module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 365
values | kind stringclasses 368
values |
|---|---|---|---|---|---|---|
Mathlib.Topology.Separation.Basic | {
"line": 1012,
"column": 4
} | {
"line": 1012,
"column": 38
} | [
{
"pp": "case insert\nX : Type u_1\ninst✝¹ : TopologicalSpace X\ninst✝ : R1Space X\nι : Type u_3\nx : ι\nt : Finset ι\nhx : x ∉ t\nih :\n ∀ {s : Set X},\n IsCompact s →\n ∀ (U : ι → Set X),\n (∀ i ∈ t, IsOpen (U i)) →\n s ⊆ ⋃ i ∈ t, U i → ∃ K, (∀ (i : ι), IsCompact (K i)) ∧ (∀ (i : ι), ... | refine ⟨update K x K₁, ?_, ?_, ?_⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 106,
"column": 6
} | {
"line": 106,
"column": 23
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝¹ : CommMonoid α\ninst✝ : TopologicalSpace α\nf : β → α\na : α\ng : β → γ\nhg : Injective g\n⊢ HasProd (extend g f 1) a ↔ HasProd f a",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Monoid.toMulOneClass",
"congrArg",
"HasP... | ← hg.hasProd_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Algebra.InfiniteSum.Basic | {
"line": 287,
"column": 4
} | {
"line": 287,
"column": 34
} | [
{
"pp": "case mp.left\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ninst✝⁶ : CommMonoid α\ninst✝⁵ : TopologicalSpace α\nL : SummationFilter β\ninst✝⁴ : CommMonoid γ\ninst✝³ : TopologicalSpace γ\ninst✝² : T2Space γ\nG : Type u_4\ninst✝¹ : FunLike G α γ\ninst✝ : MonoidHomClass G α γ\ng : G\nhg : IsInducing ⇑g\nf : β... | · exact hf.map g hg.continuous | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.Order.OrderClosed | {
"line": 315,
"column": 70
} | {
"line": 316,
"column": 68
} | [
{
"pp": "α : Type u\ninst✝³ : TopologicalSpace α\ninst✝² : LinearOrder α\ninst✝¹ : ClosedIicTopology α\nb : α\ninst✝ : PredOrder α\n⊢ 𝓝[≤] b = pure b",
"usedConstants": [
"Pure.pure",
"nhdsWithin_insert",
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"Filter.instComp... | by
rw [← Iio_insert, nhdsWithin_insert, PredOrder.nhdsLT, sup_bot_eq] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.OrderClosed | {
"line": 672,
"column": 96
} | {
"line": 676,
"column": 98
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝² : TopologicalSpace α\ninst✝¹ : Preorder α\nt : OrderClosedTopology α\ninst✝ : Preorder β\ns : Set β\n⊢ IsClosed {f | MonotoneOn f s}",
"usedConstants": [
"Eq.mpr",
"le_of_tendsto_of_tendsto_of_frequently",
"Pi.topologicalSpace",
"setOf",
... | by
simp only [isClosed_iff_clusterPt, clusterPt_principal_iff_frequently]
intro g hg a ha b hb hab
have hmain (x) : Tendsto (fun f' ↦ f' x) (𝓝 g) (𝓝 (g x)) := continuousAt_apply x _
exact le_of_tendsto_of_tendsto_of_frequently (hmain a) (hmain b) (hg.mono fun g h ↦ h ha hb hab) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Connected.Basic | {
"line": 698,
"column": 3
} | {
"line": 700,
"column": 54
} | [
{
"pp": "α : Type u\nβ : Type v\nι : Type u_1\nX : ι → Type u_2\ninst✝³ : TopologicalSpace α\ns t u v : Set α\ninst✝² : TopologicalSpace β\ninst✝¹ : PreconnectedSpace α\ninst✝ : PreconnectedSpace β\n⊢ IsPreconnected univ",
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"SProd.sprod",
... | by
rw [← univ_prod_univ]
exact isPreconnected_univ.prod isPreconnected_univ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Connected.Clopen | {
"line": 61,
"column": 6
} | {
"line": 61,
"column": 42
} | [
{
"pp": "case refine_1.inr\nι : Type u_1\nX : ι → Type u_2\nhι : Nonempty ι\ninst✝ : (i : ι) → TopologicalSpace (X i)\na : ι\nt : Set (X a)\nht : IsConnected t\nhs : IsPreconnected (mk a '' t)\nh : (mk a '' t).Nonempty\n⊢ ∃ i t_1, IsPreconnected t_1 ∧ mk a '' t = mk i '' t_1",
"usedConstants": [
"Exis... | exact ⟨a, t, ht.isPreconnected, rfl⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.Homeomorph.Lemmas | {
"line": 299,
"column": 2
} | {
"line": 299,
"column": 18
} | [
{
"pp": "case a\nX : Type u_1\ninst✝ : TopologicalSpace X\nm n : ℕ\n⊢ (Fin.appendEquiv m n).symm = (sumArrowHomeomorphProdArrow.symm.trans (piCongrLeft finSumFinEquiv)).symm",
"usedConstants": [
"Fin.natAdd",
"Equiv.instEquivLike",
"Pi.topologicalSpace",
"Fin.castAdd",
"Homeomo... | ext x i <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Topology.Compactness.Lindelof | {
"line": 372,
"column": 6
} | {
"line": 372,
"column": 26
} | [
{
"pp": "X : Type u\ninst✝¹ : TopologicalSpace X\ns : Set X\ninst✝ : DiscreteTopology X\nhs : IsLindelof s\nthis : ∀ (x : X), {x} ∈ 𝓝 x\nt : Set X\nht : t.Countable\nleft✝ : ∀ x ∈ t, x ∈ s\nhssubt : s ⊆ ⋃ x ∈ t, {x}\n⊢ s.Countable",
"usedConstants": [
"congrArg",
"Membership.mem",
"Set.bi... | biUnion_of_singleton | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Compactness.Lindelof | {
"line": 459,
"column": 87
} | {
"line": 464,
"column": 45
} | [
{
"pp": "X : Type u\ninst✝ : TopologicalSpace X\n⊢ (coclosedLindelof X).HasBasis (fun s ↦ IsClosed s ∧ IsLindelof s) compl",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"iInf",
"congrArg",
"Compl.compl",
"Filter.instInfSet",
"Filter.instCompleteLatticeFilter",
"Is... | by
simp only [Filter.coclosedLindelof, iInf_and']
refine hasBasis_biInf_principal' ?_ ⟨∅, isClosed_empty, isLindelof_empty⟩
rintro s ⟨hs₁, hs₂⟩ t ⟨ht₁, ht₂⟩
exact ⟨s ∪ t, ⟨⟨hs₁.union ht₁, hs₂.union ht₂⟩, compl_subset_compl.2 subset_union_left,
compl_subset_compl.2 subset_union_right⟩⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.Basic | {
"line": 313,
"column": 57
} | {
"line": 316,
"column": 71
} | [
{
"pp": "α : Type ua\ninst✝ : UniformSpace α\ns : SetRel α α\nhs : s ∈ 𝓤 α\n⊢ ∃ t ∈ 𝓤 α, IsOpen t ∧ SetRel.IsSymm t ∧ t ○ t ⊆ s",
"usedConstants": [
"Filter.instMembership",
"SetRel.comp_subset_comp",
"Filter.HasBasis.mem_iff",
"SetRel",
"uniformity",
"instTopologicalSp... | by
obtain ⟨t, ht₁, ht₂⟩ := comp_mem_uniformity_sets hs
obtain ⟨u, ⟨hu₁, hu₂, hu₃⟩, hu₄ : u ⊆ t⟩ := uniformity_hasBasis_open_symmetric.mem_iff.mp ht₁
exact ⟨u, hu₁, hu₂, hu₃, (SetRel.comp_subset_comp hu₄ hu₄).trans ht₂⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.UniformSpace.Basic | {
"line": 582,
"column": 21
} | {
"line": 582,
"column": 36
} | [
{
