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.Category.Profinite.Nobeling.Basic | {
"line": 425,
"column": 6
} | {
"line": 425,
"column": 14
} | [
{
"pp": "case h\nI : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\nl : Products I\nJ : I → Prop\ninst✝ : (j : I) → Decidable (J j)\nh : isGood (π C J) l\ni : I\nhi : i ∈ ↑l\nh' : ¬J i\nw✝ : I → Bool\nleft✝ : w✝ ∈ C\n⊢ (eval (π C J) l) ⟨Proj J w✝, ⋯⟩ = 0 ⟨Proj J w✝, ⋯⟩",
"usedConstants": [
"Eq.mp... | eval_eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Category.Profinite.Nobeling.Successor | {
"line": 196,
"column": 2
} | {
"line": 196,
"column": 73
} | [
{
"pp": "case h\nI : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\ny : LocallyConstant ↑(π C fun x ↦ ord I x < o) ℤ\nx : ↑(C' C ho)\n⊢ ((Linear_CC' C hsC ho) ((πs C o) y)) x = 0 x",
"... | dsimp [Linear_CC', Linear_CC'₀, Linear_CC'₁, LocallyConstant.sub_apply] | Lean.Elab.Tactic.evalDSimp | Lean.Parser.Tactic.dsimp |
Mathlib.Topology.Category.Profinite.Nobeling.Successor | {
"line": 208,
"column": 2
} | {
"line": 218,
"column": 19
} | [
{
"pp": "I : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\nx : I → Bool\nhx : x ∈ C0 C ho\n⊢ Proj (fun x ↦ ord I x < o) x = x",
"usedConstants": [
"Eq.mpr",
"Ordinal.instL... | ext i
simp only [Proj, ite_eq_left_iff, not_lt]
intro hi
rcases hi.lt_or_eq with hi | hi
· specialize hsC x hx.1 i
rw [← not_imp_not] at hsC
simp only [not_lt, Bool.not_eq_true, Order.succ_le_iff] at hsC
exact (hsC hi).symm
· simp only [C0, Set.mem_inter_iff, Set.mem_setOf_eq] at hx
rw [eq_com... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Category.Profinite.Nobeling.Successor | {
"line": 208,
"column": 2
} | {
"line": 218,
"column": 19
} | [
{
"pp": "I : Type u\nC : Set (I → Bool)\ninst✝¹ : LinearOrder I\ninst✝ : WellFoundedLT I\no : Ordinal.{u}\nhsC : contained C (Order.succ o)\nho : o < Ordinal.type fun x1 x2 ↦ x1 < x2\nx : I → Bool\nhx : x ∈ C0 C ho\n⊢ Proj (fun x ↦ ord I x < o) x = x",
"usedConstants": [
"Eq.mpr",
"Ordinal.instL... | ext i
simp only [Proj, ite_eq_left_iff, not_lt]
intro hi
rcases hi.lt_or_eq with hi | hi
· specialize hsC x hx.1 i
rw [← not_imp_not] at hsC
simp only [not_lt, Bool.not_eq_true, Order.succ_le_iff] at hsC
exact (hsC hi).symm
· simp only [C0, Set.mem_inter_iff, Set.mem_setOf_eq] at hx
rw [eq_com... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Filter | {
"line": 85,
"column": 47
} | {
"line": 87,
"column": 38
} | [
{
"pp": "ι : Sort u_1\nα : Type u_2\nl : Filter α\np : ι → Prop\ns : ι → Set α\nh : l.HasBasis p s\n⊢ (𝓝 l).HasBasis p fun i ↦ Iic (𝓟 (s i))",
"usedConstants": [
"Eq.mpr",
"congrArg",
"PartialOrder.toPreorder",
"Function.comp",
"nhds",
"id",
"Filter.nhds_eq",
... | by
rw [nhds_eq]
exact h.lift' monotone_principal.Iic | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 522,
"column": 6
} | {
"line": 524,
"column": 76
} | [
{
"pp": "case mpr.refine_4\nN✝ : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace X\nx : X\ninst✝¹ : DecidableEq N✝\nN : Type ?u.73244\ninst✝ : Unique N\na₁ a₂ : ↑(Ω^ N X x)\nH : ((genLoopEquivOfUnique N) a₁).Homotopy ((genLoopEquivOfUnique N) a₂)\n⊢ ∀ (t : ↑I),\n ∀ x_1 ∈ Cube.boundary N,\n {\n ... | rintro t y ⟨i, iH⟩
cases Unique.eq_default i
exact (H.eq_fst _ iH).trans (congr_arg a₁ (eq_const_of_unique y).symm) | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Homotopy.HomotopyGroup | {
"line": 522,
"column": 6
} | {
"line": 524,
"column": 76
} | [
{
"pp": "case mpr.refine_4\nN✝ : Type u_1\nX : Type u_2\ninst✝² : TopologicalSpace X\nx : X\ninst✝¹ : DecidableEq N✝\nN : Type ?u.73244\ninst✝ : Unique N\na₁ a₂ : ↑(Ω^ N X x)\nH : ((genLoopEquivOfUnique N) a₁).Homotopy ((genLoopEquivOfUnique N) a₂)\n⊢ ∀ (t : ↑I),\n ∀ x_1 ∈ Cube.boundary N,\n {\n ... | rintro t y ⟨i, iH⟩
cases Unique.eq_default i
exact (H.eq_fst _ iH).trans (congr_arg a₁ (eq_const_of_unique y).symm) | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.CantorSet | {
"line": 213,
"column": 4
} | {
"line": 214,
"column": 9
} | [
{
"pp": "x : ℝ\nhx : x ∈ cantorSet\n⊢ ∃ y ∈ cantorSet, y / 3 = x ∨ (2 + y) / 3 = x",
"usedConstants": [
"_private.Mathlib.Topology.Instances.CantorSet.0.cantorStep_mem_cantorSet._proof_1_2",
"Real",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddO... | rw [cantorSet_eq_union_halves] at hx
grind | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Instances.CantorSet | {
"line": 213,
"column": 4
} | {
"line": 214,
"column": 9
} | [
{
"pp": "x : ℝ\nhx : x ∈ cantorSet\n⊢ ∃ y ∈ cantorSet, y / 3 = x ∨ (2 + y) / 3 = x",
"usedConstants": [
"_private.Mathlib.Topology.Instances.CantorSet.0.cantorStep_mem_cantorSet._proof_1_2",
"Real",
"instHDiv",
"congrArg",
"Real.instDivInvMonoid",
"Nat.instAtLeastTwoHAddO... | rw [cantorSet_eq_union_halves] at hx
grind | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Instances.CantorSet | {
"line": 329,
"column": 15
} | {
"line": 340,
"column": 44
} | [
{
"pp": "⊢ Function.RightInverse (fun y ↦ ⟨ofDigits fun i ↦ bif y i then 2 else 0, ⋯⟩) fun x ↦\n match (motive := ↑cantorSet → ℕ → Bool) x with\n | ⟨x, h⟩ => (cantorToBinary x).get",
"usedConstants": [
"cond",
"Eq.mpr",
"Real",
"cantorSetEquivNatToBool.match_1",
"ofDigits... | by
intro y
simp only [Fin.isValue]
set x := @ofDigits 3 (fun i ↦ cond (y i) 2 0)
have := ofDigits_cantorToTernary (ofDigits_bool_to_fin_three_mem_cantorSet y)
apply ofDigits_zero_two_sequence_unique at this
rotate_left
· exact fun n ↦ cantorToTernary_ne_one
· grind
ext n
apply co... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.List | {
"line": 89,
"column": 4
} | {
"line": 90,
"column": 42
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\nβ : Type u_3\nf : List α → β\nb : Filter β\na : α\nl : List α\n⊢ 𝓝 (a :: l) = Filter.map (fun p ↦ p.1 :: p.2) (𝓝 a ×ˢ 𝓝 l)",
"usedConstants": [
"Filter.prod_eq",
"Eq.mpr",
"nhds_cons",
"SProd.sprod",
"congrArg",
"Filte... | simp only [nhds_cons, Filter.prod_eq, (Filter.map_def _ _).symm,
(Filter.seq_eq_filter_seq _ _).symm] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.MetricSpace.Closeds | {
