module stringlengths 16 90 | startPos dict | endPos dict | nextStartPos dict | goals listlengths 0 96 | goalsAfter listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 375
values | kind stringclasses 379
values |
|---|---|---|---|---|---|---|---|---|
Mathlib.Tactic.Algebra.Lemmas | {
"line": 95,
"column": 47
} | {
"line": 96,
"column": 43
} | {
"line": 98,
"column": 0
} | [
{
"pp": "n : ℕ\nR : Type u_3\nA : Type u_4\ninst✝³ : CommRing R\ninst✝² : CommRing A\ninst✝¹ : Algebra R A\na : A\ninst✝ : n.AtLeastTwo\n⊢ -OfNat.ofNat n • a = -OfNat.ofNat n * a",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NegZeroClass.toNeg",
"NonAssocSemi... | [] | by
simpa [← nat_rawCast_2] using! ofNat_smul | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 857,
"column": 4
} | {
"line": 862,
"column": 56
} | {
"line": 863,
"column": 4
} | [
{
"pp": "case succ.refine_1.a\na0 a' : ONote\nN0 : a0.NF\nNa' : a'.NF\nm : ℕ\nd : ω ∣ a'.repr\ne0✝ : a0.repr ≠ 0\nh : a'.repr + ↑m < ω ^ a0.repr\nn : ℕ+\nNo : (a0.oadd n a').NF\nk : ℕ\nR' : Ordinal.{0} := ⋯\nR : Ordinal.{0} := ⋯\nω0 : Ordinal.{0} := ⋯\nα' : Ordinal.{0} := ⋯\nIH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0... | [
"case succ.refine_1.a\na0 a' : ONote\nN0 : a0.NF\nNa' : a'.NF\nm : ℕ\nd : ω ∣ a'.repr\ne0✝ : a0.repr ≠ 0\nh : a'.repr + ↑m < ω ^ a0.repr\nn : ℕ+\nNo : (a0.oadd n a').NF\nk : ℕ\nR' : Ordinal.{0} := ⋯\nR : Ordinal.{0} := ⋯\nω0 : Ordinal.{0} := ⋯\nα' : Ordinal.{0} := ⋯\nIH : (k ≠ 0 → R < ω0 ^ succ ↑k) ∧ ω0 ^ ↑k * α' +... | · simp only [Nat.cast_add_one, opow_add_one, opow_mul, opow_succ, mul_assoc]
gcongr ?_ * ?_
rw [← Ordinal.opow_add]
have : _ < ω ^ (repr a0 + repr a0) := (No.below_of_lt ?_).repr_lt
· exact mul_lt_omega0_opow rr0 this (natCast_lt_omega0 _)
· simpa using (add_lt_add_iff_left (repr a0)).2 e0 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 853,
"column": 4
} | {
"line": 866,
"column": 94
} | {
"line": 867,
"column": 2
} | [
{
"pp": "case succ.refine_1\na0 a' : ONote\nN0 : a0.NF\nNa' : a'.NF\nm : ℕ\nd : ω ∣ a'.repr\ne0 : a0.repr ≠ 0\nh : a'.repr + ↑m < ω ^ a0.repr\nn : ℕ+\nNo : (a0.oadd n a').NF\nk : ℕ\nR' : Ordinal.{0} := (opowAux 0 a0 (a0.oadd n a' * ↑m) (k + 1) m).repr\nR : Ordinal.{0} := (opowAux 0 a0 (a0.oadd n a' * ↑m) k m).r... | [] | rw [RR, ← opow_mul _ _ (succ k.succ)]
have e0 := pos_iff_ne_zero.2 e0
have rr0 : 0 < repr a0 + repr a0 := lt_of_lt_of_le e0 le_add_self
apply isPrincipal_add_omega0_opow
· simp only [Nat.cast_add_one, opow_add_one, opow_mul, opow_succ, mul_assoc]
gcongr ?_ * ?_
rw [← Ordinal.opow_add]
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 853,
"column": 4
} | {
"line": 866,
"column": 94
} | {
"line": 867,
"column": 2
} | [
{
"pp": "case succ.refine_1\na0 a' : ONote\nN0 : a0.NF\nNa' : a'.NF\nm : ℕ\nd : ω ∣ a'.repr\ne0 : a0.repr ≠ 0\nh : a'.repr + ↑m < ω ^ a0.repr\nn : ℕ+\nNo : (a0.oadd n a').NF\nk : ℕ\nR' : Ordinal.{0} := (opowAux 0 a0 (a0.oadd n a' * ↑m) (k + 1) m).repr\nR : Ordinal.{0} := (opowAux 0 a0 (a0.oadd n a' * ↑m) k m).r... | [] | rw [RR, ← opow_mul _ _ (succ k.succ)]
have e0 := pos_iff_ne_zero.2 e0
have rr0 : 0 < repr a0 + repr a0 := lt_of_lt_of_le e0 le_add_self
apply isPrincipal_add_omega0_opow
· simp only [Nat.cast_add_one, opow_add_one, opow_mul, opow_succ, mul_assoc]
gcongr ?_ * ?_
rw [← Ordinal.opow_add]
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Basic | {
"line": 193,
"column": 2
} | {
"line": 195,
"column": 43
} | {
"line": 197,
"column": 0
} | [
{
"pp": "basis : Basis\n⊢ zero.Approximates",
"ppTerm": "?m.2",
"assigned": true,
"usedConstants": [
"Real",
"Tactic.ComputeAsymptotics.MultiseriesExpansion.zero",
"Tactic.ComputeAsymptotics.MultiseriesExpansion.Approximates",
"Real.instZero",
"Tactic.ComputeAsymptotics... | [] | cases basis with
| nil => simp [zero]
| cons => exact Approximates.nil (by rfl) | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Basic | {
"line": 193,
"column": 2
} | {
"line": 195,
"column": 43
} | {
"line": 197,
"column": 0
} | [
{
"pp": "basis : Basis\n⊢ zero.Approximates",
"ppTerm": "?m.2",
"assigned": true,
"usedConstants": [
"Real",
"Tactic.ComputeAsymptotics.MultiseriesExpansion.zero",
"Tactic.ComputeAsymptotics.MultiseriesExpansion.Approximates",
"Real.instZero",
"Tactic.ComputeAsymptotics... | [] | cases basis with
| nil => simp [zero]
| cons => exact Approximates.nil (by rfl) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Basic | {
"line": 193,
"column": 2
} | {
"line": 195,
"column": 43
} | {
"line": 197,
"column": 0
} | [
{
"pp": "basis : Basis\n⊢ zero.Approximates",
"ppTerm": "?m.2",
"assigned": true,
"usedConstants": [
"Real",
"Tactic.ComputeAsymptotics.MultiseriesExpansion.zero",
"Tactic.ComputeAsymptotics.MultiseriesExpansion.Approximates",
"Real.instZero",
"Tactic.ComputeAsymptotics... | [] | cases basis with
| nil => simp [zero]
| cons => exact Approximates.nil (by rfl) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 1037,
"column": 2
} | {
"line": 1037,
"column": 18
} | {
"line": 1038,
"column": 2
} | [
{
"pp": "o : ONote\n⊢ o.FundamentalSequenceProp o.fundamentalSequence",
"ppTerm": "?m.2",
"assigned": true,
"usedConstants": [
"ONote.rec",
"ONote.zero",
"ONote.fundamentalSequence",
"ONote.FundamentalSequenceProp",
"ONote",
"rfl",
"PNat"
],
"usedFVa... | [
"case oadd\na : ONote\nm : ℕ+\nb : ONote\niha : a.FundamentalSequenceProp a.fundamentalSequence\nihb : b.FundamentalSequenceProp b.fundamentalSequence\n⊢ (a.oadd m b).FundamentalSequenceProp (a.oadd m b).fundamentalSequence"
] | induction o with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 1265,
"column": 22
} | {
"line": 1265,
"column": 38
} | {
"line": 1266,
"column": 2
} | [
{
"pp": "C : NONote → Sort u_1\nH0 : C 0\nH1 : (e : NONote) → (n : ℕ+) → (a : NONote) → (h : a.below e) → C e → C a → C (e.oadd n a h)\no : ONote\nh : o.NF\n⊢ C ⟨o, h⟩",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"ONote.NF",
"ONote.rec",
"ONote.oadd",
"ONote.NF.s... | [] | induction o with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Topology.UniformSpace.Ascoli | {
"line": 385,
"column": 35
} | {
"line": 385,
"column": 52
} | {
"line": 385,
"column": 53
} | [
{
"pp": "case inr\nι : Type u_1\nX : Type u_2\nα : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : UniformSpace α\nF : ι → X → α\n𝔖 : Set (Set X)\n𝔖_compact : ∀ K ∈ 𝔖, IsCompact K\nF_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K\nH : IsClosed[UniformOnFun.topologicalSpace X α 𝔖] (range (⇑(UniformOnFun.ofFun 𝔖) ∘ ... | [
"case inr\nι : Type u_1\nX : Type u_2\nα : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : UniformSpace α\nF : ι → X → α\n𝔖 : Set (Set X)\n𝔖_compact : ∀ K ∈ 𝔖, IsCompact K\nF_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K\nH : IsClosed[UniformOnFun.topologicalSpace X α 𝔖] (range (⇑(UniformOnFun.ofFun 𝔖) ∘ F))\nh✝ : No... | ← Filter.map_top, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.UniformSpace.Ascoli | {
"line": 403,
"column": 35
} | {
"line": 403,
"column": 52
} | {
"line": 403,
"column": 53
} | [
{
"pp": "ι : Type u_1\nX : Type u_2\nα : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : UniformSpace α\nF : ι → X → α\n𝔖 : Set (Set X)\n𝔖_compact : ∀ K ∈ 𝔖, IsCompact K\n𝔖_covers : ⋃₀ 𝔖 = univ\nF_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K\n⊢ (∀ (a : X →ᵤ[𝔖] α), ClusterPt a (𝓟 (range (⇑(UniformOnFun.ofFun 𝔖... | [
"ι : Type u_1\nX : Type u_2\nα : Type u_3\ninst✝¹ : TopologicalSpace X\ninst✝ : UniformSpace α\nF : ι → X → α\n𝔖 : Set (Set X)\n𝔖_compact : ∀ K ∈ 𝔖, IsCompact K\n𝔖_covers : ⋃₀ 𝔖 = univ\nF_eqcont : ∀ K ∈ 𝔖, EquicontinuousOn F K\n⊢ (∀ (a : X →ᵤ[𝔖] α), ClusterPt a (map (⇑(UniformOnFun.ofFun 𝔖) ∘ F) ⊤) → a ∈ ra... | ← Filter.map_top, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Algebra.Module.IsWeak | {
"line": 135,
"column": 38
} | {
"line": 135,
"column": 51
} | {
"line": 135,
"column": 51
} | [
{
