module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 210,
"column": 12
} | {
"line": 210,
"column": 32
} | [
{
"pp": "case cons.inl.cons\nV : Type u_1\nG : SimpleGraph V\nv w u✝ v✝¹ w✝ : V\nph : G.Adj u✝ v✝¹\np : G.Walk v✝¹ w✝\nih : p.IsPath → ∀ (q : G.Walk v✝¹ w✝), q.IsPath → p = q\nhp : (cons ph p).IsPath\nv✝ : V\nh✝ : G.Adj u✝ v✝\nq : G.Walk v✝ w✝\nhq : (cons h✝ q).IsPath\nh : s(u✝, v✝¹) ∈ (cons h✝ q).edges\n⊢ cons... | Walk.cons_isPath_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 217,
"column": 10
} | {
"line": 217,
"column": 30
} | [
{
"pp": "case cons.inr\nV : Type u_1\nG : SimpleGraph V\nv w u✝ v✝ w✝ : V\nph : G.Adj u✝ v✝\np : G.Walk v✝ w✝\nih : p.IsPath → ∀ (q : G.Walk v✝ w✝), q.IsPath → p = q\nhp : (cons ph p).IsPath\nq : G.Walk u✝ w✝\nhq : q.IsPath\nh : s(u✝, v✝) ∈ p.edges\n⊢ cons ph p = q",
"usedConstants": [
"congrArg",
... | Walk.cons_isPath_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Acyclic | {
"line": 353,
"column": 4
} | {
"line": 353,
"column": 96
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝¹ : Fintype V\ninst✝ : Fintype ↑G.edgeSet\nhG : G.IsTree\nthis✝ : Nonempty V\ninhabited_h : Inhabited V\nthis : {default}ᶜ.card + 1 = Fintype.card V\nf : (x : V) → G.Walk x default\nhf : ∀ (x : V), (fun p ↦ p.IsPath) (f x)\nhf' : ∀ (x : V) (y : G.Walk x default), (... | simp only [mem_edgeFinset, Finset.mem_compl, Finset.mem_singleton, Sym2.forall, mem_edgeSet] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Hasse | {
"line": 68,
"column": 10
} | {
"line": 68,
"column": 21
} | [
{
"pp": "α : Type u_1\ninst✝ : Preorder α\ns : Finset α\n⊢ ¬(hasse α).IsNClique 3 s",
"usedConstants": [
"SimpleGraph.IsNClique",
"instOfNatNat",
"SimpleGraph.hasse",
"Nat",
"OfNat.ofNat"
]
}
] | ⟨hc, hcard⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.anonymousCtor |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 86,
"column": 6
} | {
"line": 86,
"column": 20
} | [
{
"pp": "n r : ℕ\nhr : 0 < r\n⊢ (turanGraph n r).CliqueFree (r + 1)",
"usedConstants": [
"Eq.mpr",
"SimpleGraph.turanGraph",
"congrArg",
"SimpleGraph.Copy",
"SimpleGraph.completeGraph",
"id",
"SimpleGraph.cliqueFree_iff",
"instOfNatNat",
"IsEmpty",
... | cliqueFree_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.Constructions | {
"line": 82,
"column": 2
} | {
"line": 82,
"column": 10
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nu : α\np : G.Walk u u\nhOdd : Odd p.length\nh : ¬3 ≤ G.chromaticNumber\nh' : G.chromaticNumber ≤ 2\nc : G.Coloring (Fin 2) := Nonempty.some ⋯\nc' : G.Coloring Bool := (G.recolorOfEquiv finTwoEquiv) c\nthis : ¬c' u = true ↔ c' u = true\n⊢ False",
"usedConstants": [
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 212,
"column": 6
} | {
"line": 212,
"column": 48
} | [
{