"line": 106,
"column": 6
} | {
"line": 106,
"column": 41
} | [
{
"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 ↑t ↑s = infEDist y ↑t + (edist x y + hausdorffEDist ↑s ↑t)",
"usedConstants": [
"Eq.mpr",
"AddMonoid.toAddSemigroup",
"congrArg",
"CommSemiring.toSe... | rw [add_assoc, hausdorffEDist_comm] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.MetricSpace.Closeds | {
"line": 106,
"column": 6
} | {
"line": 106,
"column": 41
} | [
{
"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 ↑t ↑s = infEDist y ↑t + (edist x y + hausdorffEDist ↑s ↑t)",
"usedConstants": [
"Eq.mpr",
"AddMonoid.toAddSemigroup",
"congrArg",
"CommSemiring.toSe... | rw [add_assoc, hausdorffEDist_comm] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.Closeds | {
"line": 106,
"column": 6
} | {
"line": 106,
"column": 41
} | [
{
"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 ↑t ↑s = infEDist y ↑t + (edist x y + hausdorffEDist ↑s ↑t)",
"usedConstants": [
"Eq.mpr",
"AddMonoid.toAddSemigroup",
"congrArg",
"CommSemiring.toSe... | rw [add_assoc, hausdorffEDist_comm] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sets.VietorisTopology | {
"line": 290,
"column": 2
} | {
"line": 290,
"column": 9
} | [
{
"pp": "α : Type u_1\ninst✝ : TopologicalSpace α\nS : Set (Compacts α)\nhS : IsCompact S\n⊢ ∀ {ι : Type u_1} (U : ι → Set α), (∀ (i : ι), IsOpen (U i)) → ⋃ K ∈ S, ↑K ⊆ ⋃ i, U i → ∃ t, ⋃ K ∈ S, ↑K ⊆ ⋃ i ∈ t, U i",
"usedConstants": []
}
] | intro ι | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Topology.UniformSpace.Closeds | {
"line": 717,
"column": 2
} | {
"line": 717,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝ : UniformSpace α\nx✝ : CompleteSpace (NonemptyCompacts α)\nf : Filter α\nhf : Cauchy f\nK : NonemptyCompacts α\nhK : Filter.map (fun x ↦ {x}) f ≤ 𝓝 K\nx : α\nhx : x ∈ ↑K\n⊢ ∃ x, f ≤ 𝓝 x",
"usedConstants": [
"PartialOrder.toPreorder",
"Preorder.toLE",
"nhds",
... | exists x | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticExists_,,_1» | Lean.Parser.Tactic.«tacticExists_,,» |
Mathlib.Topology.NatEmbedding | {
"line": 32,
"column": 31
} | {
"line": 40,
"column": 16
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : T2Space X\ninst✝ : Infinite X\nthis : ∃ U, (∀ (n : ℕ), (U n).Nonempty) ∧ (∀ (n : ℕ), IsOpen (U n)) ∧ Pairwise (Disjoint on U)\n⊢ ∃ U, (∀ (n : ℕ), (U n).Infinite) ∧ (∀ (n : ℕ), IsOpen (U n)) ∧ Pairwise (Disjoint on U)",
"usedConstants": [
"If... | by
rcases this with ⟨U, hne, ho, hd⟩
refine ⟨fun n ↦ ⋃ m, U (.pair n m), ?_, fun _ ↦ isOpen_iUnion fun _ ↦ ho _, ?_⟩
· refine fun n ↦ infinite_iUnion fun i j hij ↦ ?_
suffices n.pair i = n.pair j by simpa
apply hd.eq
simpa [hij, onFun] using (hne _).ne_empty
· refine fun n n' hne ↦ dis... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.MetricSpace.GromovHausdorff | {
"line": 400,
"column": 21
} | {
"line": 400,
"column": 74
} | [
{
"pp": "x : GHSpace\ny : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)\nhy : ⟦y⟧ = x\n⊢ 0 ∈ lowerBounds ((fun p ↦ hausdorffDist ↑p.1 ↑p.2) '' {a | ⟦a⟧ = x} ×ˢ {b | ⟦b⟧ = x})",
"usedConstants": [
"GromovHausdorff.IsometryRel.setoid._proof_1",
"Set.instSProd",
"TopologicalSpace.NonemptyCompacts.inst... | rintro b ⟨⟨u, v⟩, -, rfl⟩; exact hausdorffDist_nonneg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.GromovHausdorff | {
"line": 400,
"column": 21
} | {
"line": 400,
"column": 74
} | [
{
"pp": "x : GHSpace\ny : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)\nhy : ⟦y⟧ = x\n⊢ 0 ∈ lowerBounds ((fun p ↦ hausdorffDist ↑p.1 ↑p.2) '' {a | ⟦a⟧ = x} ×ˢ {b | ⟦b⟧ = x})",
"usedConstants": [
"GromovHausdorff.IsometryRel.setoid._proof_1",
"Set.instSProd",
"TopologicalSpace.NonemptyCompacts.inst... | rintro b ⟨⟨u, v⟩, -, rfl⟩; exact hausdorffDist_nonneg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.GromovHausdorff | {
"line": 406,
"column": 8
} | {
"line": 406,
"column": 61
} | [
{
"pp": "case refine_2.h₂\nx : GHSpace\ny : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)\nhy : ⟦y⟧ = x\n⊢ ∀ b ∈ (fun p ↦ hausdorffDist ↑p.1 ↑p.2) '' {a | ⟦a⟧ = x} ×ˢ {b | ⟦b⟧ = x}, 0 ≤ b",
"usedConstants": [
"GromovHausdorff.IsometryRel.setoid._proof_1",
"Set.instSProd",
"TopologicalSpace.Nonempty... | rintro b ⟨⟨u, v⟩, -, rfl⟩; exact hausdorffDist_nonneg | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.MetricSpace.GromovHausdorff | {
"line": 406,
"column": 8
} | {
"line": 406,
"column": 61
} | [
{
"pp": "case refine_2.h₂\nx : GHSpace\ny : NonemptyCompacts ↥(lp (fun n ↦ ℝ) ∞)\nhy : ⟦y⟧ = x\n⊢ ∀ b ∈ (fun p ↦ hausdorffDist ↑p.1 ↑p.2) '' {a | ⟦a⟧ = x} ×ˢ {b | ⟦b⟧ = x}, 0 ≤ b",
"usedConstants": [
"GromovHausdorff.IsometryRel.setoid._proof_1",
"Set.instSProd",
"TopologicalSpace.Nonempty... | rintro b ⟨⟨u, v⟩, -, rfl⟩; exact hausdorffDist_nonneg | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.MetricSpace.GromovHausdorff | {
"line": 649,
"column": 4
} | {
"line": 649,
"column": 49
} | [
{
"pp": "δ : ℝ\nδpos : δ > 0\nε : ℝ := 2 / 5 * δ\nεpos : 0 < ε\ns : (p : GHSpace) → Set p.Rep\nhs : ∀ (p : GHSpace), (s p).Finite ∧ univ ⊆ ⋃ x ∈ s p, ball x ε\nN : GHSpace → ℕ := fun p ↦ Nat.card ↑(s p)\nE : (p : GHSpace) → ↑(s p) ≃ Fin (Nat.card ↑(s p)) := fun p ↦ Finite.equivFin ↑(s p)\nF : GHSpace → (n : ℕ) ... | refine ghDist_le_of_approx_subsets Φ ?_ ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Order.HullKernel | {
"line": 120,
"column": 8
} | {
"line": 120,
"column": 70
} | [
{
"pp": "case h\nα : Type u_1\ninst✝³ : SemilatticeInf α\nT : Set α\ninst✝² : OrderTop α\ninst✝¹ : TopologicalSpace α\ninst✝ : IsLower α\nhT : ∀ p ∈ T, InfPrime p\nR : Set ↑T\na : α\nha' : (hull T a)ᶜ = R\n⊢ {a}.Finite ∧ (Subtype.val ⁻¹' ↑(upperClosure {a}))ᶜ = R",
"usedConstants": [
"Eq.mpr",
"... | ← (Function.Injective.preimage_image Subtype.val_injective R), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.MetricSpace.GromovHausdorffRealized | {
"line": 531,
"column": 2
} | {
"line": 533,
"column": 33
} | [
{