"pp": "𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nE' : Type u_5\nF' : Type u_6\ninst✝¹⁰ : CommSemiring 𝕜\ninst✝⁹ : TopologicalSpace 𝕜\ninst✝⁸ : AddCommMonoid E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : AddCommMonoid F\ninst✝⁵ : Module 𝕜 F\ninst : TopologicalSpace E\ninst✝⁴ : AddCommMonoid E'\ninst✝³ : Module 𝕜 E'\n... | [
"𝕜 : Type u_2\nE : Type u_3\nF : Type u_4\nE' : Type u_5\nF' : Type u_6\ninst✝¹⁰ : CommSemiring 𝕜\ninst✝⁹ : TopologicalSpace 𝕜\ninst✝⁸ : AddCommMonoid E\ninst✝⁷ : Module 𝕜 E\ninst✝⁶ : AddCommMonoid F\ninst✝⁵ : Module 𝕜 F\ninst : TopologicalSpace E\ninst✝⁴ : AddCommMonoid E'\ninst✝³ : Module 𝕜 E'\ninst✝² : Add... | induced_to_pi | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.CWComplex.Classical.Graph | {
"line": 39,
"column": 4
} | {
"line": 39,
"column": 99
} | {
"line": 40,
"column": 4
} | [
{
"pp": "X : Type u_1\ninst✝¹ : TopologicalSpace X\nC : Set X\ninst✝ : CWComplex C\ne : cell C 1\nx y z w : cell C 0\nh1 : cellFrontier 1 e = {↑(map 0 x) ![]} ∪ {↑(map 0 y) ![]}\nh2 : {↑(map 0 x) ![]} ∪ {↑(map 0 y) ![]} = {↑(map 0 z) ![]} ∪ {↑(map 0 w) ![]}\n⊢ x = z ∨ x = w",
"ppTerm": "?m.76",
"assigne... | [
"X : Type u_1\ninst✝¹ : TopologicalSpace X\nC : Set X\ninst✝ : CWComplex C\ne : cell C 1\nx y z w : cell C 0\nh1 : cellFrontier 1 e = {↑(map 0 x) ![]} ∪ {↑(map 0 y) ![]}\nh2 : y = w ∧ x = z ∨ y = z ∧ x = w\n⊢ x = z ∨ x = w"
] | simp only [(RelCWComplex.injective_map_zero C).eq_iff, union_singleton, pair_eq_pair_iff] at h2 | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.DerivedSet | {
"line": 75,
"column": 12
} | {
"line": 75,
"column": 14
} | {
"line": 76,
"column": 4
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nA : Set X\nh : derivedSet A ⊆ A\na : X\n⊢ ClusterPt a (𝓟 A) → a ∈ A",
"ppTerm": "?m.27",
"assigned": true,
"usedConstants": [
"ClusterPt",
"Filter.principal"
],
"usedFVars": [
"X",
"inst✝",
"a",
"A"
]... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nA : Set X\nh : derivedSet A ⊆ A\na : X\nha : ClusterPt a (𝓟 A)\n⊢ a ∈ A"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Category.Compactum | {
"line": 493,
"column": 2
} | {
"line": 498,
"column": 36
} | {
"line": 500,
"column": 0
} | [
{
"pp": "⊢ CreatesLimits (forget CompHaus)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"Compactum.forget",
"CategoryTheory.Equivalence.unitIso",
"CategoryTheory.Functor.createsLimitsOfIsEquivalence",
"ContinuousMap",
"Catego... | [] | let e : forget CompHaus ≅ compactumToCompHaus.inv ⋙ Compactum.forget :=
(((forget CompHaus).leftUnitor.symm ≪≫
Functor.isoWhiskerRight compactumToCompHaus.asEquivalence.symm.unitIso (forget CompHaus)) ≪≫
compactumToCompHaus.inv.associator compactumToCompHaus (forget CompHaus)) ≪≫
Functor.isoWhiskerLeft ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Category.Compactum | {
"line": 493,
"column": 2
} | {
"line": 498,
"column": 36
} | {
"line": 500,
"column": 0
} | [
{
"pp": "⊢ CreatesLimits (forget CompHaus)",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"CategoryTheory.Functor",
"Compactum.forget",
"CategoryTheory.Equivalence.unitIso",
"CategoryTheory.Functor.createsLimitsOfIsEquivalence",
"ContinuousMap",
"Catego... | [] | let e : forget CompHaus ≅ compactumToCompHaus.inv ⋙ Compactum.forget :=
(((forget CompHaus).leftUnitor.symm ≪≫
Functor.isoWhiskerRight compactumToCompHaus.asEquivalence.symm.unitIso (forget CompHaus)) ≪≫
compactumToCompHaus.inv.associator compactumToCompHaus (forget CompHaus)) ≪≫
Functor.isoWhiskerLeft ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Category.Profinite.Nobeling.ZeroLimit | {
"line": 197,
"column": 2
} | {
"line": 199,
"column": 7
} | {
"line": 201,
"column": 0
} | [
{
"pp": "case h.refine_2\nI : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no₁ o₂ : Ordinal.{u}\nh : o₁ ≤ o₂\nl : Products I\ngl : Products.isGood (π C fun x ↦ ord I x < o₁) l\n⊢ (πs C o₂) ((πs' C h) (eval (π C fun x ↦ ord I x < o₁) ⟨l, gl⟩)) = (πs C o₁) (eval (π C fun x ↦ ord I x... | [] | · rw [← LocallyConstant.coe_inj, coe_πs C o₂, ← LocallyConstant.toFun_eq_coe, coe_πs',
Function.comp_assoc, projRestricts_comp_projRestrict C _, coe_πs]
rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.CWComplex.Classical.Basic | {
"line": 616,
"column": 66
} | {
"line": 645,
"column": 37
} | {
"line": 647,
"column": 0
} | [
{
"pp": "X : Type u_1\nt : TopologicalSpace X\nC D : Set X\ninst✝ : RelCWComplex C D\nn : ℕ\ni : cell C n\n⊢ ∃ I, cellFrontier n i ⊆ D ∪ ⋃ m, ⋃ (_ : m < n), ⋃ j ∈ I m, openCell m j",
"ppTerm": "?m.42",
"assigned": true,
"usedConstants": [
"dite_cond_eq_true",
"Nat.case_strong_induction_o... | [] | by
induction n using Nat.case_strong_induction_on with
| hz => simp [cellFrontier_zero_eq_empty]
| hi n hn =>
-- We apply `cellFrontier_subset_base_union_finite_closedCell` once and then apply
-- the induction hypothesis to the finitely many cells that
-- `cellFrontier_subset_base_union_finite_closedC... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Category.Profinite.Nobeling.Successor | {
"line": 260,
"column": 2
} | {
"line": 273,
"column": 35
} | {
"line": 275,
"column": 0
} | [
{
"pp": "case refine_3\nI : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\nf : LocallyConstant ↑C ℤ\nhf :\n ⇑((LocallyConstant.comapₗ ℤ { toFun := CC'₁ C hsC ho, continuo... | [] | · ext ⟨x, hx⟩
rw [← union_C0C1_eq C ho] at hx
rcases hx with hx₀ | hx₁
· have hx₀' : ProjRestrict C (ord I · < o) ⟨x, hx⟩ = x := by
simpa only [ProjRestrict, Set.MapsTo.val_restrict_apply] using! C0_projOrd C hsC ho hx₀
simp only [C₀C, πs_apply_apply, hx₀', hx₀, LocallyConstant.piecewise'_appl... | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Category.Profinite.Nobeling.Successor | {
"line": 311,
"column": 16
} | {
"line": 311,
"column": 18
} | {
"line": 312,
"column": 8
} | [
{
"pp": "I : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\nl : Products I\nh : Products.isGood C l\nhh : term I ho ∉ ↑l\nhe :\n Products.eval (π C fun x ↦ ord I x < o) l ∈ Submodule.span... | [
"I : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\nl : Products I\nh : Products.isGood C l\nhh : term I ho ∉ ↑l\nhe :\n Products.eval (π C fun x ↦ ord I x < o) l ∈ Submodule.span ℤ (Products... | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Compactness.CompactSystem | {
"line": 140,
"column": 4
} | {
"line": 143,
"column": 45
} | {
"line": 144,
"column": 2
} | [
{
"pp": "α : Type u_1\nS : Set (Set α)\nhpi : IsPiSystem S\nh : ∀ (C : ℕ → Set α), Directed (fun x1 x2 ↦ x1 ⊇ x2) C → (∀ (i : ℕ), C i ∈ S) → ⋂ i, C i = ∅ → ∃ n, C n = ∅\nC : ℕ → Set α\nh1 : ∀ (i : ℕ), C i ∈ insert ∅ S\nh2 : ⋂ n, ⋂ m, ⋂ (_ : m ≤ n), C m = ∅\nthis : (∀ (n : ℕ), dissipate C n ∈ S ∨ dissipate C n =... | [] | by_cases! f : ∀ n, dissipate C n ∈ S
· exact h (dissipate C) directed_dissipate f this.2
· obtain ⟨n, hn⟩ := f
exact ⟨n, by simpa [hn] using this.1 n⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Compactness.CompactSystem | {
"line": 140,
"column": 4
} | {
"line": 143,
"column": 45
} | {
"line": 144,
"column": 2
} | [
{
"pp": "α : Type u_1\nS : Set (Set α)\nhpi : IsPiSystem S\nh : ∀ (C : ℕ → Set α), Directed (fun x1 x2 ↦ x1 ⊇ x2) C → (∀ (i : ℕ), C i ∈ S) → ⋂ i, C i = ∅ → ∃ n, C n = ∅\nC : ℕ → Set α\nh1 : ∀ (i : ℕ), C i ∈ insert ∅ S\nh2 : ⋂ n, ⋂ m, ⋂ (_ : m ≤ n), C m = ∅\nthis : (∀ (n : ℕ), dissipate C n ∈ S ∨ dissipate C n =... | [] | by_cases! f : ∀ n, dissipate C n ∈ S
· exact h (dissipate C) directed_dissipate f this.2
· obtain ⟨n, hn⟩ := f
exact ⟨n, by simpa [hn] using this.1 n⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Connected.CardComponents | {
"line": 52,
"column": 6
} | {
"line": 52,
"column": 59
} | {
"line": 53,
"column": 6
} | [
{
"pp": "case inl.intro.refine_1\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\nhf₁ : IsOpenMap f\nhf₂ : IsClosedMap f\ninst✝ : ConnectedSpace Y\ny : Y\nh :\n ∀ {n : ℕ} (U : Fin n → Set X),\n (∀ (i : Fin n), IsClopen (U i)) →\n (∀ (i : Fin n), (U i).No... | [
