (∑ b, if g b = a then ↑(sgn b) else 0) = ... | simp [hβ, h] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Sign.Basic | {
"line": 193,
"column": 15
} | {
"line": 193,
"column": 27
} | [
{
"pp": "α : Type u\ninst✝¹ : Nonempty α\ninst✝ : DecidableEq α\ns : Finset α\nf : α → ℤ\nn : ℕ\nh : ∑ i ∈ s, (f i).natAbs ≤ n\nβ : Type u\nw✝ : Fintype β\nsgn : β → SignType\ng : β → α\nhg : ∀ (b : β), g b ∈ s\nhβ : Fintype.card β = ∑ a ∈ s, (f a).natAbs\nhf : ∀ a ∈ s, (∑ b, if g b = a then ↑(sgn b) else 0) = ... | simp [hβ, h] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Sign.Basic | {
"line": 193,
"column": 15
} | {
"line": 193,
"column": 27
} | [
{
"pp": "α : Type u\ninst✝¹ : Nonempty α\ninst✝ : DecidableEq α\ns : Finset α\nf : α → ℤ\nn : ℕ\nh : ∑ i ∈ s, (f i).natAbs ≤ n\nβ : Type u\nw✝ : Fintype β\nsgn : β → SignType\ng : β → α\nhg : ∀ (b : β), g b ∈ s\nhβ : Fintype.card β = ∑ a ∈ s, (f a).natAbs\nhf : ∀ a ∈ s, (∑ b, if g b = a then ↑(sgn b) else 0) = ... | simp [hβ, h] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.EReal.Basic | {
"line": 708,
"column": 4
} | {
"line": 708,
"column": 53
} | [
{
"pp": "x : EReal\nhx : x ≤ 0\n⊢ ENNReal.ofReal x.toReal = 0",
"usedConstants": [
"EReal.toReal_nonpos",
"ENNReal.ofReal_of_nonpos",
"EReal.toReal"
]
}
] | exact ENNReal.ofReal_of_nonpos (toReal_nonpos hx) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Data.EReal.Basic | {
"line": 708,
"column": 4
} | {
"line": 708,
"column": 53
} | [
{
"pp": "x : EReal\nhx : x ≤ 0\n⊢ ENNReal.ofReal x.toReal = 0",
"usedConstants": [
"EReal.toReal_nonpos",
"ENNReal.ofReal_of_nonpos",
"EReal.toReal"
]
}
] | exact ENNReal.ofReal_of_nonpos (toReal_nonpos hx) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.EReal.Basic | {
"line": 708,
"column": 4
} | {
"line": 708,
"column": 53
} | [
{
"pp": "x : EReal\nhx : x ≤ 0\n⊢ ENNReal.ofReal x.toReal = 0",
"usedConstants": [
"EReal.toReal_nonpos",
"ENNReal.ofReal_of_nonpos",
"EReal.toReal"
]
}
] | exact ENNReal.ofReal_of_nonpos (toReal_nonpos hx) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.ENNReal.Inv | {
"line": 270,
"column": 2
} | {
"line": 270,
"column": 26
} | [
{
"pp": "a b c : ℝ≥0∞\nh : 0 < b → b < a → c ≠ 0\n⊢ (a - b) / c = a / c - b / c",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"HSub.hSub",
"id",
"MulOne.toMul",
"HDiv.hDiv",
... | simp_rw [div_eq_mul_inv] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Data.ENNReal.Inv | {
"line": 361,
"column": 27
} | {
"line": 361,
"column": 39
} | [
{
"pp": "a b : ℝ≥0∞\nh₀ : a ≠ 0\nh₁ : a ≠ ∞\n⊢ a = 0 → b = 0",
"usedConstants": [
"False",
"eq_false",
"instZeroENNReal",
"implies_congr",
"True",
"ENNReal",
"of_eq_true",
"Zero.toOfNat0",
"Eq.refl",
"instIsEmptyFalse",
"OfNat.ofNat",
"... | by simp [h₀] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.EvenFunction | {
"line": 143,
"column": 36
} | {
"line": 143,
"column": 41
} | [
{
"pp": "case h\nα : Type u_3\nβ : Type u_4\ninst✝² : AddCommGroup β\ninst✝¹ : IsAddTorsionFree β\nf : α → β\ninst✝ : Neg α\nhe : Function.Even f\nho : Function.Odd f\nr : α\n⊢ -f r = f r",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Eq.mpr",
"NegZeroClass.toNeg",
"congrArg",... | ← ho, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.ENNReal.Inv | {
"line": 521,
"column": 6
} | {
"line": 521,
"column": 26
} | [
{
"pp": "a b c d : ℝ≥0∞\nha : a ≠ 0\nha' : a ≠ ∞\nhb : b ≠ 0\nhb' : b ≠ ∞\n⊢ a * (c / b) = d ↔ b * d = a * c",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"congrArg",
"CommSemiring.toSemiring",
"id",
"HDiv.hDiv",
"ENNReal.instCommSemiring",
"If... | ← eq_div_iff hb hb', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.EReal.Operations | {
"line": 418,
"column": 4
} | {
"line": 419,
"column": 44
} | [
{
"pp": "case neg\nx y : ℝ\nhy : 0 ≤ ↑y\nhxy : ¬x ≤ y\n⊢ (↑x - ↑y).toENNReal = (↑x).toENNReal - (↑y).toENNReal",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"Preorder.toLT",
"EReal.toENNReal_of_ne_top",