"pp": "case right\nV : Type u_1\ninst✝² : Fintype V\nG : SimpleGraph V\ninst✝¹ : DecidableRel G.Adj\nr : ℕ\nh : G.IsTuranMaximal r\ninst✝ : DecidableEq V\nfp : Finpartition univ := h.finpartition\nlarge : Finset V\nhl : large ∈ fp.parts\nsmall : Finset V\nhs : small ∈ fp.parts\nineq : #small + 1 < #large\nw :... | G.card_edgeFinset_replaceVertex_of_adj ha, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Coloring.EdgeLabeling | {
"line": 126,
"column": 4
} | {
"line": 126,
"column": 39
} | [
{
"pp": "V : Type u_1\nV' : Type u_2\nG : SimpleGraph V\nG' : SimpleGraph V'\nK : Type u_3\nK' : Type u_4\nC : G.EdgeLabeling K\nf : (x y : V) → G.Adj x y → K\nf_symm : ∀ (x y : V) (H : G.Adj x y), f y x ⋯ = f x y H\ne : Sym2 V\n⊢ e ∈ G.edgeSet → K",
"usedConstants": [
"Membership.mem",
"SimpleG... | refine Sym2.hrec f (fun a b ↦ ?_) e | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Combinatorics.SimpleGraph.Prod | {
"line": 178,
"column": 76
} | {
"line": 178,
"column": 84
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nG : SimpleGraph α\nH : SimpleGraph β\na₁ a₂✝ : α\nb₁ b₂✝ : β\ninst✝³ : DecidableEq α\ninst✝² : DecidableEq β\ninst✝¹ : DecidableRel G.Adj\ninst✝ : DecidableRel H.Adj\nw : (G □ H).Walk (a₁, b₁) (a₂✝, b₂✝)\nb₂ : β\na₂ : α\nc : α × β\nx : (G □ H).Adj (a₁, b₁) c\nw' : (G □ H).Wa... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 82,
"column": 6
} | {
"line": 82,
"column": 14
} | [
{
"pp": "ι : Type u_1\nV : ι → Type u_2\n⊢ ∀ (a b c : (i : ι) × V i),\n ¬(completeMultipartiteGraph V).Adj a b →\n ¬(completeMultipartiteGraph V).Adj b c → ¬(completeMultipartiteGraph V).Adj a c",
"usedConstants": [
"False",
"SimpleGraph.comap_adj._simp_1",
"Classical.not_not._simp... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 82,
"column": 6
} | {
"line": 82,
"column": 14
} | [
{
"pp": "ι : Type u_1\nV : ι → Type u_2\n⊢ ∀ (a b c : (i : ι) × V i),\n ¬(completeMultipartiteGraph V).Adj a b →\n ¬(completeMultipartiteGraph V).Adj b c → ¬(completeMultipartiteGraph V).Adj a c",
"usedConstants": [
"False",
"SimpleGraph.comap_adj._simp_1",
"Classical.not_not._simp... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 82,
"column": 6
} | {
"line": 82,
"column": 14
} | [
{
"pp": "ι : Type u_1\nV : ι → Type u_2\n⊢ ∀ (a b c : (i : ι) × V i),\n ¬(completeMultipartiteGraph V).Adj a b →\n ¬(completeMultipartiteGraph V).Adj b c → ¬(completeMultipartiteGraph V).Adj a c",
"usedConstants": [
"False",
"SimpleGraph.comap_adj._simp_1",
"Classical.not_not._simp... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 332,
"column": 8
} | {
"line": 332,
"column": 22
} | [
{
"pp": "n r c : ℕ\nhr : r > 0\nthis : #({i ∈ range r | c % r = (n + i) % r}) = 1\n⊢ (∑ w ∈ range r, if c % r ≠ (n + w) % r then 1 else 0) = r - 1",
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"congrArg",
"HSub.hSub",
"id",
"Nat.instMod",
"instHMod",
