module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 34
} | {
"line": 239,
"column": 2
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nd : ℕ\nhdp : d ≤ rootMultiplicity 1 p\nhp : p = 0\n⊢ p.hilbertPoly d = 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Polynomial.hilbertPoly_zero_left",
... | [] | rw [hp, hilbertPoly_zero_left] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Polynomial.HilbertPoly | {
"line": 238,
"column": 4
} | {
"line": 238,
"column": 34
} | {
"line": 239,
"column": 2
} | [
{
"pp": "case pos\nF : Type u_1\ninst✝¹ : Field F\ninst✝ : CharZero F\np : F[X]\nd : ℕ\nhdp : d ≤ rootMultiplicity 1 p\nhp : p = 0\n⊢ p.hilbertPoly d = 0",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"Polynomial.hilbertPoly_zero_left",
... | [] | rw [hp, hilbertPoly_zero_left] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
"line": 36,
"column": 2
} | {
"line": 37,
"column": 49
} | {
"line": 38,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na b c x1 x2 : R\nhroots : (C a * X ^ 2 + C b * X + C c).roots = {x1, x2}\np : R[X] := C a * X ^ 2 + C b * X + C c\nhp_natDegree : p.natDegree = 2\n⊢ b = -a * (x1 + x2)",
"ppTerm": "?m.134",
"assigned": true,
"usedConstants": [
"Eq... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na b c x1 x2 : R\nhroots : (C a * X ^ 2 + C b * X + C c).roots = {x1, x2}\np : R[X] := C a * X ^ 2 + C b * X + C c\nhp_natDegree : p.natDegree = 2\nhp_roots_card : p.roots.card = p.natDegree\n⊢ b = -a * (x1 + x2)"
] | have hp_roots_card : p.roots.card = p.natDegree := by
rw [hp_natDegree, hroots, Multiset.card_pair] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.Polynomial.SmallDegreeVieta | {
"line": 48,
"column": 2
} | {
"line": 49,
"column": 49
} | {
"line": 50,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na b c x1 x2 : R\nhroots : (C a * X ^ 2 + C b * X + C c).roots = {x1, x2}\np : R[X] := C a * X ^ 2 + C b * X + C c\nhp_natDegree : p.natDegree = 2\n⊢ c = a * x1 * x2",
"ppTerm": "?m.132",
"assigned": true,
"usedConstants": [
"Eq.mp... | [
"R : Type u_1\ninst✝¹ : CommRing R\ninst✝ : IsDomain R\na b c x1 x2 : R\nhroots : (C a * X ^ 2 + C b * X + C c).roots = {x1, x2}\np : R[X] := C a * X ^ 2 + C b * X + C c\nhp_natDegree : p.natDegree = 2\nhp_roots_card : p.roots.card = p.natDegree\n⊢ c = a * x1 * x2"
] | have hp_roots_card : p.roots.card = p.natDegree := by
rw [hp_natDegree, hroots, Multiset.card_pair] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.PowerSeries.Restricted | {
"line": 80,
"column": 91
} | {
"line": 90,
"column": 44
} | {
"line": 93,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : NormedRing R\nc : ℝ\nf : R⟦X⟧\nhf : IsRestricted c f\nr : R\n⊢ IsRestricted c (r • f)",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"abs_nonneg._simp_1",
"Iff.mpr",
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Norm... | [] | by
if h : r = 0 then simpa [h] using zero c else
simp_rw [isRestricted_iff, norm_mul, norm_pow, Real.norm_eq_abs, abs_norm] at ⊢ hf
intro ε _
obtain ⟨n, hn⟩ := hf (ε / ‖r‖) (by positivity)
refine ⟨n, fun N hN ↦ ?_⟩
calc _ ≤ ‖r‖ * ‖(coeff N) f‖ * |c| ^ N :=
mul_le_mul_of_nonneg (norm_mul_le _ _) (by ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.PolynomialLaw.Basic | {
"line": 463,
"column": 2
} | {
"line": 464,
"column": 56
} | {
"line": 465,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommSemiring R\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nS : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nt : S ⊗[R] M\ns : S\nA : Subalgebra R S\nhA : A.FG\nht : t ∈ (rTensor M A.val.toLinearMap).range\n⊢ ∃ n ψ p q, (rTensor M ψ.toLinearMap) p = t ∧ ψ ... | [
"R : Type u\ninst✝⁴ : CommSemiring R\nM : Type u_1\ninst✝³ : AddCommMonoid M\ninst✝² : Module R M\nS : Type v\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\nt : S ⊗[R] M\ns : S\nA : Subalgebra R S\nhA : A.FG\nht : t ∈ (rTensor M A.val.toLinearMap).range\nhB : (A ⊔ Algebra.adjoin R ↑{s}).FG\n⊢ ∃ n ψ p q, (rTensor M ... | have hB : Subalgebra.FG (A ⊔ Algebra.adjoin R ({s} : Finset S)) :=
Subalgebra.FG.sup hA (Subalgebra.fg_adjoin_finset _) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.RingTheory.PowerSeries.Restricted | {
"line": 147,
"column": 2
} | {
"line": 153,
"column": 15
} | {
"line": 154,
"column": 2
} | [
{
"pp": "case calc.step.inl\nR : Type u_1\ninst✝¹ : NormedRing R\nc : ℝ\ninst✝ : IsUltrametricDist R\nf g : R⟦X⟧\na : ℝ\nha : 1 ≤ a\nb : ℝ\nhb : 1 ≤ b\nfBound1 : ∀ (a_1 : ℕ), ‖(coeff a_1) f‖ * |c| ^ a_1 ≤ a\ngBound1 : ∀ (a : ℕ), ‖(coeff a) g‖ * |c| ^ a ≤ b\nhf : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖(coef... | [
"case calc.step.inr\nR : Type u_1\ninst✝¹ : NormedRing R\nc : ℝ\ninst✝ : IsUltrametricDist R\nf g : R⟦X⟧\na : ℝ\nha : 1 ≤ a\nb : ℝ\nhb : 1 ≤ b\nfBound1 : ∀ (a_1 : ℕ), ‖(coeff a_1) f‖ * |c| ^ a_1 ≤ a\ngBound1 : ∀ (a : ℕ), ‖(coeff a) g‖ * |c| ^ a ≤ b\nhf : ∀ (ε : ℝ), 0 < ε → ∃ N, ∀ (n : ℕ), N ≤ n → ‖(coeff n) f‖ * |c... | · calc _ < ε / max a b * b := by
grw [gBound1 snd]
gcongr
exact fBound2 fst (by omega)
_ ≤ ε := by
rw [div_mul_comm, mul_le_iff_le_one_left ‹_›]
bound | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.RingTheory.RegularLocalRing.Polynomial | {
"line": 49,
"column": 71
} | {
"line": 49,
"column": 88
} | {
"line": 50,
"column": 8
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsRegularLocalRing R\np : Ideal R[X]\ninst✝ : p.IsPrime\nmax : comap C p = maximalIdeal R\nq : Ideal R[X] := Ideal.map C (maximalIdeal R)\nqle : q ≤ p\nreg : ↑(Submodule.spanFinrank (maximalIdeal R)) = ringKrullDim R\nfg' : (maximalIdeal R).FG\nfg : (Submodul... | [
"R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsRegularLocalRing R\np : Ideal R[X]\ninst✝ : p.IsPrime\nmax : comap C p = maximalIdeal R\nq : Ideal R[X] := Ideal.map C (maximalIdeal R)\nqle : q ≤ p\nreg : ↑(Submodule.spanFinrank (maximalIdeal R)) = ringKrullDim R\nfg' : (maximalIdeal R).FG\nfg : (Submodule.generators... | RingHom.coe_comp, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.PowerSeries.WeierstrassPreparation | {
"line": 764,
"column": 4
} | {
"line": 764,
"column": 65
} | {
"line": 765,
"column": 4
} | [
{
"pp": "A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : IsLocalRing A\ng q : A⟦X⟧\nr : A[X]\nhg : (map (IsLocalRing.residue A)) g ≠ 0\nH✝ : (X ^ ((map (IsLocalRing.residue A)) g).order.toNat).IsWeierstrassDivision g q r\nn : ℕ := ((map (IsLocalRing.residue A)) g).order.toNat\nH : (X ^ n).IsWeierstrassDivision g q r... | [
"A : Type u_1\ninst✝¹ : CommRing A\ninst✝ : IsLocalRing A\ng q : A⟦X⟧\nr : A[X]\nhg : (map (IsLocalRing.residue A)) g ≠ 0\nH✝ : (X ^ ((map (IsLocalRing.residue A)) g).order.toNat).IsWeierstrassDivision g q r\nn : ℕ := ((map (IsLocalRing.residue A)) g).order.toNat\nH : (X ^ n).IsWeierstrassDivision g q r\nf : A[X] :... | suffices f.degree = n by rw [Polynomial.natDegree, this]; rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticSuffices__1 | Lean.Parser.Tactic.tacticSuffices_ |
Mathlib.RingTheory.SimpleModule.Isotypic | {
"line": 248,
"column": 2
} | {
"line": 249,
"column": 73
} | {
"line": 251,
"column": 0
} | [
{
"pp": "R : Type u_2\nM : Type u\ninst✝³ : Ring R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\nN : Submodule R M\ninst✝ : IsSimpleModule R ↥N\ns : Set (Submodule R M)\nhs : ∀ (m : ↑s), IsSemisimpleModule R ↥↑m\nhN : N ≤ ⨆ a ∈ s, a\ne : ↥N ≃ₗ[R] ↥(inclusion hN).range\nthis✝ : IsSimpleModule R ↥(inclusion hN).... | [] | exact ⟨m, hm, _, map_le_iff_le_comap.mpr le,
⟨(e.trans e').trans (equivMapOfInjective _ (subtype_injective _) _)⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.RingTheory.SimpleRing.Congr | {
"line": 36,
"column": 83
} | {
"line": 40,
"column": 27
} | {
"line": 41,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝ : Ring R\n⊢ IsSimpleRing R ↔ Nontrivial R ∧ ∀ (I : Ideal R), I.IsTwoSided → I = ⊥ ∨ I = ⊤",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Nontrivial",