"LinearOrder.toDecidableEq",
"Real.instZero",
"EN... | rw [toENNReal_of_ne_top (ne_of_beq_false rfl).symm, ← coe_sub, toReal_coe,
ofReal_sub x (EReal.coe_nonneg.mp hy)] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.EReal.Operations | {
"line": 441,
"column": 2
} | {
"line": 451,
"column": 66
} | [
{
"pp": "a b c : EReal\nhb : b ≠ ⊥ ∨ c ≠ ⊥\nht : b ≠ ⊤ ∨ c ≠ ⊤\n⊢ a ≤ c - b ↔ a + b ≤ c",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"_private.Mathlib.Data.EReal.Operations.0.EReal.le_sub_iff_add_le._simp_1_3",
"EReal.sub_top",
"False",
"Real",
"EReal.addLECancellable_... | induction b with
| bot =>
simp only [ne_eq, not_true_eq_false, false_or] at hb
simp only [sub_bot hb, le_top, add_bot, bot_le]
| coe b =>
rw [← (addLECancellable_coe b).add_le_add_iff_right, sub_add_cancel]
| top =>
simp only [ne_eq, not_true_eq_false, false_or, sub_top, le_bot_iff] at ht ⊢
re... | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Data.EReal.Operations | {
"line": 441,
"column": 2
} | {
"line": 451,
"column": 66
} | [
{
"pp": "a b c : EReal\nhb : b ≠ ⊥ ∨ c ≠ ⊥\nht : b ≠ ⊤ ∨ c ≠ ⊤\n⊢ a ≤ c - b ↔ a + b ≤ c",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"_private.Mathlib.Data.EReal.Operations.0.EReal.le_sub_iff_add_le._simp_1_3",
"EReal.sub_top",
"False",
"Real",
"EReal.addLECancellable_... | induction b with
| bot =>
simp only [ne_eq, not_true_eq_false, false_or] at hb
simp only [sub_bot hb, le_top, add_bot, bot_le]
| coe b =>
rw [← (addLECancellable_coe b).add_le_add_iff_right, sub_add_cancel]
| top =>
simp only [ne_eq, not_true_eq_false, false_or, sub_top, le_bot_iff] at ht ⊢
re... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.EReal.Operations | {
"line": 441,
"column": 2
} | {
"line": 451,
"column": 66
} | [
{
"pp": "a b c : EReal\nhb : b ≠ ⊥ ∨ c ≠ ⊥\nht : b ≠ ⊤ ∨ c ≠ ⊤\n⊢ a ≤ c - b ↔ a + b ≤ c",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"_private.Mathlib.Data.EReal.Operations.0.EReal.le_sub_iff_add_le._simp_1_3",
"EReal.sub_top",
"False",
"Real",
"EReal.addLECancellable_... | induction b with
| bot =>
simp only [ne_eq, not_true_eq_false, false_or] at hb
simp only [sub_bot hb, le_top, add_bot, bot_le]
| coe b =>
rw [← (addLECancellable_coe b).add_le_add_iff_right, sub_add_cancel]
| top =>
simp only [ne_eq, not_true_eq_false, false_or, sub_top, le_bot_iff] at ht ⊢
re... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.EReal.Operations | {
"line": 717,
"column": 20
} | {
"line": 717,
"column": 30
} | [
{
"pp": "a b : EReal\n⊢ 0 ≤ -(a * b) ↔ 0 ≤ a ∧ b ≤ 0 ∨ a ≤ 0 ∧ 0 ≤ b",
"usedConstants": [
"Eq.mpr",
"HMul.hMul",
"congrArg",
"PartialOrder.toPreorder",
"EReal.instNeg",
"EReal",
"Preorder.toLE",
"id",
"instZeroEReal",
"LE.le",
"SubNegZeroMono... | ← mul_neg, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.EReal.Inv | {
"line": 432,
"column": 2
} | {
"line": 432,
"column": 84
} | [
{
"pp": "b : EReal\nh : b < 0\nh' : b ≠ ⊥\na a' : EReal\na_lt_a' : a < a'\n⊢ a' / b < a / b",
"usedConstants": [
"EReal.instDivInvMonoid",
"instHDiv",
"PartialOrder.toPreorder",
"EReal",
"le_of_lt",
"HDiv.hDiv",
"instZeroEReal",
"EReal.div_le_div_right_of_nonp... | apply lt_of_le_of_ne <| div_le_div_right_of_nonpos (le_of_lt h) (le_of_lt a_lt_a') | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Order.Filter.CountablyGenerated | {
"line": 166,
"column": 2
} | {
"line": 168,
"column": 82
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Sort u_5\nf g : Filter α\ninst✝¹ : f.IsCountablyGenerated\ninst✝ : g.IsCountablyGenerated\n⊢ (f ⊔ g).IsCountablyGenerated",
"usedConstants": [
"Filter.HasCountableBasis.mk",
"Filter.HasAntitoneBasis.toHasBasis",
"setOf",... | rcases f.exists_antitone_basis with ⟨s, hs⟩
rcases g.exists_antitone_basis with ⟨t, ht⟩
exact HasCountableBasis.isCountablyGenerated ⟨hs.1.sup ht.1, Set.to_countable _⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.CountablyGenerated | {
"line": 166,
"column": 2
} | {
"line": 168,
"column": 82