"pp": "α : Type ua\nβ : Type ub\nf : α → β\nu₁ : Set (UniformSpace α)\nu₂ : UniformSpace β\nu : UniformSpace α\nh₁ : u ∈ u₁\nhf : UniformContinuous f\n⊢ Tendsto (fun x ↦ (f x.1, f x.2)) (𝓤 α) (𝓤 β)",
"usedConstants": [
"UniformSpace",
"Eq.mpr",
"iInf",
"congrArg",
"Filter.i... | iInf_uniformity | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.Basic | {
"line": 593,
"column": 6
} | {
"line": 593,
"column": 21
} | [
{
"pp": "α : Type ua\nβ : Type ub\nι : Sort u_1\nf : α → β\nu₁ : ι → UniformSpace α\nu₂ : UniformSpace β\ni : ι\nhf : UniformContinuous f\n⊢ Tendsto (fun x ↦ (f x.1, f x.2)) (𝓤 α) (𝓤 β)",
"usedConstants": [
"UniformSpace",
"Eq.mpr",
"iInf",
"congrArg",
"Filter.instInfSet",
... | iInf_uniformity | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.Cauchy | {
"line": 306,
"column": 2
} | {
"line": 309,
"column": 29
} | [
{
"pp": "case refine_2\nα : Type u\nβ : Type v\nuniformSpace : UniformSpace α\nγ : Sort u_1\ninst✝¹ : Nonempty β\ninst✝ : SemilatticeSup β\nu : β → α\np : γ → Prop\ns : γ → SetRel α α\nH : (𝓤 α).HasBasis p s\nh : ∀ (i : γ), p i → ∃ N, ∀ n ≥ N, (u n, u N) ∈ s i\ni : γ\nhi : p i\n⊢ ∃ N, ∀ (m : β), N ≤ m → ∀ (n :... | · rcases comp_symm_of_uniformity (H.mem_of_mem hi) with ⟨t, ht, ht', hts⟩
rcases H.mem_iff.1 ht with ⟨j, hj, hjt⟩
refine (h j hj).imp fun N hN m hm n hn => hts ⟨u N, hjt ?_, ht' <| hjt ?_⟩
exacts [hN m hm, hN n hn] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.UniformSpace.Basic | {
"line": 686,
"column": 6
} | {
"line": 686,
"column": 38
} | [
{
"pp": "α : Type ua\nβ : Type ub\ninst✝¹ : UniformSpace α\ninst✝ : UniformSpace β\nf : α → β\ns : Set α\nh : UniformContinuousOn f s\n⊢ ContinuousOn f s",
"usedConstants": [
"UniformContinuous",
"congrArg",
"instUniformSpaceSubtype",
"uniformContinuousOn_iff_restrict",
"Member... | uniformContinuousOn_iff_restrict | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Filter.NAry | {
"line": 248,
"column": 2
} | {
"line": 249,
"column": 53
} | [
{
"pp": "α : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nδ : Type u_7\nm : α → β → γ\nf : Filter α\ng : Filter β\nn : γ → δ\nm' : β' → α' → δ\nn₁ : β → β'\nn₂ : α → α'\nh_antidistrib : ∀ (a : α) (b : β), n (m a b) = m' (n₁ b) (n₂ a)\n⊢ map n (map₂ m f g) = map₂ m' (map n₁ g) (map n₂ f)",... | rw [map₂_swap m]
exact map_map₂_distrib fun _ _ => h_antidistrib _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Filter.NAry | {
"line": 248,
"column": 2
} | {
"line": 249,
"column": 53
} | [
{
"pp": "α : Type u_1\nα' : Type u_2\nβ : Type u_3\nβ' : Type u_4\nγ : Type u_5\nδ : Type u_7\nm : α → β → γ\nf : Filter α\ng : Filter β\nn : γ → δ\nm' : β' → α' → δ\nn₁ : β → β'\nn₂ : α → α'\nh_antidistrib : ∀ (a : α) (b : β), n (m a b) = m' (n₁ b) (n₂ a)\n⊢ map n (map₂ m f g) = map₂ m' (map n₁ g) (map n₂ f)",... | rw [map₂_swap m]
exact map_map₂_distrib fun _ _ => h_antidistrib _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Rel | {
"line": 207,
"column": 2
} | {
"line": 207,
"column": 86
} | [
{
"pp": "case neg\nα : Type u_1\nβ : Type u_2\nγ : Type u_3\ns : Set α\nt₁ t₂ : Set β\nu : Set γ\ninst✝ : Decidable (Disjoint t₁ t₂)\nhst : ¬Disjoint t₁ t₂\n⊢ s ×ˢ t₁ ○ t₂ ×ˢ u = s ×ˢ u",
"usedConstants": [
"Set.instSProd",
"Eq.mpr",
"SetRel.prod_comp_prod_of_inter_nonempty",
"SetRel... | · rw [prod_comp_prod_of_inter_nonempty <| Set.not_disjoint_iff_nonempty_inter.1 hst] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 174,
"column": 58
} | {
"line": 174,
"column": 92
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b : α\n⊢ (Ioo a b)⁻¹ = Ioo b⁻¹ a⁻¹",
"usedConstants": [
"Set.Ioi",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"PartialOrder.toPreorder",
... | simp [← Ioi_inter_Iio, inter_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 174,
"column": 58
} | {
"line": 174,
"column": 92
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b : α\n⊢ (Ioo a b)⁻¹ = Ioo b⁻¹ a⁻¹",
"usedConstants": [
"Set.Ioi",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"PartialOrder.toPreorder",
... | simp [← Ioi_inter_Iio, inter_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 174,
"column": 58
} | {
"line": 174,
"column": 92
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b : α\n⊢ (Ioo a b)⁻¹ = Ioo b⁻¹ a⁻¹",
"usedConstants": [
"Set.Ioi",
"DivisionCommMonoid.toDivisionMonoid",
"DivInvOneMonoid.toInvOneClass",
"congrArg",
"PartialOrder.toPreorder",
... | simp [← Ioi_inter_Iio, inter_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 318,
"column": 2
} | {
"line": 318,
"column": 36
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b c : α\n⊢ (fun x ↦ a / x) ⁻¹' Ioo b c = Ioo (a / c) (a / b)",
"usedConstants": [
"Set.preimage_const_div_Iio",
"Set.Ioi",
"instHDiv",
"congrArg",
"PartialOrder.toPreorder",
... | simp [← Ioi_inter_Iio, inter_comm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 318,
"column": 2
} | {
"line": 318,
"column": 36
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b c : α\n⊢ (fun x ↦ a / x) ⁻¹' Ioo b c = Ioo (a / c) (a / b)",
"usedConstants": [
"Set.preimage_const_div_Iio",
"Set.Ioi",
"instHDiv",
"congrArg",
"PartialOrder.toPreorder",
... | simp [← Ioi_inter_Iio, inter_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Order.Group.Pointwise.Interval | {
"line": 318,
"column": 2
} | {
"line": 318,
"column": 36
} | [
{
"pp": "α : Type u_1\ninst✝² : CommGroup α\ninst✝¹ : PartialOrder α\ninst✝ : IsOrderedMonoid α\na b c : α\n⊢ (fun x ↦ a / x) ⁻¹' Ioo b c = Ioo (a / c) (a / b)",
"usedConstants": [
"Set.preimage_const_div_Iio",
"Set.Ioi",
"instHDiv",
"congrArg",
"PartialOrder.toPreorder",
... | simp [← Ioi_inter_Iio, inter_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Algebra.ConstMulAction | {
"line": 380,
"column": 89
} | {
"line": 386,
"column": 29
} | [
{
"pp": "G₀ : Type u_4\ninst✝⁵ : GroupWithZero G₀\nE : Type u_5\ninst✝⁴ : Zero E\ninst✝³ : MulActionWithZero G₀ E\ninst✝² : TopologicalSpace E\ninst✝¹ : T1Space E\ninst✝ : ContinuousConstSMul G₀ E\nc : G₀\ns : Set E\n⊢ closure (c • s) = c • closure s",
"usedConstants": [
"Eq.mpr",
"GroupWithZero... | by