"NonUnitalNonAssocRing",
"Eq.mpr",
"Semiring.toModule",
"Equiv.instEquivLike",
... | [] | by
let e := orderIsoIsTwoSided (R := R)
simp_rw [isSimpleRing_iff, isSimpleOrder_iff, orderIsoRingCon.toEquiv.nontrivial_congr,
RingCon.nontrivial_iff, e.forall_congr_left, Subtype.forall, ← e.injective.eq_iff]
simp [e, Subtype.ext_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.WittVector.FrobeniusFractionField | {
"line": 94,
"column": 2
} | {
"line": 94,
"column": 34
} | {
"line": 95,
"column": 2
} | [
{
"pp": "p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : IsDomain k\nn : ℕ\na₁ a₂ : 𝕎 k\nbs : Fin (n + 1) → k\nha₁ : a₁.coeff 0 ≠ 0\nha₂ : a₂.coeff 0 ≠ 0\nthis✝ : (X ^ p * C (a₁.coeff 0 ^ p ^ (n + 1))).degree = ↑p\nthis : (X ^ p * C (a₁.coeff 0 ^ p ^ (n + 1)) - X ... | [
"p : ℕ\nhp : Fact (Nat.Prime p)\nk : Type u_1\ninst✝² : CommRing k\ninst✝¹ : CharP k p\ninst✝ : IsDomain k\nn : ℕ\na₁ a₂ : 𝕎 k\nbs : Fin (n + 1) → k\nha₁ : a₁.coeff 0 ≠ 0\nha₂ : a₂.coeff 0 ≠ 0\nthis✝ : (X ^ p * C (a₁.coeff 0 ^ p ^ (n + 1))).degree = ↑p\nthis : (X ^ p * C (a₁.coeff 0 ^ p ^ (n + 1)) - X * C (a₂.coef... | apply lt_of_le_of_lt degree_C_le | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.SetTheory.Cardinal.NatCount | {
"line": 34,
"column": 40
} | {
"line": 34,
"column": 65
} | {
"line": 34,
"column": 65
} | [
{
"pp": "p : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh : {k | p k}.Finite\n⊢ ↑(count p n) ≤ ↑{k | p k}.encard.toNat",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.encard_eq_top_iff._simp_1",
"Set.encard",
"ENat.instNatCast",
"instTopENat",
... | [
"p : ℕ → Prop\ninst✝ : DecidablePred p\nn : ℕ\nh : {k | p k}.Finite\n⊢ ↑(count p n) ≤ {k | p k}.encard"
] | ENat.coe_toNat (by simpa) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 124,
"column": 2
} | {
"line": 124,
"column": 80
} | {
"line": 125,
"column": 2
} | [
{
"pp": "case inl\nA : Type u_1\nT : ↥(tree A)\nx y : List A\nh : y ∈ pullSub (subAt T x) x\nh' : y.length ≤ x.length\n⊢ y ∈ ↑T",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Descriptive.Tree.mem_of_prefix",
"congrArg",
"Membership.mem",
"Eq.mp",
"Subtype",
... | [
"case inr\nA : Type u_1\nT : ↥(tree A)\nx y : List A\nh : y ∈ pullSub (subAt T x) x\nh' : x.length ≤ y.length\n⊢ y ∈ ↑T"
] | · rw [mem_pullSub_short h'] at h; exact mem_of_prefix h.1 (by simpa using h.2) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 123,
"column": 2
} | {
"line": 125,
"column": 67
} | {
"line": 127,
"column": 0
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\nx : List A\n⊢ pullSub (subAt T x) x ≤ T",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Descriptive.Tree.mem_of_prefix",
"Descriptive.Tree.mem_pullSub_long",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"M... | [] | intro y (h : y ∈ pullSub _ x); rcases le_total y.length x.length with h' | h'
· rw [mem_pullSub_short h'] at h; exact mem_of_prefix h.1 (by simpa using h.2)
· rw [mem_pullSub_long h'] at h; obtain ⟨_, h, rfl⟩ := h; exact h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Descriptive.Tree | {
"line": 123,
"column": 2
} | {
"line": 125,
"column": 67
} | {
"line": 127,
"column": 0
} | [
{
"pp": "A : Type u_1\nT : ↥(tree A)\nx : List A\n⊢ pullSub (subAt T x) x ≤ T",
"ppTerm": "?m.7",
"assigned": true,
"usedConstants": [
"Descriptive.Tree.mem_of_prefix",
"Descriptive.Tree.mem_pullSub_long",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"M... | [] | intro y (h : y ∈ pullSub _ x); rcases le_total y.length x.length with h' | h'
· rw [mem_pullSub_short h'] at h; exact mem_of_prefix h.1 (by simpa using h.2)
· rw [mem_pullSub_long h'] at h; obtain ⟨_, h, rfl⟩ := h; exact h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
"line": 97,
"column": 2
} | {
"line": 98,
"column": 55
} | {
"line": 100,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : CompleteLattice α\nf : α →o α\nx : α\na b : Ordinal.{u}\nh : a < b\n⊢ f (lfpApprox f x a) ≤ lfpApprox f x b",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"le_rfl",
"ChainComp... | [] | nth_rw 2 [lfpApprox]
exact le_sup_of_le_right <| le_iSup₂_of_le a h le_rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
"line": 97,
"column": 2
} | {
"line": 98,
"column": 55
} | {
"line": 100,
"column": 0
} | [
{
"pp": "α : Type u\ninst✝ : CompleteLattice α\nf : α →o α\nx : α\na b : Ordinal.{u}\nh : a < b\n⊢ f (lfpApprox f x a) ≤ lfpApprox f x b",
"ppTerm": "?m.12",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Lattice.toSemilatticeSup",
"le_rfl",
"ChainComp... | [] | nth_rw 2 [lfpApprox]
exact le_sup_of_le_right <| le_iSup₂_of_le a h le_rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Lists | {
"line": 290,
"column": 4
} | {
"line": 290,
"column": 33
} | {
"line": 291,
"column": 2
} | [
{
"pp": "case C0\nα : Type u_1\ntrans : Lists α → Prop := fun l₁ ↦ ∀ ⦃l₂ l₃ : Lists α⦄, l₁ ~ l₂ → l₂ ~ l₃ → l₁ ~ l₃\na : α\nl₂ l₃ : Lists α\nh₁ : atom a ~ l₂\nh₂ : l₂ ~ l₃\n⊢ atom a ~ l₃",
"ppTerm": "?C0",
"assigned": true,
"usedConstants": [
"congrArg",
"Lists",
"Eq.mp",
"If... | [] | rwa [← equiv_atom.1 h₁] at h₂ | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.SetTheory.Ordinal.CantorNormalForm | {
"line": 166,
"column": 17
} | {
"line": 166,
"column": 27
} | {
"line": 166,
"column": 28
} | [
{
"pp": "b o : Ordinal.{u_1}\n⊢ (map Prod.toSigma (CNF b o)).keys.Nodup",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Prod.toSigma",
"congrArg",
"List.map",
"List.keys",
"id",
"Sigma.fst",
"List.Nodup",
"List",
"Ordin... | [
"b o : Ordinal.{u_1}\n⊢ (map Sigma.fst (map Prod.toSigma (CNF b o))).Nodup"
] | List.keys, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.FixedPointApproximants | {
"line": 153,
"column": 20
} | {
"line": 153,
"column": 35
} | {
"line": 153,
"column": 35
} | [
{
"pp": "case h.left\nα : Type u\ninst✝ : CompleteLattice α\nf : α →o α\nx : α\nh_ninj :\n ∃ a b,\n (Set.Iio (SuccOrder.succ #α).ord).restrict (lfpApprox f x) a =\n (Set.Iio (SuccOrder.succ #α).ord).restrict (lfpApprox f x) b ∧\n a ≠ b\na b : ↑(Set.Iio (SuccOrder.succ #α).ord)\nh_fab :\n (Set.I... | [
"case h.left\nα : Type u\ninst✝ : CompleteLattice α\nf : α →o α\nx : α\nh_ninj :\n ∃ a b,\n (Set.Iio (SuccOrder.succ #α).ord).restrict (lfpApprox f x) a =\n (Set.Iio (SuccOrder.succ #α).ord).restrict (lfpApprox f x) b ∧\n a ≠ b\na b : ↑(Set.Iio (SuccOrder.succ #α).ord)\nh_fab :\n (Set.Iio (SuccOrde... | Subtype.coe_inj | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.SuccPred | {
"line": 61,
"column": 2
} | {
"line": 62,
"column": 7
} | {
"line": 64,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : SuccOrder α\ninst✝ : NoMaxOrder α\na : α\nh : IsMin a\n⊢ 𝓝 a = pure a",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Eq.mpr",
"False",
"Pre... | [] | rw [nhds_eq_pure, isSuccLimit_iff]
tauto | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.SuccPred | {
"line": 61,
"column": 2
} | {
"line": 62,
"column": 7
} | {
"line": 64,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : LinearOrder α\ninst✝³ : TopologicalSpace α\ninst✝² : OrderTopology α\ninst✝¹ : SuccOrder α\ninst✝ : NoMaxOrder α\na : α\nh : IsMin a\n⊢ 𝓝 a = pure a",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"Pure.pure",
"Eq.mpr",
"False",
"Pre... | [] | rw [nhds_eq_pure, isSuccLimit_iff]
tauto | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 108,
"column": 2
} | {
"line": 108,
"column": 30
} | {
"line": 109,
"column": 2
} | [
{
"pp": "f : Ordinal.{u} → Ordinal.{u}\no₁ o₂ a : Ordinal.{u}\nhf : IsNormal f\nh : o₁ ≤ o₂\nh' : veblenWith f o₂ a = a\n⊢ veblenWith f o₁ a = a",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"LE.le.eq_or_lt",
"Preorder.toLT",
"Ordinal.partialOrder",
"PartialOrder.t... | [
"case inl\nf : Ordinal.{u} → Ordinal.{u}\no₁ a : Ordinal.{u}\nhf : IsNormal f\nh : o₁ ≤ o₁\nh' : veblenWith f o₁ a = a\n⊢ veblenWith f o₁ a = a",