"pp": "case refine_1\nX : Type u\nY : Type v\ninst✝⁵ : MetricSpace X\ninst✝⁴ : CompactSpace X\ninst✝³ : Nonempty X\ninst✝² : MetricSpace Y\ninst✝¹ : CompactSpace Y\ninst✝ : Nonempty Y\nf : Cb X Y\nh : f ∈ candidatesB X Y\nr : ℝ\nhr : HD (optimalGHDist X Y) < r\nA : ∀ x ∈ range (optimalGHInjl X Y), ∃ y ∈ range... | · inhabit X
rcases A _ (mem_range_self default) with ⟨y, -, hy⟩
exact le_trans dist_nonneg hy | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.MetricSpace.GromovHausdorff | {
"line": 783,
"column": 4
} | {
"line": 783,
"column": 49
} | [
{
"pp": "t : Set GHSpace\nC : ℝ\nu : ℕ → ℝ\nK : ℕ → ℕ\nulim : Tendsto u atTop (𝓝 0)\nhdiam : ∀ p ∈ t, diam univ ≤ C\nhcov : ∀ p ∈ t, ∀ (n : ℕ), ∃ s, #↑s ≤ ↑(K n) ∧ univ ⊆ ⋃ x ∈ s, ball x (u n)\nδ : ℝ\nδpos : δ > 0\nε : ℝ := 1 / 5 * δ\nεpos : 0 < ε\nn : ℕ\nhn : ∀ n_1 ≥ n, dist (u n_1) 0 < ε\nu_le_ε : u n ≤ ε\ns... | refine ghDist_le_of_approx_subsets Φ ?_ ?_ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.Separation.DisjointCover | {
"line": 108,
"column": 62
} | {
"line": 113,
"column": 62
} | [
{
"pp": "X : Type u_1\ninst✝³ : TopologicalSpace X\nS : Set (X × X)\ninst✝² : CompactSpace X\ninst✝¹ : TotallyDisconnectedSpace X\ninst✝ : T2Space X\nhS : S ∈ 𝓝ˢ (diagonal X)\n⊢ ∃ n D,\n (∀ (i : Fin n), D i ≠ ⊥) ∧\n (∀ (i : Fin n), ∀ y ∈ D i, ∀ z ∈ D i, (y, z) ∈ S) ∧ univ ⊆ ⋃ i, ↑(D i) ∧ Pairwise (Disj... | by
obtain ⟨t, U, hUc, hUS⟩ := exists_finite_open_cover_prod_subset_of_mem_nhds_diagonal_of_compact hS
-- Now refine it to a disjoint covering.
obtain ⟨n, W, hW₁, hW₂, hW₃⟩ := hUc.exists_finite_nonempty_disjoint_clopen_cover
refine ⟨n, W, fun j ↦ (hW₁ j).1, fun j y hy z hz ↦ ?_, hW₂, hW₃⟩
exact match (hW₁ j).2... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Sheaves.Alexandrov | {
"line": 158,
"column": 4
} | {
"line": 159,
"column": 45
} | [
{
"pp": "case w\nX✝ : Type v\ninst✝⁶ : TopologicalSpace X✝\ninst✝⁵ : Preorder X✝\ninst✝⁴ : Topology.IsUpperSet X✝\nC : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasLimits C\nF✝ : X✝ ⥤ C\nX : TopCat\ninst✝¹ : Preorder ↑X\ninst✝ : Topology.IsUpperSet ↑X\nF : ↑X ⥤ C\nα : Type v\nUs : α → Opens ↑X\nS : Cone ((Ob... | simp only [lowerCone_pt, comp_obj, limit.lift_π, lowerCone_π_app, const_obj_obj, projSup_obj,
op_obj, pointwiseRightKanExtension_obj] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.Sheaves.Skyscraper | {
"line": 63,
"column": 64
} | {
"line": 63,
"column": 90
} | [
{
"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 : C\nU V : (Opens ↑X)ᵒᵖ\ni : U ⟶ V\nh : p₀ ∈ unop V\n⊢ p₀ ∈ unop U",
"usedConstants": [
"CategoryTheory.CategoryStruct.toQuiver",
"TopologicalSpace.Opens.i... | by simpa using i.unop.le h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Sheaves.Alexandrov | {
"line": 172,
"column": 2
} | {
"line": 172,
"column": 9
} | [
{
"pp": "C : Type u\ninst✝³ : Category.{v, u} C\ninst✝² : HasLimits C\nX : TopCat\ninst✝¹ : Preorder ↑X\ninst✝ : Topology.IsUpperSet ↑X\nF : ↑X ⥤ C\n⊢ IsSheafOpensLeCover (principalsKanExtension F)",
"usedConstants": []
}
] | intro ι | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Topology.Sheaves.Alexandrov | {
"line": 198,
"column": 2
} | {
"line": 198,
"column": 27
} | [
{
"pp": "X : Type v\ninst✝⁵ : TopologicalSpace X\ninst✝⁴ : Preorder X\ninst✝³ : Topology.IsUpperSet X\nC : Type u\ninst✝² : Category.{v, u} C\ninst✝¹ : HasLimits C\nF : X ⥤ C\nP : (Opens X)ᵒᵖ ⥤ C\nη : principals X ⋙ P ⟶ F\ninst✝ : P.IsRightKanExtension η\nγ : principals X ⋙ principalsKanExtension F ⟶ F := (prin... | rw [isSheaf_iso_iff this] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.Sheaves.Skyscraper | {
"line": 372,
"column": 6
} | {
"line": 373,
"column": 9
} | [
{
"pp": "case pos\nX : TopCat\np₀ : ↑X\ninst✝³ : (U : Opens ↑X) → Decidable (p₀ ∈ U)\nC : Type v\ninst✝² : Category.{u, v} C\nA : C\ninst✝¹ : HasTerminal C\ninst✝ : HasColimits C\nY : C\nU✝ : Opens ↑X\nh : p₀ ∈ U✝\n⊢ ((skyscraperPresheaf p₀ Y).germ U✝ p₀ ⋯ ≫ eqToHom ⋯) ≫\n eqToHom ⋯ ≫\n (colimit.iso... | simp [Presheaf.germ]
rfl | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.Sheaves.Skyscraper | {
"line": 372,
"column": 6
} | {
"line": 373,
"column": 9
} | [
{
"pp": "case pos\nX : TopCat\np₀ : ↑X\ninst✝³ : (U : Opens ↑X) → Decidable (p₀ ∈ U)\nC : Type v\ninst✝² : Category.{u, v} C\nA : C\ninst✝¹ : HasTerminal C\ninst✝ : HasColimits C\nY : C\nU✝ : Opens ↑X\nh : p₀ ∈ U✝\n⊢ ((skyscraperPresheaf p₀ Y).germ U✝ p₀ ⋯ ≫ eqToHom ⋯) ≫\n eqToHom ⋯ ≫\n (colimit.iso... | simp [Presheaf.germ]
rfl | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.Sheaves.Abelian | {
"line": 94,
"column": 2
} | {
"line": 97,
"column": 74
} | [
{
"pp": "C : Type v\ninst✝⁶ : Category.{u, v} C\ninst✝⁵ : HasColimits C\ninst✝⁴ : HasLimits C\nFC : C → C → Type u_1\nCC : C → Type u\ninst✝³ : (X Y : C) → FunLike (FC X Y) (CC X) (CC Y)\ninstCC : ConcreteCategory C FC\ninst✝² : PreservesFilteredColimits (CategoryTheory.forget C)\ninst✝¹ : PreservesLimits (Cate... | have : IsIso f := by
rw[Presheaf.isIso_iff_stalkFunctor_map_iso]
exact fun x => isIso_of_source_target_iso_zero _ (h x).isoZero
((forget C X ⋙ stalkFunctor C x).map_isZero (isZero_zero _)).isoZero | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.Spectral.ConstructibleTopology | {
"line": 120,
"column": 4
} | {
"line": 120,
"column": 48
} | [
{
"pp": "case pos\nX : Type u_1\ninst✝⁴ : TopologicalSpace X\ninst✝³ : CompactSpace X\ninst✝² : QuasiSober X\ninst✝¹ : PrespectralSpace X\ninst✝ : QuasiSeparatedSpace X\n𝔅 : Set (Set X) := constructibleTopologySubbasis X\n𝒮 : Set (Set (Set X)) := {P | P ⊆ 𝔅 ∧ (∀ Q ⊆ P, Q.Finite → (⋂₀ Q).Nonempty) ∧ ⋂₀ P = ∅}... | · exact Set.sInter_subset_of_mem hiB' hη.mem | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Topology.UniformSpace.Dini | {
"line": 79,
"column": 6
} | {
"line": 79,
"column": 68
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nG : Type u_3\ninst✝⁵ : Preorder ι\ninst✝⁴ : TopologicalSpace α\ninst✝³ : NormedAddCommGroup G\ninst✝² : Lattice G\ninst✝¹ : HasSolidNorm G\ninst✝ : IsOrderedAddMonoid G\nF : ι → α → G\nf : α → G\ns : Set α\nhF_cont : ∀ (i : ι), ContinuousOn (F i) s\nhF_mono : ∀ x ∈ s, Monoto... | tendstoLocallyUniformlyOn_iff_tendstoLocallyUniformly_comp_coe | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.Sheaves.Flasque | {