"case inl.intro.refine_2\nX : Type u_1\nY : Type u_2\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\nf : X → Y\nhf₁ : IsOpenMap f\nhf₂ : IsClosedMap f\ninst✝ : ConnectedSpace Y\ny : Y\nh :\n ∀ {n : ℕ} (U : Fin n → Set X),\n (∀ (i : Fin n), IsClopen (U i)) →\n (∀ (i : Fin n), (U i).Nonempty) → Pa... | · exact (isClopen_discrete _).preimage continuous_coe | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Category.Profinite.Nobeling.Successor | {
"line": 520,
"column": 2
} | {
"line": 528,
"column": 84
} | {
"line": 530,
"column": 0
} | [
{
"pp": "I : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\nq : ↑(GoodProducts (π C fun x ↦ ord I x < o))\nl : ↑(MaxProducts C ho)\n⊢ List.Lex (fun x1 x2 ↦ x1 < x2) ↑↑q ↑↑l",
"ppTerm":... | [] | have : Inhabited I := ⟨term I ho⟩
by_cases h : q.val.val = []
· rw [h, max_eq_o_cons_tail C hsC ho l]
exact List.Lex.nil
· rw [← List.cons_head!_tail h, max_eq_o_cons_tail C hsC ho l]
apply List.Lex.rel
rw [← Ordinal.typein_lt_typein (· < ·)]
simp only [term, Ordinal.typein_enum]
exact Product... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Category.Profinite.Nobeling.Successor | {
"line": 520,
"column": 2
} | {
"line": 528,
"column": 84
} | {
"line": 530,
"column": 0
} | [
{
"pp": "I : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\nq : ↑(GoodProducts (π C fun x ↦ ord I x < o))\nl : ↑(MaxProducts C ho)\n⊢ List.Lex (fun x1 x2 ↦ x1 < x2) ↑↑q ↑↑l",
"ppTerm":... | [] | have : Inhabited I := ⟨term I ho⟩
by_cases h : q.val.val = []
· rw [h, max_eq_o_cons_tail C hsC ho l]
exact List.Lex.nil
· rw [← List.cons_head!_tail h, max_eq_o_cons_tail C hsC ho l]
apply List.Lex.rel
rw [← Ordinal.typein_lt_typein (· < ·)]
simp only [term, Ordinal.typein_enum]
exact Product... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Category.Profinite.Nobeling.Successor | {
"line": 596,
"column": 4
} | {
"line": 596,
"column": 88
} | {
"line": 597,
"column": 4
} | [
{
"pp": "case h₂\nI : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\nl : ↑(MaxProducts C ho)\nh₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x ↦ ord I x < o)))\nthis ... | [
"case h₂\nI : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhC : IsClosed C\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\nl : ↑(MaxProducts C ho)\nh₁ : ⊤ ≤ Submodule.span ℤ (Set.range (eval (π C fun x ↦ ord I x < o)))\nthis : Inhabited ... | refine ⟨⟨term I ho :: q.val, isChain_cons_of_lt C hsC ho l q (hmmem hq)⟩, ⟨?_, rfl⟩⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Convenient.HomSpace | {
"line": 81,
"column": 2
} | {
"line": 86,
"column": 13
} | {
"line": 87,
"column": 2
} | [
{
"pp": "case refine_1\nι : Type t\nX : ι → Type u\ninst✝⁴ : (i : ι) → TopologicalSpace (X i)\nY : Type v\ninst✝³ : TopologicalSpace Y\nZ : Type v'\ninst✝² : TopologicalSpace Z\nT : Type v''\ninst✝¹ : TopologicalSpace T\ninst✝ : ∀ (i j : ι), IsGeneratedBy X (X i × X j)\ng : Y × Z → T\nh : ContinuousGeneratedBy ... | [
"case refine_2\nι : Type t\nX : ι → Type u\ninst✝⁴ : (i : ι) → TopologicalSpace (X i)\nY : Type v\ninst✝³ : TopologicalSpace Y\nZ : Type v'\ninst✝² : TopologicalSpace Z\nT : Type v''\ninst✝¹ : TopologicalSpace T\ninst✝ : ∀ (i j : ι), IsGeneratedBy X (X i × X j)\ng : Y × Z → T\nh :\n ∀ (i₁ : ι) (f₁ : C(X i₁, Z)) (i... | · rw [IsGeneratedBy.continuous_iff X]
intro j p
let φ : X i₁ × X i₂ → Y × Z := fun (x₁, x₂) ↦ (f₂ x₂, f₁ x₁)
replace h := h.comp (show Continuous φ by fun_prop).continuousGeneratedBy
rw [continuousGeneratedBy_def] at h
exact h p | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Germ | {
"line": 165,
"column": 10
} | {
"line": 165,
"column": 12
} | {
"line": 166,
"column": 2
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝ : TopologicalSpace X\nf : X → Y\nh : ∀ (x : X), (↑f).IsConstant\ns : Set Y\na : X\n⊢ a ∈ f ⁻¹' s → f ⁻¹' s ∈ 𝓝 a",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Membership.mem",
"Set.preimage",
"Set.instMembership",
"Set"... | [
"X : Type u_1\nY : Type u_2\ninst✝ : TopologicalSpace X\nf : X → Y\nh : ∀ (x : X), (↑f).IsConstant\ns : Set Y\na : X\nha : a ∈ f ⁻¹' s\n⊢ f ⁻¹' s ∈ 𝓝 a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Homotopy.HSpaces | {
"line": 125,
"column": 47
} | {
"line": 125,
"column": 66
} | {
"line": 125,
"column": 66
} | [
{
"pp": "M : Type u\ninst✝² : MulOneClass M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\n⊢ { toFun := Function.uncurry Mul.mul, continuous_toFun := ⋯ }.comp ((const M 1).prodMk (ContinuousMap.id M)) =\n ContinuousMap.id M",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"... | [] | ext1; apply one_mul | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Homotopy.HSpaces | {
"line": 125,
"column": 47
} | {
"line": 125,
"column": 66
} | {
"line": 125,
"column": 66
} | [
{
"pp": "M : Type u\ninst✝² : MulOneClass M\ninst✝¹ : TopologicalSpace M\ninst✝ : ContinuousMul M\n⊢ { toFun := Function.uncurry Mul.mul, continuous_toFun := ⋯ }.comp ((const M 1).prodMk (ContinuousMap.id M)) =\n ContinuousMap.id M",
"ppTerm": "?m.61",
"assigned": true,
"usedConstants": [
"... | [] | ext1; apply one_mul | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.List | {
"line": 77,
"column": 6
} | {
"line": 77,
"column": 16
} | {
"line": 77,
"column": 17
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\na : α\nl : List α\n⊢ Tendsto (fun p ↦ p.1 :: p.2) (𝓝 a ×ˢ 𝓝 l) (𝓝 (a :: l))",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"nhds_cons",
"SProd.sprod",
"congrArg",
"nhds",
"id",
"Seq.s... | [
"α : Type u_1\ninst✝ : TopologicalSpace α\na : α\nl : List α\n⊢ Tendsto (fun p ↦ p.1 :: p.2) (𝓝 a ×ˢ 𝓝 l) (cons <$> 𝓝 a <*> 𝓝 l)"
] | nhds_cons, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Maps.Proper.UniversallyClosed | {
"line": 35,
"column": 4
} | {
"line": 35,
"column": 33
} | {
"line": 37,
"column": 4
} | [
{
"pp": "case mpr\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nH : Continuous f ∧ IsClosedMap (Prod.map f id)\n⊢ Continuous f ∧ ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (nhds y) → ∃ x, f x = y ∧ ↑𝒰 ≤ nhds x",
"ppTerm": "?mpr",
"assigned": true,
... | [
"case mpr\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nH : Continuous f ∧ IsClosedMap (Prod.map f id)\n𝒰 : Ultrafilter X\ny : Y\nhy : Tendsto f (↑𝒰) (nhds y)\n⊢ ∃ x, f x = y ∧ ↑𝒰 ≤ nhds x"
] | refine ⟨H.1, fun 𝒰 y hy ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Maps.Proper.UniversallyClosed | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 33
} | {
"line": 79,
"column": 4
} | [
{
"pp": "case mpr\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nH : Continuous f ∧ IsClosedMap (Prod.map f id)\n⊢ Continuous f ∧ ∀ ⦃𝒰 : Ultrafilter X⦄ ⦃y : Y⦄, Tendsto f (↑𝒰) (nhds y) → ∃ x, f x = y ∧ ↑𝒰 ≤ nhds x",
"ppTerm": "?mpr",
"assigned": true,
... | [
"case mpr\nX : Type u\nY : Type v\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X → Y\nH : Continuous f ∧ IsClosedMap (Prod.map f id)\n𝒰 : Ultrafilter X\ny : Y\nhy : Tendsto f (↑𝒰) (nhds y)\n⊢ ∃ x, f x = y ∧ ↑𝒰 ≤ nhds x"
] | refine ⟨H.1, fun 𝒰 y hy ↦ ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.MetricSpace.BundledFun | {
"line": 191,
"column": 2
} | {
"line": 191,
"column": 34
} | {
"line": 192,
"column": 2
} | [
{
"pp": "case inr\nX : Type u_1\nR : Type u_2\nY : Type u_3\ninst✝³ : AddCommMonoid R\ninst✝² : LinearOrder R\ninst✝¹ : AddLeftStrictMono R\ninst✝ : IsOrderedAddMonoid R\nf : Y → PseudoMetric X R\ns : Finset Y\nh : ∀ d ∈ s, (f d).IsUltra\nx y z : X\nhs : s.Nonempty\ni : Y\nhi : i ∈ s\n⊢ (∃ b ∈ s, (f i) x z ≤ (f... | [
"case inr\nX : Type u_1\nR : Type u_2\nY : Type u_3\ninst✝³ : AddCommMonoid R\ninst✝² : LinearOrder R\ninst✝¹ : AddLeftStrictMono R\ninst✝ : IsOrderedAddMonoid R\nf : Y → PseudoMetric X R\ns : Finset Y\nh✝ : ∀ d ∈ s, (f d).IsUltra\nx y z : X\nhs : s.Nonempty\ni : Y\nhi : i ∈ s\nh : (f i) x z ≤ max ((f i) x y) ((f i... | have h := (h i hi).le_sup' x y z | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 522,
"column": 4