rcases eq_or_ne c 0 with (rfl | hc)
· rcases eq_empty_or_nonempty s with (rfl | hs)
· simp
· rw [zero_smul_set hs, zero_smul_set hs.closure]
exact closure_singleton
· exact closure_smul₀' hc s | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Maps.Proper.Basic | {
"line": 140,
"column": 2
} | {
"line": 140,
"column": 41
} | [
{
"pp": "X : Type u_1\nY : Type u_2\nZ : Type u_3\ninst✝² : TopologicalSpace X\ninst✝¹ : TopologicalSpace Y\ninst✝ : TopologicalSpace Z\nf : X → Y\ng : Y → Z\nhf : Continuous f\nhg : Continuous g\nhgf : IsProperMap (g ∘ f)\nf_surj : Surjective f\nℱ : Filter Y\nx : X\nhx : ClusterPt x (comap f ℱ)\nh : MapCluster... | rw [← ℱ.map_comap_of_surjective f_surj] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 78,
"column": 98
} | {
"line": 80,
"column": 5
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : SeparatelyContinuousMul G\na : G\n⊢ (Homeomorph.mulLeft a).symm = Homeomorph.mulLeft a⁻¹",
"usedConstants": [
"Homeomorph.ext",
"DivInvOneMonoid.toInvOneClass",
"Group.toDivisionMonoid",
"DivisionMonoid.toDiv... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Group.Basic | {
"line": 112,
"column": 62
} | {
"line": 114,
"column": 5
} | [
{
"pp": "G : Type w\ninst✝² : TopologicalSpace G\ninst✝¹ : Group G\ninst✝ : SeparatelyContinuousMul G\na : G\n⊢ (Homeomorph.mulRight a).symm = Homeomorph.mulRight a⁻¹",
"usedConstants": [
"Homeomorph.ext",
"DivInvOneMonoid.toInvOneClass",
"Homeomorph.mulRight",
"Group.toDivisionMonoi... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 317,
"column": 2
} | {
"line": 319,
"column": 5
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : UniformSpace α\ninst✝³ : CommGroup α\ninst✝² : IsUniformGroup α\ninst✝¹ : CompleteSpace α\ninst✝ : T2Space α\nf : β → α\nhf : Multipliable f\ns : Finset β\n⊢ (∏ x ∈ s, f x) * ∏' (x : { x // x ∉ s }), f ↑x = ∏' (x : β), f x",
"usedConstants": [
"Eq.mpr",
... | rw [← hf.tprod_subtype_mul_tprod_subtype_compl s]
simp only [Finset.tprod_subtype', mul_right_inj]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Algebra.InfiniteSum.Group | {
"line": 317,
"column": 2
} | {
"line": 319,
"column": 5
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝⁴ : UniformSpace α\ninst✝³ : CommGroup α\ninst✝² : IsUniformGroup α\ninst✝¹ : CompleteSpace α\ninst✝ : T2Space α\nf : β → α\nhf : Multipliable f\ns : Finset β\n⊢ (∏ x ∈ s, f x) * ∏' (x : { x // x ∉ s }), f ↑x = ∏' (x : β), f x",
"usedConstants": [
"Eq.mpr",
... | rw [← hf.tprod_subtype_mul_tprod_subtype_compl s]
simp only [Finset.tprod_subtype', mul_right_inj]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Order.MonotoneConvergence | {
"line": 235,
"column": 4
} | {
"line": 235,
"column": 61
} | [
{
"pp": "case mpr\nι₁ : Type u_3\nι₂ : Type u_4\nα : Type u_5\ninst✝⁶ : SemilatticeSup ι₁\ninst✝⁵ : Preorder ι₂\ninst✝⁴ : Nonempty ι₁\ninst✝³ : TopologicalSpace α\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : OrderTopology α\ninst✝ : NoMaxOrder α\nf : ι₂ → α\nφ : ι₁ → ι₂\nl : α\nhf : Monotone f\nhg : T... | rcases tendsto_atTop_of_monotone hf with (h' | ⟨l', hl'⟩) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.Order.Basic | {
"line": 406,
"column": 98
} | {
"line": 409,
"column": 58
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝³ : TopologicalSpace β\ninst✝² : Preorder β\ninst✝¹ : OrderTop β\ninst✝ : OrderTopology β\nl : Filter α\nf g : α → β\nhf : Tendsto f l (𝓝 ⊤)\nhg : f ≤ᶠ[l] g\n⊢ Tendsto g l (𝓝 ⊤)",
"usedConstants": [
"Eq.mpr",
"Set.Ioi",
"Preorder.toLT",
"iInf",... | by
simp only [nhds_top_order, tendsto_iInf, tendsto_principal] at hf ⊢
intro x hx
filter_upwards [hf x hx, hg] with _ using lt_of_lt_of_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.InfiniteSum.Order | {
"line": 158,
"column": 11
} | {
"line": 158,
"column": 24
} | [
{
"pp": "case neg\nι : Type u_1\nα : Type u_3\nL : SummationFilter ι\ninst✝³ : CommMonoid α\ninst✝² : Preorder α\ninst✝¹ : TopologicalSpace α\ninst✝ : OrderClosedTopology α\nf : ι → α\na₂ : α\nha₂ : 1 ≤ a₂\nh : ∀ (s : Finset ι), ∏ i ∈ s, f i ≤ a₂\nhL : ¬L.NeBot\nhf : ¬(mulSupport f).Finite\n⊢ ∏'[L] (i : ι), f i... | tprod_bot hL, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.IsLUB | {
"line": 220,
"column": 34
} | {
"line": 220,
"column": 74
} | [
{
"pp": "γ : Type u_2\nα : Type u_3\ninst✝³ : TopologicalSpace α\ninst✝² : ConditionallyCompleteLinearOrder α\ninst✝¹ : ClosedIicTopology α\nf : γ → α\ninst✝ : TopologicalSpace γ\nS : Set γ\nhS : Dense S\nhf : Continuous f\nh : ¬BddAbove (range fun x ↦ f ↑x)\nthis : ¬BddAbove (range f)\n⊢ ⨆ s, f ↑s = ⨆ i, f i",... | by simp [ciSup_of_not_bddAbove, this, h] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Order.IsLUB | {
"line": 403,
"column": 2
} | {
"line": 405,
"column": 80
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : TopologicalSpace α\ninst✝³ : LinearOrder α\ninst✝² : OrderTopology α\ninst✝¹ : DenselyOrdered α\ninst✝ : FirstCountableTopology α\nx y : α\nh : x < y\nu : ℕ → α\nhu_anti : StrictAnti u\nhu_mem : ∀ (n : ℕ), u n ∈ Ioo x y\nhux : Tendsto u atTop (𝓝 x)\nv : ℕ → α\nhv_mono : StrictMo... | exact
⟨u, v, hu_anti, hv_mono, hu_mem, fun l => ⟨(hu_mem 0).1.trans (hv_mem l).1, (hv_mem l).2⟩,
fun k l => (hu_anti.antitone (zero_le k)).trans_lt (hv_mem l).1, hux, hvy⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.UniformSpace.Compact | {
"line": 133,
"column": 6
} | {
"line": 135,
"column": 52
} | [
{
"pp": "case mpr\nα : Type ua\nι : Sort u_1\ninst✝ : UniformSpace α\nK : Set α\np : ι → Prop\nV : ι → Set (α × α)\nhbasis : (𝓤 α).HasBasis p V\nhK : IsCompact K\nU : Set α\n⊢ (∃ i, p i ∧ ⋃ x ∈ K, ball x (V i) ⊆ U) → U ∈ 𝓝ˢ K",
"usedConstants": [
"Filter.instMembership",
"UniformSpace.ball_mem... | rintro ⟨i, hpi, hi⟩
refine mem_of_superset (bUnion_mem_nhdsSet fun x _ ↦ ?_) hi
exact ball_mem_nhds _ <| hbasis.mem_of_mem hpi | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UniformSpace.Compact | {
"line": 133,
"column": 6
} | {
"line": 135,
"column": 52
} | [
{