"Fin... | ← card_filter, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 333,
"column": 8
} | {
"line": 333,
"column": 40
} | [
{
"pp": "case h\nn r c : ℕ\nhr : r > 0\nthis : #({i ∈ range r | c % r = (n + i) % r}) = 1\n⊢ #({i ∈ range r | c % r = (n + i) % r}) + #({i ∈ range r | c % r ≠ (n + i) % r}) = r",
"usedConstants": [
"Eq.mpr",
"instDecidableNot",
"congrArg",
"id",
"Nat.instMod",
"instHMod",... | card_filter_add_card_filter_not, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite | {
"line": 433,
"column": 6
} | {
"line": 433,
"column": 22
} | [
{
"pp": "case neg.refine_5\nα : Type u\nG✝ : SimpleGraph α\ns : Set α\nV : Type u_1\nG : SimpleGraph V\nr t : ℕ\nK : G.CompleteEquipartiteSubgraph r t\nf : (completeEquipartiteGraph r t).Copy G\nht : ¬t = 0\nx✝³ x✝² : Finset V\nhne : x✝³ ≠ x✝²\nx✝¹ : V\nh₁' : x✝¹ ∈ ↑x✝³\nx✝ : V\nh₂' : x✝ ∈ ↑x✝²\nw✝¹ : Fin r\nh₁... | rw [← h₁] at h₁' | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Extremal.Turan | {
"line": 391,
"column": 6
} | {
"line": 392,
"column": 75
} | [
{
"pp": "case inr.inr\nn r : ℕ\nhr : r > 0\nring₁ : ∀ (n : ℕ), (n ^ 2 - (n % r) ^ 2) * (r - 1) / (2 * r) = n % r * (n / r) * (r - 1) + r * (r - 1) * (n / r) ^ 2 / 2\nh : r ≤ n\nn' : ℕ := n - r\nn'r : n = n' + r\n⊢ #(turanGraph n r).edgeFinset = (n ^ 2 - (n % r) ^ 2) * (r - 1) / (2 * r) + (n % r).choose 2",
... | rw [n'r, card_edgeFinset_turanGraph_add, card_edgeFinset_turanGraph, ring₁, ring₁,
add_rotate, ← add_assoc, Nat.add_mod_right, Nat.add_div_right _ hr] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 76,
"column": 64
} | {
"line": 77,
"column": 79
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nu : α\n⊢ 0 < G.eccent u ↔ Nontrivial α",
"usedConstants": [
"Nontrivial",
"Eq.mpr",
"Preorder.toLT",
"instCompleteLinearOrderENat",
"instAddMonoidWithOneENat",
"congrArg",
"CommSemiring.toSemiring",
"instIsBotZeroClass... | by
rw [pos_iff_ne_zero, ← not_subsingleton_iff_nontrivial, ← eccent_eq_zero_iff] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 281,
"column": 4
} | {
"line": 281,
"column": 62
} | [
{
"pp": "case inr\nα : Type u_1\nG : SimpleGraph α\nh : ¬G.Connected\nh✝ : Nonempty α\n⊢ G.diam = 0",
"usedConstants": [
"Eq.mpr",
"instTopENat",
"congrArg",
"id",
"SimpleGraph.ediam",
"instOfNatNat",
"Nat",
"SimpleGraph.ediam_eq_top_of_not_connected",
"... | rw [diam, ediam_eq_top_of_not_connected h, ENat.toNat_top] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 281,
"column": 4
} | {
"line": 281,
"column": 62
} | [
{
"pp": "case inr\nα : Type u_1\nG : SimpleGraph α\nh : ¬G.Connected\nh✝ : Nonempty α\n⊢ G.diam = 0",
"usedConstants": [
"Eq.mpr",
"instTopENat",
"congrArg",
"id",
"SimpleGraph.ediam",
"instOfNatNat",
"Nat",
"SimpleGraph.ediam_eq_top_of_not_connected",
"... | rw [diam, ediam_eq_top_of_not_connected h, ENat.toNat_top] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 281,