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nι : Type u_4\nι' : Sort u_5\nf g : Filter α\ninst✝¹ : f.IsCountablyGenerated\ninst✝ : g.IsCountablyGenerated\n⊢ (f ⊔ g).IsCountablyGenerated",
"usedConstants": [
"Filter.HasCountableBasis.mk",
"Filter.HasAntitoneBasis.toHasBasis",
"setOf",... | rcases f.exists_antitone_basis with ⟨s, hs⟩
rcases g.exists_antitone_basis with ⟨t, ht⟩
exact HasCountableBasis.isCountablyGenerated ⟨hs.1.sup ht.1, Set.to_countable _⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Filter.Pi | {
"line": 322,
"column": 2
} | {
"line": 322,
"column": 55
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\nf : (i : ι) → Filter (α i)\nβ : ι → Type u_3\nm : (i : ι) → α i → β i\ns : Set ((i : ι) → β i)\nh : ∀ (i : ι), ∃ t₁, m i ⁻¹' t₁ ∈ f i ∧ eval i ⁻¹' t₁ ⊆ s\ni : ι\nt : Set (β i)\nH : m i ⁻¹' t ∈ f i\nhH : eval i ⁻¹' t ⊆ s\n⊢ ∃ t₁ ∈ f i, eval i ⁻¹' t₁ ⊆ (fun k i ↦ m i (k i)... | exact ⟨{ x : α i | m i x ∈ t }, H, fun x hx => hH hx⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.Filter.Prod | {
"line": 300,
"column": 72
} | {
"line": 302,
"column": 45
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nf : Filter α\ng : Filter β\nh : Filter γ\n⊢ map (⇑(Equiv.prodAssoc α β γ).symm) (f ×ˢ g ×ˢ h) = (f ×ˢ g) ×ˢ h",
"usedConstants": [
"Eq.mpr",
"Equiv.instEquivLike",
"CompleteLattice.toLattice",
"SProd.sprod",
"congrArg",
"... | by
simp_rw [map_equiv_symm, prod_eq_inf, comap_inf, comap_comap, inf_assoc,
Function.comp_def, Equiv.prodAssoc_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Filter.IsBounded | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 84
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Preorder β\ninst✝ : IsDirectedOrder β\nf : α → β\nb : β\nhb : ∀ᶠ (x : β) in map f cofinite, (fun x1 x2 ↦ x1 ≤ x2) x b\nthis : Nonempty β\n⊢ BddAbove (range f)",
"usedConstants": [
"Eq.mpr",
"Set.image_univ",
"bddAbove_union",
"congrArg",
... | rw [← image_univ, ← union_compl_self { x | f x ≤ b }, image_union, bddAbove_union] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Filter.Lift | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 52
} | [
{
"pp": "α : Type u_1\nγ : Type u_3\nι : Type u_6\np : ι → Prop\ns : ι → Set α\nf : Filter α\nhf : f.HasBasis p s\nβ : ι → Type u_5\npg : (i : ι) → β i → Prop\nsg : (i : ι) → β i → Set γ\ng : Set α → Filter γ\nhg : ∀ (i : ι), (g (s i)).HasBasis (pg i) (sg i)\ngm : Monotone g\n⊢ (f.lift g).HasBasis (fun i ↦ p i.... | refine ⟨fun t => (hf.mem_lift_iff hg gm).trans ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Order.Filter.Lift | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 87
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : Filter α\ng : Set α → Filter β\nhm : Monotone g\n⊢ (f.lift g).NeBot ↔ ∀ s ∈ f, (g s).NeBot",
"usedConstants": [
"Filter.instMembership",
"congrArg",
"_private.Mathlib.Order.Filter.Lift.0.Filter.lift_neBot_iff._simp_1_1",
"Filter.NeBot",
... | simp only [neBot_iff, Ne, ← empty_mem_iff_bot, mem_lift_sets hm, not_exists, not_and] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Order.Filter.Lift | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 87
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : Filter α\ng : Set α → Filter β\nhm : Monotone g\n⊢ (f.lift g).NeBot ↔ ∀ s ∈ f, (g s).NeBot",
"usedConstants": [
"Filter.instMembership",
"congrArg",
"_private.Mathlib.Order.Filter.Lift.0.Filter.lift_neBot_iff._simp_1_1",
"Filter.NeBot",
... | simp only [neBot_iff, Ne, ← empty_mem_iff_bot, mem_lift_sets hm, not_exists, not_and] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.Lift | {
"line": 156,
"column": 2
} | {
"line": 156,
"column": 87
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : Filter α\ng : Set α → Filter β\nhm : Monotone g\n⊢ (f.lift g).NeBot ↔ ∀ s ∈ f, (g s).NeBot",
"usedConstants": [
"Filter.instMembership",
"congrArg",
"_private.Mathlib.Order.Filter.Lift.0.Filter.lift_neBot_iff._simp_1_1",
"Filter.NeBot",