"line": 165,
"column": 10
} | {
"line": 167,
"column": 15
} | [
{
"pp": "X : TopCat\nU : Opens ↑X\nS : ShortComplex (Sheaf AddCommGrpCat X)\nhS : S.ShortExact\ninst✝ : S.X₁.IsFlasque\ns : ↑(S.X₃.obj.obj (op U))\nt : Under S.g s\nht : ∀ (a : Under S.g s), Nonempty (a ⟶ t) → Nonempty (t ⟶ a)\ntle : unop t.right.fst ≤ U\ntcomp : (s |_ unop t.right.fst) tle = (ConcreteCategory.... | have : (S.f.hom.app (op W) ≫ S.g.hom.app (op W)) = 0 := by
rw [← NatTrans.comp_app, ← ObjectProperty.FullSubcategory.comp_hom, S.zero]
rfl | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Topology.UniformSpace.OfCompactT2 | {
"line": 91,
"column": 8
} | {
"line": 91,
"column": 17
} | [
{
"pp": "case refine_2.refl\nγ : Type u_1\ninst✝² : TopologicalSpace γ\ninst✝¹ : CompactSpace γ\ninst✝ : R1Space γ\n𝓝Δ : Filter (γ × γ) := 𝓝ˢ (diagonal γ)\nF : Filter (γ × γ) := 𝓝Δ.lift' fun s ↦ s ○ s\nV : Set (γ × γ)\nV_in : V ∈ 𝓝Δ\nH : ¬F ⊓ 𝓟 Vᶜ = ⊥\nthis : (F ⊓ 𝓟 Vᶜ).NeBot\nx y : γ\nhxy : ClusterPt (x,... | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Topology.UniformSpace.OfCompactT2 | {
"line": 101,
"column": 4
} | {
"line": 101,
"column": 13
} | [
{
"pp": "γ : Type u_1\ninst✝² : TopologicalSpace γ\ninst✝¹ : CompactSpace γ\ninst✝ : R1Space γ\n𝓝Δ : Filter (γ × γ) := 𝓝ˢ (diagonal γ)\nF : Filter (γ × γ) := 𝓝Δ.lift' fun s ↦ s ○ s\nV : Set (γ × γ)\nV_in : V ∈ 𝓝Δ\nH : ¬F ⊓ 𝓟 Vᶜ = ⊥\nthis✝ : (F ⊓ 𝓟 Vᶜ).NeBot\nx y : γ\nhxy : ClusterPt (x, y) (F ⊓ 𝓟 Vᶜ)\ncl... | tauto_set | Mathlib.Tactic.TautoSet._aux_Mathlib_Tactic_TautoSet___macroRules_Mathlib_Tactic_TautoSet_tacticTauto_set_1 | Mathlib.Tactic.TautoSet.tacticTauto_set |
Mathlib.Logic.Function.Basic | {
"line": 503,
"column": 2
} | {
"line": 503,
"column": 64
} | [
{
"pp": "α : Sort u\nβ : Sort v\nγ : Sort w\ninst✝ : Nonempty α\ng : β → γ\n⊢ (Surjective fun x ↦ g ∘ x) ↔ Surjective g",
"usedConstants": [
"Function.Surjective.comp_left",
"Nonempty.elim",
"Function.comp",
"Exists",
"Iff.intro",
"Eq",
"Function.Surjective"
]
... | refine ⟨fun h c ↦ Nonempty.elim ‹_› fun a ↦ ?_, (·.comp_left)⟩ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Algebra.Notation.Pi.Basic | {
"line": 49,
"column": 58
} | {
"line": 51,
"column": 44
} | [
{
"pp": "ι : Type u_1\nM : ι → Type u_6\ninst✝¹ : (i : ι) → One (M i)\ninst✝ : DecidableEq ι\ni : ι\nx : M i\n⊢ mulSingle i x = 1 ↔ x = 1",
"usedConstants": [
"Eq.mpr",
"Pi.mulSingle_eq_same",
"congrArg",
"Pi.one_apply",
"Pi.mulSingle_one",
"Eq.rec",
"id",
"Pi... | by
refine ⟨fun h => ?_, fun h => h.symm ▸ mulSingle_one i⟩
rw [← mulSingle_eq_same i x, h, one_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Option.Basic | {
"line": 112,
"column": 62
} | {
"line": 112,
"column": 88
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nf₁ : α → β\nf₂ : α → γ\ng₁ : β → δ\ng₂ : γ → δ\nh : g₁ ∘ f₁ = g₂ ∘ f₂\na : α\n⊢ Option.map g₁ (Option.map f₁ (some a)) = Option.map g₂ (Option.map f₂ (some a))",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Option.some",
"... | rw [map_map, h, ← map_map] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Option.Basic | {
"line": 112,
"column": 62
} | {
"line": 112,
"column": 88
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nf₁ : α → β\nf₂ : α → γ\ng₁ : β → δ\ng₂ : γ → δ\nh : g₁ ∘ f₁ = g₂ ∘ f₂\na : α\n⊢ Option.map g₁ (Option.map f₁ (some a)) = Option.map g₂ (Option.map f₂ (some a))",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Option.some",
"... | rw [map_map, h, ← map_map] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Option.Basic | {
"line": 112,
"column": 62
} | {
"line": 112,
"column": 88
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nδ : Type u_4\nf₁ : α → β\nf₂ : α → γ\ng₁ : β → δ\ng₂ : γ → δ\nh : g₁ ∘ f₁ = g₂ ∘ f₂\na : α\n⊢ Option.map g₁ (Option.map f₁ (some a)) = Option.map g₂ (Option.map f₂ (some a))",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Option.some",
"... | rw [map_map, h, ← map_map] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Basic | {
"line": 986,
"column": 4
} | {
"line": 991,
"column": 58
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\nπ : ι → Type u_4\ninst✝³ : Preorder α\ninst✝² : Preorder β\ninst✝¹ : DenselyOrdered α\ninst✝ : DenselyOrdered β\na b : α × β\n⊢ a < b → ∃ a_2, a < a_2 ∧ a_2 < b",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"le_rfl",
"congrArg",
... | simp_rw [Prod.lt_iff]
rintro (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩)
· obtain ⟨c, ha, hb⟩ := exists_between h₁
exact ⟨(c, _), Or.inl ⟨ha, h₂⟩, Or.inl ⟨hb, le_rfl⟩⟩
· obtain ⟨c, ha, hb⟩ := exists_between h₂
exact ⟨(_, c), Or.inr ⟨h₁, ha⟩, Or.inr ⟨le_rfl, hb⟩⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Basic | {
"line": 986,
"column": 4
} | {
"line": 991,
"column": 58
} | [
{
"pp": "ι : Type u_1\nα : Type u_2\nβ : Type u_3\nπ : ι → Type u_4\ninst✝³ : Preorder α\ninst✝² : Preorder β\ninst✝¹ : DenselyOrdered α\ninst✝ : DenselyOrdered β\na b : α × β\n⊢ a < b → ∃ a_2, a < a_2 ∧ a_2 < b",
"usedConstants": [
"Eq.mpr",
"Preorder.toLT",
"le_rfl",
"congrArg",
... | simp_rw [Prod.lt_iff]
rintro (⟨h₁, h₂⟩ | ⟨h₁, h₂⟩)
· obtain ⟨c, ha, hb⟩ := exists_between h₁
exact ⟨(c, _), Or.inl ⟨ha, h₂⟩, Or.inl ⟨hb, le_rfl⟩⟩
· obtain ⟨c, ha, hb⟩ := exists_between h₂
exact ⟨(_, c), Or.inr ⟨h₁, ha⟩, Or.inr ⟨le_rfl, hb⟩⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.RelClasses | {
"line": 462,
"column": 76
} | {
"line": 462,
"column": 81
} | [
{
"pp": "α : Type u\ninst✝ : HasSubset α\na b c : α\nhab : a ⊆ b\nhbc : b = c\n⊢ a ⊆ c",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"HasSubset.Subset",
"Eq.symm",
"Eq"
]
}
] | ← hbc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.RelClasses | {
"line": 515,
"column": 78
} | {
"line": 515,
"column": 83
} | [
{
"pp": "α : Type u\ninst✝ : HasSSubset α\na b c : α\nhab : a ⊂ b\nhbc : b = c\n⊢ a ⊂ c",