"case inr\nf : Ordinal.{u} → Ordinal.{u}\no₁ o₂ a : Ordinal.{u}\nhf : IsNormal f\nh✝ : o₁ ≤ o₂\nh' : veblenWith f o₂ a = a\nh : o₁ < o₂\n⊢ veblenWith f o₁ a = a"
] | obtain rfl | h := h.eq_or_lt | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 127,
"column": 6
} | {
"line": 127,
"column": 43
} | {
"line": 128,
"column": 4
} | [
{
"pp": "case mpr.inl\nf : Ordinal.{u} → Ordinal.{u}\nhf : IsNormal f\na b : Ordinal.{u}\nha : veblenWith f b a = a\nhb : b < b + 1\n⊢ a ∈ Function.fixedPoints (veblenWith f ↑⟨b, hb⟩)",
"ppTerm": "?mpr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.partialOrder",
"co... | [] | rw [Function.mem_fixedPoints_iff, ha] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 127,
"column": 6
} | {
"line": 127,
"column": 43
} | {
"line": 128,
"column": 4
} | [
{
"pp": "case mpr.inl\nf : Ordinal.{u} → Ordinal.{u}\nhf : IsNormal f\na b : Ordinal.{u}\nha : veblenWith f b a = a\nhb : b < b + 1\n⊢ a ∈ Function.fixedPoints (veblenWith f ↑⟨b, hb⟩)",
"ppTerm": "?mpr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.partialOrder",
"co... | [] | rw [Function.mem_fixedPoints_iff, ha] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 127,
"column": 6
} | {
"line": 127,
"column": 43
} | {
"line": 128,
"column": 4
} | [
{
"pp": "case mpr.inl\nf : Ordinal.{u} → Ordinal.{u}\nhf : IsNormal f\na b : Ordinal.{u}\nha : veblenWith f b a = a\nhb : b < b + 1\n⊢ a ∈ Function.fixedPoints (veblenWith f ↑⟨b, hb⟩)",
"ppTerm": "?mpr.inl",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Ordinal.partialOrder",
"co... | [] | rw [Function.mem_fixedPoints_iff, ha] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Topology | {
"line": 227,
"column": 65
} | {
"line": 230,
"column": 69
} | {
"line": 232,
"column": 0
} | [
{
"pp": "o : Ordinal.{u_1}\nS : Set (Set Ordinal.{u_1})\nh : ∀ C ∈ S, IsClosedBelow C o\n⊢ IsClosedBelow (⋂₀ S) o",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Ordinal.IsAcc",
"AccPt.mono",
"Eq.mpr",
"Preorder.toLT",
"Ordinal.partialOrder",
"congrArg",... | [] | by
rw [isClosedBelow_iff]
exact fun p plto pAcc C CmemS ↦ (h C CmemS).forall_lt p plto <|
AccPt.mono pAcc (monotone_principal (sInter_subset_of_mem CmemS)) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.SetTheory.Ordinal.Veblen | {
"line": 179,
"column": 42
} | {
"line": 179,
"column": 79
} | {
"line": 179,
"column": 79
} | [
{
"pp": "f : Ordinal.{u} → Ordinal.{u}\nhf : IsNormal f\nhp : 0 < f 0\no₁ o₂ : Ordinal.{u}\nh : o₁ < o₂\n⊢ veblenWith f o₁ 0 < veblenWith f o₁ (veblenWith f o₂ 0)",
"ppTerm": "?m.30",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"Ordinal.partialOrder",
"con... | [
"f : Ordinal.{u} → Ordinal.{u}\nhf : IsNormal f\nhp : 0 < f 0\no₁ o₂ : Ordinal.{u}\nh : o₁ < o₂\n⊢ 0 < veblenWith f o₂ 0"
] | veblenWith_lt_veblenWith_iff_right hf | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 734,
"column": 4
} | {
"line": 738,
"column": 47
} | {
"line": 740,
"column": 0
} | [
{
"pp": "case succ\ne a0 a : ONote\ninst✝² : e.NF\ninst✝¹ : a0.NF\ninst✝ : a.NF\nk m : ℕ\n⊢ (match k, m + 1 with\n | x, 0 => 0\n | 0, m.succ => e.oadd m.succPNat 0\n | k.succ, m => (e + a0.mulNat k).scale a + e.opowAux a0 a k m).NF",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
... | [] | cases k with
| zero => exact NF.oadd_zero _ _
| succ k =>
haveI := nf_opowAux e a0 a k
simp only [mulNat_eq_mul]; infer_instance | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 734,
"column": 4
} | {
"line": 738,
"column": 47
} | {
"line": 740,
"column": 0
} | [
{
"pp": "case succ\ne a0 a : ONote\ninst✝² : e.NF\ninst✝¹ : a0.NF\ninst✝ : a.NF\nk m : ℕ\n⊢ (match k, m + 1 with\n | x, 0 => 0\n | 0, m.succ => e.oadd m.succPNat 0\n | k.succ, m => (e + a0.mulNat k).scale a + e.opowAux a0 a k m).NF",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
... | [] | cases k with
| zero => exact NF.oadd_zero _ _
| succ k =>
haveI := nf_opowAux e a0 a k
simp only [mulNat_eq_mul]; infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 734,
"column": 4
} | {
"line": 738,
"column": 47
} | {
"line": 740,
"column": 0
} | [
{
"pp": "case succ\ne a0 a : ONote\ninst✝² : e.NF\ninst✝¹ : a0.NF\ninst✝ : a.NF\nk m : ℕ\n⊢ (match k, m + 1 with\n | x, 0 => 0\n | 0, m.succ => e.oadd m.succPNat 0\n | k.succ, m => (e + a0.mulNat k).scale a + e.opowAux a0 a k m).NF",
"ppTerm": "?succ",
"assigned": true,
"usedConstants": [
... | [] | cases k with
| zero => exact NF.oadd_zero _ _
| succ k =>
haveI := nf_opowAux e a0 a k
simp only [mulNat_eq_mul]; infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 792,
"column": 42
} | {
"line": 812,
"column": 22
} | {
"line": 814,
"column": 0
} | [
{
"pp": "e a : ONote\nNe : e.NF\nNa : a.NF\na' : Ordinal.{0}\ne0 : e.repr ≠ 0\nh : a' < ω ^ e.repr\naa : a.repr = a'\nn : ℕ+\n⊢ (ω ^ e.repr * ↑↑n + a') ^ ω = (ω ^ e.repr) ^ ω",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Ordinal.opow_mul",
"PNat.val",
"Iff.mpr",
... | [] | by
subst aa
have No := Ne.oadd n (Na.below_of_lt' h)
have := omega0_le_oadd e n a
rw [repr] at this
refine le_antisymm ?_ (opow_le_opow_left _ this)
apply (opow_le_of_isSuccLimit ((opow_pos _ omega0_pos).trans_le this).ne' isSuccLimit_omega0).2
intro b l
have := (No.below_of_lt (lt_succ _)).repr_lt
rw... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Majorized | {
"line": 110,
"column": 2
} | {
"line": 110,
"column": 52
} | {
"line": 111,
"column": 2
} | [
{
"pp": "f g b : ℝ → ℝ\nf_exp g_exp : ℝ\nhf : ∀ exp' > f_exp, f =o[atTop] (b ^ exp')\nhg : ∀ exp' > g_exp, g =o[atTop] (b ^ exp')\nh_pos : ∀ᶠ (t : ℝ) in atTop, 0 < b t\nexp : ℝ\nh_exp : exp > f_exp + g_exp\nε : ℝ := (exp - f_exp - g_exp) / 2\n⊢ (f * g) =o[atTop] (b ^ exp)",
"ppTerm": "?m.40",
"assigned"... | [
"f g b : ℝ → ℝ\nf_exp g_exp : ℝ\nhg : ∀ exp' > g_exp, g =o[atTop] (b ^ exp')\nh_pos : ∀ᶠ (t : ℝ) in atTop, 0 < b t\nexp : ℝ\nh_exp : exp > f_exp + g_exp\nε : ℝ := (exp - f_exp - g_exp) / 2\nhf : f =o[atTop] (b ^ (f_exp + ε))\n⊢ (f * g) =o[atTop] (b ^ exp)"
] | specialize hf (f_exp + ε) (by dsimp [ε]; linarith) | Lean.Elab.Tactic.evalSpecialize | Lean.Parser.Tactic.specialize |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Majorized | {
"line": 116,
"column": 69
} | {
"line": 116,
"column": 76
} | {
"line": 116,
"column": 76
} | [
{
"pp": "f g b : ℝ → ℝ\nf_exp g_exp : ℝ\nh_pos : ∀ᶠ (t : ℝ) in atTop, 0 < b t\nexp : ℝ\nh_exp : exp > f_exp + g_exp\nε : ℝ := (exp - f_exp - g_exp) / 2\nhg : g =o[atTop] (b ^ (g_exp + ε))\nhf : f =o[atTop] (b ^ (f_exp + ε))\nt : ℝ\nhx : 0 < b t\n⊢ exp = f_exp + (exp - f_exp - g_exp) / 2 + (g_exp + (exp - f_exp ... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs | {
"line": 362,
"column": 6
} | {
"line": 362,
"column": 18
} | {
"line": 362,
"column": 18
} | [
{
"pp": "basis_hd : ℝ → ℝ\nbasis_tl : List (ℝ → ℝ)\nms : MultiseriesExpansion (basis_hd :: basis_tl)\ns : Multiseries basis_hd basis_tl\nf : ℝ → ℝ\n⊢ mk ms.seq ms.toFun = mk s f ↔ ms.seq = s ∧ ms.toFun = f",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
... | [
"basis_hd : ℝ → ℝ\nbasis_tl : List (ℝ → ℝ)\nms : MultiseriesExpansion (basis_hd :: basis_tl)\ns : Multiseries basis_hd basis_tl\nf : ℝ → ℝ\n⊢ ms.seq = s ∧ ms.toFun = f ↔ ms.seq = s ∧ ms.toFun = f"
] | mk_eq_mk_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Monomial.Basic | {
"line": 329,
"column": 6
} | {
"line": 329,
"column": 23
} | {
"line": 330,
"column": 6
} | [
{
"pp": "case cons.cons\nexps_hd : ℝ\nexps_tl : List ℝ\nh_allZero : AllZero (exps_hd :: exps_tl)\nbasis_hd : ℝ → ℝ\nbasis_tl : List (ℝ → ℝ)\n⊢ Tendsto (toFun (exps_hd :: exps_tl) (basis_hd :: basis_tl)) atTop (𝓝 1)",
"ppTerm": "?cons.cons",
"assigned": true,
"usedConstants": [
"Real",
"... | [
"case cons.cons\nexps_hd : ℝ\nexps_tl : List ℝ\nbasis_hd : ℝ → ℝ\nbasis_tl : List (ℝ → ℝ)\nh_allZero : exps_hd = 0 ∧ AllZero exps_tl\n⊢ Tendsto (toFun (exps_hd :: exps_tl) (basis_hd :: basis_tl)) atTop (𝓝 1)"
] | simp at h_allZero | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs | {
"line": 580,
"column": 2
} | {
"line": 580,
"column": 43
} | {
"line": 582,
"column": 0
} | [
{
"pp": "basis_hd : ℝ → ℝ\nbasis_tl : Basis\nexp : ℝ\ncoef : MultiseriesExpansion basis_tl\nf : ℝ → ℝ\nh_coef : coef.Sorted\n⊢ (mk (Multiseries.cons exp coef Multiseries.nil) f).Sorted",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Tactic.ComputeAsymptotics.MultiseriesExpansion.sort... | [] | simp [Multiseries.Sorted.cons_nil h_coef] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs | {
"line": 580,
"column": 2
} | {
"line": 580,
"column": 43
} | {
"line": 582,
"column": 0
} | [
{
"pp": "basis_hd : ℝ → ℝ\nbasis_tl : Basis\nexp : ℝ\ncoef : MultiseriesExpansion basis_tl\nf : ℝ → ℝ\nh_coef : coef.Sorted\n⊢ (mk (Multiseries.cons exp coef Multiseries.nil) f).Sorted",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Tactic.ComputeAsymptotics.MultiseriesExpansion.sort... | [] | simp [Multiseries.Sorted.cons_nil h_coef] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Tactic.ComputeAsymptotics.Multiseries.Defs | {
"line": 580,
"column": 2
} | {
"line": 580,
"column": 43
} | {
"line": 582,
"column": 0
} | [
{
"pp": "basis_hd : ℝ → ℝ\nbasis_tl : Basis\nexp : ℝ\ncoef : MultiseriesExpansion basis_tl\nf : ℝ → ℝ\nh_coef : coef.Sorted\n⊢ (mk (Multiseries.cons exp coef Multiseries.nil) f).Sorted",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Tactic.ComputeAsymptotics.MultiseriesExpansion.sort... | [] | simp [Multiseries.Sorted.cons_nil h_coef] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Tactic.DeriveEncodable | {
"line": 128,
"column": 8
} | {
"line": 128,
"column": 26
} | {
"line": 129,
"column": 8
} | [
{
"pp": "case a\nn : ℕ\nih : ∀ m < n, (S.decode m).encode = m\nh : ¬(Nat.unpair n).1 = 0\n⊢ (Nat.unpair n).1 - 1 < n",
"ppTerm": "?a✝",
"assigned": true,
"usedConstants": [
"Nat.unpair",
"HSub.hSub",
"instSubNat",
"instOfNatNat",
"Prod.fst",
"Nat.casesAuxOn",
... | [
"case a.zero\nih : ∀ m < 0, (S.decode m).encode = m\nh : ¬(Nat.unpair 0).1 = 0\n⊢ (Nat.unpair 0).1 - 1 < 0",
"case a.succ\nn' : ℕ\nih : ∀ m < n' + 1, (S.decode m).encode = m\nh : ¬(Nat.unpair (n' + 1)).1 = 0\n⊢ (Nat.unpair (n' + 1)).1 - 1 < n' + 1"
] | obtain _ | n' := n | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 1058,
"column": 6
} | {
"line": 1058,
"column": 32
} | {
"line": 1059,
"column": 4
} | [
{
"pp": "case oadd.inl.none.inl.none.succ\na : ONote\nm : ℕ+\nb : ONote\niha : a = 0\nihb : b = 0\ne✝ : b.fundamentalSequence = Sum.inl none\ne : a.fundamentalSequence = Sum.inl none\nm' : ℕ\ne' : m.natPred = m' + 1\n⊢ 1 * ↑m' + 1 + 1 = succ (1 * ↑m' + 1) ∧ ((oadd 0 (m' + 1).succPNat 0).NF → (oadd 0 m'.succPNat... | [] | exact ⟨rfl, inferInstance⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 1058,
"column": 6
} | {
"line": 1058,
"column": 32
} | {
"line": 1059,
"column": 4
} | [
{
"pp": "case oadd.inl.none.inl.none.succ\na : ONote\nm : ℕ+\nb : ONote\niha : a = 0\nihb : b = 0\ne✝ : b.fundamentalSequence = Sum.inl none\ne : a.fundamentalSequence = Sum.inl none\nm' : ℕ\ne' : m.natPred = m' + 1\n⊢ 1 * ↑m' + 1 + 1 = succ (1 * ↑m' + 1) ∧ ((oadd 0 (m' + 1).succPNat 0).NF → (oadd 0 m'.succPNat... | [] | exact ⟨rfl, inferInstance⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 1058,
"column": 6
} | {
"line": 1058,
"column": 32
} | {
"line": 1059,
"column": 4
} | [
{
"pp": "case oadd.inl.none.inl.none.succ\na : ONote\nm : ℕ+\nb : ONote\niha : a = 0\nihb : b = 0\ne✝ : b.fundamentalSequence = Sum.inl none\ne : a.fundamentalSequence = Sum.inl none\nm' : ℕ\ne' : m.natPred = m' + 1\n⊢ 1 * ↑m' + 1 + 1 = succ (1 * ↑m' + 1) ∧ ((oadd 0 (m' + 1).succPNat 0).NF → (oadd 0 m'.succPNat... | [] | exact ⟨rfl, inferInstance⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 1074,
"column": 12
} | {
"line": 1074,
"column": 17
} | {
"line": 1074,
"column": 18
} | [
{
"pp": "case oadd.inl.none.inl.some.succ.refine_2\na : ONote\nm : ℕ+\nb : ONote\nihb : b = 0\ne✝ : b.fundamentalSequence = Sum.inl none\na' : ONote\niha : a.repr = succ a'.repr ∧ (a.NF → a'.NF)\ne : a.fundamentalSequence = Sum.inl (some a')\nm' : ℕ\ne' : m.natPred = m' + 1\nthis : 0 < ω ^ a'.repr\ni : ℕ\nH : (... | [
"case oadd.inl.none.inl.some.succ.refine_2\na : ONote\nm : ℕ+\nb : ONote\nihb : b = 0\ne✝ : b.fundamentalSequence = Sum.inl none\na' : ONote\niha : a.repr = succ a'.repr ∧ (a.NF → a'.NF)\ne : a.fundamentalSequence = Sum.inl (some a')\nm' : ℕ\ne' : m.natPred = m' + 1\nthis : 0 < ω ^ a'.repr\ni : ℕ\nH : (a.oadd (m' +... | repr, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 1088,
"column": 11
} | {
"line": 1088,
"column": 16
} | {
"line": 1088,
"column": 17
} | [
{
"pp": "case oadd.inl.none.inr.succ\na : ONote\nm : ℕ+\nb : ONote\nihb : b = 0\ne✝ : b.fundamentalSequence = Sum.inl none\nf : ℕ → ONote\ne : a.fundamentalSequence = Sum.inr f\nm' : ℕ\ne' : m.natPred = m' + 1\nh1 : IsSuccLimit a.repr\nh2 : ∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (a.NF → (f i).NF)\nh3 : ∀ a_1 < ... | [
"case oadd.inl.none.inr.succ\na : ONote\nm : ℕ+\nb : ONote\nihb : b = 0\ne✝ : b.fundamentalSequence = Sum.inl none\nf : ℕ → ONote\ne : a.fundamentalSequence = Sum.inr f\nm' : ℕ\ne' : m.natPred = m' + 1\nh1 : IsSuccLimit a.repr\nh2 : ∀ (i : ℕ), f i < f (i + 1) ∧ f i < a ∧ (a.NF → (f i).NF)\nh3 : ∀ a_1 < a.repr, ∃ i,... | repr, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.SetTheory.Ordinal.Notation | {
"line": 1091,
"column": 10
} | {
"line": 1091,
"column": 15
} | {
"line": 1091,
"column": 16
} | [
{
"pp": "case oadd.inl.some.refine_1\na : ONote\nm : ℕ+\nb : ONote\niha : a.FundamentalSequenceProp a.fundamentalSequence\nb' : ONote\nihb : b.repr = succ b'.repr ∧ (b.NF → b'.NF)\ne : b.fundamentalSequence = Sum.inl (some b')\n⊢ (a.oadd m b).repr = succ (a.oadd m b').repr",
"ppTerm": "?oadd.inl.some.refine... | [
"case oadd.inl.some.refine_1\na : ONote\nm : ℕ+\nb : ONote\niha : a.FundamentalSequenceProp a.fundamentalSequence\nb' : ONote\nihb : b.repr = succ b'.repr ∧ (b.NF → b'.NF)\ne : b.fundamentalSequence = Sum.inl (some b')\n⊢ ω ^ a.repr * ↑↑m + b.repr = succ (a.oadd m b').repr"
] | repr, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Tactic.Sat.FromLRAT | {
"line": 171,
"column": 4
} | {
"line": 171,
"column": 42
} | {
"line": 172,
"column": 4
} | [
{
"pp": "case cons\np : Prop\nas✝ : List Prop\na : Prop\nas : List Prop\nih : ∀ (as₁ : List Prop), as✝ = as₁.reverseAux as → (mk as✝).implies p as as₁.length → p\nas₁ : List Prop\n⊢ as✝ = as₁.reverseAux (a :: as) → (mk as✝).implies p (a :: as) as₁.length → p",
"ppTerm": "?cons",
"assigned": true,
"u... | [
"case cons\np : Prop\nas✝ : List Prop\na : Prop\nas : List Prop\nih : ∀ (as₁ : List Prop), as✝ = as₁.reverseAux as → (mk as✝).implies p as as₁.length → p\nas₁ : List Prop\ne : as✝ = as₁.reverseAux (a :: as)\nH : (mk as✝).implies p (a :: as) as₁.length\n⊢ mk as✝ as₁.length ↔ a"
] | refine fun e H ↦ @ih (a::as₁) e (H ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Algebra.Group.SubmonoidClosure | {
"line": 33,
"column": 70
} | {
"line": 33,
"column": 82
} | {
"line": 33,
"column": 82
} | [
{
"pp": "G : Type u_1\ninst✝¹ : DivInvMonoid G\ninst✝ : TopologicalSpace G\nx y : G\n⊢ ClusterPt x (map (fun x ↦ y ^ ↑x) atTop) ↔ ClusterPt x (map (fun x ↦ y ^ x) atTop)",
"ppTerm": "?m.25",
"assigned": true,
"usedConstants": [
"zpow_natCast",
"congrArg",
"Filter.map",
"DivIn... | [] | zpow_natCast | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Algebra.Nonarchimedean.TotallyDisconnected | {
"line": 58,
"column": 4
} | {
"line": 58,
"column": 57
} | {
"line": 59,
"column": 4
} | [
{
"pp": "G : Type u_1\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : NonarchimedeanGroup G\ninst✝ : T2Space G\nx : G\nx✝¹ : x ∈ Set.univ\ny : G\nx✝ : y ∈ Set.univ\nhxy : x ≠ y\n⊢ (fun x y ↦ ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Set.univ ⊆ u ∪ v ∧ Disjoint u v) x y",
"ppTerm": "?m.12",
"... | [