... | simp only [neBot_iff, Ne, ← empty_mem_iff_bot, mem_lift_sets hm, not_exists, not_and] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Basic | {
"line": 87,
"column": 2
} | {
"line": 87,
"column": 33
} | [
{
"pp": "X : Type u\nα : Type u_1\ns : Set X\ninst✝ : TopologicalSpace X\nf : α → Set X\nho : ∀ (i : α), IsOpen (f i)\nhU : ⋃ i, f i = univ\nh : ∀ (i : α), IsOpen (f i ∩ s)\n⊢ IsOpen (⋃ i, f i ∩ s)",
"usedConstants": [
"Set.instInter",
"Inter.inter",
"isOpen_iUnion",
"Set"
]
}
... | exact isOpen_iUnion fun i ↦ h i | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Closure | {
"line": 446,
"column": 4
} | {
"line": 446,
"column": 15
} | [
{
"pp": "case mpr\nX : Type u\ninst✝ : TopologicalSpace X\nx : X\nho : ¬IsOpen {x}\nhU : IsOpen {x}\nhne : {x}.Nonempty\nhUx : {x} ⊆ {x}\n⊢ False",
"usedConstants": []
}
] | exact ho hU | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Order.LiminfLimsup | {
"line": 1153,
"column": 6
} | {
"line": 1153,
"column": 29
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : ConditionallyCompleteLinearOrder β\nf : Filter α\nF : ι → α → β\ns : Finset ι\nhs : s.Nonempty\nh₁ : ∀ i ∈ s, IsCoboundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (F i)\nh₂ : ∀ i ∈ s, IsBoundedUnder (fun x1 x2 ↦ x1 ≤ x2) f (F i)\nbddsup : IsBoundedUnder (fun x1 x2 ... | rcases hs with ⟨i, i_s⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.ClusterPt | {
"line": 250,
"column": 2
} | {
"line": 251,
"column": 51
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\nf : Filter X\n⊢ IsClosed {x | ClusterPt x f}",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"isClosed_biInter",
"congrArg",
"Set.iInter",
"setOf",
"Membership.mem",
"_private.Mathlib.Topology.ClusterPt.... | simp only [clusterPt_iff_forall_mem_closure, setOf_forall]
exact isClosed_biInter fun _ _ ↦ isClosed_closure | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ClusterPt | {
"line": 250,
"column": 2
} | {
"line": 251,
"column": 51
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\nf : Filter X\n⊢ IsClosed {x | ClusterPt x f}",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"isClosed_biInter",
"congrArg",
"Set.iInter",
"setOf",
"Membership.mem",
"_private.Mathlib.Topology.ClusterPt.... | simp only [clusterPt_iff_forall_mem_closure, setOf_forall]
exact isClosed_biInter fun _ _ ↦ isClosed_closure | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order | {
"line": 968,
"column": 2
} | {
"line": 968,
"column": 25
} | [
{
"pp": "α : Type u_1\nl : Filter α\np : α → Prop\nq : Prop\n⊢ Tendsto p l (𝓝 q) ↔ q → ∀ᶠ (x : α) in l, p x",
"usedConstants": [
"Pure.pure",
"False",
"eq_false",
"nhds_false",
"congrArg",
"Filter.Eventually",
"nhds_true",
"Filter.tendsto_pure._simp_1",
... | by_cases q <;> simp [*] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Topology.Order | {
"line": 968,
"column": 2
} | {
"line": 968,
"column": 25
} | [
{
"pp": "α : Type u_1\nl : Filter α\np : α → Prop\nq : Prop\n⊢ Tendsto p l (𝓝 q) ↔ q → ∀ᶠ (x : α) in l, p x",
"usedConstants": [
"Pure.pure",
"False",
"eq_false",
"nhds_false",
"congrArg",
"Filter.Eventually",
"nhds_true",
"Filter.tendsto_pure._simp_1",
... | by_cases q <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order | {
"line": 968,
"column": 2
} | {
"line": 968,
"column": 25
} | [
{
"pp": "α : Type u_1\nl : Filter α\np : α → Prop\nq : Prop\n⊢ Tendsto p l (𝓝 q) ↔ q → ∀ᶠ (x : α) in l, p x",
"usedConstants": [
"Pure.pure",
"False",
"eq_false",
"nhds_false",
"congrArg",
"Filter.Eventually",
"nhds_true",
"Filter.tendsto_pure._simp_1",
... | by_cases q <;> simp [*] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Homeomorph.Defs | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 48
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nh : X ≃ₜ Y\n⊢ IsQuotientMap (⇑h ∘ ⇑h.symm)",
"usedConstants": [
"congrArg",
"Function.comp",
"id",
"_private.Mathlib.Topology.Homeomorph.Defs.0.Homeomorph.isQuotientMap._simp_1_1",