"column": 4
} | {
"line": 281,
"column": 62
} | [
{
"pp": "case inr\nα : Type u_1\nG : SimpleGraph α\nh : ¬G.Connected\nh✝ : Nonempty α\n⊢ G.diam = 0",
"usedConstants": [
"Eq.mpr",
"instTopENat",
"congrArg",
"id",
"SimpleGraph.ediam",
"instOfNatNat",
"Nat",
"SimpleGraph.ediam_eq_top_of_not_connected",
"... | rw [diam, ediam_eq_top_of_not_connected h, ENat.toNat_top] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 370,
"column": 68
} | {
"line": 373,
"column": 50
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\ninst✝ : Nontrivial α\n⊢ G.radius ≠ 0",
"usedConstants": [
"Eq.mpr",
"False",
"instCompleteLinearOrderENat",
"instAddMonoidWithOneENat",
"eq_false",
"congrArg",
"CommSemiring.toSemiring",
"CompletelyDistribLattice.toCom... | by
rw [← ENat.one_le_iff_ne_zero]
apply le_iInf
simp [ENat.one_le_iff_ne_zero, G.eccent_ne_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Diam | {
"line": 410,
"column": 53
} | {
"line": 419,
"column": 94
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\ninst✝ : Nonempty α\n⊢ G.radius = G.ediam ↔ ∃ e, ∀ (u : α), G.eccent u = e",
"usedConstants": [
"Eq.mpr",
"Classical.ofNonempty",
"Eq.ge",
"instCompleteLinearOrderENat",
"congrArg",
"CompletelyDistribLattice.toCompleteLattice",
... | by
refine ⟨fun h ↦ ?_, fun h ↦ ?_⟩
· use G.radius
intro u
exact le_antisymm (h ▸ eccent_le_ediam) radius_le_eccent
· obtain ⟨e, h⟩ := h
have ediam_eq : G.ediam = e :=
le_antisymm (iSup_le fun u ↦ (h u).le) ((h Classical.ofNonempty) ▸ eccent_le_ediam)
rw [ediam_eq]
exact le_antisymm ((h C... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 42,
"column": 4
} | {
"line": 42,
"column": 12
} | [
{
"pp": "α : Type u_1\ninst✝ : DecidableEq α\nt u : Finset α\nhdtu : Disjoint t u\nhctu : #t = #u\n⊢ Even #(t ∪ u)",
"usedConstants": [
"Finset.instUnion",
"congrArg",
"Finset",
"instHAdd",
"Even.add_self._simp_1",
"HAdd.hAdd",
"Nat",
"True",
"Even",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Data.Set.Card.Arithmetic | {
"line": 75,
"column": 8
} | {
"line": 75,
"column": 22
} | [
{
"pp": "α : Type u_1\ns : Set α\nh : s.Infinite\nthis : Infinite ↑s\n⊢ Nonempty (↑s ≃ ↑s ⊕ ↑s)",
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"Cardinal.mk",
"Set.Elem",
"Sum",
"id",
"Equiv",
"Cardinal.eq",
"propext",
"Nonempty",
... | ← Cardinal.eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Hall | {
"line": 75,
"column": 25
} | {
"line": 75,
"column": 33
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : G.LocallyFinite\np₁ p₂ : Set V\nh₁ : G.IsBipartiteWith p₁ p₂\nh₂ : ∀ s ⊆ p₁, s.ncard ≤ (⋃ x ∈ s, G.neighborSet x).ncard\nf : ↑p₁ → V\nhf₁ : Injective f\nhf₂ : ∀ (x : ↑p₁), f x ∈ G.neighborFinset ↑x\nthis : ∀ (x : ↑p₁), f x ∉ p₁\nv : V\nh' : v ∈ p₁\n⊢ (fun w ↦ if... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.Hall | {
"line": 75,