} | {
"line": 522,
"column": 77
} | {
"line": 523,
"column": 4
} | [
{
"pp": "case mpr.refine_2\nN✝ : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace X\nx : X\ninst✝¹ : DecidableEq N✝\nN : Type ?u.15\ninst✝ : Unique N\na₁ a₂ : ↑(Ω^ N X x)\nH : ((genLoopEquivOfUnique N) a₁).Homotopy ((genLoopEquivOfUnique N) a₂)\ny : N → ↑I\n⊢ { toFun := fun tx ↦ H (tx.1, tx.2 default), continu... | [
"case mpr.refine_3\nN✝ : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace X\nx : X\ninst✝¹ : DecidableEq N✝\nN : Type ?u.15\ninst✝ : Unique N\na₁ a₂ : ↑(Ω^ N X x)\nH : ((genLoopEquivOfUnique N) a₁).Homotopy ((genLoopEquivOfUnique N) a₂)\ny : N → ↑I\n⊢ { toFun := fun tx ↦ H (tx.1, tx.2 default), continuous_toFun :=... | · exact (H.apply_zero _).trans (congr_arg a₁ (eq_const_of_unique y).symm) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.MetricSpace.Closeds | {
"line": 167,
"column": 12
} | {
"line": 167,
"column": 76
} | {
"line": 168,
"column": 12
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝² : EMetricSpace α\ninst✝¹ : EMetricSpace β\ns✝ : Set α\ninst✝ : CompleteSpace α\nB : ℕ → ℝ≥0∞ := fun n ↦ 2⁻¹ ^ n\nB_pos : ∀ (n : ℕ), 0 < B n\nB_ne_top : ∀ (n : ℕ), B n ≠ ∞\ns : ℕ → Closeds α\nhs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N\nt0 : Set α := ⋂ ... | [
"α : Type u_1\nβ : Type u_2\ninst✝² : EMetricSpace α\ninst✝¹ : EMetricSpace β\ns✝ : Set α\ninst✝ : CompleteSpace α\nB : ℕ → ℝ≥0∞ := fun n ↦ 2⁻¹ ^ n\nB_pos : ∀ (n : ℕ), 0 < B n\nB_ne_top : ∀ (n : ℕ), B n ≠ ∞\ns : ℕ → Closeds α\nhs : ∀ (N n m : ℕ), N ≤ n → N ≤ m → edist (s n) (s m) < B N\nt0 : Set α := ⋂ n, closure[P... | simp only [exists_prop, Set.mem_iUnion, Filter.eventually_atTop] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.HausdorffAlexandroff | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 73
} | {
"line": 95,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝² : Nonempty X\ninst✝¹ : MetricSpace X\ninst✝ : CompactSpace X\nthis : TopologicalSpace.SeparableSpace X := TopologicalSpace.SecondCountableTopology.to_separableSpace\nemb : X → ℕ → ↑unitInterval\nh_emb : Topology.IsEmbedding emb\nKH : Set (ℕ → ↑unitInterval) := Set.range emb\ng : X ... | [
"X : Type u_1\ninst✝² : Nonempty X\ninst✝¹ : MetricSpace X\ninst✝ : CompactSpace X\nthis : TopologicalSpace.SeparableSpace X := TopologicalSpace.SecondCountableTopology.to_separableSpace\nemb : X → ℕ → ↑unitInterval\nh_emb : Topology.IsEmbedding emb\nKH : Set (ℕ → ↑unitInterval) := Set.range emb\ng : X ≃ₜ ↑KH := h_... | let f' : (ℕ → Bool) → KC := Subtype.coind f (by simp [← hf_surjective]) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Topology.MetricSpace.CoveringNumbers | {
"line": 399,
"column": 68
} | {
"line": 406,
"column": 13
} | {
"line": 407,
"column": 4
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : PseudoEMetricSpace X\ninst✝ : PseudoEMetricSpace Y\nA : Set X\nε : ℝ≥0\nf : X → Y\nhf : Isometry f\nhf_inj : InjOn f A\nC : Set Y\nhC_subset : C ⊆ f '' A\nhC_cover : IsCover ε (f '' A) C\n⊢ ∃ C' ⊆ A, C = f '' C'",
"ppTerm": "?m.103",
"assigned": true,
"u... | [] | by
have (x : C) : ∃ y ∈ A, f y = x := by simpa using hC_subset x.2
choose g hg_mem hg using this
refine ⟨Set.range g, ?_, ?_⟩
· rwa [Set.range_subset_iff]
· ext
simp
grind | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.HolderNorm | {
"line": 205,
"column": 12
} | {
"line": 205,
"column": 14
} | {
"line": 205,
"column": 15
} | [
{
"pp": "case mp\nX : Type u_1\nY : Type u_2\ninst✝¹ : MetricSpace X\ninst✝ : EMetricSpace Y\nr : ℝ≥0\nf : X → Y\nh : eHolderNorm r f = 0\n⊢ ∀ (x₁ x₂ : X), f x₁ = f x₂",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"X"
],
"usedGoals": [
{
"ne... | [
"case mp\nX : Type u_1\nY : Type u_2\ninst✝¹ : MetricSpace X\ninst✝ : EMetricSpace Y\nr : ℝ≥0\nf : X → Y\nh : eHolderNorm r f = 0\nx₁ : X\n⊢ ∀ (x₂ : X), f x₁ = f x₂"
] | x₁ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.MetricSpace.HolderNorm | {
"line": 227,
"column": 33
} | {
"line": 227,
"column": 45
} | {
"line": 227,
"column": 45
} | [
{
"pp": "case neg\nX : Type u_1\nY : Type u_2\ninst✝¹ : MetricSpace X\ninst✝ : EMetricSpace Y\nr : ℝ≥0\nf : X → Y\nhf : MemHolder r f\nx₁ x₂ : X\nhx : ¬x₁ = x₂\n⊢ edist (f x₁) (f x₂) ≤ ↑(⨅ C, ⨅ (_ : HolderWith C r f), ↑C).toNNReal * edist x₁ x₂ ^ ↑r",
"ppTerm": "?neg✝",
"assigned": true,
"usedConsta... | [
"case neg\nX : Type u_1\nY : Type u_2\ninst✝¹ : MetricSpace X\ninst✝ : EMetricSpace Y\nr : ℝ≥0\nf : X → Y\nhf : MemHolder r f\nx₁ x₂ : X\nhx : ¬x₁ = x₂\n⊢ edist (f x₁) (f x₂) ≤ (⨅ C, ⨅ (_ : HolderWith C r f), ↑C) * edist x₁ x₂ ^ ↑r",
"case neg\nX : Type u_1\nY : Type u_2\ninst✝¹ : MetricSpace X\ninst✝ : EMetricSp... | coe_toNNReal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.HolderNorm | {
"line": 245,
"column": 20
} | {
"line": 245,
"column": 32
} | {
"line": 245,
"column": 32
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : MetricSpace X\ninst✝ : EMetricSpace Y\nr : ℝ≥0\nf : X → Y\nhf : MemHolder r f\n⊢ ↑(eHolderNorm r f).toNNReal = eHolderNorm r f",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"ENNReal.coe_toNNReal",
"Eq.mpr",
"ENNReal.ofNNReal... | [
"X : Type u_1\nY : Type u_2\ninst✝¹ : MetricSpace X\ninst✝ : EMetricSpace Y\nr : ℝ≥0\nf : X → Y\nhf : MemHolder r f\n⊢ eHolderNorm r f = eHolderNorm r f",
"X : Type u_1\nY : Type u_2\ninst✝¹ : MetricSpace X\ninst✝ : EMetricSpace Y\nr : ℝ≥0\nf : X → Y\nhf : MemHolder r f\n⊢ eHolderNorm r f ≠ ∞"
] | coe_toNNReal | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Infsep | {
"line": 376,
"column": 15
} | {
"line": 376,
"column": 25
} | {
"line": 376,
"column": 26
} | [
{
"pp": "α : Type u_1\ninst✝¹ : PseudoMetricSpace α\ns : Set α\ninst✝ : Decidable s.Nontrivial\nhs : s.Nontrivial\nd : ℝ\nh : d ∈ uncurry dist '' s.offDiag\n⊢ 0 ≤ d",
"ppTerm": "?m.54",
"assigned": true,
"usedConstants": [
"Real",
"Set.offDiag",
"Membership.mem",
"Exists",
... | [
"α : Type u_1\ninst✝¹ : PseudoMetricSpace α\ns : Set α\ninst✝ : Decidable s.Nontrivial\nhs : s.Nontrivial\nd : ℝ\nh : ∃ x ∈ s.offDiag, uncurry dist x = d\n⊢ 0 ≤ d"
] | mem_image, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.MetricSpace.Infsep | {
"line": 391,
"column": 2
} | {
"line": 393,
"column": 47
} | {
"line": 394,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝² : PseudoMetricSpace α\ns : Set α\ninst✝¹ : Decidable s.Nontrivial\ninst✝ : Fintype ↑s\nhs : s.Nontrivial\n⊢ s.infsep = s.offDiag.toFinset.inf' ⋯ (uncurry dist)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PseudoEMetricSpace.... | [
"case neg\nα : Type u_1\ninst✝² : PseudoMetricSpace α\ns : Set α\ninst✝¹ : Decidable s.Nontrivial\ninst✝ : Fintype ↑s\nhs : ¬s.Nontrivial\n⊢ s.infsep = 0"
] | · refine eq_of_forall_le_iff fun _ => ?_
simp_rw [hs.le_infsep_iff, imp_forall_iff, Finset.le_inf'_iff, mem_toFinset, mem_offDiag,
Prod.forall, uncurry_apply_pair, and_imp] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.MetricSpace.Infsep | {
"line": 420,
"column": 4
} | {
"line": 420,
"column": 95
} | {
"line": 421,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : PseudoMetricSpace α\ns : Finset α\nH : (↑s).Nontrivial ↔ s.offDiag.Nonempty\nhs : s.offDiag.Nonempty\n⊢ (↑s).infsep = s.offDiag.inf' hs (uncurry dist)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"PseudoEMet... | [] | classical simp_rw [(H.mpr hs).infsep_of_fintype, ← Finset.coe_offDiag, Finset.toFinset_coe] | Lean.Elab.Tactic.evalClassical | Lean.Parser.Tactic.classical |
Mathlib.Topology.MetricSpace.Infsep | {
"line": 420,
"column": 4
} | {
"line": 420,
"column": 95
} | {
"line": 421,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : PseudoMetricSpace α\ns : Finset α\nH : (↑s).Nontrivial ↔ s.offDiag.Nonempty\nhs : s.offDiag.Nonempty\n⊢ (↑s).infsep = s.offDiag.inf' hs (uncurry dist)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"PseudoEMet... | [] | classical simp_rw [(H.mpr hs).infsep_of_fintype, ← Finset.coe_offDiag, Finset.toFinset_coe] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Infsep | {