"G : Type u_1\ninst✝³ : TopologicalSpace G\ninst✝² : Group G\ninst✝¹ : NonarchimedeanGroup G\ninst✝ : T2Space G\nx : G\nx✝¹ : x ∈ Set.univ\ny : G\nx✝ : y ∈ Set.univ\nhxy : x ≠ y\nV : OpenSubgroup G\ndxy : Disjoint (x • ↑V) (y • ↑V)\n⊢ ∃ u v, IsOpen u ∧ IsOpen v ∧ x ∈ u ∧ y ∈ v ∧ Set.univ ⊆ u ∪ v ∧ Disjoint u v"
] | obtain ⟨V, dxy⟩ := exists_openSubgroup_separating hxy | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Topology.CWComplex.Classical.Subcomplex | {
"line": 160,
"column": 29
} | {
"line": 162,
"column": 26
} | {
"line": 164,
"column": 0
} | [
{
"pp": "X : Type u_1\nt : TopologicalSpace X\nC D : Set X\ninst✝² : T2Space X\ninst✝¹ : RelCWComplex C D\ninst✝ : FiniteDimensional C\nE : Subcomplex C\n⊢ ∀ᶠ (n : ℕ) in Filter.atTop, IsEmpty (cell (↑E) n)",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"setOf",
"Topology.RelCW... | [] | by
filter_upwards [FiniteDimensional.eventually_isEmpty_cell (C := C) (D := D)] with n hn
simp [isEmpty_subtype] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Category.Profinite.Nobeling.Span | {
"line": 110,
"column": 8
} | {
"line": 110,
"column": 34
} | {
"line": 110,
"column": 35
} | [
{
"pp": "case neg\nI : Type u\nC : Set (I → Bool)\ninst✝ : LinearOrder I\ns : Finset I\nx y : ↑(π C fun x ↦ x ∈ s)\nh : y = x\nb : I\nleft✝ : b ∈ s.sort fun x1 x2 ↦ x1 ≥ x2\nhh : ¬↑x b = true\n⊢ (1 - e (π C fun x ↦ x ∈ s) b) x = 1",
"ppTerm": "?neg✝",
"assigned": true,
"usedConstants": [
"Eq.m... | [
"case neg\nI : Type u\nC : Set (I → Bool)\ninst✝ : LinearOrder I\ns : Finset I\nx y : ↑(π C fun x ↦ x ∈ s)\nh : y = x\nb : I\nleft✝ : b ∈ s.sort fun x1 x2 ↦ x1 ≥ x2\nhh : ¬↑x b = true\n⊢ 1 x - (e (π C fun x ↦ x ∈ s) b) x = 1"
] | LocallyConstant.sub_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Category.Profinite.Nobeling.Span | {
"line": 137,
"column": 48
} | {
"line": 137,
"column": 74
} | {
"line": 138,
"column": 6
} | [
{
"pp": "case false\nI : Type u\nC : Set (I → Bool)\ninst✝ : LinearOrder I\ns : Finset I\nx y : ↑(π C fun x ↦ x ∈ s)\nh : y ≠ x\na : I\nha : ↑y a = true\nhx : ↑x a = false\n⊢ (1 - { toFun := fun f ↦ if ↑f a = true then 1 else 0, isLocallyConstant := ⋯ }) y = 0",
"ppTerm": "?false",
"assigned": true,
... | [
"case false\nI : Type u\nC : Set (I → Bool)\ninst✝ : LinearOrder I\ns : Finset I\nx y : ↑(π C fun x ↦ x ∈ s)\nh : y ≠ x\na : I\nha : ↑y a = true\nhx : ↑x a = false\n⊢ 1 y - { toFun := fun f ↦ if ↑f a = true then 1 else 0, isLocallyConstant := ⋯ } y = 0"
] | LocallyConstant.sub_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Category.Profinite.Nobeling.ZeroLimit | {
"line": 74,
"column": 30
} | {
"line": 74,
"column": 99
} | {
"line": 75,
"column": 2
} | [
{
"pp": "I : Type u_1\ninst✝ : LinearOrder I\nf : LocallyConstant ↑{fun x ↦ false} ℤ\na✝ : f ∈ ⊤\nx : ↑{fun x ↦ false}\n⊢ x = default",
"ppTerm": "?m.64",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Topology.Category.Profinite.Nobeling.ZeroLimit.0.Profinite.NobelingProof.Products.sp... | [] | by simp only [Set.default_coe_singleton, eq_iff_true_of_subsingleton] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Category.Profinite.Nobeling.Successor | {
"line": 342,
"column": 34
} | {
"line": 342,
"column": 80
} | {
"line": 344,
"column": 0
} | [
{
"pp": "I : 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\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\n⊢ Set.range (sum_to C ho) = Good... | [] | by rw [sum_to_range C ho, union_succ C hsC ho] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Category.Profinite.Nobeling.Successor | {
"line": 432,
"column": 2
} | {
"line": 432,
"column": 48
} | {
"line": 433,
"column": 2
} | [
{
"pp": "I : Type u\nC : Set (I → Bool)\ninst✝² : LinearOrder I\ninst✝¹ : WellFoundedLT I\no : Ordinal.{u}\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\ninst✝ : Inhabited I\nl : ↑(MaxProducts C ho)\nhm : term I ho ∈ ↑↑l\nthis : ord I (↑↑l).head! ≤ o\n⊢ ord I (↑↑l).head! = o",
... | [
"I : Type u\nC : Set (I → Bool)\ninst✝² : LinearOrder I\ninst✝¹ : WellFoundedLT I\no : Ordinal.{u}\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\ninst✝ : Inhabited I\nl : ↑(MaxProducts C ho)\nhm : term I ho ∈ ↑↑l\nthis : ord I (↑↑l).head! ≤ o\n⊢ o ≤ ord I (↑↑l).head!"
] | refine eq_of_le_of_not_lt this (not_lt.mpr ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Compactness.CompactSystem | {
"line": 92,
"column": 2
} | {
"line": 92,
"column": 48
} | {
"line": 93,
"column": 2
} | [
{
"pp": "case neg\nα : Type u_1\nS : Set (Set α)\nh : ∀ (C : ℕ → Set α), (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → (⋂ i, C i).Nonempty\nh✝ : Nonempty α\ns : ℕ → Set α\nh' : ∀ (i : ℕ), s i ∈ insert Set.univ S\nhd : ∀ (n : ℕ), (dissipate s n).Nonempty\nh₀ : ∃ n, s n ∈ S\nn : ℕ := Nat.find h₀\... | [
"case neg\nα : Type u_1\nS : Set (Set α)\nh : ∀ (C : ℕ → Set α), (∀ (i : ℕ), C i ∈ S) → (∀ (n : ℕ), (dissipate C n).Nonempty) → (⋂ i, C i).Nonempty\nh✝ : Nonempty α\ns : ℕ → Set α\nh' : ∀ (i : ℕ), s i ∈ insert Set.univ S\nhd : ∀ (n : ℕ), (dissipate s n).Nonempty\nh₀ : ∃ n, s n ∈ S\nn : ℕ := Nat.find h₀\ns' : ℕ → Se... | let s' := fun i ↦ if s i ∈ S then s i else s n | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.Topology.ContinuousMap.SecondCountableSpace | {
"line": 55,
"column": 6
} | {
"line": 55,
"column": 60
} | {
"line": 56,
"column": 6
} | [
{
"pp": "X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nS : Set (Set X)\nT : Set (Set Y)\nhS₁ : ∀ K ∈ S, IsCompact K\nhT : IsTopologicalBasis T\nhS₂ : ∀ (f : C(X, Y)) (x : X), ∀ V ∈ T, f x ∈ V → ∃ K ∈ S, K ∈ 𝓝 x ∧ MapsTo (⇑f) K V\nf : C(X, Y)\nK : Set X\nhK : IsCompact K\n... | [
"X : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nS : Set (Set X)\nT : Set (Set Y)\nhS₁ : ∀ K ∈ S, IsCompact K\nhT : IsTopologicalBasis T\nhS₂ : ∀ (f : C(X, Y)) (x : X), ∀ V ∈ T, f x ∈ V → ∃ K ∈ S, K ∈ 𝓝 x ∧ MapsTo (⇑f) K V\nf : C(X, Y)\nK : Set X\nhK : IsCompact K\nU : Set Y\nh... | rcases hK.elim_nhds_subcover L hLmem with ⟨s, hsK, hs⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Homotopy.LocallyContractible | {
"line": 160,
"column": 17
} | {
"line": 160,
"column": 25
} | {
"line": 160,
"column": 26
} | [
{
"pp": "case refine_2\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\nx✝ y✝ : X\nι : Type u_3\ninst✝¹ : StronglyLocallyContractibleSpace X\ninst✝ : StronglyLocallyContractibleSpace Y\nx : X\ny : Y\n⊢ ∀ (i : Set X × Set Y),\n (match (x, y) with\n | (x, y) =>\n ... | [
"case refine_2\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\nx✝ y✝ : X\nι : Type u_3\ninst✝¹ : StronglyLocallyContractibleSpace X\ninst✝ : StronglyLocallyContractibleSpace Y\nx : X\ny : Y\nUx : Set X\nUy : Set Y\n⊢ (match (x, y) with\n | (x, y) =>\n match (Ux, Uy) wi... | (Ux, Uy) | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.tuple |
Mathlib.Topology.Homotopy.LocallyContractible | {
"line": 163,
"column": 4
} | {
"line": 163,
"column": 55
} | {
"line": 165,
"column": 0
} | [
{
"pp": "case refine_2\nX : Type u_1\nY : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : TopologicalSpace Y\nx✝ y✝ : X\nι : Type u_3\ninst✝¹ : StronglyLocallyContractibleSpace X\ninst✝ : StronglyLocallyContractibleSpace Y\nx : X\ny : Y\nUx : Set X\nUy : Set Y\nhUx : Ux ∈ 𝓝 x ∧ ContractibleSpace ↑Ux\nhUy : Uy ... | [] | exact (Homeomorph.Set.prod Ux Uy).contractibleSpace | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 324,
"column": 2
} | {
"line": 324,
"column": 46
} | {
"line": 325,
"column": 2
} | [
{
"pp": "T : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝ : DecidableEq T\n⊢ (Finset.range #J).biUnion (pairSetSeq J a c) ⊆ J ×ˢ J",
"ppTerm": "?m.11",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.biUnion_subset_iff_forall_subset",
"instDecidab... | [
"T : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝ : DecidableEq T\n⊢ ∀ x ∈ Finset.range #J, pairSetSeq J a c x ⊆ J ×ˢ J"