"usedConstants": [
"Eq.mpr",
"congrArg",
"HasSSubset.SSubset",
"id",
"Eq.symm",
"Eq"
]
}
] | ← hbc | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.Monotone.Basic | {
"line": 464,
"column": 2
} | {
"line": 464,
"column": 32
} | [
{
"pp": "α : Type u\nβ : Type v\ninst✝¹ : LinearOrder α\ninst✝ : LinearOrder β\nf : α → β\n⊢ ¬Monotone f ∧ ¬Antitone f ↔ ∃ a b c, a ≤ b ∧ b ≤ c ∧ (f a < f b ∧ f c < f b ∨ f b < f a ∧ f b < f c)",
"usedConstants": [
"Preorder.toLT",
"PartialOrder.toPreorder",
"Monotone",
"Preorder.toL... | simp only [Monotone, Antitone] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Data.Set.Subsingleton | {
"line": 133,
"column": 2
} | {
"line": 133,
"column": 25
} | [
{
"pp": "α : Type u\na : α\ns✝ t : Set α\ninst✝ : Subsingleton α\ns : Set α\n⊢ Subsingleton ↑s",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.Elem",
"id",
"Set.Subsingleton",
"propext",
"Set.subsingleton_coe",
"Subsingleton",
"Eq"
]
}
] | rw [s.subsingleton_coe] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Set.Subsingleton | {
"line": 321,
"column": 2
} | {
"line": 323,
"column": 6
} | [
{
"pp": "α : Type u\n⊢ univ = {∅} ↔ IsEmpty α",
"usedConstants": [
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"Membership.mem",
"Eq.mp",
"Set.instSingletonSet",
"id",
"Set.univ_eq_empty_iff._simp_1",
"IsEmpty",
"Set.univ_set_of_isEmpty",
... | refine ⟨fun h ↦ ?_, fun _ ↦ by simp⟩
suffices @univ α ∈ univ by aesop
simp | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Subsingleton | {
"line": 321,
"column": 2
} | {
"line": 323,
"column": 6
} | [
{
"pp": "α : Type u\n⊢ univ = {∅} ↔ IsEmpty α",
"usedConstants": [
"congrArg",
"Set.mem_univ._simp_1",
"Set.univ",
"Membership.mem",
"Eq.mp",
"Set.instSingletonSet",
"id",
"Set.univ_eq_empty_iff._simp_1",
"IsEmpty",
"Set.univ_set_of_isEmpty",
... | refine ⟨fun h ↦ ?_, fun _ ↦ by simp⟩
suffices @univ α ∈ univ by aesop
simp | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Basic | {
"line": 897,
"column": 24
} | {
"line": 897,
"column": 36
} | [
{
"pp": "α : Type u\ns : Set α\np q : α → Prop\n⊢ (∀ (x : α), x ∈ {x | x ∈ s ∧ p x} ↔ x ∈ {x | x ∈ s ∧ q x}) ↔ ∀ (x : α), x ∈ s → (p x ↔ q x)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.mem_sep_iff._simp_1",
"setOf",
"Membership.mem",
"id",
"And",
"Iff",
... | mem_sep_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Set.Basic | {
"line": 910,
"column": 24
} | {
"line": 910,
"column": 36
} | [
{
"pp": "α : Type u\ns : Set α\np : α → Prop\n⊢ (∀ (x : α), x ∈ {x | x ∈ s ∧ p x} ↔ x ∈ s) ↔ ∀ (x : α), x ∈ s → p x",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.mem_sep_iff._simp_1",
"setOf",
"Membership.mem",
"id",
"And",
"Iff",
"congrFun'",
"E... | mem_sep_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Data.Set.Basic | {
"line": 914,
"column": 24
} | {
"line": 914,
"column": 36
} | [
{
"pp": "α : Type u\ns : Set α\np : α → Prop\n⊢ (∀ (x : α), x ∈ {x | x ∈ s ∧ p x} ↔ x ∈ ∅) ↔ ∀ (x : α), x ∈ s → ¬p x",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.mem_sep_iff._simp_1",
"setOf",
"Membership.mem",
"id",
"And",
"Iff",
"Set.instEmptyCollec... | mem_sep_iff, | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.GaloisConnection.Defs | {
"line": 121,
"column": 4
} | {
"line": 121,
"column": 16
} | [
{
"pp": "case mp\nα : Type u\nβ : Type v\ninst✝¹ : PartialOrder α\ninst✝ : Preorder β\nl : α → β\nu : β → α\ngc : GaloisConnection l u\nz : α\ny : β\n⊢ u y = z → ∀ (x : α), x ≤ z ↔ l x ≤ y",
"usedConstants": [
"Eq"
]
}
] | rintro rfl x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRIntro | Lean.Parser.Tactic.rintro |
Mathlib.Data.Set.Insert | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 43
} | [
{
"pp": "α : Type u_1\ns : Set α\na : α\n⊢ s ∩ {a} = ∅ ↔ ¬a ∈ s",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.singleton_inter_eq_empty",
"Iff.rfl",
"Membership.mem",
"Set.instSingletonSet",
"id",
"Set.instInter",
"Inter.inter",
"Set.inter_comm",
... | rw [inter_comm, singleton_inter_eq_empty] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Set.Insert | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 43
} | [
{
"pp": "α : Type u_1\ns : Set α\na : α\n⊢ s ∩ {a} = ∅ ↔ ¬a ∈ s",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.singleton_inter_eq_empty",
"Iff.rfl",
"Membership.mem",
"Set.instSingletonSet",
"id",
"Set.instInter",
"Inter.inter",
"Set.inter_comm",
... | rw [inter_comm, singleton_inter_eq_empty] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Insert | {
"line": 239,
"column": 2
} | {
"line": 239,
"column": 43
} | [
{
"pp": "α : Type u_1\ns : Set α\na : α\n⊢ s ∩ {a} = ∅ ↔ ¬a ∈ s",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Set.singleton_inter_eq_empty",
"Iff.rfl",
"Membership.mem",
"Set.instSingletonSet",
"id",
"Set.instInter",
"Inter.inter",
"Set.inter_comm",
... | rw [inter_comm, singleton_inter_eq_empty] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Heyting.Basic | {
"line": 653,
"column": 2
} | {
"line": 653,
"column": 62
} | [
{
"pp": "α : Type u_2\ninst✝ : HeytingAlgebra α\na b : α\n⊢ a ≤ bᶜ ↔ b ≤ aᶜ",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Iff.rfl",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Disjoint",
"SemilatticeInf.toPartialOrder",
"id",
"LE.le",... | rw [le_compl_iff_disjoint_right, le_compl_iff_disjoint_left] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.Heyting.Basic | {
"line": 653,
"column": 2
} | {
"line": 653,
"column": 62
} | [
{
"pp": "α : Type u_2\ninst✝ : HeytingAlgebra α\na b : α\n⊢ a ≤ bᶜ ↔ b ≤ aᶜ",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Iff.rfl",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Disjoint",
"SemilatticeInf.toPartialOrder",
"id",
"LE.le",... | rw [le_compl_iff_disjoint_right, le_compl_iff_disjoint_left] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.Heyting.Basic | {
"line": 653,
"column": 2
} | {
"line": 653,
"column": 62
} | [
{
"pp": "α : Type u_2\ninst✝ : HeytingAlgebra α\na b : α\n⊢ a ≤ bᶜ ↔ b ≤ aᶜ",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Compl.compl",