"pp": "case mpr\nα : Type ua\nι : Sort u_1\ninst✝ : UniformSpace α\nK : Set α\np : ι → Prop\nV : ι → Set (α × α)\nhbasis : (𝓤 α).HasBasis p V\nhK : IsCompact K\nU : Set α\n⊢ (∃ i, p i ∧ ⋃ x ∈ K, ball x (V i) ⊆ U) → U ∈ 𝓝ˢ K",
"usedConstants": [
"Filter.instMembership",
"UniformSpace.ball_mem... | rintro ⟨i, hpi, hi⟩
refine mem_of_superset (bUnion_mem_nhdsSet fun x _ ↦ ?_) hi
exact ball_mem_nhds _ <| hbasis.mem_of_mem hpi | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UniformSpace.UniformConvergence | {
"line": 178,
"column": 44
} | {
"line": 178,
"column": 86
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\np : Filter ι\np' : Filter α\nF' : ι → α → β\nhf : Tendsto (fun q ↦ (f q.2, F q.1 q.2)) (p ×ˢ p') (𝓤 β)\nhff' : ∀ᶠ (n : ι × α) in p ×ˢ p', Inseparable (F n.1 n.2) (F' n.1 n.2)\n⊢ Tendsto (fun q ↦ (f q.2, F' q.1 ... | uniformity_hasBasis_open.tendsto_right_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.UniformConvergence | {
"line": 210,
"column": 38
} | {
"line": 210,
"column": 80
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\np : Filter ι\ng : α → β\nhf : Tendsto (fun q ↦ (f q.2, F q.1 q.2)) (p ×ˢ 𝓟 s) (𝓤 β)\nhfg : ∀ x ∈ s, Inseparable (f x) (g x)\n⊢ Tendsto (fun q ↦ (g q.2, F q.1 q.2)) (p ×ˢ 𝓟 s) (𝓤 β)",
"usedCons... | uniformity_hasBasis_open.tendsto_right_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Order.IntermediateValue | {
"line": 387,
"column": 4
} | {
"line": 387,
"column": 42
} | [
{
"pp": "α : Type u\ninst✝³ : ConditionallyCompleteLinearOrder α\ninst✝² : TopologicalSpace α\ninst✝¹ : OrderTopology α\ninst✝ : DenselyOrdered α\na b : α\ns : Set α\nhs : IsClosed (s ∩ Icc a b)\nha : a ∈ s\nh : ∀ t ∈ Ico a b, Icc a t ⊆ s → s ∈ 𝓝[>] t\nhab : a ≤ b\nA : Set α := {t | t ∈ Icc a b ∧ Icc a t ⊆ s}\... | rcases le_or_gt t' t₁ with h't' | h't' | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 308,
"column": 6
} | {
"line": 308,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\np : Filter ι\nG : ι → α → β\nhf✝ : ∀ x ∈ s, Tendsto (fun y ↦ (f y.2, F y.1 y.2)) (p ×ˢ 𝓝[s] x) (𝓤 β)\nhg✝ : ∀ᶠ (n : ι) in p, ∀ x ∈ s, Inseparable (F n x) (G n x)\nhg : ∀... | uniformity_hasBasis_open.tendsto_right_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UniformSpace.LocallyUniformConvergence | {
"line": 324,
"column": 6
} | {
"line": 324,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nι : Type u_4\ninst✝¹ : TopologicalSpace α\ninst✝ : UniformSpace β\nF : ι → α → β\nf : α → β\ns : Set α\np : Filter ι\ng : α → β\nhf✝ : ∀ x ∈ s, Tendsto (fun y ↦ (f y.2, F y.1 y.2)) (p ×ˢ 𝓝[s] x) (𝓤 β)\nhg✝ : ∀ x ∈ s, Inseparable (f x) (g x)\nhg : ∀ᶠ (x : ι × α) in p ×ˢ 𝓟 ... | uniformity_hasBasis_open.tendsto_right_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 576,
"column": 6
} | {
"line": 576,
"column": 28
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nx y : α\nε₁ ε₂ : ℝ\nh : dist x y ≤ ε₂ - ε₁\nz : α\nzx : z ∈ ball x ε₁\n⊢ z ∈ ball y ε₂",
"usedConstants": [
"Eq.mpr",
"Real",
"congrArg",
"AddMonoid.toAddZeroClass",
"HSub.hSub",
"AddCommGroup.toAddGroup",
"Membershi... | ← add_sub_cancel ε₁ ε₂ | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1022,
"column": 85
} | {
"line": 1024,
"column": 5
} | [
{
"pp": "α : Type u_3\nU : UniformSpace α\nm : PseudoMetricSpace α\nH : 𝓤 α = 𝓤 α\n⊢ m.replaceUniformity H = m",
"usedConstants": [
"Real",
"PseudoMetricSpace.ext",
"funext",
"Dist.ext",
"Eq.refl",
"Dist.dist",
"PseudoMetricSpace.toDist",
"PseudoMetricSpace.... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1042,
"column": 79
} | {
"line": 1044,
"column": 5
} | [
{
"pp": "γ : Type u_3\nU : TopologicalSpace γ\nm : PseudoMetricSpace γ\nH : U = toUniformSpace.toTopologicalSpace\n⊢ m.replaceTopology H = m",
"usedConstants": [
"Real",
"PseudoMetricSpace.ext",
"funext",
"Dist.ext",
"Eq.refl",
"Dist.dist",
"PseudoMetricSpace.toDist... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Pseudo.Defs | {
"line": 1088,
"column": 50
} | {
"line": 1090,
"column": 5
} | [
{
"pp": "α : Type u_3\nm : PseudoMetricSpace α\nB : Bornology α\nH : ∀ (s : Set α), Bornology.IsBounded s ↔ Bornology.IsBounded s\n⊢ m.replaceBornology H = m",
"usedConstants": [
"Real",
"PseudoMetricSpace.ext",
"PseudoMetricSpace.replaceBornology",
"funext",
"Dist.ext",
... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Defs | {
"line": 176,
"column": 44
} | {
"line": 178,
"column": 5
} | [
{
"pp": "α : Type u_3\nm : MetricSpace α\nB : Bornology α\nH : ∀ (s : Set α), Bornology.IsBounded s ↔ Bornology.IsBounded s\n⊢ m.replaceBornology H = m",
"usedConstants": [
"Real",
"MetricSpace.ext",
"funext",
"Dist.ext",
"MetricSpace.replaceBornology",
"Eq.refl",
"... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 82
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nc : α\n⊢ (cobounded α).HasBasis (fun x ↦ True) fun i ↦ (fun x ↦ dist x c) ⁻¹' Ici i",
"usedConstants": [
"Real",
"PseudoMetricSpace.toBornology",
"congrArg",
"Compl.compl",
"PartialOrder.toPreorder",
"setOf",
"Real.i... | simpa only [compl_def, mem_ball, not_lt] using hasBasis_cobounded_compl_ball c | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 82
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nc : α\n⊢ (cobounded α).HasBasis (fun x ↦ True) fun i ↦ (fun x ↦ dist x c) ⁻¹' Ici i",
"usedConstants": [
"Real",
"PseudoMetricSpace.toBornology",
"congrArg",
"Compl.compl",
"PartialOrder.toPreorder",
"setOf",
"Real.i... | simpa only [compl_def, mem_ball, not_lt] using hasBasis_cobounded_compl_ball c | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Bounded | {
"line": 151,
"column": 4
} | {
"line": 151,
"column": 82
} | [
{
"pp": "α : Type u\ninst✝ : PseudoMetricSpace α\nc : α\n⊢ (cobounded α).HasBasis (fun x ↦ True) fun i ↦ (fun x ↦ dist x c) ⁻¹' Ici i",
"usedConstants": [
"Real",
"PseudoMetricSpace.toBornology",
"congrArg",
"Compl.compl",
"PartialOrder.toPreorder",
"setOf",
"Real.i... | simpa only [compl_def, mem_ball, not_lt] using hasBasis_cobounded_compl_ball c | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.DirectedInverseSystem | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 61
} | [
{