"Ho... | simp only [self_comp_symm, IsQuotientMap.id] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Homeomorph.Defs | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 48
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nh : X ≃ₜ Y\n⊢ IsQuotientMap (⇑h ∘ ⇑h.symm)",
"usedConstants": [
"congrArg",
"Function.comp",
"id",
"_private.Mathlib.Topology.Homeomorph.Defs.0.Homeomorph.isQuotientMap._simp_1_1",
"Ho... | simp only [self_comp_symm, IsQuotientMap.id] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Homeomorph.Defs | {
"line": 222,
"column": 4
} | {
"line": 222,
"column": 48
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nh : X ≃ₜ Y\n⊢ IsQuotientMap (⇑h ∘ ⇑h.symm)",
"usedConstants": [
"congrArg",
"Function.comp",
"id",
"_private.Mathlib.Topology.Homeomorph.Defs.0.Homeomorph.isQuotientMap._simp_1_1",
"Ho... | simp only [self_comp_symm, IsQuotientMap.id] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Maps.Basic | {
"line": 502,
"column": 6
} | {
"line": 502,
"column": 18
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nf : X → Y\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\n⊢ IsClosedMap f ↔ ∀ {u : Set X}, IsOpen u → IsOpen (kernImage f u)",
"usedConstants": [
"Eq.mpr",
"Set.kernImage",
"congrArg",
"id",
"IsClosedMap.eq_1",
"IsClosedMap",... | IsClosedMap, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Constructions | {
"line": 306,
"column": 4
} | {
"line": 306,
"column": 68
} | [
{
"pp": "case h.mp\nX : Type u\nx : CofiniteTopology X\nU V : Set (CofiniteTopology X)\nhVU : V ⊆ U\nV_op : IsOpen V\nhaV : x ∈ V\n⊢ U ∈ pure x ⊔ cofinite",
"usedConstants": [
"Pure.pure",
"Filter.instMembership",
"Iff.mpr",
"Membership.mem",
"Filter.instPure",
"And",
... | exact mem_sup.mpr ⟨hVU haV, mem_of_superset (V_op ⟨_, haV⟩) hVU⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.NhdsWithin | {
"line": 571,
"column": 78
} | {
"line": 572,
"column": 82
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\ns t : Set α\n⊢ 𝓟 (s ∩ t) ≤ 𝓝ˢ[t] s",
"usedConstants": [
"Filter.instMembership",
"Eq.mpr",
"and_true",
"Set.inter_subset_right._simp_1",
"congrArg",
"Filter.inf_principal",
"Filter.instCompleteLatticeFilter",
... | by
simpa [nhdsSetWithin] using inf_le_of_left_le (b := 𝓟 t) <| principal_le_nhdsSet | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Constructions.SumProd | {
"line": 974,
"column": 2
} | {
"line": 974,
"column": 78
} | [
{
"pp": "X : Type u\nY : Type v\nZ : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : X → Z\ng : Y → Z\nhf : IsInducing f\nhg : IsInducing g\nhFg : Disjoint (𝓟 (range f)) (𝓝ˢ (range g))\nhfG : Disjoint (𝓝ˢ (range f)) (𝓟 (range g))\nx : X ⊕ Y\n⊢ 𝓝 x = comap... | apply le_antisymm ((hf.continuous.sumElim hg.continuous).tendsto x).le_comap | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.Bases | {
"line": 478,
"column": 2
} | {
"line": 478,
"column": 56
} | [
{
"pp": "α : Type u\nt : TopologicalSpace α\nι : Sort u_2\ninst✝ : Countable ι\ns c : ι → Set α\nhc : ∀ (i : ι), (c i).Countable\nh'c : ∀ (i : ι), s i ⊆ closure (c i)\ni : ι\n⊢ s i ⊆ closure (⋃ i, c i)",
"usedConstants": [
"HasSubset.Subset.trans",
"Set.instIsTransSubset",
"closure_mono",
... | exact (h'c i).trans (closure_mono (subset_iUnion _ i)) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Compactness.SigmaCompact | {
"line": 119,
"column": 6
} | {
"line": 121,
"column": 47
} | [
{
"pp": "case mpr.refine_1\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ns : Set X\nhf : IsInducing f\nL : ℕ → Set Y\nhcomp : ∀ (n : ℕ), IsCompact (L n)\nhcov : ⋃ n, L n = f '' s\nn : ℕ\n⊢ IsCompact ((fun n ↦ f ⁻¹' L n ∩ s) n)",
"usedConstants": [
"Se... | have : f '' (f ⁻¹' (L n) ∩ s) = L n := by
rw [image_preimage_inter, inter_eq_left.mpr]
exact (subset_iUnion _ n).trans hcov.le | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Compactness.Compact | {
"line": 604,
"column": 4
} | {
"line": 604,
"column": 63