"line": 420,
"column": 4
} | {
"line": 420,
"column": 95
} | {
"line": 421,
"column": 2
} | [
{
"pp": "case pos\nα : Type u_1\ninst✝ : PseudoMetricSpace α\ns : Finset α\nH : (↑s).Nontrivial ↔ s.offDiag.Nonempty\nhs : s.offDiag.Nonempty\n⊢ (↑s).infsep = s.offDiag.inf' hs (uncurry dist)",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"PseudoEMet... | [] | classical simp_rw [(H.mpr hs).infsep_of_fintype, ← Finset.coe_offDiag, Finset.toFinset_coe] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.Infsep | {
"line": 425,
"column": 6
} | {
"line": 425,
"column": 24
} | {
"line": 425,
"column": 25
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoMetricSpace α\ns : Finset α\nhs : s.offDiag.Nonempty\n⊢ (↑s).infsep = s.offDiag.inf' hs (uncurry dist)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"Real",
"Real.instZer... | [
"α : Type u_1\ninst✝ : PseudoMetricSpace α\ns : Finset α\nhs : s.offDiag.Nonempty\n⊢ (if hs : s.offDiag.Nonempty then s.offDiag.inf' hs (uncurry dist) else 0) = s.offDiag.inf' hs (uncurry dist)"
] | Finset.coe_infsep, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Infsep | {
"line": 430,
"column": 6
} | {
"line": 430,
"column": 24
} | {
"line": 430,
"column": 25
} | [
{
"pp": "α : Type u_1\ninst✝ : PseudoMetricSpace α\ns : Finset α\nhs : ¬s.offDiag.Nonempty\n⊢ (↑s).infsep = 0",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"PseudoEMetricSpace.toWeakPseudoEMetricSpace",
"Real",
"Real.instZero",
"Finset.inf'",
... | [
"α : Type u_1\ninst✝ : PseudoMetricSpace α\ns : Finset α\nhs : ¬s.offDiag.Nonempty\n⊢ (if hs : s.offDiag.Nonempty then s.offDiag.inf' hs (uncurry dist) else 0) = 0"
] | Finset.coe_infsep, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.LowerUpperTopology | {
"line": 562,
"column": 29
} | {
"line": 562,
"column": 39
} | {
"line": 562,
"column": 40
} | [
{
"pp": "a : Prop\n⊢ (Iic a)ᶜ ∈ {∅, {True}}",
"ppTerm": "?m.51",
"assigned": true,
"usedConstants": [
"Compl.compl",
"PartialOrder.toPreorder",
"Classical.propDecidable",
"Membership.mem",
"Set.instSingletonSet",
"Insert.insert",
"Set.instCompl",
"dite... | [
"case pos\na : Prop\nh✝ : a\n⊢ (Iic a)ᶜ ∈ {∅, {True}}",
"case neg\na : Prop\nh✝ : ¬a\n⊢ (Iic a)ᶜ ∈ {∅, {True}}"
] | by_cases a | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__1» | «tacticBy_cases_:_» |
Mathlib.Topology.Order.Completion | {
"line": 81,
"column": 6
} | {
"line": 81,
"column": 36
} | {
"line": 82,
"column": 6
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : LinearOrder α\nx : α\nq : ℚ\nhx₁ : IsSuccPrelimit x → 0 ≤ q\nhx₂ : IsPredPrelimit x → q ≤ 0\ny : α\nr : ℚ\nhy₁ : IsSuccPrelimit y → 0 ≤ r\nhy₂ : IsPredPrelimit y → r ≤ 0\nh : x < y\n⊢ ∃ a, ⟨toLex (x, q), ⋯⟩ < a ∧ a < ⟨toLex (y, r), ⋯⟩",
"ppTerm": "?inl",
"assigne... | [
"case pos\nα : Type u_1\ninst✝ : LinearOrder α\nx : α\nq : ℚ\nhx₁ : IsSuccPrelimit x → 0 ≤ q\nhx₂ : IsPredPrelimit x → q ≤ 0\ny : α\nr : ℚ\nhy₁ : IsSuccPrelimit y → 0 ≤ r\nhy₂ : IsPredPrelimit y → r ≤ 0\nh : x < y\nhx : IsPredPrelimit x\n⊢ ∃ a, ⟨toLex (x, q), ⋯⟩ < a ∧ a < ⟨toLex (y, r), ⋯⟩",
"case neg\nα : Type u... | by_cases hx : IsPredPrelimit x | «_aux_Init_ByCases___macroRules_tacticBy_cases_:__2» | «tacticBy_cases_:_» |
Mathlib.Topology.Order.Completion | {
"line": 104,
"column": 6
} | {
"line": 104,
"column": 33
} | {
"line": 105,
"column": 6
} | [
{
"pp": "case refine_1.inr\nα : Type u_1\ninst✝² : LinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nx : α\nq : ℚ\nx✝ : (IsSuccPrelimit x → 0 ≤ q) ∧ (IsPredPrelimit x → q ≤ 0)\nhx₁ : IsSuccPrelimit x → 0 ≤ q\nhx₂ : IsPredPrelimit x → q ≤ 0\nhq : q < 0\ny : α\nhy : y ⋖ x\n⊢ IsOpen (⇑some ⁻¹' I... | [
"α : Type u_1\ninst✝² : LinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderTopology α\nx : α\nq : ℚ\nx✝ : (IsSuccPrelimit x → 0 ≤ q) ∧ (IsPredPrelimit x → q ≤ 0)\nhx₁ : IsSuccPrelimit x → 0 ≤ q\nhx₂ : IsPredPrelimit x → q ≤ 0\nhq : q < 0\ny : α\nhy : y ⋖ x\n⊢ ⇑some ⁻¹' Ioi ⟨toLex (x, q), ⋯⟩ = Ioi y"
] | convert isOpen_Ioi (a := y) | Mathlib.Tactic._aux_Mathlib_Tactic_Convert___elabRules_Mathlib_Tactic_convert_1 | Mathlib.Tactic.convert |
Mathlib.Topology.Separation.PerfectlyNormal | {
"line": 69,
"column": 42
} | {
"line": 69,
"column": 60
} | {
"line": 69,
"column": 60
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nh✝ : PerfectlyNormalSpace X\ns : Set X\nhs : IsClosed[inst✝] s\nU : ℕ → Set X\nho : ∀ (n : ℕ), IsOpen[inst✝] (U n)\nhu : s = ⋂ n, U n\nthis : ∀ (n : ℕ), Disjoint s (U n)ᶜ\nf : ℕ → C(X, ℝ)\nhfs : ∀ (n : ℕ), EqOn (⇑(f n)) 0 s\nhfu : ∀ (n : ℕ), EqOn (⇑(f n)) 1 (U ... | [] | simp [(hfr n x).2] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Separation.PerfectlyNormal | {
"line": 69,
"column": 42
} | {
"line": 69,
"column": 60
} | {
"line": 69,
"column": 60
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nh✝ : PerfectlyNormalSpace X\ns : Set X\nhs : IsClosed[inst✝] s\nU : ℕ → Set X\nho : ∀ (n : ℕ), IsOpen[inst✝] (U n)\nhu : s = ⋂ n, U n\nthis : ∀ (n : ℕ), Disjoint s (U n)ᶜ\nf : ℕ → C(X, ℝ)\nhfs : ∀ (n : ℕ), EqOn (⇑(f n)) 0 s\nhfu : ∀ (n : ℕ), EqOn (⇑(f n)) 1 (U ... | [] | simp [(hfr n x).2] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Separation.PerfectlyNormal | {
"line": 69,
"column": 42
} | {
"line": 69,
"column": 60
} | {
"line": 69,
"column": 60
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nh✝ : PerfectlyNormalSpace X\ns : Set X\nhs : IsClosed[inst✝] s\nU : ℕ → Set X\nho : ∀ (n : ℕ), IsOpen[inst✝] (U n)\nhu : s = ⋂ n, U n\nthis : ∀ (n : ℕ), Disjoint s (U n)ᶜ\nf : ℕ → C(X, ℝ)\nhfs : ∀ (n : ℕ), EqOn (⇑(f n)) 0 s\nhfu : ∀ (n : ℕ), EqOn (⇑(f n)) 1 (U ... | [] | simp [(hfr n x).2] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sheaves.Skyscraper | {
"line": 70,
"column": 6
} | {
"line": 71,
"column": 55
} | {
"line": 72,
"column": 4
} | [
{
"pp": "case pos\nX : TopCat\np₀ : ↑X\ninst✝² : (U : Opens ↑X) → Decidable (p₀ ∈ U)\nC : Type v\ninst✝¹ : Category.{w, v} C\ninst✝ : HasTerminal C\nA : C\nU V W : (Opens ↑X)ᵒᵖ\niVU : U ⟶ V\niWV : V ⟶ W\nhW : p₀ ∈ unop W\n⊢ (if h : p₀ ∈ unop W then eqToHom ⋯ else (⋯ ▸ terminalIsTerminal).from (if p₀ ∈ unop U th... | [] | have hV : p₀ ∈ unop V := leOfHom iWV.unop hW
simp only [dif_pos hW, dif_pos hV, eqToHom_trans] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sheaves.Skyscraper | {
"line": 70,
"column": 6
} | {
"line": 71,
"column": 55
} | {
"line": 72,
"column": 4
} | [
{
"pp": "case pos\nX : TopCat\np₀ : ↑X\ninst✝² : (U : Opens ↑X) → Decidable (p₀ ∈ U)\nC : Type v\ninst✝¹ : Category.{w, v} C\ninst✝ : HasTerminal C\nA : C\nU V W : (Opens ↑X)ᵒᵖ\niVU : U ⟶ V\niWV : V ⟶ W\nhW : p₀ ∈ unop W\n⊢ (if h : p₀ ∈ unop W then eqToHom ⋯ else (⋯ ▸ terminalIsTerminal).from (if p₀ ∈ unop U th... | [] | have hV : p₀ ∈ unop V := leOfHom iWV.unop hW
simp only [dif_pos hW, dif_pos hV, eqToHom_trans] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sheaves.Abelian | {
"line": 112,
"column": 2
} | {
"line": 113,
"column": 72
} | {
"line": 115,
"column": 0
} | [
{
"pp": "case mpr\nC : Type v\ninst✝⁶ : Category.{u, v} C\ninst✝⁵ : HasColimits C\ninst✝⁴ : HasLimits C\nFC : C → C → Type u_1\nCC : C → Type u\ninst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝² : PreservesFilteredColimits (CategoryTheory.forget C)\ninst✝¹ : PreservesLi... | [] | exact fun x => (h x).of_iso
(ShortComplex.mapHomologyIso S (forget C X ⋙ stalkFunctor C x)).symm | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Sheaves.Alexandrov | {
"line": 142,
"column": 4
} | {
"line": 142,
"column": 23
} | {
"line": 143,
"column": 4
} | [
{
"pp": "X✝ : Type v\ninst✝⁶ : TopologicalSpace X✝\ninst✝⁵ : Preorder X✝\ninst✝⁴ : Topology.IsUpperSet X✝\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasLimits C\nF✝ : X✝ ⥤ C\nX : TopCat\ninst✝¹ : Preorder ↑X\ninst✝ : Topology.IsUpperSet ↑X\nF : ↑X ⥤ C\nα : Type v\nUs : α → Opens ↑X\n⊢ ∀ (s : Cone ((Objec... | [