"column": 25
} | {
"line": 75,
"column": 33
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : G.LocallyFinite\np₁ p₂ : Set V\nh₁ : G.IsBipartiteWith p₁ p₂\nh₂ : ∀ s ⊆ p₁, s.ncard ≤ (⋃ x ∈ s, G.neighborSet x).ncard\nf : ↑p₁ → V\nhf₁ : Injective f\nhf₂ : ∀ (x : ↑p₁), f x ∈ G.neighborFinset ↑x\nthis : ∀ (x : ↑p₁), f x ∉ p₁\nv : V\nh' : v ∈ p₁\n⊢ (fun w ↦ if... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Hall | {
"line": 75,
"column": 25
} | {
"line": 75,
"column": 33
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : G.LocallyFinite\np₁ p₂ : Set V\nh₁ : G.IsBipartiteWith p₁ p₂\nh₂ : ∀ s ⊆ p₁, s.ncard ≤ (⋃ x ∈ s, G.neighborSet x).ncard\nf : ↑p₁ → V\nhf₁ : Injective f\nhf₂ : ∀ (x : ↑p₁), f x ∈ G.neighborFinset ↑x\nthis : ∀ (x : ↑p₁), f x ∉ p₁\nv : V\nh' : v ∈ p₁\n⊢ (fun w ↦ if... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Hall | {
"line": 111,
"column": 2
} | {
"line": 111,
"column": 56
} | [
{
"pp": "case refine_3\nV : Type u_1\nG : SimpleGraph V\ninst✝ : G.LocallyFinite\np₁ p₂ : Set V\nh₁ : G.IsBipartiteWith p₁ p₂\nh₂ : ∀ (s : Set V), s.ncard ≤ (⋃ x ∈ s, G.neighborSet x).ncard\nf : V → V\nhf₁ : Injective f\nhf₂ : ∀ (x : V), f x ∈ G.neighborFinset x\nthis✝¹ : ∀ x ∈ p₁, f x ∉ p₁\nthis✝ : ∀ x ∈ p₂, f... | · exact fun v ↦ mem_neighborFinset _ _ _ |>.mp (hf₂ v) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 132,
"column": 2
} | {
"line": 132,
"column": 14
} | [
{
"pp": "case inl\nV : Type u_1\nG : SimpleGraph V\nM M' : G.Subgraph\nhM : M.IsMatching\nhM' : M'.IsMatching\nhd : Disjoint M.support M'.support\nv : V\nhv : v ∈ (M ⊔ M').verts\naux : ∀ {N N' : G.Subgraph}, N.IsMatching → Disjoint N.support N'.support → v ∈ N.verts → ∃! w, (N ⊔ N').Adj v w\nhmM : v ∈ M.verts\n... | | inl hmM => | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 294,
"column": 8
} | {
"line": 294,
"column": 22
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nt u : Set V\nhc : G.IsClique (t ∪ u)\nhu : (t ∪ u).Finite\nh : Even (t ∪ u).ncard\nhd : Disjoint t u\nhcard : t.ncard = u.ncard\n⊢ Nonempty (↑t ≃ ↑u)",
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"Cardinal.mk",
"Set.Elem",
... | ← Cardinal.eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.StronglyRegular | {
"line": 75,
"column": 4
} | {
"line": 75,
"column": 49
} | [
{
"pp": "case a\nV : Type u\ninst✝² : Fintype V\nG : SimpleGraph V\ninst✝¹ : DecidableRel G.Adj\nn k ℓ μ : ℕ\ninst✝ : Nontrivial V\nh : G.IsSRGWith n k ℓ μ\nht : G ≠ ⊤\nhm : μ ≠ 0\nu v : V\nhc : 2 < G.edist u v\n⊢ False",
"usedConstants": [
"ENat.instNatCast",
"SimpleGraph.Adj",
"Nat.instA... | obtain ⟨hn, ha, he⟩ := two_lt_edist_iff.mp hc | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 428,
"column": 6
} | {
"line": 428,
"column": 20