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\ny : Y\nhf : Tendsto f (cocompact X) (𝓝 y)\nhfc : Continuous f\nl : Filter Y\nhne : l.NeBot\nhle : l ≤ 𝓟 (insert y (range f))\ns : Set Y\nhsy : s ∈ 𝓝 y\nt : Set Y\nhtl : t ∈ l\nhd : Disjoint s t\nK : Set X\nhK... | filter_upwards [htl, le_principal_iff.1 hle] with y hyt hyf | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.Topology.GDelta.Basic | {
"line": 189,
"column": 37
} | {
"line": 189,
"column": 94
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\ns : Set X\n⊢ (∃ S ⊆ {t | IsOpen t ∧ Dense t}, S.Countable ∧ ⋂₀ S ⊆ s) ↔\n ∃ S, (∀ t ∈ S, IsOpen t) ∧ (∀ t ∈ S, Dense t) ∧ S.Countable ∧ ⋂₀ S ⊆ s",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Topology.GDelta.Basic.0.mem_residual_iff._simp_1... | by simp_rw [subset_def, mem_setOf, forall_and, and_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Compactness.Compact | {
"line": 1145,
"column": 6
} | {
"line": 1145,
"column": 74
} | [
{
"pp": "X✝ : Type u\nY : Type v\nι : Type u_1\ninst✝³ : TopologicalSpace X✝\ninst✝² : TopologicalSpace Y\ns t : Set X✝\nf : X✝ → Y\nX : ι → Type u_2\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\ninst✝ : ∀ (i : ι), CompactSpace (X i)\n⊢ IsCompact univ",
"usedConstants": [
"Eq.mpr",
"Pi.topological... | rw [← pi_univ univ]; exact isCompact_univ_pi fun i => isCompact_univ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.Compact | {
"line": 1145,
"column": 6
} | {
"line": 1145,
"column": 74
} | [
{
"pp": "X✝ : Type u\nY : Type v\nι : Type u_1\ninst✝³ : TopologicalSpace X✝\ninst✝² : TopologicalSpace Y\ns t : Set X✝\nf : X✝ → Y\nX : ι → Type u_2\ninst✝¹ : (i : ι) → TopologicalSpace (X i)\ninst✝ : ∀ (i : ι), CompactSpace (X i)\n⊢ IsCompact univ",
"usedConstants": [
"Eq.mpr",
"Pi.topological... | rw [← pi_univ univ]; exact isCompact_univ_pi fun i => isCompact_univ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Compactness.Compact | {
"line": 1217,
"column": 22
} | {
"line": 1217,
"column": 27
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ninst✝ : CompactSpace X\nS : Set X\nhS : IsClosed S\nhne : S.Nonempty\nopens : Set (Set X) := {U | Sᶜ ⊆ U ∧ IsOpen U ∧ Uᶜ.Nonempty}\nU : Set X\nh : Maximal (fun x ↦ x ∈ opens) U\nUc : Sᶜ ⊆ U\nUo : IsOpen U\nUcne : Uᶜ.Nonempty\nV' : Set X\nV'sub : V' ⊆ Uᶜ\nV'ne : ... | V'cls | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Irreducible | {
"line": 240,
"column": 2
} | {
"line": 240,
"column": 53
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ns : Set X\ninst✝ : PreirreducibleSpace ↑s\n⊢ IsPreirreducible s",
"usedConstants": [
"IsPreirreducible",
"Eq.mpr",
"Set.image_univ",
"congrArg",
"Set.univ",
"Membership.mem",
"id",
"Subtype",
"Subtype.r... | rw [← Subtype.range_coe (s := s), ← Set.image_univ] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Irreducible | {
"line": 261,
"column": 4
} | {
"line": 264,
"column": 97
} | [
{
"pp": "X✝ : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\ns t : Set X✝\nX : Type u_3\ninst✝ : Infinite X\nu v : Set (CofiniteTopology X)\n⊢ IsOpen u → IsOpen v → (univ ∩ u).Nonempty → (univ ∩ v).Nonempty → (univ ∩ (u ∩ v)).Nonempty",
"usedConstants": [
"Set.Finit... | haveI : Infinite (CofiniteTopology X) := ‹_›
simp only [CofiniteTopology.isOpen_iff, univ_inter]
intro hu hv hu' hv'
simpa only [compl_union, compl_compl] using ((hu hu').union (hv hv')).infinite_compl.nonempty | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Irreducible | {
"line": 261,
"column": 4
} | {
"line": 264,
"column": 97
} | [
{
"pp": "X✝ : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X✝\ninst✝¹ : TopologicalSpace Y\ns t : Set X✝\nX : Type u_3\ninst✝ : Infinite X\nu v : Set (CofiniteTopology X)\n⊢ IsOpen u → IsOpen v → (univ ∩ u).Nonempty → (univ ∩ v).Nonempty → (univ ∩ (u ∩ v)).Nonempty",
"usedConstants": [
"Set.Finit... | haveI : Infinite (CofiniteTopology X) := ‹_›
simp only [CofiniteTopology.isOpen_iff, univ_inter]
intro hu hv hu' hv'