"X✝ : Type v\ninst✝⁶ : TopologicalSpace X✝\ninst✝⁵ : Preorder X✝\ninst✝⁴ : Topology.IsUpperSet X✝\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasLimits C\nF✝ : X✝ ⥤ C\nX : TopCat\ninst✝¹ : Preorder ↑X\ninst✝ : Topology.IsUpperSet ↑X\nF : ↑X ⥤ C\nα : Type v\nUs : α → Opens ↑X\nS : Cone ((ObjectProperty.ι fun V... | rintro S ⟨V, i, hV⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Topology.MetricSpace.GromovHausdorff | {
"line": 694,
"column": 8
} | {
"line": 694,
"column": 45
} | {
"line": 695,
"column": 6
} | [
{
"pp": "δ : ℝ\nδpos : δ > 0\nε : ℝ := 2 / 5 * δ\nεpos : 0 < ε\ns : (p : GHSpace) → Set p.Rep\nhs : ∀ (p : GHSpace), (s p).Finite ∧ univ ⊆ ⋃ x ∈ s p, ball x ε\nN : GHSpace → ℕ := fun p ↦ Nat.card ↑(s p)\nE : (p : GHSpace) → ↑(s p) ≃ Fin (Nat.card ↑(s p)) := fun p ↦ Finite.equivFin ↑(s p)\nF : GHSpace → (n : ℕ) ... | [] | simp only [F, (E p).symm_apply_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Sheaves.SheafCondition.EqualizerProducts | {
"line": 164,
"column": 2
} | {
"line": 164,
"column": 17
} | {
"line": 165,
"column": 2
} | [
{
"pp": "C : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\nG : Presheaf C X\nα : F ≅ G\n⊢ fork F U ≅ (Cone.postcompose (diagram.isoOfIso U α).inv).obj (fork G U)",
"ppTerm": "?m.40",
"assigned": true,
"usedConstants": [
... | [
"case i\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasProducts C\nX : TopCat\nF : Presheaf C X\nι : Type v'\nU : ι → Opens ↑X\nG : Presheaf C X\nα : F ≅ G\n⊢ (fork F U).pt ≅ ((Cone.postcompose (diagram.isoOfIso U α).inv).obj (fork G U)).pt",
"case w\nC : Type u\ninst✝¹ : Category.{v, u} C\ninst✝ : HasProduc... | fapply Fork.ext | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_tacticFapply__1 | Batteries.Tactic.tacticFapply_ |
Mathlib.Topology.MetricSpace.GromovHausdorff | {
"line": 838,
"column": 44
} | {
"line": 838,
"column": 81
} | {
"line": 839,
"column": 10
} | [
{
"pp": "t : Set GHSpace\nC : ℝ\nu : ℕ → ℝ\nK : ℕ → ℕ\nulim : Tendsto u atTop (𝓝 0)\nhdiam : ∀ p ∈ t, diam univ ≤ C\nhcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)\nδ : ℝ\nδpos : δ > 0\nε : ℝ := 1 / 5 * δ\nεpos : 0 < ε\nn : ℕ\nhn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε\nu_le_ε : u n ≤ ε\ns... | [] | simp only [F, (E p).symm_apply_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.GromovHausdorff | {
"line": 838,
"column": 44
} | {
"line": 838,
"column": 81
} | {
"line": 839,
"column": 10
} | [
{
"pp": "t : Set GHSpace\nC : ℝ\nu : ℕ → ℝ\nK : ℕ → ℕ\nulim : Tendsto u atTop (𝓝 0)\nhdiam : ∀ p ∈ t, diam univ ≤ C\nhcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)\nδ : ℝ\nδpos : δ > 0\nε : ℝ := 1 / 5 * δ\nεpos : 0 < ε\nn : ℕ\nhn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε\nu_le_ε : u n ≤ ε\ns... | [] | simp only [F, (E p).symm_apply_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.GromovHausdorff | {
"line": 838,
"column": 44
} | {
"line": 838,
"column": 81
} | {
"line": 839,
"column": 10
} | [
{
"pp": "t : Set GHSpace\nC : ℝ\nu : ℕ → ℝ\nK : ℕ → ℕ\nulim : Tendsto u atTop (𝓝 0)\nhdiam : ∀ p ∈ t, diam univ ≤ C\nhcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)\nδ : ℝ\nδpos : δ > 0\nε : ℝ := 1 / 5 * δ\nεpos : 0 < ε\nn : ℕ\nhn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε\nu_le_ε : u n ≤ ε\ns... | [] | simp only [F, (E p).symm_apply_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.GromovHausdorffRealized | {
"line": 514,
"column": 2
} | {
"line": 523,
"column": 64
} | {
"line": 524,
"column": 2
} | [
{
"pp": "X : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : Cb X Y\nh : f ∈ candidatesB X Y\nr : ℝ\nhr : HD (optimalGHDist X Y) < r\n⊢ hausdorffDist (range (optimalGHInjl X Y)) (range (optimalGHIn... | [
"X : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : Cb X Y\nh : f ∈ candidatesB X Y\nr : ℝ\nhr : HD (optimalGHDist X Y) < r\nA : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist ... | have A : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range (optimalGHInjr X Y), dist x y ≤ r := by
rintro _ ⟨z, rfl⟩
have I1 : (⨆ x, ⨅ y, optimalGHDist X Y (inl x, inr y)) < r :=
lt_of_le_of_lt (le_max_left _ _) hr
have I2 :
⨅ y, optimalGHDist X Y (inl z, inr y) ≤ ⨆ x, ⨅ y, optimalGHDist X Y (inl x... | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.SmallInductiveDimension | {
"line": 112,
"column": 10
} | {
"line": 112,
"column": 12
} | {
"line": 113,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝ : TopologicalSpace X\nn : ℕ\nh : ¬HasSmallInductiveDimensionLT X n\na : WithBot ℕ∞\n⊢ (∀ (i : ℕ), a < ↑i → HasSmallInductiveDimensionLT X i) → ↑n ≤ a",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"WithBot.instPreorder",
"WithBot",
"Preorder.... | [
"X : Type u_1\ninst✝ : TopologicalSpace X\nn : ℕ\nh : ¬HasSmallInductiveDimensionLT X n\na : WithBot ℕ∞\nha : ∀ (i : ℕ), a < ↑i → HasSmallInductiveDimensionLT X i\n⊢ ↑n ≤ a"
] | ha | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.Topology.Sion | {
"line": 215,
"column": 9
} | {
"line": 215,
"column": 41
} | {
"line": 215,
"column": 42
} | [
{
"pp": "E : Type u_1\nF : Type u_2\nβ : Type u_3\ninst✝⁸ : LinearOrder β\nX : Set E\nY : Set F\nf : E → F → β\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : IsTopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\nne_X : X.Nonempty\nkX : IsCompact X\nhfy : ∀ y ∈ Y, LowerSemic... | [
"E : Type u_1\nF : Type u_2\nβ : Type u_3\ninst✝⁸ : LinearOrder β\nX : Set E\nY : Set F\nf : E → F → β\ninst✝⁷ : TopologicalSpace E\ninst✝⁶ : AddCommGroup E\ninst✝⁵ : Module ℝ E\ninst✝⁴ : IsTopologicalAddGroup E\ninst✝³ : ContinuousSMul ℝ E\nne_X : X.Nonempty\nkX : IsCompact X\nhfy : ∀ y ∈ Y, LowerSemicontinuousOn ... | monotone_sublevelLeft _ hbb'.le, | Mathlib.Tactic.GRewrite.evalGRewriteSeq | null |
Mathlib.Topology.VectorBundle.ContinuousAlternatingMap | {
"line": 185,
"column": 30
} | {
"line": 185,
"column": 87
} | {
"line": 186,
"column": 6
} | [
{
"pp": "𝕜 : Type u_1\nι : Type u_2\ninst✝²¹ : NontriviallyNormedField 𝕜\nB : Type u_3\ninst✝²⁰ : TopologicalSpace B\nF₁ : Type u_4\ninst✝¹⁹ : NormedAddCommGroup F₁\ninst✝¹⁸ : NormedSpace 𝕜 F₁\nE₁ : B → Type u_5\ninst✝¹⁷ : (x : B) → AddCommGroup (E₁ x)\ninst✝¹⁶ : (x : B) → Module 𝕜 (E₁ x)\ninst✝¹⁵ : Topolog... | [] | simp [continuousAlternatingMap, Pretrivialization.toFun'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.VectorBundle.ContinuousAlternatingMap | {
"line": 186,
"column": 30
} | {
"line": 186,
"column": 87
} | {
"line": 186,
"column": 88
} | [
{
"pp": "𝕜 : Type u_1\nι : Type u_2\ninst✝²¹ : NontriviallyNormedField 𝕜\nB : Type u_3\ninst✝²⁰ : TopologicalSpace B\nF₁ : Type u_4\ninst✝¹⁹ : NormedAddCommGroup F₁\ninst✝¹⁸ : NormedSpace 𝕜 F₁\nE₁ : B → Type u_5\ninst✝¹⁷ : (x : B) → AddCommGroup (E₁ x)\ninst✝¹⁶ : (x : B) → Module 𝕜 (E₁ x)\ninst✝¹⁵ : Topolog... | [] | simp [continuousAlternatingMap, Pretrivialization.toFun'] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Logic.Function.Basic | {
"line": 316,
"column": 9
} | {
"line": 316,
"column": 57
} | {
"line": 318,
"column": 0
} | [
{
"pp": "case refine_1\nα : Sort u_1\nβ : Sort u_2\nf : α → β\nγ : Type u_4\ninst✝ : Nontrivial γ\nnot_surj : ¬Surjective f\ninj : Injective fun g ↦ g ∘ f\nc c' : γ\nb₀ : β\nhb : ¬∃ a, f a = b₀\na : α\n⊢ (fun g ↦ g ∘ f) (fun x ↦ c) a = (fun g ↦ g ∘ f) (fun x ↦ if x = b₀ then c' else c) a",
"ppTerm": "?refin... | [] | simp only [comp_apply, if_neg fun h ↦ hb ⟨a, h⟩] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Subtype | {
"line": 127,
"column": 50
} | {
"line": 129,
"column": 86
} | {
"line": 131,
"column": 0
} | [
{
"pp": "α : Sort u_5\nβ : α → Type u_4\nne : ∀ (a : α), Nonempty (β a)\np : α → Prop\n⊢ Surjective fun f ↦ restrict p f",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Classical.propDecidable",
"Subtype",
"Nonempty.some",
"dite",
"Subtype.mk",
"Subtype.... | [] | by
classical
exact fun f ↦ ⟨fun x ↦ if h : p x then f ⟨x, h⟩ else Nonempty.some (ne x), by grind⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Logic.Relation | {