"Iff.rfl",
"PartialOrder.toPreorder",
"Preorder.toLE",
"Disjoint",
"SemilatticeInf.toPartialOrder",
"id",
"LE.le",... | rw [le_compl_iff_disjoint_right, le_compl_iff_disjoint_left] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.Heyting.Basic | {
"line": 875,
"column": 43
} | {
"line": 875,
"column": 57
} | [
{
"pp": "α : Type u_2\ninst✝ : CoheytingAlgebra α\na b : α\n⊢ ￢￢￢a ≤ b ↔ ￢a ≤ b",
"usedConstants": [
"CoheytingAlgebra.toHNot",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"LE.le",
"HNot.hnot",
"iff_self",
"If... | hnot_hnot_hnot | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.Heyting.Basic | {
"line": 879,
"column": 42
} | {
"line": 879,
"column": 56
} | [
{
"pp": "α : Type u_2\ninst✝ : CoheytingAlgebra α\na b : α\n⊢ ￢￢￢b ≤ a ↔ ￢b ≤ a",
"usedConstants": [
"CoheytingAlgebra.toHNot",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.toLE",
"SemilatticeInf.toPartialOrder",
"LE.le",
"HNot.hnot",
"iff_self",
"If... | hnot_hnot_hnot | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Order.BooleanAlgebra.Basic | {
"line": 300,
"column": 59
} | {
"line": 300,
"column": 94
} | [
{
"pp": "α : Type u\nx y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ (x \\ y) \\ z = x \\ y ⊓ x \\ z",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"sdiff_sup",
"SemilatticeSup.toMax",
"id",
"SemilatticeInf.toMin",
"GeneralizedBoolean... | by rw [sdiff_sdiff_left, sdiff_sup] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.BooleanAlgebra.Basic | {
"line": 331,
"column": 10
} | {
"line": 331,
"column": 100
} | [
{
"pp": "α : Type u\nx y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ y ⊓ z ⊔ x \\ z ⊓ y \\ z = (x ⊓ y ⊓ (z ⊔ x) ⊔ x \\ z) ⊓ (x ⊓ y ⊓ z ⊔ y \\ z)",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"SemilatticeSup.toMax",
"DistribLattice.toLattice",
... | rw [sup_inf_left, sup_inf_right, sup_sdiff_self_right, inf_sup_right, inf_sdiff_sup_right] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Order.BooleanAlgebra.Basic | {
"line": 331,
"column": 10
} | {
"line": 331,
"column": 100
} | [
{
"pp": "α : Type u\nx y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ y ⊓ z ⊔ x \\ z ⊓ y \\ z = (x ⊓ y ⊓ (z ⊔ x) ⊔ x \\ z) ⊓ (x ⊓ y ⊓ z ⊔ y \\ z)",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"SemilatticeSup.toMax",
"DistribLattice.toLattice",
... | rw [sup_inf_left, sup_inf_right, sup_sdiff_self_right, inf_sup_right, inf_sdiff_sup_right] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Order.BooleanAlgebra.Basic | {
"line": 331,
"column": 10
} | {
"line": 331,
"column": 100
} | [
{
"pp": "α : Type u\nx y z : α\ninst✝ : GeneralizedBooleanAlgebra α\n⊢ x ⊓ y ⊓ z ⊔ x \\ z ⊓ y \\ z = (x ⊓ y ⊓ (z ⊔ x) ⊔ x \\ z) ⊓ (x ⊓ y ⊓ z ⊔ y \\ z)",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"SemilatticeSup.toMax",
"DistribLattice.toLattice",
... | rw [sup_inf_left, sup_inf_right, sup_sdiff_self_right, inf_sup_right, inf_sdiff_sup_right] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Order.SymmDiff | {
"line": 151,
"column": 6
} | {
"line": 151,
"column": 20
} | [
{
"pp": "α : Type u_2\ninst✝ : GeneralizedCoheytingAlgebra α\na b : α\n⊢ a ∆ b \\ (a ⊓ b) = a ∆ b",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"SemilatticeSup.toMax",
"id",
"SemilatticeInf.toMin",
"SDiff.sdiff",
"Max.max",
"symm... | symmDiff_sdiff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Logic.Equiv.Basic | {
"line": 275,
"column": 67
} | {
"line": 277,
"column": 5
} | [
{
"pp": "α : Sort u_1\np : α → Prop\nh : ∀ (a : α), p a ↔ p ((Equiv.refl α) a)\n⊢ (Equiv.refl α).subtypeEquiv h = Equiv.refl { a // p a }",
"usedConstants": [
"Equiv.instEquivLike",
"Equiv.subtypeEquiv",
"Subtype",
"Equiv.Perm",
"Eq.refl",
"Subtype.val",
"Equiv.refl... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Order.SymmDiff | {
"line": 391,
"column": 53
} | {
"line": 391,
"column": 67
} | [
{
"pp": "α : Type u_2\ninst✝ : GeneralizedBooleanAlgebra α\na b c : α\n⊢ a ∆ b \\ c ⊔ (c \\ (a ⊔ b) ⊔ c ⊓ a ⊓ b) = a \\ (b ⊔ c) ⊔ b \\ (a ⊔ c) ⊔ (c \\ (a ⊔ b) ⊔ c ⊓ a ⊓ b)",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"SemilatticeSup.toMax",
"id",
... | symmDiff_sdiff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Order.SymmDiff | {
"line": 399,
"column": 53
} | {
"line": 399,
"column": 67
} | [
{
"pp": "α : Type u_2\ninst✝ : GeneralizedBooleanAlgebra α\na b c : α\n⊢ a \\ (b ⊔ c) ⊔ a ⊓ b ⊓ c ⊔ b ∆ c \\ a = a \\ (b ⊔ c) ⊔ a ⊓ b ⊓ c ⊔ (b \\ (c ⊔ a) ⊔ c \\ (b ⊔ a))",
"usedConstants": [
"Eq.mpr",
"Lattice.toSemilatticeSup",
"congrArg",
"SemilatticeSup.toMax",
"id",
"... | symmDiff_sdiff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Data.Set.Image | {
"line": 445,
"column": 39
} | {
"line": 445,
"column": 86
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nf : α → β\ns : Set α\nt : Set β\n⊢ f '' (f ⁻¹' t ∩ s) = t ∩ f '' s",
"usedConstants": [
"congrArg",
"Set.image_inter_preimage",
"Set.instInter",
"Inter.inter",
"Set.inter_comm",
"Set.preimage",
"True",
"eq_self",
"of_... | by simp only [inter_comm, image_inter_preimage] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Function | {
"line": 387,
"column": 35
} | {
"line": 388,
"column": 87
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ns s₁ s₂ : Set α\nf : α → β\nh : InjOn f s\nh₁ : s₁ ⊆ s\nh₂ : s₂ ⊆ s\n⊢ f '' s₁ ⊂ f '' s₂ ↔ s₁ ⊂ s₂",
"usedConstants": [
"Eq.mpr",
"congrArg",
"HasSSubset.SSubset",
"Set.instHasSSubset",
"id",
"HasSubset.Subset",
"iff_self",
... | by
simp_rw [ssubset_def, h.image_subset_image_iff h₁ h₂, h.image_subset_image_iff h₂ h₁] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Prod | {
"line": 312,
"column": 2
} | {
"line": 314,
"column": 27
} | [
{
"pp": "case inr.refine_3\nα : Type u_1\nβ : Type u_2\ns s₁ : Set α\nt t₁ : Set β\nh : (s ×ˢ t).Nonempty\nst : s.Nonempty ∧ t.Nonempty\n⊢ s ⊆ s₁ ∧ t ⊆ t₁ ∨ s = ∅ ∨ t = ∅ → s ×ˢ t ⊆ s₁ ×ˢ t₁",
"usedConstants": [
"False",
"eq_false",
"congrArg",
"Set.Nonempty.ne_empty",
"Eq.mp",... | · intro H
simp only [st.1.ne_empty, st.2.ne_empty, or_false] at H