] | rw [Finset.biUnion_subset_iff_forall_subset] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.EMetricSpace.PairReduction | {
"line": 374,
"column": 2
} | {
"line": 374,
"column": 46
} | {
"line": 375,
"column": 2
} | [
{
"pp": "case a\nT✝ : Type u_1\ninst✝³ : PseudoEMetricSpace T✝\na✝ c✝ : ℝ≥0∞\nJ✝ : Finset T✝\ninst✝² : DecidableEq T✝\nT : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝ : DecidableEq T\nha : 1 < a\nhJ : J.Nonempty\n⊢ ((Finset.range #J).biUnion fun x ↦\n {x_1 ∈ (logSizeBallSeq J h... | [
"case a\nT✝ : Type u_1\ninst✝³ : PseudoEMetricSpace T✝\na✝ c✝ : ℝ≥0∞\nJ✝ : Finset T✝\ninst✝² : DecidableEq T✝\nT : Type u_1\ninst✝¹ : PseudoEMetricSpace T\na c : ℝ≥0∞\nJ : Finset T\ninst✝ : DecidableEq T\nha : 1 < a\nhJ : J.Nonempty\n⊢ ∀ x ∈ Finset.range #J,\n {x_1 ∈ (logSizeBallSeq J hJ a c x).finset |\n ... | rw [Finset.biUnion_subset_iff_forall_subset] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 292,
"column": 4
} | {
"line": 292,
"column": 98
} | {
"line": 293,
"column": 4
} | [
{
"pp": "case inl\nN : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\nx : X\ninst✝ : DecidableEq N\ny : N → ↑I\nj : N\nHj : y j = 0 ∨ y j = 1\np : Ω (↑(Ω^ { j_1 // j_1 ≠ j } X x)) const\n⊢ (↑(p.toContinuousMap (y j)) fun j_1 ↦ y ↑j_1) = x",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": ... | [
"case inr\nN : Type u_1\nX : Type u_2\ninst✝¹ : TopologicalSpace X\nx : X\ninst✝ : DecidableEq N\ni : N\np : Ω (↑(Ω^ { j // j ≠ i } X x)) const\ny : N → ↑I\nj : N\nHj : y j = 0 ∨ y j = 1\nHne : j ≠ i\n⊢ (↑(p.toContinuousMap (y i)) fun j ↦ y ↑j) = x"
] | · rcases Hj with Hj | Hj <;> simp only [Hj, p.coe_toContinuousMap, p.source, p.target] <;> rfl | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.MetricSpace.Closeds | {
"line": 104,
"column": 14
} | {
"line": 104,
"column": 50
} | {
"line": 105,
"column": 4
} | [
{
"pp": "case h₁\nα : Type u_1\ninst✝ : EMetricSpace α\nx : α\ns : Closeds α\ny : α\nt : Closeds α\n⊢ infEDist x ↑t ≤ infEDist y ↑t + edist x y",
"ppTerm": "?h₁",
"assigned": true,
"usedConstants": [
"Metric.infEDist_le_infEDist_add_edist",
"TopologicalSpace.Closeds.instSetLike",
"... | [] | apply infEDist_le_infEDist_add_edist | Lean.Elab.Tactic.evalApply | Lean.Parser.Tactic.apply |
Mathlib.Topology.MetricSpace.Closeds | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 56
} | {
"line": 109,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : EMetricSpace α\nx : α\ns : Closeds α\ny : α\nt : Closeds α\n⊢ infEDist y ↑t + (edist x y + hausdorffEDist ↑s ↑t) ≤ infEDist y ↑t + (edist (x, s) (y, t) + edist (x, s) (y, t))",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
"le_max_right",
"ENNRea... | [] | gcongr <;> apply_rules [le_max_left, le_max_right] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Topology.MetricSpace.Closeds | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 56
} | {
"line": 109,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : EMetricSpace α\nx : α\ns : Closeds α\ny : α\nt : Closeds α\n⊢ infEDist y ↑t + (edist x y + hausdorffEDist ↑s ↑t) ≤ infEDist y ↑t + (edist (x, s) (y, t) + edist (x, s) (y, t))",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
"le_max_right",
"ENNRea... | [] | gcongr <;> apply_rules [le_max_left, le_max_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Closeds | {
"line": 108,
"column": 6
} | {
"line": 108,
"column": 56
} | {
"line": 109,
"column": 4
} | [
{
"pp": "α : Type u_1\ninst✝ : EMetricSpace α\nx : α\ns : Closeds α\ny : α\nt : Closeds α\n⊢ infEDist y ↑t + (edist x y + hausdorffEDist ↑s ↑t) ≤ infEDist y ↑t + (edist (x, s) (y, t) + edist (x, s) (y, t))",
"ppTerm": "?m.120",
"assigned": true,
"usedConstants": [
"le_max_right",
"ENNRea... | [] | gcongr <;> apply_rules [le_max_left, le_max_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.CoveringNumbers | {
"line": 372,
"column": 42
} | {
"line": 372,
"column": 49
} | {
"line": 373,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝ : PseudoEMetricSpace X\nA B : Set X\nε : ℝ≥0\nh : A ⊆ B\n⊢ packingNumber ε A = packingNumber (2 * (ε / 2)) A",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidW... | [] | ring_nf | Mathlib.Tactic.RingNF._aux_Mathlib_Tactic_Ring_RingNF___elabRules_Mathlib_Tactic_RingNF_ringNF_1 | Mathlib.Tactic.RingNF.ringNF |
Mathlib.Topology.MetricSpace.CoveringNumbers | {
"line": 372,
"column": 42
} | {
"line": 372,
"column": 49
} | {
"line": 373,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝ : PseudoEMetricSpace X\nA B : Set X\nε : ℝ≥0\nh : A ⊆ B\n⊢ packingNumber ε A = packingNumber (2 * (ε / 2)) A",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidW... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.CoveringNumbers | {
"line": 372,
"column": 42
} | {
"line": 372,
"column": 49
} | {
"line": 373,
"column": 2
} | [
{
"pp": "X : Type u_1\ninst✝ : PseudoEMetricSpace X\nA B : Set X\nε : ℝ≥0\nh : A ⊆ B\n⊢ packingNumber ε A = packingNumber (2 * (ε / 2)) A",
"ppTerm": "?m.48",
"assigned": true,
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidW... | [] | ring_nf | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 77,
"column": 2
} | {
"line": 78,
"column": 74
} | {
"line": 80,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\n⊢ IsClopen {∅}",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.powerset_empty",
"TopologicalSpace.vietoris",
"Set.powerset",
"Set.instSingletonSet",
"id",
"isClosed_... | [] | rw [← powerset_empty]
exact ⟨isClosed_empty.powerset_vietoris, isOpen_empty.powerset_vietoris⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 77,
"column": 2
} | {
"line": 78,
"column": 74
} | {
"line": 80,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\n⊢ IsClopen {∅}",
"ppTerm": "?m.8",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.powerset_empty",
"TopologicalSpace.vietoris",
"Set.powerset",
"Set.instSingletonSet",
"id",
"isClosed_... | [] | rw [← powerset_empty]
exact ⟨isClosed_empty.powerset_vietoris, isOpen_empty.powerset_vietoris⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 37
} | {
"line": 90,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝ : TopologicalSpace α\nu : Set (Set α)\nhu₁ : u.Finite\nhu₂ : ∀ U ∈ u, IsOpen[inst✝] U\n⊢ IsOpen[generateFrom (powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V})]\n {s | s ⊆ ⋃₀ u ∧ ∀ U ∈ u, (s ∩ U).Nonempty}",
"ppTerm"... | [
"case refine_1\nα : Type u_1\ninst✝ : TopologicalSpace α\nu : Set (Set α)\nhu₁ : u.Finite\nhu₂ : ∀ U ∈ u, IsOpen[inst✝] U\n⊢ IsOpen[generateFrom (powerset '' {U | IsOpen[inst✝] U} ∪ (fun V ↦ {s | (s ∩ V).Nonempty}) '' {V | IsOpen[inst✝] V})]\n ({a | a ⊆ ⋃₀ u} ∩ ⋂ i ∈ u, {x | (x ∩ i).Nonempty})"
] | simp_rw [setOf_and, setOf_forall] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.UniformSpace.Closeds | {
"line": 75,
"column": 6
} | {
"line": 75,
"column": 50
} | {
"line": 76,
"column": 4
} | [
{
"pp": "case a.refine_1\nα : Type u_1\nU V : SetRel α α\ns t : Set α\nhst : s ⊆ (U ○ V).preimage t\nhts : t ⊆ (U ○ V).image s\nx : α\nhx : x ∈ s\nz : α\nhz : z ∈ t\ny : α\nhxy : (x, y) ∈ U\nhyz : (y, z) ∈ V\n⊢ x ∈ U.preimage (U.image s ∩ V.preimage t)",
"ppTerm": "?a.refine_1✝",
"assigned": true,
"... | [] | exact ⟨y, ⟨⟨x, hx, hxy⟩, ⟨z, hz, hyz⟩⟩, hxy⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 37
} | {
"line": 127,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝ : TopologicalSpace α\nB : Set (Set α)\nhB : IsTopologicalBasis B\nV : Set α\nu : Set (Set α)\nhV : IsOpen[inst✝] (V, u).1\nhu : (V, u).2.Finite\nhuB : (V, u).2 ⊆ B\nright✝ : ∀ U ∈ (V, u).2, U ⊆ (V, u).1\n⊢ IsOpen[TopologicalSpace.vietoris α] {s | s ⊆ (V, u).1 ∧ ∀ U ∈ ... | [
"case refine_1\nα : Type u_1\ninst✝ : TopologicalSpace α\nB : Set (Set α)\nhB : IsTopologicalBasis B\nV : Set α\nu : Set (Set α)\nhV : IsOpen[inst✝] (V, u).1\nhu : (V, u).2.Finite\nhuB : (V, u).2 ⊆ B\nright✝ : ∀ U ∈ (V, u).2, U ⊆ (V, u).1\n⊢ IsOpen[TopologicalSpace.vietoris α] ({a | a ⊆ V} ∩ ⋂ i ∈ u, {x | (x ∩ i).N... | simp_rw [setOf_and, setOf_forall] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.UniformSpace.Closeds | {