"pp": "ι : Type u_1\ninst✝³ : Preorder ι\nF : ι → Type u_4\nT : ⦃i j : ι⦄ → i ≤ j → Sort u_8\nf : (i j : ι) → (h : i ≤ j) → T h\ninst✝² : ⦃i j : ι⦄ → (h : i ≤ j) → FunLike (T h) (F i) (F j)\ninst✝¹ : DirectedSystem F fun x1 x2 x3 ↦ ⇑(f x1 x2 x3)\ninst✝ : IsDirectedOrder ι\nC : DirectLimit F f → DirectLimit F ... | obtain ⟨_, _, _, rfl, rfl⟩ := exists_eq_mk₂ f x y; apply ih | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.DirectedInverseSystem | {
"line": 139,
"column": 2
} | {
"line": 139,
"column": 61
} | [
{
"pp": "ι : Type u_1\ninst✝³ : Preorder ι\nF : ι → Type u_4\nT : ⦃i j : ι⦄ → i ≤ j → Sort u_8\nf : (i j : ι) → (h : i ≤ j) → T h\ninst✝² : ⦃i j : ι⦄ → (h : i ≤ j) → FunLike (T h) (F i) (F j)\ninst✝¹ : DirectedSystem F fun x1 x2 x3 ↦ ⇑(f x1 x2 x3)\ninst✝ : IsDirectedOrder ι\nC : DirectLimit F f → DirectLimit F ... | obtain ⟨_, _, _, rfl, rfl⟩ := exists_eq_mk₂ f x y; apply ih | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Colimit.Module | {
"line": 261,
"column": 2
} | {
"line": 265,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Semiring R\nι : Type u_2\ninst✝⁵ : Preorder ι\nG : ι → Type u_3\ninst✝⁴ : (i : ι) → AddCommMonoid (G i)\ninst✝³ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\ninst✝² : DecidableEq ι\ninst✝¹ : DirectedSystem G fun x1 x2 x3 ↦ ⇑(f x1 x2 x3)\ninst✝ : IsDirectedOrd... | have := Nonempty.intro i
apply_fun linearEquiv _ _ at h
simp_rw [linearEquiv_of] at h
have ⟨j, h⟩ := Quotient.exact h
exact ⟨j, h.1, h.2.2⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Colimit.Module | {
"line": 261,
"column": 2
} | {
"line": 265,
"column": 23
} | [
{
"pp": "R : Type u_1\ninst✝⁶ : Semiring R\nι : Type u_2\ninst✝⁵ : Preorder ι\nG : ι → Type u_3\ninst✝⁴ : (i : ι) → AddCommMonoid (G i)\ninst✝³ : (i : ι) → Module R (G i)\nf : (i j : ι) → i ≤ j → G i →ₗ[R] G j\ninst✝² : DecidableEq ι\ninst✝¹ : DirectedSystem G fun x1 x2 x3 ↦ ⇑(f x1 x2 x3)\ninst✝ : IsDirectedOrd... | have := Nonempty.intro i
apply_fun linearEquiv _ _ at h
simp_rw [linearEquiv_of] at h
have ⟨j, h⟩ := Quotient.exact h
exact ⟨j, h.1, h.2.2⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Nilpotent.Defs | {
"line": 154,
"column": 2
} | {
"line": 154,
"column": 7
} | [
{
"pp": "R : Type u_1\nx y : R\ninst✝ : Semiring R\nh_comm : Commute x y\nn : ℕ\nhn : x ^ n = 0\n⊢ IsNilpotent (x * y)",
"usedConstants": [
"HMul.hMul",
"NonUnitalNonAssocSemiring.toMulZeroClass",
"Monoid.toPow",
"HPow.hPow",
"Distrib.toMul",
"NonAssocSemiring.toNonUnital... | use n | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 253,
"column": 23
} | {
"line": 253,
"column": 60
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : AddCommGroup P\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nQ : Type u_5\ninst✝¹ : AddCommGroup Q\ninst✝ : Module R Q\... | lTensor.inverse_of_rightInverse_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.LinearAlgebra.TensorProduct.RightExactness | {
"line": 276,
"column": 8
} | {
"line": 276,
"column": 45
} | [
{
"pp": "R : Type u_1\nM : Type u_2\nN : Type u_3\nP : Type u_4\ninst✝⁸ : CommRing R\ninst✝⁷ : AddCommGroup M\ninst✝⁶ : AddCommGroup N\ninst✝⁵ : AddCommGroup P\ninst✝⁴ : Module R M\ninst✝³ : Module R N\ninst✝² : Module R P\nf : M →ₗ[R] N\ng : N →ₗ[R] P\nQ : Type u_5\ninst✝¹ : AddCommGroup Q\ninst✝ : Module R Q\... | lTensor.inverse_of_rightInverse_apply | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Colimit.DirectLimit | {
"line": 265,
"column": 48
} | {
"line": 265,
"column": 64
} | [
{
"pp": "R : Type u_1\nι : Type u_2\ninst✝⁶ : Preorder ι\nG : ι → Type u_3\nT : ⦃i j : ι⦄ → i ≤ j → Type u_4\nf : (x x_1 : ι) → (h : x ≤ x_1) → T h\ninst✝⁵ : (i j : ι) → (h : i ≤ j) → FunLike (T h) (G i) (G j)\ninst✝⁴ : DirectedSystem G fun x1 x2 x3 ↦ ⇑(f x1 x2 x3)\ninst✝³ : IsDirectedOrder ι\ninst✝² : Nonempty... | Int.cast_natCast | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Finset.Density | {
"line": 104,
"column": 71
} | {
"line": 110,
"column": 44
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝¹ : Fintype α\ns : Finset α\ninst✝ : Fintype β\nf : α ↪ β\n⊢ (map f s).dens ≤ s.dens",
"usedConstants": [
"Fintype.card_le_of_injective",
"div_le_div₀",
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"LinearOrderedCommGroupWithZero.toLinea... | by
cases isEmpty_or_nonempty α
· simp [Subsingleton.elim s ∅]
simp_rw [dens, card_map]
gcongr
· exact mod_cast Fintype.card_pos
· exact Fintype.card_le_of_injective _ f.2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Int.Interval | {
"line": 122,
"column": 6
} | {
"line": 122,
"column": 15
} | [
{
"pp": "a b : ℤ\nh : a ≤ b\n⊢ ↑(#(Ioc a b)) = b - a",
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"HSub.hSub",
"Int.card_Ioc",
"SemilatticeInf.toPartialOrder",
"id",
"Int",
"Int.instLocallyFiniteOrder",
"Nat.cast",
"i... | card_Ioc, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Int.Interval | {
"line": 158,
"column": 38
} | {
"line": 158,
"column": 80
} | [
{
"pp": "n a : ℤ\nh : 0 ≤ a\nha : 0 < a\ni : ℤ\nhi₀ : 0 ≤ i\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : i < n % a\n⊢ 0 + n % a + a * (n / a) ≤ i + a + a * (n / a)",
"usedConstants": [
"Int.instDiv",
"instHDiv",
"HMul.hMul",
"covariant_swap_add_of_covariant_add",
"add_le_ad... | gcongr; exact (Int.emod_lt_of_pos n ha).le | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Int.Interval | {
"line": 158,
"column": 38
} | {
"line": 158,
"column": 80
} | [
{
"pp": "n a : ℤ\nh : 0 ≤ a\nha : 0 < a\ni : ℤ\nhi₀ : 0 ≤ i\nhia : i < a\nhn : n % a + a * (n / a) = n\nhi : i < n % a\n⊢ 0 + n % a + a * (n / a) ≤ i + a + a * (n / a)",
"usedConstants": [
"Int.instDiv",
"instHDiv",
"HMul.hMul",
"covariant_swap_add_of_covariant_add",
"add_le_ad... | gcongr; exact (Int.emod_lt_of_pos n ha).le | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Ring.Nat | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 64
} | [
{
"pp": "ι : Type u_1\ns : Finset ι\nf : ι → ℕ\n⊢ Odd (∑ i ∈ s, f i) ↔ Odd #({x ∈ s | Odd (f x)})",
"usedConstants": [
"instDecidableNot",
"congrArg",
"_private.Mathlib.Algebra.BigOperators.Ring.Nat.0.Finset.odd_sum_iff_odd_card_odd._simp_1_1",