exact prod_mono H.1 H.2 | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Data.Set.Image | {
"line": 1297,
"column": 35
} | {
"line": 1297,
"column": 82
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\nhf : Bijective f\ns : Set β\nt : Set α\n⊢ f ⁻¹' s = t ↔ s = f '' t",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
"id",
"Set.image_eq_image",
"Iff",
"And.right",
"Set.preimage",
"And.left",
"propex... | rw [← image_eq_image hf.1, hf.2.image_preimage] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Set.Image | {
"line": 1297,
"column": 35
} | {
"line": 1297,
"column": 82
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\nhf : Bijective f\ns : Set β\nt : Set α\n⊢ f ⁻¹' s = t ↔ s = f '' t",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
"id",
"Set.image_eq_image",
"Iff",
"And.right",
"Set.preimage",
"And.left",
"propex... | rw [← image_eq_image hf.1, hf.2.image_preimage] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Image | {
"line": 1297,
"column": 35
} | {
"line": 1297,
"column": 82
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\nhf : Bijective f\ns : Set β\nt : Set α\n⊢ f ⁻¹' s = t ↔ s = f '' t",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
"id",
"Set.image_eq_image",
"Iff",
"And.right",
"Set.preimage",
"And.left",
"propex... | rw [← image_eq_image hf.1, hf.2.image_preimage] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Image | {
"line": 1300,
"column": 35
} | {
"line": 1300,
"column": 82
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\nhf : Bijective f\ns : Set α\nt : Set β\n⊢ s = f ⁻¹' t ↔ f '' s = t",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
"id",
"Set.image_eq_image",
"Iff",
"And.right",
"Set.preimage",
"And.left",
"propex... | rw [← image_eq_image hf.1, hf.2.image_preimage] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Data.Set.Image | {
"line": 1300,
"column": 35
} | {
"line": 1300,
"column": 82
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\nhf : Bijective f\ns : Set α\nt : Set β\n⊢ s = f ⁻¹' t ↔ f '' s = t",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
"id",
"Set.image_eq_image",
"Iff",
"And.right",
"Set.preimage",
"And.left",
"propex... | rw [← image_eq_image hf.1, hf.2.image_preimage] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Data.Set.Image | {
"line": 1300,
"column": 35
} | {
"line": 1300,
"column": 82
} | [
{
"pp": "α : Type u\nβ : Type v\nf : α → β\nhf : Bijective f\ns : Set α\nt : Set β\n⊢ s = f ⁻¹' t ↔ f '' s = t",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Iff.rfl",
"id",
"Set.image_eq_image",
"Iff",
"And.right",
"Set.preimage",
"And.left",
"propex... | rw [← image_eq_image hf.1, hf.2.image_preimage] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Set.Piecewise | {
"line": 43,
"column": 78
} | {
"line": 49,
"column": 40
} | [
{
"pp": "α : Type u_1\nδ : α → Sort u_7\ns : Set α\nf g : (i : α) → δ i\ninst✝² : (j : α) → Decidable (j ∈ s)\ninst✝¹ : DecidableEq α\nj : α\ninst✝ : (i : α) → Decidable (i ∈ insert j s)\n⊢ (insert j s).piecewise f g = update (s.piecewise f g) j (f j)",
"usedConstants": [
"Eq.mpr",
"False",
... | by
simp +unfoldPartialApp only [piecewise, mem_insert_iff]
ext i
by_cases h : i = j
· rw [h]
simp
· by_cases h' : i ∈ s <;> simp [h, h'] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Group.Hom.Defs | {
"line": 482,
"column": 18
} | {
"line": 482,
"column": 58
} | [
{
"pp": "G : Type u_7\nH : Type u_8\nF : Type u_9\ninst✝³ : FunLike F G H\ninst✝² : DivInvMonoid G\ninst✝¹ : DivInvMonoid H\ninst✝ : MonoidHomClass F G H\nf : F\nhf : ∀ (x : G), f x⁻¹ = (f x)⁻¹\na : G\nn : ℕ\n⊢ f (a ^ ↑n) = f a ^ ↑n",
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"congrArg"... | rw [zpow_natCast, map_pow, zpow_natCast] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Group.Hom.Defs | {
"line": 482,
"column": 18
} | {
"line": 482,
"column": 58
} | [
{
"pp": "G : Type u_7\nH : Type u_8\nF : Type u_9\ninst✝³ : FunLike F G H\ninst✝² : DivInvMonoid G\ninst✝¹ : DivInvMonoid H\ninst✝ : MonoidHomClass F G H\nf : F\nhf : ∀ (x : G), f x⁻¹ = (f x)⁻¹\na : G\nn : ℕ\n⊢ f (a ^ ↑n) = f a ^ ↑n",
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"congrArg"... | rw [zpow_natCast, map_pow, zpow_natCast] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Group.Hom.Defs | {
"line": 482,
"column": 18
} | {
"line": 482,
"column": 58
} | [
{
"pp": "G : Type u_7\nH : Type u_8\nF : Type u_9\ninst✝³ : FunLike F G H\ninst✝² : DivInvMonoid G\ninst✝¹ : DivInvMonoid H\ninst✝ : MonoidHomClass F G H\nf : F\nhf : ∀ (x : G), f x⁻¹ = (f x)⁻¹\na : G\nn : ℕ\n⊢ f (a ^ ↑n) = f a ^ ↑n",
"usedConstants": [
"zpow_natCast",
"Eq.mpr",
"congrArg"... | rw [zpow_natCast, map_pow, zpow_natCast] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Algebra.Group.Units.Defs | {
"line": 457,
"column": 62
} | {
"line": 457,
"column": 67
} | [
{
"pp": "M : Type u_1\ninst✝ : Monoid M\na : M\nu : Mˣ\nx✝ : IsUnit (a * ↑u)\nv : Mˣ\nhv : ↑v = a * ↑u\n⊢ ↑(v * u⁻¹) = a * ↑u * ↑u⁻¹",
"usedConstants": [
"Units.val",
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Units",
"id",
"MulOne.toMul",
... | ← hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Units.Defs | {
"line": 468,
"column": 64
} | {
"line": 468,
"column": 69
} | [
{
"pp": "M : Type u_3\ninst✝ : Monoid M\nu : Mˣ\na : M\nx✝ : IsUnit (↑u * a)\nv : Mˣ\nhv : ↑v = ↑u * a\n⊢ ↑(u⁻¹ * v) = ↑u⁻¹ * (↑u * a)",
"usedConstants": [
"Units.val",
"Eq.mpr",
"HMul.hMul",
"Monoid.toMulOneClass",
"congrArg",
"Units",
"id",
"MulOne.toMul",
... | ← hv, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Group.Units.Defs | {
"line": 607,
"column": 82
} | {
"line": 608,
"column": 39
} | [
{
"pp": "α : Type u\ninst✝ : DivisionCommMonoid α\na : α\nh : IsUnit a\nb : α\n⊢ a / (a * b) = b⁻¹",
"usedConstants": [
"Eq.mpr",
"DivInvMonoid.toInv",
"instHDiv",
"HMul.hMul",
"DivisionCommMonoid.toDivisionMonoid",
"CommMonoid.toCommSemigroup",
"Monoid.toMulOneClas... | by
rw [mul_comm, h.div_mul_cancel_right] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.GroupWithZero.Basic | {