simpa only [compl_union, compl_compl] using ((hu hu').union (hv hv')).infinite_compl.nonempty | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Separation.Hausdorff | {
"line": 683,
"column": 4
} | {
"line": 683,
"column": 91
} | [
{
"pp": "case mp\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : R1Space X\nS : Set X\nh : IsPreirreducible S\nx : X\nhx : x ∈ S\ny : X\nhy : y ∈ S\ne : y ∉ closure {x}\nU V : Set X\nhU : IsOpen U\nhV : IsOpen V\nhxU : x ∈ U\nhyV : y ∈ V\nh' : Disjoint U V\n⊢ False",
"usedConstants": [
"Membership... | exact ((h U V hU hV ⟨x, hx, hxU⟩ ⟨y, hy, hyV⟩).mono inter_subset_right).not_disjoint h' | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.DiscreteSubset | {
"line": 266,
"column": 6
} | {
"line": 266,
"column": 15
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝ : TopologicalSpace X\ns U : Set X\nhs : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \\ s))\nhU : IsClosed U\nx : X\nhx : x ∈ (U \\ s)ᶜ\nh₁x : x ∈ U\na : X\nha : a ∈ {x}ᶜ → a ∉ U \\ s\nh₂a : a = x\n⊢ a ∈ (U \\ s)ᶜ",
"usedConstants": [
"Eq.mpr",
"False",
"eq_fal... | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Topology.DiscreteSubset | {
"line": 266,
"column": 6
} | {
"line": 266,
"column": 15
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝ : TopologicalSpace X\ns U : Set X\nhs : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \\ s))\nhU : IsClosed U\nx : X\nhx : x ∈ (U \\ s)ᶜ\nh₁x : x ∈ U\na : X\nha : a ∈ {x}ᶜ → a ∉ U \\ s\nh₂a : a = x\n⊢ a ∈ (U \\ s)ᶜ",
"usedConstants": [
"Eq.mpr",
"False",
"eq_fal... | tauto_set | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.DiscreteSubset | {
"line": 266,
"column": 6
} | {
"line": 266,
"column": 15
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝ : TopologicalSpace X\ns U : Set X\nhs : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \\ s))\nhU : IsClosed U\nx : X\nhx : x ∈ (U \\ s)ᶜ\nh₁x : x ∈ U\na : X\nha : a ∈ {x}ᶜ → a ∉ U \\ s\nh₂a : a = x\n⊢ a ∈ (U \\ s)ᶜ",
"usedConstants": [
"Eq.mpr",
"False",
"eq_fal... | tauto_set | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.DiscreteSubset | {
"line": 268,
"column": 6
} | {
"line": 268,
"column": 15
} | [
{
"pp": "case neg\nX : Type u_1\ninst✝ : TopologicalSpace X\ns U : Set X\nhs : ∀ x ∈ U, Disjoint (𝓝[≠] x) (𝓟 (U \\ s))\nhU : IsClosed U\nx : X\nhx : x ∈ (U \\ s)ᶜ\nh₁x : x ∈ U\na : X\nh₂a : ¬a = x\nha : a ∉ U \\ s\n⊢ a ∈ (U \\ s)ᶜ",
"usedConstants": [
"Eq.mpr",
"False",
"eq_false",
... | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Topology.DiscreteSubset | {
"line": 272,
"column": 4
} | {
"line": 272,
"column": 13
} | [
{
"pp": "case right\nX : Type u_1\ninst✝ : TopologicalSpace X\ns U : Set X\nhs : s ∈ codiscreteWithin U\nhU : IsClosed U\nx : X\nhx : x ∈ (U \\ s)ᶜ\nh₁x : x ∉ U\ny : X\nhy : y ∈ Uᶜ\n⊢ y ∈ (U \\ s)ᶜ",
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"False",
"eq_false",
"congrAr... | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Topology.DiscreteSubset | {
"line": 284,
"column": 4
} | {
"line": 284,
"column": 13
} | [
{
"pp": "case h.right\nX : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\nx : X\nU s : Set X\nhs : Finite ↑s\nt : Set X\nht : IsOpen t\nh₁ts : x ∈ t\nh₂ts : t ∩ {x}ᶜ ⊆ U\n⊢ x ∈ t \\ (s \\ {x}) ∧ t \\ (s \\ {x}) ∩ {x}ᶜ ⊆ U \\ s",
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"... | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Topology.DiscreteSubset | {
"line": 284,
"column": 4
} | {
"line": 284,
"column": 13
} | [
{
"pp": "case h.right\nX : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\nx : X\nU s : Set X\nhs : Finite ↑s\nt : Set X\nht : IsOpen t\nh₁ts : x ∈ t\nh₂ts : t ∩ {x}ᶜ ⊆ U\n⊢ x ∈ t \\ (s \\ {x}) ∧ t \\ (s \\ {x}) ∩ {x}ᶜ ⊆ U \\ s",
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"... | tauto_set | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.DiscreteSubset | {
"line": 284,
"column": 4
} | {
"line": 284,
"column": 13
} | [
{