"line": 910,
"column": 8
} | {
"line": 910,
"column": 23
} | {
"line": 911,
"column": 10
} | [
{
"pp": "case tail.single\nα : Type u_1\nr : α → α → Prop\na✝² b✝¹ c : α\nh : ∀ (a b c : α), r a b → r a c → ∃ d, ReflGen r b d ∧ ReflTransGen r c d\nhac : ReflTransGen r a✝² c\nd e : α\na✝¹ : ReflTransGen r a✝² d\nhde : r d e\nb✝ f b : α\na✝ : ReflTransGen r d f\nhfb : r f b\na : α\nhea : ReflTransGen r e a\nh... | [] | | single hfa => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.Algebra.Group.Action.Faithful | {
"line": 120,
"column": 2
} | {
"line": 120,
"column": 42
} | {
"line": 121,
"column": 2
} | [
{
"pp": "R : Type u_4\nS : Type u_5\nT : Type u_6\ninst✝⁸ : Monoid S\ninst✝⁷ : MulOneClass T\ninst✝⁶ : SMul R S\ninst✝⁵ : SMul R T\ninst✝⁴ : MulAction S T\ninst✝³ : IsScalarTower R S S\ninst✝² : IsScalarTower R T T\ninst✝¹ : IsScalarTower R S T\ninst✝ : FaithfulSMul R T\n⊢ FaithfulSMul R S",
"ppTerm": "?m.1... | [
"R : Type u_4\nS : Type u_5\nT : Type u_6\ninst✝⁸ : Monoid S\ninst✝⁷ : MulOneClass T\ninst✝⁶ : SMul R S\ninst✝⁵ : SMul R T\ninst✝⁴ : MulAction S T\ninst✝³ : IsScalarTower R S S\ninst✝² : IsScalarTower R T T\ninst✝¹ : IsScalarTower R S T\ninst✝ : FaithfulSMul R T\n⊢ Injective fun r ↦ r • 1"
] | rw [faithfulSMul_iff_injective_smul_one] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Option.Basic | {
"line": 212,
"column": 56
} | {
"line": 212,
"column": 74
} | {
"line": 214,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nh : α → β\nf : γ → α\nx : α\ni : Option γ\n⊢ (i.elim (h x) fun j ↦ h (f j)) = h (i.elim x f)",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Option.casesOn",
"Option.some",
"Option.none",
"Eq.ndrec",
"Eq.refl"... | [] | by cases i <;> rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.Basic | {
"line": 1044,
"column": 2
} | {
"line": 1044,
"column": 81
} | {
"line": 1046,
"column": 0
} | [
{
"pp": "case inl.inl.inl\nα : Type u_2\ninst✝ : LinearOrder α\nh : ∀ ⦃x y z : α⦄, x < y → y < z → False\nx y z : α\nhne : x ≠ y ∧ y ≠ z ∧ x ≠ z\nh₁ : x < y\nh₂ : y < z\nh₃ : x < z\n⊢ False",
"ppTerm": "?inl.inl.inl",
"assigned": true,
"usedConstants": [],
"usedFVars": [
"h",
"x",
... | [] | exacts [h h₁ h₂, h h₂ h₃, h h₃ h₂, h h₃ h₁, h h₁ h₃, h h₂ h₃, h h₁ h₃, h h₂ h₁] | Batteries.Tactic._aux_Batteries_Tactic_Init___elabRules_Batteries_Tactic_exacts_1 | Batteries.Tactic.exacts |
Mathlib.Data.Nat.Basic | {
"line": 89,
"column": 9
} | {
"line": 89,
"column": 22
} | {
"line": 89,
"column": 23
} | [
{
"pp": "case refl\nC : ℕ → Sort u_1\nn m : ℕ\nnext : {k : ℕ} → C k → C (k + 1)\nHnext : ∀ (n : ℕ), Injective next\nx y : C n\nH : leRecOn ⋯ (fun {k} ↦ next) x = leRecOn ⋯ (fun {k} ↦ next) y\n⊢ x = y",
"ppTerm": "?refl",
"assigned": true,
"usedConstants": [
"Nat.le_refl",
"congrArg",
... | [
"case refl\nC : ℕ → Sort u_1\nn m : ℕ\nnext : {k : ℕ} → C k → C (k + 1)\nHnext : ∀ (n : ℕ), Injective next\nx y : C n\nH : x = leRecOn ⋯ (fun {k} ↦ next) y\n⊢ x = y"
] | leRecOn_self, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Compare | {
"line": 101,
"column": 4
} | {
"line": 102,
"column": 24
} | {
"line": 103,
"column": 2
} | [
{
"pp": "case inr.inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na b : α\na' b' : β\nh : ∀ {o : Ordering}, o.Compares a b → o.Compares a' b'\no : Ordering\nho : o.Compares a' b'\nhab : a = b\n⊢ o.Compares a b",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
... | [] | have hab : Compares Ordering.eq a b := hab
rwa [ho.inj (h hab)] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Compare | {
"line": 101,
"column": 4
} | {
"line": 102,
"column": 24
} | {
"line": 103,
"column": 2
} | [
{
"pp": "case inr.inl\nα : Type u_1\nβ : Type u_2\ninst✝¹ : LinearOrder α\ninst✝ : Preorder β\na b : α\na' b' : β\nh : ∀ {o : Ordering}, o.Compares a b → o.Compares a' b'\no : Ordering\nho : o.Compares a' b'\nhab : a = b\n⊢ o.Compares a b",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConstants": [
... | [] | have hab : Compares Ordering.eq a b := hab
rwa [ho.inj (h hab)] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Disjoint | {
"line": 533,
"column": 2
} | {
"line": 535,
"column": 28
} | {
"line": 537,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : Lattice α\ninst✝¹ : BoundedOrder α\ninst✝ : Subsingleton α\n⊢ ComplementedLattice α",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Codisjoint",
"ComplementedLattice.mk",
"congrArg",
"disjoint_bot_right",
"OrderB... | [] | refine ⟨fun a ↦ ⟨⊥, disjoint_bot_right, ?_⟩⟩
rw [Subsingleton.elim ⊥ ⊤]
exact codisjoint_top_right | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Disjoint | {
"line": 533,
"column": 2
} | {
"line": 535,
"column": 28
} | {
"line": 537,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝² : Lattice α\ninst✝¹ : BoundedOrder α\ninst✝ : Subsingleton α\n⊢ ComplementedLattice α",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Codisjoint",
"ComplementedLattice.mk",
"congrArg",
"disjoint_bot_right",
"OrderB... | [] | refine ⟨fun a ↦ ⟨⊥, disjoint_bot_right, ?_⟩⟩
rw [Subsingleton.elim ⊥ ⊤]
exact codisjoint_top_right | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Basic | {
"line": 1050,
"column": 29
} | {
"line": 1050,
"column": 34
} | {
"line": 1050,
"column": 35
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : Set α → Set β\nhf : Injective f\nh2 : f ∅ = ∅\ns : Set α\n⊢ f s ≠ ∅ ↔ s.Nonempty",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Ne",
"Iff",
"Set.Nonempty",
"Set.instEmptyCollect... | [
"α : Type u_1\nβ : Type u_2\nf : Set α → Set β\nhf : Injective f\nh2 : f ∅ = ∅\ns : Set α\n⊢ f s ≠ f ∅ ↔ s.Nonempty"
] | ← h2, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Control.EquivFunctor | {
"line": 90,
"column": 17
} | {
"line": 92,
"column": 30
} | {
"line": 93,
"column": 2
} | [
{
"pp": "f : Type u₀ → Type u₁\ninst✝¹ : Functor f\ninst✝ : LawfulFunctor f\nα : Type u₀\n⊢ Functor.map ⇑(Equiv.refl α) = id",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Equiv.instEquivLike",
"id",
"Equiv",
"funext",
"LawfulFunctor.id_map",
"Equiv.re... | [] | by
ext
apply LawfulFunctor.id_map | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Logic.Equiv.Option | {
"line": 192,
"column": 4
} | {
"line": 193,
"column": 7
} | {
"line": 195,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : DecidableEq β\nx : β\ne : α ≃ { y // y ≠ x }\n⊢ (fun e ↦ { toFun := fun a ↦ ⟨↑e (some a), ⋯⟩, invFun := fun b ↦ ((↑e).symm ↑b).get ⋯, left_inv := ⋯, right_inv := ⋯ })\n ((fun e ↦\n ⟨{ toFun := fun a ↦ a.casesOn' x (Subtype.val ∘ ⇑e),\n ... | [] | ext a
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Equiv.Option | {
"line": 192,
"column": 4
} | {
"line": 193,
"column": 7
} | {
"line": 195,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝ : DecidableEq β\nx : β\ne : α ≃ { y // y ≠ x }\n⊢ (fun e ↦ { toFun := fun a ↦ ⟨↑e (some a), ⋯⟩, invFun := fun b ↦ ((↑e).symm ↑b).get ⋯, left_inv := ⋯, right_inv := ⋯ })\n ((fun e ↦\n ⟨{ toFun := fun a ↦ a.casesOn' x (Subtype.val ∘ ⇑e),\n ... | [] | ext a
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.BooleanAlgebra.Basic | {
"line": 225,
"column": 2
} | {
"line": 227,
"column": 31
} | {
"line": 229,
"column": 0
} | [
{
"pp": "α : Type u\nx y : α\ninst✝ : GeneralizedBooleanAlgebra α\nhx : y ≤ x\nhy : y ≠ ⊥\n⊢ x \\ y < x",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"OrderBot.toBot",
"PartialOrder.toPreorde... | [] | refine sdiff_le.lt_of_ne fun h => hy ?_
rw [sdiff_eq_left, disjoint_iff] at h
rw [← h, inf_eq_right.mpr hx] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.BooleanAlgebra.Basic | {
"line": 225,
"column": 2
} | {
"line": 227,
"column": 31
} | {
"line": 229,
"column": 0
} | [
{
"pp": "α : Type u\nx y : α\ninst✝ : GeneralizedBooleanAlgebra α\nhx : y ≤ x\nhy : y ≠ ⊥\n⊢ x \\ y < x",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"OrderBot.toBot",