"line": 446,
"column": 60
} | {
"line": 447,
"column": 42
} | [
{
"pp": "G₀ : Type u_2\ninst✝ : GroupWithZero G₀\na : G₀\nh : a ≠ 0\n⊢ 1 / a ≠ 0",
"usedConstants": [
"Eq.mpr",
"GroupWithZero.toMonoidWithZero",
"DivInvMonoid.toInv",
"instHDiv",
"GroupWithZero.toDivisionMonoid",
"InvOneClass.toOne",
"GroupWithZero.toDivInvMonoid",... | by
simpa only [one_div] using inv_ne_zero h | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Int.Basic | {
"line": 64,
"column": 90
} | {
"line": 64,
"column": 98
} | [
{
"pp": "case e_a.refine_1\nm n✝ : ℤ\nP : ℤ → Sort u_1\nlt : (n : ℤ) → n < m → P n\nge : (n : ℤ) → n ≥ m → ((k : ℤ) → k < n → P k) → P n\nn : ℤ\nx✝¹ : n ≥ m\nhn : m ≤ n\nx✝ : ∀ (k : ℤ), k < m → ∀ (hn : m ≤ k), Int.strongRec lt ge k = ge k hn fun k_1 x ↦ Int.strongRec lt ge k_1\n⊢ (Int.inductionOn' (motive := fu... | ext l hl | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Data.Int.Basic | {
"line": 64,
"column": 90
} | {
"line": 64,
"column": 98
} | [
{
"pp": "case e_a.refine_2\nm n✝ : ℤ\nP : ℤ → Sort u_1\nlt : (n : ℤ) → n < m → P n\nge : (n : ℤ) → n ≥ m → ((k : ℤ) → k < n → P k) → P n\nn : ℤ\nx✝ : n ≥ m\nhn : m ≤ n\nk : ℤ\nhmk : m ≤ k\nih' :\n (∀ (k_1 : ℤ), k_1 < k → ∀ (hn : m ≤ k_1), Int.strongRec lt ge k_1 = ge k_1 hn fun k x ↦ Int.strongRec lt ge k) →\n... | ext l hl | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Data.Int.Basic | {
"line": 64,
"column": 90
} | {
"line": 64,
"column": 98
} | [
{
"pp": "case e_a.refine_3\nm n✝ : ℤ\nP : ℤ → Sort u_1\nlt : (n : ℤ) → n < m → P n\nge : (n : ℤ) → n ≥ m → ((k : ℤ) → k < n → P k) → P n\nn : ℤ\nx✝¹ : n ≥ m\nhn : m ≤ n\nk : ℤ\nhkm : k ≤ m\nih' :\n (∀ (k_1 : ℤ), k_1 < k → ∀ (hn : m ≤ k_1), Int.strongRec lt ge k_1 = ge k_1 hn fun k x ↦ Int.strongRec lt ge k) →\... | ext l hl | _private.Lean.Elab.Tactic.Ext.0.Lean.Elab.Tactic.Ext.evalExt | Lean.Elab.Tactic.Ext.ext |
Mathlib.Data.Int.Basic | {
"line": 69,
"column": 8
} | {
"line": 69,
"column": 33
} | [
{
"pp": "case e_a.refine_3.h.h\nm n✝ : ℤ\nP : ℤ → Sort u_1\nlt : (n : ℤ) → n < m → P n\nge : (n : ℤ) → n ≥ m → ((k : ℤ) → k < n → P k) → P n\nn : ℤ\nx✝¹ : n ≥ m\nhn : m ≤ n\nk : ℤ\nhkm : k ≤ m\nih' :\n (∀ (k_1 : ℤ), k_1 < k → ∀ (hn : m ≤ k_1), Int.strongRec lt ge k_1 = ge k_1 hn fun k x ↦ Int.strongRec lt ge k... | inductionOn'_sub_one hkm, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Algebra.Ring.Int.Defs | {
"line": 60,
"column": 2
} | {
"line": 62,
"column": 37
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : NonAssocRing α\nm : ℕ\n⊢ ∀ (n : ℤ), ↑(↑m * n) = ↑↑m * ↑n",
"usedConstants": [
"add_mul",
"AddGroup.toSubtractionMonoid",
"Int.cast",
"Int.cast_natCast",
"Nat.recAux",
"NonUnitalCommRing.toNonUnitalNonAssocCommRing",
"HMul.hMu... | · induction m with
| zero => simp
| succ m ih => simp_all [add_mul] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Ring.Int.Defs | {
"line": 63,
"column": 2
} | {
"line": 65,
"column": 37
} | [
{
"pp": "case inr\nα : Type u_1\ninst✝ : NonAssocRing α\nm : ℕ\n⊢ ∀ (n : ℤ), ↑(-↑m * n) = ↑(-↑m) * ↑n",
"usedConstants": [
"neg_add_rev",
"Int.instAddCommGroup",
"add_mul",
"AddGroup.toSubtractionMonoid",
"Int.cast_neg",
"Int.cast",
"NegZeroClass.toNeg",
"Int.... | · induction m with
| zero => simp
| succ m ih => simp_all [add_mul] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Algebra.Ring.Basic | {
"line": 276,
"column": 51
} | {
"line": 276,
"column": 63
} | [
{
"pp": "R : Type u_1\ninst✝¹ : DivisionMonoid R\ninst✝ : HasDistribNeg R\na b : R\n⊢ -(b / a) = -b / a",
"usedConstants": [
"Eq.mpr",
"neg_div",
"instHDiv",
"Monoid.toMulOneClass",
"congrArg",
"id",
"MulOne.toMul",
"HDiv.hDiv",
"DivInvMonoid.toMonoid",
... | rw [neg_div] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Algebra.Ring.Basic | {
"line": 276,
"column": 51
} | {
"line": 276,
"column": 63
} | [
{
"pp": "R : Type u_1\ninst✝¹ : DivisionMonoid R\ninst✝ : HasDistribNeg R\na b : R\n⊢ -(b / a) = -b / a",
"usedConstants": [
"Eq.mpr",
"neg_div",
"instHDiv",
"Monoid.toMulOneClass",
"congrArg",
"id",
"MulOne.toMul",
"HDiv.hDiv",
"DivInvMonoid.toMonoid",
... | rw [neg_div] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Algebra.Ring.Basic | {
"line": 276,
"column": 51
} | {
"line": 276,
"column": 63
} | [
{
"pp": "R : Type u_1\ninst✝¹ : DivisionMonoid R\ninst✝ : HasDistribNeg R\na b : R\n⊢ -(b / a) = -b / a",
"usedConstants": [
"Eq.mpr",
"neg_div",
"instHDiv",
"Monoid.toMulOneClass",
"congrArg",
"id",
"MulOne.toMul",
"HDiv.hDiv",
"DivInvMonoid.toMonoid",
... | rw [neg_div] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Logic.Nontrivial.Basic | {
"line": 30,
"column": 88
} | {
"line": 32,
"column": 48
} | [
{
"pp": "α : Type u_3\ninst✝¹ : Nontrivial α\ninst✝ : LinearOrder α\n⊢ ∃ x y, x < y",
"usedConstants": [
"lt_or_gt_of_ne",
"Preorder.toLT",
"PartialOrder.toPreorder",
"Exists",
"Ne",
"Or.casesOn",
"Exists.casesOn",
"LT.lt",
"Exists.intro",
"Eq.refl... | by
rcases exists_pair_ne α with ⟨x, y, hxy⟩
cases lt_or_gt_of_ne hxy <;> exact ⟨_, _, ‹_›⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Ring.Hom.Defs | {
"line": 242,
"column": 64
} | {
"line": 244,
"column": 5
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\nγ : Type u_4\ninst✝² : NonUnitalNonAssocSemiring α\ninst✝¹ : NonUnitalNonAssocSemiring β\ninst✝ : NonUnitalNonAssocSemiring γ\nf : α →ₙ+* β\n⊢ comp 0 f = 0",
"usedConstants": [
"NonUnitalRingHom.ext",
"NonUnitalRingHom.instZero",
"NonUnitalRingHom.instF... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Algebra.Ring.Hom.Defs | {
"line": 628,
"column": 68
} | {
"line": 630,
"column": 5
} | [
{
"pp": "α : Type u_2\nβ : Type u_3\ninst✝² : CommRing α\ninst✝¹ : IsDomain α\ninst✝ : CommRing β\nf : β →+ α\nh : ∀ (x : β), f (x * x) = f x * f x\nh_two : 2 ≠ 0\nh_one : f 1 = 1\n⊢ ↑(f.mkRingHomOfMulSelfOfTwoNeZero h h_two h_one) = f",
"usedConstants": [
"RingHom.instRingHomClass",
"RingHomCla... | by
ext
rfl | [anonymous] | Lean.Parser.Term.byTactic |
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