"line": 475,
"column": 2
} | {
"line": 485,
"column": 23
} | {
"line": 487,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : UniformSpace α\n⊢ CompactSpace (Closeds α) ↔ CompactSpace α",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"TopologicalSpace.Closeds.uniformSpace",
"isClosed_biInter",
"CompleteLattice.toLattice",
"con... | [] | refine ⟨fun _ => compactSpace_of_finite_subfamily_closed fun {ι} F hF₁ hF₂ => ?_,
fun _ => inferInstance⟩
have := isClopen_singleton_bot.compl.isClosed.isCompact.elim_finite_subfamily_closed
(fun i => {C : Closeds α | ↑C ⊆ F i})
(fun i => isClosed_subsets_of_isClosed (hF₁ i))
simp_rw [← Set.disjoint_iff... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Closeds | {
"line": 475,
"column": 2
} | {
"line": 485,
"column": 23
} | {
"line": 487,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : UniformSpace α\n⊢ CompactSpace (Closeds α) ↔ CompactSpace α",
"ppTerm": "?m.4",
"assigned": true,
"usedConstants": [
"Lattice.toSemilatticeSup",
"TopologicalSpace.Closeds.uniformSpace",
"isClosed_biInter",
"CompleteLattice.toLattice",
"con... | [] | refine ⟨fun _ => compactSpace_of_finite_subfamily_closed fun {ι} F hF₁ hF₂ => ?_,
fun _ => inferInstance⟩
have := isClopen_singleton_bot.compl.isClosed.isCompact.elim_finite_subfamily_closed
(fun i => {C : Closeds α | ↑C ⊆ F i})
(fun i => isClosed_subsets_of_isClosed (hF₁ i))
simp_rw [← Set.disjoint_iff... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.GromovHausdorff | {
"line": 270,
"column": 6
} | {
"line": 270,
"column": 34
} | {
"line": 271,
"column": 6
} | [
{
"pp": "X : Type u\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\nY : Type v\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\ninhabited_h✝ : Inhabited X\ninhabited_h : Inhabited Y\np q : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)\nhp : ⟦p⟧ = toGHSpace X\nhq : ⟦q⟧ = toGHSp... | [
"X : Type u\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\nY : Type v\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\ninhabited_h✝ : Inhabited X\ninhabited_h : Inhabited Y\np q : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)\nhp : ⟦p⟧ = toGHSpace X\nhq : ⟦q⟧ = toGHSpace Y\nbound... | rcases this with ⟨y, hy, dy⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 387,
"column": 2
} | {
"line": 391,
"column": 49
} | {
"line": 393,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\ns : Set α\n⊢ closure {K | (↑K).Finite ∧ ↑K ⊆ s} = {K | ↑K ⊆ closure[inst✝] s}",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"TopologicalSpace.Compacts.isEmbedding_coe",
"congrArg",
"TopologicalSpace.viet... | [] | change closure (SetLike.coe ⁻¹' {K : Set α | K.Finite ∧ K ⊆ s}) =
SetLike.coe ⁻¹' (closure s).powerset
rw [isEmbedding_coe.closure_eq_preimage_closure_image, image_preimage_eq_of_subset ?_,
vietoris.closure_finite_subsets]
exact fun K ⟨hK, _⟩ => ⟨⟨K, hK.isCompact⟩, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 387,
"column": 2
} | {
"line": 391,
"column": 49
} | {
"line": 393,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\ns : Set α\n⊢ closure {K | (↑K).Finite ∧ ↑K ⊆ s} = {K | ↑K ⊆ closure[inst✝] s}",
"ppTerm": "?m.21",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"TopologicalSpace.Compacts.isEmbedding_coe",
"congrArg",
"TopologicalSpace.viet... | [] | change closure (SetLike.coe ⁻¹' {K : Set α | K.Finite ∧ K ⊆ s}) =
SetLike.coe ⁻¹' (closure s).powerset
rw [isEmbedding_coe.closure_eq_preimage_closure_image, image_preimage_eq_of_subset ?_,
vietoris.closure_finite_subsets]
exact fun K ⟨hK, _⟩ => ⟨⟨K, hK.isCompact⟩, rfl⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 407,
"column": 4
} | {
"line": 407,
"column": 37
} | {
"line": 408,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝ : TopologicalSpace α\nB : Set (Set α)\nhB : IsTopologicalBasis B\nu : Set (Set α)\nhu : u.Finite\nhuB : u ⊆ B\n⊢ IsOpen {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrA... | [
"case refine_1\nα : Type u_1\ninst✝ : TopologicalSpace α\nB : Set (Set α)\nhB : IsTopologicalBasis B\nu : Set (Set α)\nhu : u.Finite\nhuB : u ⊆ B\n⊢ IsOpen ({a | ↑a ⊆ ⋃₀ u} ∩ ⋂ i ∈ u, {x | (↑x ∩ i).Nonempty})"
] | simp_rw [setOf_and, setOf_forall] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.MetricSpace.Infsep | {
"line": 286,
"column": 2
} | {
"line": 286,
"column": 41
} | {
"line": 288,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : EDist α\ns : Set α\n⊢ s.infsep = 0 ↔ s.einfsep = 0 ∨ s.einfsep = ∞",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Iff.rfl",
"id",
"Set.infsep.eq_1",
"Iff",
... | [] | rw [infsep, ENNReal.toReal_eq_zero_iff] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.MetricSpace.Infsep | {
"line": 286,
"column": 2
} | {
"line": 286,
"column": 41
} | {
"line": 288,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : EDist α\ns : Set α\n⊢ s.infsep = 0 ↔ s.einfsep = 0 ∨ s.einfsep = ∞",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Iff.rfl",
"id",
"Set.infsep.eq_1",
"Iff",
... | [] | rw [infsep, ENNReal.toReal_eq_zero_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Infsep | {
"line": 286,
"column": 2
} | {
"line": 286,
"column": 41
} | {
"line": 288,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : EDist α\ns : Set α\n⊢ s.infsep = 0 ↔ s.einfsep = 0 ∨ s.einfsep = ∞",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Real",
"Real.instZero",
"congrArg",
"Iff.rfl",
"id",
"Set.infsep.eq_1",
"Iff",
... | [] | rw [infsep, ENNReal.toReal_eq_zero_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 432,
"column": 4
} | {
"line": 432,
"column": 24
} | {
"line": 434,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\nB : Set (Set α)\nhB : IsTopologicalBasis B\nV : Set α\nu : Set (Set α)\nhV : IsOpen[inst✝] V\nhu : u.Finite\nhuB : u ⊆ B\nhuV : ∀ U ∈ u, U ⊆ V\nK : Compacts α\nhKu : ∀ U ∈ u, (↑K ∩ U).Nonempty\nw : Set (Set α)\nhKV : ↑K ⊆ ⋃₀ w\nhwB : w ⊆ B\nhwV : ⋃₀ w ⊆ V\nhwK ... | [] | grind (splits := 12) | Lean.Elab.Tactic.evalGrind | Lean.Parser.Tactic.grind |
Mathlib.Topology.MetricSpace.Snowflaking | {
"line": 488,
"column": 2
} | {
"line": 489,
"column": 52
} | {
"line": 491,
"column": 0
} | [
{
"pp": "X : Type u_1\nα : ℝ\nhα₀ : 0 < α\nhα₁ : α ≤ 1\ninst✝ : PseudoMetricSpace X\nx : X\nr : ℝ\nhr : 0 ≤ r\n⊢ ⇑ofSnowflaking ⁻¹' closedBall x r = closedBall (toSnowflaking x) (r ^ α)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Set.ext",
"Metric.Snowflaking.instPseudoMet... | [] | ext ⟨y⟩
simp (disch := positivity) [Real.rpow_le_rpow_iff] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Snowflaking | {
"line": 488,
"column": 2
} | {
"line": 489,
"column": 52
} | {
"line": 491,
"column": 0
} | [
{
"pp": "X : Type u_1\nα : ℝ\nhα₀ : 0 < α\nhα₁ : α ≤ 1\ninst✝ : PseudoMetricSpace X\nx : X\nr : ℝ\nhr : 0 ≤ r\n⊢ ⇑ofSnowflaking ⁻¹' closedBall x r = closedBall (toSnowflaking x) (r ^ α)",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Set.ext",
"Metric.Snowflaking.instPseudoMet... | [] | ext ⟨y⟩
simp (disch := positivity) [Real.rpow_le_rpow_iff] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 742,
"column": 4
} | {
"line": 742,
"column": 37
} | {
"line": 743,
"column": 4
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝ : TopologicalSpace α\nB : Set (Set α)\nhB : IsTopologicalBasis B\nu : Set (Set α)\nhu : u.Finite\nhuB : u ⊆ B\n⊢ IsOpen {K | ↑K ⊆ ⋃₀ u ∧ ∀ U ∈ u, (↑K ∩ U).Nonempty}",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [
"TopologicalSpace.Nonem... | [
"case refine_1\nα : Type u_1\ninst✝ : TopologicalSpace α\nB : Set (Set α)\nhB : IsTopologicalBasis B\nu : Set (Set α)\nhu : u.Finite\nhuB : u ⊆ B\n⊢ IsOpen ({a | ↑a ⊆ ⋃₀ u} ∩ ⋂ i ∈ u, {x | (↑x ∩ i).Nonempty})"
] | simp_rw [setOf_and, setOf_forall] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Topology.Order.UpperLowerSetTopology | {