"Finset",
"Odd",
"Finset.fi... | simp only [← Nat.not_even_iff_odd, even_sum_iff_even_card_odd] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Algebra.BigOperators.Ring.Nat | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 64
} | [
{
"pp": "ι : Type u_1\ns : Finset ι\nf : ι → ℕ\n⊢ Odd (∑ i ∈ s, f i) ↔ Odd #({x ∈ s | Odd (f x)})",
"usedConstants": [
"instDecidableNot",
"congrArg",
"_private.Mathlib.Algebra.BigOperators.Ring.Nat.0.Finset.odd_sum_iff_odd_card_odd._simp_1_1",
"Finset",
"Odd",
"Finset.fi... | simp only [← Nat.not_even_iff_odd, even_sum_iff_even_card_odd] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.BigOperators.Ring.Nat | {
"line": 35,
"column": 2
} | {
"line": 35,
"column": 64
} | [
{
"pp": "ι : Type u_1\ns : Finset ι\nf : ι → ℕ\n⊢ Odd (∑ i ∈ s, f i) ↔ Odd #({x ∈ s | Odd (f x)})",
"usedConstants": [
"instDecidableNot",
"congrArg",
"_private.Mathlib.Algebra.BigOperators.Ring.Nat.0.Finset.odd_sum_iff_odd_card_odd._simp_1_1",
"Finset",
"Odd",
"Finset.fi... | simp only [← Nat.not_even_iff_odd, even_sum_iff_even_card_odd] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.BigOperators.Ring.Nat | {
"line": 43,
"column": 6
} | {
"line": 43,
"column": 62
} | [
{
"pp": "ι : Type u_1\nM : Type u_2\nf : ι → M\ns : Finset M\nhb : ∀ b ∈ s, {a | f a = b}.Finite\nt : Finset M := ⋯.toFinset\n⊢ Nat.card ↑(f ⁻¹' ↑s) = ∑ b ∈ s, Nat.card { a // f a = b }",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"Set.univ",
"Set.inter_univ",
"Cl... | show Nat.card (f ⁻¹' s) = Nat.card (f ⁻¹' t) by simp [t] | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.BigOperators.ModEq | {
"line": 186,
"column": 58
} | {
"line": 189,
"column": 31
} | [
{
"pp": "α : Type u_1\nn : ℤ\nf : α → ℤ\ninst✝ : DecidableEq α\ns : Finset α\na : α\nhf : ∀ x ∈ s, x ≠ a → f x ≡ 1 [ZMOD n]\n⊢ ∏ x ∈ s, f x ≡ if a ∈ s then f a else 1 [ZMOD n]",
"usedConstants": [
"Int.instCommMonoid",
"Int.cast",
"Eq.mpr",
"ZMod.commRing",
"congrArg",
"F... | by
simp only [← modEq_natAbs (n := n), ← ZMod.intCast_eq_intCast_iff, cast_one, cast_prod,
apply_ite Int.cast] at *
exact Finset.prod_eq_ite _ hf | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.Quiver.Basic | {
"line": 106,
"column": 2
} | {
"line": 107,
"column": 9
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\nX Y X' Y' : V\nhX : X = X'\nhY : Y = Y'\nf g : X ⟶ Y\nh : homOfEq f hX hY = homOfEq g hX hY\n⊢ f = g",
"usedConstants": [
"Quiver.Hom",
"Quiver.homOfEq",
"Eq.rec",
"Eq.refl",
"Eq"
]
}
] | subst hX hY
exact h | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Quiver.Basic | {
"line": 106,
"column": 2
} | {
"line": 107,
"column": 9
} | [
{
"pp": "V : Type u_1\ninst✝ : Quiver V\nX Y X' Y' : V\nhX : X = X'\nhY : Y = Y'\nf g : X ⟶ Y\nh : homOfEq f hX hY = homOfEq g hX hY\n⊢ f = g",
"usedConstants": [
"Quiver.Hom",
"Quiver.homOfEq",
"Eq.rec",
"Eq.refl",
"Eq"
]
}
] | subst hX hY
exact h | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Category.Basic | {
"line": 303,
"column": 33
} | {
"line": 303,
"column": 71
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nf g : X ⟶ Y\nw : (fun {Z} h ↦ f ≫ h) = fun {Z} h ↦ g ≫ h\nZ : C\nh : Y ⟶ Z\n⊢ f ≫ h = g ≫ h",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.CategoryStruct.comp",
"Eq.... | by convert congr_fun (congr_fun w Z) h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Sym.Sym2 | {
"line": 999,
"column": 14
} | {
"line": 999,
"column": 31
} | [
{
"pp": "case h.mk\nα : Type u_1\nι : Type u_4\nf : ι → Set α\nx✝ : Sym2 α\nx y : α\n⊢ Quot.mk (Rel α) (x, y) ∈ (⋂ i, f i).sym2 ↔ Quot.mk (Rel α) (x, y) ∈ ⋂ i, (f i).sym2",
"usedConstants": [
"Sym2.Rel",
"Sym2.mk",
"congrArg",
"Set.iInter",
"Membership.mem",
"Prod.mk",
... | simp [forall_and] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.RingTheory.IsTensorProduct | {
"line": 475,
"column": 10
} | {
"line": 475,
"column": 12
} | [
{
"pp": "case h\nR : Type u_1\nM : Type v₁\nN : Type v₂\ninst✝⁴ : AddCommMonoid M\ninst✝³ : AddCommMonoid N\ninst✝² : CommSemiring R\ninst✝¹ : Module R M\ninst✝ : Module R N\ne : M ≃ₗ[R] N\nQ : Type (max v₁ v₂ u_1)\n⊢ ∀ [inst : AddCommMonoid Q] [inst_1 : Module R Q] [inst_2 : Module R Q] [inst_3 : IsScalarTower... | I₁ | Lean.Elab.Tactic.evalIntro | ident |
Mathlib.RingTheory.IsTensorProduct | {
"line": 719,
"column": 34
} | {
"line": 719,
"column": 58
} | [
{
"pp": "R : Type u_1\nS : Type v₃\ninst✝²⁷ : CommSemiring R\ninst✝²⁶ : CommSemiring S\ninst✝²⁵ : Algebra R S\nT : Type u_4\ninst✝²⁴ : CommSemiring T\ninst✝²³ : Algebra R T\ninst✝²² : Algebra S T\ninst✝²¹ : IsScalarTower R S T\nR' : Type u_6\nS' : Type u_7\ninst✝²⁰ : CommSemiring R'\ninst✝¹⁹ : CommSemiring S'\n... | algebraMap_apply R S S', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Products.Basic | {
"line": 102,
"column": 39
} | {
"line": 110,
"column": 51
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nP Q : C\nS T : D\nf : (P, S) ⟶ (Q, T)\n⊢ IsIso f ↔ IsIso f.1 ∧ IsIso f.2",
"usedConstants": [
"CategoryTheory.IsIso",
"CategoryTheory.prod",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver... | by
constructor
· rintro ⟨g, hfg, hgf⟩
rcases Prod.hom_ext_iff.1 hfg with ⟨hfg₁, hfg₂⟩
rcases Prod.hom_ext_iff.1 hgf with ⟨hgf₁, hgf₂⟩
exact ⟨⟨⟨g.1, hfg₁, hgf₁⟩⟩, ⟨⟨g.2, hfg₂, hgf₂⟩⟩⟩
· rintro ⟨⟨g₁, hfg₁, hgf₁⟩, ⟨g₂, hfg₂, hgf₂⟩⟩
dsimp at hfg₁ hgf₁ hfg₂ hgf₂
refine ⟨⟨(g₁, g₂), by aesop_cat, by ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Pi.Basic | {
"line": 223,
"column": 2
} | {
"line": 227,
"column": 9
} | [
{
"pp": "case h_map\nI : Type w₀\nC : I → Type u₁\ninst✝¹ : (i : I) → Category.{v₁, u₁} (C i)\nA : Type u₃\ninst✝ : Category.{v₃, u₃} A\nf f' : A ⥤ ((i : I) → C i)\nh : ∀ (i : I), f ⋙ Pi.eval C i = f' ⋙ Pi.eval C i\n⊢ autoParam (∀ (X Y : A) (f_1 : X ⟶ Y), f.map f_1 = eqToHom ⋯ ≫ f'.map f_1 ≫ eqToHom ⋯) ext._aut... | · intro X Y g
funext i
specialize h i
have := congr_hom h g
simpa | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Functor.ReflectsIso.Basic | {
"line": 88,
"column": 41
} | {
"line": 88,
"column": 66
} | [
{