"pp": "case h.right\nX : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : T1Space X\nx : X\nU s : Set X\nhs : Finite ↑s\nt : Set X\nht : IsOpen t\nh₁ts : x ∈ t\nh₂ts : t ∩ {x}ᶜ ⊆ U\n⊢ x ∈ t \\ (s \\ {x}) ∧ t \\ (s \\ {x}) ∩ {x}ᶜ ⊆ U \\ s",
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"... | tauto_set | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Separation.Basic | {
"line": 205,
"column": 4
} | {
"line": 205,
"column": 54
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : T0Space X\ns : Finset X\nihs : ∀ t ⊂ s, (↑t).Nonempty → IsOpen ↑t → ∃ x ∈ ↑t, IsOpen {x}\nhne : (↑s).Nonempty\nho : IsOpen ↑s\nht : ¬∃ t ⊂ s, t.Nonempty ∧ IsOpen ↑t\nt : Set X\nhts : t ⊆ ↑s\nhtne : t.Nonempty\nhto : IsOpen t\nhts' : ¬t = ↑s\n⊢ ∃ t ⊂ s,... | lift t to Finset X using s.finite_toSet.subset hts | Mathlib.Tactic._aux_Mathlib_Tactic_Lift___elabRules_Mathlib_Tactic_lift_1 | Mathlib.Tactic.lift |
Mathlib.Topology.Separation.Basic | {
"line": 303,
"column": 2
} | {
"line": 307,
"column": 36
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : R0Space X\nx : X\n⊢ IsCompact (closure {x})",
"usedConstants": [
"Eq.mpr",
"Specializes",
"Iff.of_eq",
"congrArg",
"subset_closure",
"Specializes.mem_open",
"Finset",
"Set.mem_iUnion",
"speciali... | refine isCompact_of_finite_subcover fun U hUo hxU ↦ ?_
obtain ⟨i, hi⟩ : ∃ i, x ∈ U i := mem_iUnion.1 <| hxU <| subset_closure rfl
refine ⟨{i}, fun y hy ↦ ?_⟩
rw [← specializes_iff_mem_closure, specializes_comm] at hy
simpa using hy.mem_open (hUo i) hi | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Separation.Basic | {
"line": 303,
"column": 2
} | {
"line": 307,
"column": 36
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : R0Space X\nx : X\n⊢ IsCompact (closure {x})",
"usedConstants": [
"Eq.mpr",
"Specializes",
"Iff.of_eq",
"congrArg",
"subset_closure",
"Specializes.mem_open",
"Finset",
"Set.mem_iUnion",
"speciali... | refine isCompact_of_finite_subcover fun U hUo hxU ↦ ?_
obtain ⟨i, hi⟩ : ∃ i, x ∈ U i := mem_iUnion.1 <| hxU <| subset_closure rfl
refine ⟨{i}, fun y hy ↦ ?_⟩
rw [← specializes_iff_mem_closure, specializes_comm] at hy
simpa using hy.mem_open (hUo i) hi | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Separation.Basic | {
"line": 665,
"column": 2
} | {
"line": 665,
"column": 36
} | [
{
"pp": "case h.e'_3.h.e'_4\nX : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T1Space X\ninst✝ : ∀ (x : X), (𝓝[≠] x).NeBot\ns : Set X\nhs : Dense s\nt : Set X\nht : t.Finite\n⊢ t = ↑ht.toFinset",
"usedConstants": [
"Finset",
"Set.Finite.coe_toFinset",
"SetLike.coe",
"Finset.instS... | exact (Finite.coe_toFinset _).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Connected.Basic | {
"line": 128,
"column": 2
} | {
"line": 128,
"column": 29
} | [
{
"pp": "α : Type u\ninst✝ : TopologicalSpace α\ns t : Set α\nhs : IsPreconnected s\nht : IsPreconnected t\nx : α\nhxs : x ∈ s\nhxt : x ∈ t\n⊢ IsPreconnected (s ∪ t)",
"usedConstants": [
"IsPreconnected.union"
]
}
] | exact hs.union x hxs hxt ht | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 190,
"column": 76
} | {
"line": 192,
"column": 86
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nf : β → α\na : α\ng : γ → α\ne : ↑(mulSupport f) ≃ ↑(mulSupport g)\nhe : ∀ (x : ↑(mulSupport f)), g ↑(e x) = f ↑x\n⊢ HasProd f a ↔ HasProd g a",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
... | by
have : (g ∘ (↑)) ∘ e = f ∘ (↑) := funext he
rw [← hasProd_subtype_mulSupport, ← this, e.hasProd_iff, hasProd_subtype_mulSupport] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 318,
"column": 2
} | {
"line": 318,
"column": 33
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : CommMonoid α\ninst✝¹ : TopologicalSpace α\nf g : β → α\na b : α\nL : SummationFilter β\ninst✝ : ContinuousMul α\nhf : HasProd f a L\nhg : HasProd g b L\n⊢ HasProd (fun b ↦ f b * g b) (a * b) L",
"usedConstants": [
"HMul.hMul",
"Monoid.toMulOneClass",... | dsimp only [HasProd] at hf hg ⊢ | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
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