"PartialOrder.toPreorde... | [] | refine sdiff_le.lt_of_ne fun h => hy ?_
rw [sdiff_eq_left, disjoint_iff] at h
rw [← h, inf_eq_right.mpr hx] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.BooleanAlgebra.Basic | {
"line": 261,
"column": 6
} | {
"line": 266,
"column": 67
} | {
"line": 268,
"column": 0
} | [] | [] | x ⊓ y \ z ⊓ (z ⊓ x ⊔ x \ y) = x ⊓ y \ z ⊓ (z ⊓ x) ⊔ x ⊓ y \ z ⊓ x \ y := by rw [inf_sup_left]
_ = x ⊓ (y \ z ⊓ z ⊓ x) ⊔ x ⊓ y \ z ⊓ x \ y := by ac_rfl
_ = x ⊓ y \ z ⊓ x \ y := by rw [inf_sdiff_self_left, bot_inf_eq, inf_bot_eq, bot_sup_eq]
_ = x ⊓ (y \ z ⊓ y) ⊓ x \ y := by conv_lhs => rw [← inf_sdiff_... | Lean.Elab.Tactic.evalCalc | Lean.calcSteps |
Mathlib.Order.BooleanAlgebra.Basic | {
"line": 339,
"column": 18
} | {
"line": 339,
"column": 67
} | {
"line": 339,
"column": 67
} | [
{
"pp": "α : Type u\nx y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ y ⊓ (z ⊓ x \\ z) ⊓ y \\ z = ⊥",
"ppTerm": "?m.138",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"OrderBot.toBot",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.... | [] | rw [inf_sdiff_self_right, inf_bot_eq, bot_inf_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.BooleanAlgebra.Basic | {
"line": 339,
"column": 18
} | {
"line": 339,
"column": 67
} | {
"line": 339,
"column": 67
} | [
{
"pp": "α : Type u\nx y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ y ⊓ (z ⊓ x \\ z) ⊓ y \\ z = ⊥",
"ppTerm": "?m.138",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"OrderBot.toBot",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.... | [] | rw [inf_sdiff_self_right, inf_bot_eq, bot_inf_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.BooleanAlgebra.Basic | {
"line": 339,
"column": 18
} | {
"line": 339,
"column": 67
} | {
"line": 339,
"column": 67
} | [
{
"pp": "α : Type u\nx y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ y ⊓ (z ⊓ x \\ z) ⊓ y \\ z = ⊥",
"ppTerm": "?m.138",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"OrderBot.toBot",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.... | [] | rw [inf_sdiff_self_right, inf_bot_eq, bot_inf_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.BooleanAlgebra.Basic | {
"line": 376,
"column": 78
} | {
"line": 380,
"column": 62
} | {
"line": 382,
"column": 0
} | [
{
"pp": "α : Type u\nx y z : α\ninst✝ : GeneralizedBooleanAlgebra α\nh : x < z \\ y\nhyz : y ≤ z\n⊢ x ⊔ y < z",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"sdiff_sup_cancel",
"Eq.mpr",
"le_refl",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"sup_le_su... | [] | by
rw [← sdiff_sup_cancel hyz]
refine lt_of_le_not_ge (by grw [h]) fun h' => h.not_ge ?_
rw [← sdiff_idem]
exact (sdiff_le_sdiff_of_sup_le_sup_right h').trans sdiff_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Logic.Equiv.Basic | {
"line": 454,
"column": 35
} | {
"line": 454,
"column": 52
} | {
"line": 455,
"column": 2
} | [
{
"pp": "α✝ : Sort u_1\nα₁ : Sort u_2\nα₂ : Sort u_3\nβ✝ : Sort u_4\nβ₁ : Sort u_5\nβ₂ : Sort u_6\nγ✝ : Sort u_7\nδ : Sort u_8\nα : Type u_9\nβ : α → Type u_10\nγ : (a : α) → β a → Type u_11\np : (a : α) × β a → Prop\nuniq : Unique { ab // p ab }\na : α\nb : β a\nh : p ⟨a, b⟩\ns : (a : α) × (b : β a) × γ a b\n⊢... | [] | simp [sigmaAssoc] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Logic.Equiv.Basic | {
"line": 454,
"column": 35
} | {
"line": 454,
"column": 52
} | {
"line": 455,
"column": 2
} | [
{
"pp": "α✝ : Sort u_1\nα₁ : Sort u_2\nα₂ : Sort u_3\nβ✝ : Sort u_4\nβ₁ : Sort u_5\nβ₂ : Sort u_6\nγ✝ : Sort u_7\nδ : Sort u_8\nα : Type u_9\nβ : α → Type u_10\nγ : (a : α) → β a → Type u_11\np : (a : α) × β a → Prop\nuniq : Unique { ab // p ab }\na : α\nb : β a\nh : p ⟨a, b⟩\ns : (a : α) × (b : β a) × γ a b\n⊢... | [] | simp [sigmaAssoc] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Logic.Equiv.Basic | {
"line": 454,
"column": 35
} | {
"line": 454,
"column": 52
} | {
"line": 455,
"column": 2
} | [
{
"pp": "α✝ : Sort u_1\nα₁ : Sort u_2\nα₂ : Sort u_3\nβ✝ : Sort u_4\nβ₁ : Sort u_5\nβ₂ : Sort u_6\nγ✝ : Sort u_7\nδ : Sort u_8\nα : Type u_9\nβ : α → Type u_10\nγ : (a : α) → β a → Type u_11\np : (a : α) × β a → Prop\nuniq : Unique { ab // p ab }\na : α\nb : β a\nh : p ⟨a, b⟩\ns : (a : α) × (b : β a) × γ a b\n⊢... | [] | simp [sigmaAssoc] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Image | {
"line": 150,
"column": 17
} | {
"line": 150,
"column": 71
} | {
"line": 152,
"column": 0
} | [
{
"pp": "case succ\nα : Type u_1\nf : α → α\nn : ℕ\nih : preimage f^[n] = (preimage f)^[n]\n⊢ preimage f^[n + 1] = (preimage f)^[n + 1]",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Function.iterate_succ",
"congrArg",
"Function.iterate_succ'",
"Fu... | [] | rw [iterate_succ, iterate_succ', preimage_comp_eq, ih] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Set.Image | {
"line": 150,
"column": 17
} | {
"line": 150,
"column": 71
} | {
"line": 152,
"column": 0
} | [
{
"pp": "case succ\nα : Type u_1\nf : α → α\nn : ℕ\nih : preimage f^[n] = (preimage f)^[n]\n⊢ preimage f^[n + 1] = (preimage f)^[n + 1]",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Function.iterate_succ",
"congrArg",
"Function.iterate_succ'",
"Fu... | [] | rw [iterate_succ, iterate_succ', preimage_comp_eq, ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Image | {
"line": 150,
"column": 17
} | {
"line": 150,
"column": 71
} | {
"line": 152,
"column": 0
} | [
{
"pp": "case succ\nα : Type u_1\nf : α → α\nn : ℕ\nih : preimage f^[n] = (preimage f)^[n]\n⊢ preimage f^[n + 1] = (preimage f)^[n + 1]",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Function.iterate_succ",
"congrArg",
"Function.iterate_succ'",
"Fu... | [] | rw [iterate_succ, iterate_succ', preimage_comp_eq, ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Prod | {
"line": 605,
"column": 2
} | {
"line": 610,
"column": 36
} | {
"line": 612,
"column": 0
} | [
{
"pp": "α : Type u_1\ns t : Set α\nh : Disjoint s t\n⊢ (s ∪ t).offDiag = s.offDiag ∪ t.offDiag ∪ s ×ˢ t ∪ t ×ˢ s",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Set.ext",
"Eq.mpr",
"False",
"eq_false",
"and_true",
"SProd.sprod",
... | [] | ext x
simp only [mem_offDiag, mem_union, ne_eq, mem_prod]
constructor
· rintro ⟨h0 | h0, h1 | h1, h2⟩ <;> simp [h0, h1, h2]
· rintro (((⟨h0, h1, h2⟩ | ⟨h0, h1, h2⟩) | ⟨h0, h1⟩) | ⟨h0, h1⟩) <;>
simp [*, h.ne_of_mem, Ne.symm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Prod | {
"line": 605,
"column": 2
} | {
"line": 610,
"column": 36
} | {
"line": 612,
"column": 0
} | [
{
"pp": "α : Type u_1\ns t : Set α\nh : Disjoint s t\n⊢ (s ∪ t).offDiag = s.offDiag ∪ t.offDiag ∪ s ×ˢ t ∪ t ×ˢ s",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Set.instSProd",
"Set.ext",
"Eq.mpr",
"False",
"eq_false",
"and_true",
"SProd.sprod",
... | [] | ext x
simp only [mem_offDiag, mem_union, ne_eq, mem_prod]
constructor
· rintro ⟨h0 | h0, h1 | h1, h2⟩ <;> simp [h0, h1, h2]
· rintro (((⟨h0, h1, h2⟩ | ⟨h0, h1, h2⟩) | ⟨h0, h1⟩) | ⟨h0, h1⟩) <;>
simp [*, h.ne_of_mem, Ne.symm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Prod | {
"line": 688,
"column": 63
} | {
"line": 688,
"column": 85
} | {
"line": 690,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ns : Set ι\nt : (i : ι) → Set (α i)\ninst✝ : ∀ (i : ι), Nonempty (α i)\n⊢ s.pi t = ∅ ↔ ∃ i, i ∈ s ∧ t i = ∅",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"Membership.mem",
"Exists",
"not_isEmpty_of... | [] | simp [pi_eq_empty_iff] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Set.Prod | {
"line": 688,
"column": 63
} | {
"line": 688,
"column": 85
} | {
"line": 690,
"column": 0
} | [
{
"pp": "ι : Type u_1\nα : ι → Type u_2\ns : Set ι\nt : (i : ι) → Set (α i)\ninst✝ : ∀ (i : ι), Nonempty (α i)\n⊢ s.pi t = ∅ ↔ ∃ i, i ∈ s ∧ t i = ∅",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"False",
"congrArg",
"Membership.mem",
"Exists",
"not_isEmpty_of... | [] | simp [pi_eq_empty_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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