"pp": "C : Type u_1\ninst✝² : Category.{v_1, u_1} C\nD : Type u_2\ninst✝¹ : Category.{v_2, u_2} D\nF G : C ⥤ D\nα : F ≅ G\ninst✝ : F.ReflectsIsomorphisms\nA✝ B✝ : C\nf : A✝ ⟶ B✝\nx✝ : IsIso (G.map f)\n⊢ IsIso (F.map f)",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.Functor",
"CategoryThe... | ← NatIso.naturality_2 α f | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Category.Preorder | {
"line": 273,
"column": 4
} | {
"line": 275,
"column": 18
} | [
{
"pp": "case mpr\nX : Type u\ninst✝ : PartialOrder X\na b : X\nf : a ⟶ b\n⊢ a = b → IsIso f",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.IsIso",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"PartialOrder.toPreorder",
"CategoryTheory.Categor... | rintro rfl
rw [Subsingleton.elim f (𝟙 _)]
infer_instance | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Category.Preorder | {
"line": 273,
"column": 4
} | {
"line": 275,
"column": 18
} | [
{
"pp": "case mpr\nX : Type u\ninst✝ : PartialOrder X\na b : X\nf : a ⟶ b\n⊢ a = b → IsIso f",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.IsIso",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"PartialOrder.toPreorder",
"CategoryTheory.Categor... | rintro rfl
rw [Subsingleton.elim f (𝟙 _)]
infer_instance | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.IsTerminal | {
"line": 479,
"column": 59
} | {
"line": 479,
"column": 88
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝ : Category.{v_2, u_2} D\nX : D\nhX : IsTerminal X\n⊢ ∀ (X_1 : C ⥤ D) (m : X_1 ⟶ (const C).obj X), m = (fun Y ↦ { app := fun Z ↦ hX.from (Y.obj Z), naturality := ⋯ }) X_1",
"usedConstants": [
"CategoryTheory.Limits.IsTerminal.fr... | intros; ext; apply hX.hom_ext | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.IsTerminal | {
"line": 479,
"column": 59
} | {
"line": 479,
"column": 88
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝ : Category.{v_2, u_2} D\nX : D\nhX : IsTerminal X\n⊢ ∀ (X_1 : C ⥤ D) (m : X_1 ⟶ (const C).obj X), m = (fun Y ↦ { app := fun Z ↦ hX.from (Y.obj Z), naturality := ⋯ }) X_1",
"usedConstants": [
"CategoryTheory.Limits.IsTerminal.fr... | intros; ext; apply hX.hom_ext | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.IsTerminal | {
"line": 488,
"column": 57
} | {
"line": 488,
"column": 86
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝ : Category.{v_2, u_2} D\nX : D\nhX : IsInitial X\n⊢ ∀ (Y : C ⥤ D) (m : (const C).obj X ⟶ Y), m = (fun Y ↦ { app := fun Z ↦ hX.to (Y.obj Z), naturality := ⋯ }) Y",
"usedConstants": [
"CategoryTheory.Limits.IsInitial.hom_ext",
... | intros; ext; apply hX.hom_ext | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.IsTerminal | {
"line": 488,
"column": 57
} | {
"line": 488,
"column": 86
} | [
{
"pp": "C : Type u_1\ninst✝¹ : Category.{v_1, u_1} C\nD : Type u_2\ninst✝ : Category.{v_2, u_2} D\nX : D\nhX : IsInitial X\n⊢ ∀ (Y : C ⥤ D) (m : (const C).obj X ⟶ Y), m = (fun Y ↦ { app := fun Z ↦ hX.to (Y.obj Z), naturality := ⋯ }) Y",
"usedConstants": [
"CategoryTheory.Limits.IsInitial.hom_ext",
... | intros; ext; apply hX.hom_ext | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | {
"line": 133,
"column": 10
} | {
"line": 133,
"column": 55
} | [
{
"pp": "J : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nF : WidePullbackShape J ⥤ C\nX : C\nf✝ : X ⟶ F.obj none\nπ : (j : J) → X ⟶ F.obj (some j)\nw : ∀ (j : J), π j ≫ F.map (Hom.term j) = f✝\nj j' : WidePullbackShape J\nf : j ⟶ j'\n⊢ (((Functor.const (WidePullbackShape J)).obj X).map f ≫\n match j' wi... | cases j <;> cases j' <;> cases f <;> simp [w] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | {
"line": 133,
"column": 10
} | {
"line": 133,
"column": 55
} | [
{
"pp": "J : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nF : WidePullbackShape J ⥤ C\nX : C\nf✝ : X ⟶ F.obj none\nπ : (j : J) → X ⟶ F.obj (some j)\nw : ∀ (j : J), π j ≫ F.map (Hom.term j) = f✝\nj j' : WidePullbackShape J\nf : j ⟶ j'\n⊢ (((Functor.const (WidePullbackShape J)).obj X).map f ≫\n match j' wi... | cases j <;> cases j' <;> cases f <;> simp [w] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | {
"line": 133,
"column": 10
} | {
"line": 133,
"column": 55
} | [
{
"pp": "J : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nF : WidePullbackShape J ⥤ C\nX : C\nf✝ : X ⟶ F.obj none\nπ : (j : J) → X ⟶ F.obj (some j)\nw : ∀ (j : J), π j ≫ F.map (Hom.term j) = f✝\nj j' : WidePullbackShape J\nf : j ⟶ j'\n⊢ (((Functor.const (WidePullbackShape J)).obj X).map f ≫\n match j' wi... | cases j <;> cases j' <;> cases f <;> simp [w] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | {
"line": 261,
"column": 10
} | {
"line": 261,
"column": 55
} | [
{
"pp": "J : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nF : WidePushoutShape J ⥤ C\nX : C\nf✝ : F.obj none ⟶ X\nι : (j : J) → F.obj (some j) ⟶ X\nw : ∀ (j : J), F.map (Hom.init j) ≫ ι j = f✝\nj j' : WidePushoutShape J\nf : j ⟶ j'\n⊢ (F.map f ≫\n match j' with\n | none => f✝\n | some j => ι j)... | cases j <;> cases j' <;> cases f <;> simp [w] | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | {
"line": 261,
"column": 10
} | {
"line": 261,
"column": 55
} | [
{
"pp": "J : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nF : WidePushoutShape J ⥤ C\nX : C\nf✝ : F.obj none ⟶ X\nι : (j : J) → F.obj (some j) ⟶ X\nw : ∀ (j : J), F.map (Hom.init j) ≫ ι j = f✝\nj j' : WidePushoutShape J\nf : j ⟶ j'\n⊢ (F.map f ≫\n match j' with\n | none => f✝\n | some j => ι j)... | cases j <;> cases j' <;> cases f <;> simp [w] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.Limits.Shapes.WidePullbacks | {
"line": 261,
"column": 10
} | {
"line": 261,
"column": 55
} | [
{
"pp": "J : Type w\nC : Type u\ninst✝ : Category.{v, u} C\nF : WidePushoutShape J ⥤ C\nX : C\nf✝ : F.obj none ⟶ X\nι : (j : J) → F.obj (some j) ⟶ X\nw : ∀ (j : J), F.map (Hom.init j) ≫ ι j = f✝\nj j' : WidePushoutShape J\nf : j ⟶ j'\n⊢ (F.map f ≫\n match j' with\n | none => f✝\n | some j => ι j)... | cases j <;> cases j' <;> cases f <;> simp [w] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Comma.StructuredArrow.Basic | {
"line": 388,
"column": 20
} | {