"line": 348,
"column": 49
} | {
"line": 348,
"column": 71
} | {
"line": 348,
"column": 72
} | [
{
"pp": "α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : Topology.IsLowerSet α\ns : Set α\n⊢ IsLowerSet sᶜ ↔ IsUpperSet s",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"IsUpperSet",
"Preorder.toLE"... | [
"α : Type u_1\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : Topology.IsLowerSet α\ns : Set α\n⊢ IsUpperSet sᶜᶜ ↔ IsUpperSet s"
] | isUpperSet_compl.symm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.LowerUpperTopology | {
"line": 504,
"column": 4
} | {
"line": 505,
"column": 47
} | {
"line": 507,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁷ : Preorder α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : IsUpper α\ninst✝⁴ : OrderTop α\ninst✝³ : Preorder β\ninst✝² : TopologicalSpace β\ninst✝¹ : IsUpper β\ninst✝ : OrderTop β\n⊢ instTopologicalSpaceProd = upper (α × β)",
"ppTerm": "?m.12",
"assigned": true,
... | [] | suffices IsLower (α × β)ᵒᵈ from IsLower.topology_eq_lowerTopology (α := (α × β)ᵒᵈ)
exact instIsLowerProd (α := αᵒᵈ) (β := βᵒᵈ) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Order.LowerUpperTopology | {
"line": 504,
"column": 4
} | {
"line": 505,
"column": 47
} | {
"line": 507,
"column": 0
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁷ : Preorder α\ninst✝⁶ : TopologicalSpace α\ninst✝⁵ : IsUpper α\ninst✝⁴ : OrderTop α\ninst✝³ : Preorder β\ninst✝² : TopologicalSpace β\ninst✝¹ : IsUpper β\ninst✝ : OrderTop β\n⊢ instTopologicalSpaceProd = upper (α × β)",
"ppTerm": "?m.12",
"assigned": true,
... | [] | suffices IsLower (α × β)ᵒᵈ from IsLower.topology_eq_lowerTopology (α := (α × β)ᵒᵈ)
exact instIsLowerProd (α := αᵒᵈ) (β := βᵒᵈ) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.HullKernel | {
"line": 197,
"column": 77
} | {
"line": 199,
"column": 5
} | {
"line": 201,
"column": 0
} | [
{
"pp": "α : Type u_1\ninst✝ : CompleteLattice α\nT : Set α\nhG : OrderGenerates T\na : α\n⊢ kernel (hull T a) = a",
"ppTerm": "?m.10",
"assigned": true,
"usedConstants": [
"OrderDual.toDual",
"Eq.mpr",
"Equiv.instEquivLike",
"PrimitiveSpectrum.hull",
"OrderDual.ofDual"... | [] | by
conv_rhs => rw [← OrderDual.ofDual_toDual a, ← (gi hG).l_u_eq a]
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.NhdsSet | {
"line": 62,
"column": 30
} | {
"line": 62,
"column": 44
} | {
"line": 62,
"column": 44
} | [
{
"pp": "α : Type u_1\ninst✝² : LinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\na b : α\n⊢ Ici b ⊆ Ioi a ↔ a < b",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Set.Ioi",
"Preorder.toLT",
"Set.Ici",
"congrArg",
"Part... | [
"α : Type u_1\ninst✝² : LinearOrder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\na b : α\n⊢ a < b ↔ a < b"
] | Ici_subset_Ioi | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Partial | {
"line": 75,
"column": 2
} | {
"line": 75,
"column": 27
} | {
"line": 76,
"column": 2
} | [
{
"pp": "case mpr\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X →. Y\nhf : ∀ {x : X} {y : Y}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set Y\nos : IsOpen[inst✝] s\nx : X\ny : Y\nys : y ∈ s\nfxy : (x, y) ∈ f.graph'\nt : Set X\nh : f.preimage s ⊆ t\nh' : ∀ s ∈ 𝓝 y, f... | [
"case mpr\nX : Type u_1\nY : Type u_2\ninst✝¹ : TopologicalSpace X\ninst✝ : TopologicalSpace Y\nf : X →. Y\nhf : ∀ {x : X} {y : Y}, y ∈ f x → PTendsto' f (𝓝 x) (𝓝 y)\ns : Set Y\nos : IsOpen[inst✝] s\nx : X\ny : Y\nys : y ∈ s\nfxy : (x, y) ∈ f.graph'\nt : Set X\nh : f.preimage s ⊆ t\nh' : ∀ s ∈ 𝓝 y, f.preimage s ... | change f.preimage s ∈ 𝓝 x | Lean.Elab.Tactic.evalChange | Lean.Parser.Tactic.change |
Mathlib.Topology.Separation.LinearUpperLowerSetTopology | {
"line": 23,
"column": 4
} | {
"line": 23,
"column": 20
} | {
"line": 24,
"column": 4
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : Topology.IsUpperSet α\nt : Set α\nhcst : Disjoint (closure ∅) t\nhsct : Disjoint ∅ (closure t)\n⊢ Disjoint (nhdsSet ∅) (nhdsSet t)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"nhdsSet_emp... | [
"case inr\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : Topology.IsUpperSet α\ns t : Set α\nhcst : Disjoint (closure s) t\nhsct : Disjoint s (closure t)\na : α\nha : a ∈ s\n⊢ Disjoint (nhdsSet s) (nhdsSet t)"
] | case inl => simp | Lean.Elab.Tactic.evalCase | Lean.Parser.Tactic.case |
Mathlib.Topology.Separation.LinearUpperLowerSetTopology | {
"line": 25,
"column": 4
} | {
"line": 25,
"column": 20
} | {
"line": 26,
"column": 4
} | [
{
"pp": "case inr.inl\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : Topology.IsUpperSet α\ns : Set α\na : α\nha : a ∈ s\nhcst : Disjoint (closure s) ∅\nhsct : Disjoint s (closure ∅)\n⊢ Disjoint (nhdsSet s) (nhdsSet ∅)",
"ppTerm": "?inr.inl",
"assigned": true,
"usedConst... | [
"case inr.inr\nα : Type u_1\ninst✝² : TopologicalSpace α\ninst✝¹ : LinearOrder α\ninst✝ : Topology.IsUpperSet α\ns t : Set α\nhcst : Disjoint (closure s) t\nhsct : Disjoint s (closure t)\na : α\nha : a ∈ s\nb : α\nhb : b ∈ t\n⊢ Disjoint (nhdsSet s) (nhdsSet t)"
] | case inl => simp | Lean.Elab.Tactic.evalCase | Lean.Parser.Tactic.case |
Mathlib.Topology.Sheaves.Skyscraper | {
"line": 94,
"column": 22
} | {
"line": 100,
"column": 63
} | {
"line": 102,
"column": 0
} | [
{
"pp": "X : TopCat\np₀ : ↑X\ninst✝² : (U : Opens ↑X) → Decidable (p₀ ∈ U)\nC : Type v\ninst✝¹ : Category.{w, v} C\ninst✝ : HasTerminal C\nA a b : C\nf : a ⟶ b\nU V : (Opens ↑X)ᵒᵖ\ni : U ⟶ V\n⊢ ((skyscraperPresheaf p₀ a).map i ≫\n if h : p₀ ∈ unop V then eqToHom ⋯ ≫ f ≫ eqToHom ⋯\n else (⋯ ▸ terminalI... | [] | by
simp only [skyscraperPresheaf_map]
by_cases hV : p₀ ∈ V.unop
· have hU : p₀ ∈ U.unop := leOfHom i.unop hV
simp only [skyscraperPresheaf_obj, hU, hV, ↓reduceDIte, eqToHom_trans_assoc, Category.assoc,
eqToHom_trans]
· apply ((if_neg hV).symm.ndrec terminalIsTerminal).hom_ext | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Sheaves.Skyscraper | {
"line": 163,
"column": 8
} | {
"line": 163,
"column": 51
} | {
"line": 163,
"column": 51
} | [
{
"pp": "X : TopCat\np₀ : ↑X\ninst✝² : (U : Opens ↑X) → Decidable (p₀ ∈ U)\nC : Type v\ninst✝¹ : Category.{u, v} C\nA : C\ninst✝ : HasTerminal C\ny : ↑X\nh : p₀ ⤳ y\nc : Cocone ((OpenNhds.inclusion y).op ⋙ skyscraperPresheaf p₀ A)\nU : (OpenNhds y)ᵒᵖ\n⊢ eqToHom ⋯ ≫ eqToHom ⋯ ≫ c.ι.app (op ⊤) = c.ι.app U",
"... | [
"X : TopCat\np₀ : ↑X\ninst✝² : (U : Opens ↑X) → Decidable (p₀ ∈ U)\nC : Type v\ninst✝¹ : Category.{u, v} C\nA : C\ninst✝ : HasTerminal C\ny : ↑X\nh : p₀ ⤳ y\nc : Cocone ((OpenNhds.inclusion y).op ⋙ skyscraperPresheaf p₀ A)\nU : (OpenNhds y)ᵒᵖ\n⊢ eqToHom ⋯ ≫\n eqToHom ⋯ ≫ ((OpenNhds.inclusion y).op ⋙ skyscraper... | ← c.w (homOfLE <| (le_top : unop U ≤ _)).op | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Sheaves.MayerVietoris | {
"line": 54,
"column": 8
} | {
"line": 54,
"column": 67
} | {
"line": 54,
"column": 67
} | [
{
"pp": "case inr\nT : Type u\ninst✝ : TopologicalSpace T\nsq : Square (Opens T)\nh₄ : sq.X₄ = sq.X₂ ⊔ sq.X₃\nh₁ : sq.X₁ = sq.X₂ ⊓ sq.X₃\nx : T\nhx : x ∈ ↑sq.X₃\n⊢ ∃ U f, (Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄).arrows f ∧ x ∈ U",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"CategoryTheory... | [] | exact ⟨_, _, ⟨Sieve.ofArrows_mk _ _ WalkingPair.right, hx⟩⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Sheaves.MayerVietoris | {
"line": 54,
"column": 8
} | {
"line": 54,
"column": 67
} | {
"line": 54,
"column": 67
} | [
{
"pp": "case inr\nT : Type u\ninst✝ : TopologicalSpace T\nsq : Square (Opens T)\nh₄ : sq.X₄ = sq.X₂ ⊔ sq.X₃\nh₁ : sq.X₁ = sq.X₂ ⊓ sq.X₃\nx : T\nhx : x ∈ ↑sq.X₃\n⊢ ∃ U f, (Sieve.ofTwoArrows sq.f₂₄ sq.f₃₄).arrows f ∧ x ∈ U",
"ppTerm": "?inr",
"assigned": true,
"usedConstants": [
"CategoryTheory... | [] | exact ⟨_, _, ⟨Sieve.ofArrows_mk _ _ WalkingPair.right, hx⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
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