"line": 388,
"column": 34
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nS : D\nT : C ⥤ D\nf : StructuredArrow S T\nh : f.IsUniversal\nc : C\nη η' : f.right ⟶ c\nw : f.hom ≫ T.map η = f.hom ≫ T.map η'\n⊢ h.desc (mk (f.hom ≫ T.map η)) = η'",
"usedConstants": [
"Eq.mpr",
"Cate... | h.hom_desc η', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Comma.StructuredArrow.Basic | {
"line": 731,
"column": 20
} | {
"line": 731,
"column": 34
} | [
{
"pp": "C : Type u₁\ninst✝¹ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝ : Category.{v₂, u₂} D\nT : D\nS : C ⥤ D\nf : CostructuredArrow S T\nh : f.IsUniversal\nc : C\nη η' : c ⟶ f.left\nw : S.map η ≫ f.hom = S.map η' ≫ f.hom\n⊢ h.lift (mk (S.map η ≫ f.hom)) = η'",
"usedConstants": [
"Eq.mpr",
"Cat... | h.hom_desc η', | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.CategoryTheory.Limits.Shapes.BinaryProducts | {
"line": 397,
"column": 2
} | {
"line": 400,
"column": 68
} | [
{
"pp": "case mpr\nC : Type u\ninst✝ : Category.{v, u} C\nX Y : C\nh : IsTerminal Y\nc : BinaryFan X Y\n⊢ IsIso c.fst → Nonempty (IsLimit c)",
"usedConstants": [
"CategoryTheory.Category.assoc",
"CategoryTheory.Functor",
"CategoryTheory.Limits.BinaryFan.fst",
"CategoryTheory.IsIso",
... | · intro
exact
⟨BinaryFan.IsLimit.mk _ (fun f _ => f ≫ inv c.fst) (fun _ _ => by simp)
(fun _ _ => h.hom_ext _ _) fun _ _ _ e _ => by simp [← e]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.CategoryTheory.Limits.Shapes.Pullback.PullbackCone | {
"line": 335,
"column": 23
} | {
"line": 336,
"column": 84
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nW✝ X Y Z : C\nf : X ⟶ Y\ng : X ⟶ Z\nW : C\ninl : Y ⟶ W\ninr : Z ⟶ W\neq : f ≫ inl = g ≫ inr\n⊢ ∀ ⦃X_1 Y_1 : WalkingSpan⦄ (f_1 : X_1 ⟶ Y_1),\n ((span f g).map f_1 ≫ Option.casesOn Y_1 (f ≫ inl) fun j' ↦ WalkingPair.casesOn j' inl inr) =\n (Option.casesOn X_... | by
rintro (⟨⟩ | ⟨⟨⟩⟩) (⟨⟩ | ⟨⟨⟩⟩) <;> intro f <;> cases f <;> dsimp <;> aesop | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.MorphismProperty.Factorization | {
"line": 97,
"column": 2
} | {
"line": 105,
"column": 37
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nW₁ W₂ : MorphismProperty C\n⊢ W₁.comp W₂ = ⊤ ↔ W₁.HasFactorization W₂",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.MorphismProperty... | constructor
· intro h
refine ⟨fun f => ?_⟩
have : W₁.comp W₂ f := by simp only [h, top_apply]
exact ⟨this.some⟩
· intro
ext X Y f
simp only [top_apply, iff_true]
exact ⟨factorizationData W₁ W₂ f⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.CategoryTheory.MorphismProperty.Factorization | {
"line": 97,
"column": 2
} | {
"line": 105,
"column": 37
} | [
{
"pp": "C : Type u_1\ninst✝ : Category.{v_1, u_1} C\nW₁ W₂ : MorphismProperty C\n⊢ W₁.comp W₂ = ⊤ ↔ W₁.HasFactorization W₂",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.MorphismProperty",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"CategoryTheory.MorphismProperty... | constructor
· intro h
refine ⟨fun f => ?_⟩
have : W₁.comp W₂ f := by simp only [h, top_apply]
exact ⟨this.some⟩
· intro
ext X Y f
simp only [top_apply, iff_true]
exact ⟨factorizationData W₁ W₂ f⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.CategoryTheory.Limits.Shapes.Equalizers | {
"line": 154,
"column": 6
} | {
"line": 154,
"column": 28
} | [
{
"pp": "i j : WalkingParallelPairᵒᵖ\nf : i ⟶ j\n⊢ (walkingParallelPairOp.leftOp ⋙ walkingParallelPairOp).map f ≫ ((fun j ↦ eqToIso ⋯) j).hom =\n ((fun j ↦ eqToIso ⋯) i).hom ≫ (𝟭 WalkingParallelPairᵒᵖ).map f",
"usedConstants": [
"Opposite",
"CategoryTheory.CategoryStruct.toQuiver",
"Qu... | induction i with | _ i | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.CategoryTheory.Limits.Shapes.Images | {
"line": 773,
"column": 56
} | {
"line": 774,
"column": 49
} | [
{
"pp": "C : Type u\ninst✝² : Category.{v, u} C\nf g : Arrow C\ninst✝¹ : HasImage f.hom\ninst✝ : HasImage g.hom\nsq : f ⟶ g\nF G : ImageMap sq\n⊢ F.map = G.map",
"usedConstants": [
"CategoryTheory.Limits.ImageMap.map_uniq_aux",
"CategoryTheory.Limits.ImageMap.map_ι",
"CategoryTheory.Limits... | by
apply ImageMap.map_uniq_aux _ F.map_ι _ G.map_ι | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 81,
"column": 2
} | {
"line": 84,
"column": 28
} | [
{
"pp": "C : Type u\ninst✝ : Category.{v, u} C\nI J : HasZeroMorphisms C\nw : ∀ (X Y : C), Zero.zero = Zero.zero\n⊢ I = J",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
"congrArg",
"Zero.zero",
"CategoryTheory.Limits.HasZeroMorphisms.zero",
"... | have : I.zero = J.zero := by
funext X Y
specialize w X Y
apply congrArg Zero.mk w | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.CategoryTheory.Limits.Preserves.Shapes.Zero | {
"line": 58,
"column": 24
} | {
"line": 60,
"column": 33
} | [
{
"pp": "C : Type u₁\ninst✝⁴ : Category.{v₁, u₁} C\nD : Type u₂\ninst✝³ : Category.{v₂, u₂} D\ninst✝² : HasZeroMorphisms C\ninst✝¹ : HasZeroMorphisms D\nF : C ⥤ D\ninst✝ : F.PreservesZeroMorphisms\nX : C\nhX : IsZero X\n⊢ IsZero (F.obj X)",
"usedConstants": [
"Eq.mpr",
"CategoryTheory.CategorySt... | by
simp only [IsZero.iff_id_eq_zero] at hX ⊢
rw [← F.map_id, hX, F.map_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 449,
"column": 2
} | {
"line": 449,
"column": 38
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nX Y : C\n⊢ IsIso 0 ≃ (X ≅ 0) × (Y ≅ 0)",
"usedConstants": [
"CategoryTheory.IsIso",
"CategoryTheory.CategoryStruct.toQuiver",
"Quiver.Hom",
... | refine (isIsoZeroEquiv X Y).trans ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.CategoryTheory.Limits.Shapes.ZeroMorphisms | {
"line": 458,
"column": 6
} | {
"line": 458,
"column": 37
} | [
{
"pp": "case invFun.fst\nC : Type u\ninst✝³ : Category.{v, u} C\nD : Type u'\ninst✝² : Category.{v', u'} D\ninst✝¹ : HasZeroMorphisms C\ninst✝ : HasZeroObject C\nX Y : C\nhX : 𝟙 X = 0\nhY : 𝟙 Y = 0\n⊢ X ≅ 0",
"usedConstants": [
"Equiv.instEquivLike",
"CategoryTheory.CategoryStruct.toQuiver",
... | exact (idZeroEquivIsoZero X) hX | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
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