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nv : V\ninst✝ : G.LocallyFinite\nu : V\np : G.Walk u u\nhp : p.IsCycle\nhcyc : G.IsCycles\nhv : v ∈ p.toSubgraph.verts\nw : V\nthis : (p.toSubgraph.neighborSet v).ncard = 2\n⊢ Nonempty (↑(G.neighborSet v) ≃ ↑(p.toSubgraph.neighborSet v))",
"usedConstants": [
"E... | ← Cardinal.eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 463,
"column": 8
} | {
"line": 463,
"column": 22
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nv w : V\nhcyc : G.IsCycles\np : G.Walk v w\nhp : p.IsPath\nn : ℕ\nhnl : n ≤ p.length\nhw : w ≠ p.getVert n\nhadj : G.Adj v (p.getVert n)\nhn : ¬(n = 0 ∨ n = p.length)\n⊢ Nonempty (↑(G.neighborSet (p.getVert n)) ≃ ↑(p.toSubgraph.neighborSet (p.getVert n... | ← Cardinal.eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 496,
"column": 8
} | {
"line": 496,
"column": 28
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nv w : V\nhcyc : G.IsCycles\np : G.Walk v w\nhp : p.IsPath\nhvw : ¬v = w\nhpn : ¬p.Nil\nw' : V\nhwu : ∀ (y : V), (fun w' ↦ p.snd ≠ w' ∧ G.Adj v w') y → y = w'\nhw'1 : p.snd ≠ w'\nhw'2 : G.Adj v w'\nhnpvw' : ¬p.toSubgraph.Adj v w'\nhww' : ¬w = w'\nhle : ... | Walk.cons_isPath_iff | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Trails | {
"line": 92,
"column": 16
} | {
"line": 92,
"column": 57
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v : V\np : G.Walk u v\nh : p.IsEulerian\ne : Sym2 V\nhe : e ∈ G.edgeSet\n⊢ e ∈ p.edges",
"usedConstants": [
"List.count_pos_iff._simp_1",
"Eq.ge",
"instLawfulBEq",
"Preorder.toLE",
"Membership.mem",
"Eq.mp... | simpa [Nat.succ_le_iff] using (h e he).ge | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Combinatorics.SimpleGraph.Trails | {
"line": 92,
"column": 16
} | {
"line": 92,
"column": 57
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v : V\np : G.Walk u v\nh : p.IsEulerian\ne : Sym2 V\nhe : e ∈ G.edgeSet\n⊢ e ∈ p.edges",
"usedConstants": [
"List.count_pos_iff._simp_1",
"Eq.ge",
"instLawfulBEq",
"Preorder.toLE",
"Membership.mem",
"Eq.mp... | simpa [Nat.succ_le_iff] using (h e he).ge | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Trails | {
"line": 92,
"column": 16
} | {
"line": 92,
"column": 57
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v : V\np : G.Walk u v\nh : p.IsEulerian\ne : Sym2 V\nhe : e ∈ G.edgeSet\n⊢ e ∈ p.edges",
"usedConstants": [
"List.count_pos_iff._simp_1",
"Eq.ge",
"instLawfulBEq",
"Preorder.toLE",
"Membership.mem",
"Eq.mp... | simpa [Nat.succ_le_iff] using (h e he).ge | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 546,
"column": 10
} | {
"line": 546,
"column": 24
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nv✝ : V\ninst✝ : Finite V\nc : G.ConnectedComponent\nh : G.IsCycles\nhv✝ : v✝ ∈ c.supp\nw✝ : V\nhw : w✝ ∈ G.neighborSet v✝\nu : V\np : G.Walk u u\nhp : p.IsCycle ∧ s(v✝, w✝) ∈ p.edges\nhvp : v✝ ∈ p.support\nv : V\nhv : v ∈ p.toSubgraph.verts\nw : V\nhadj : G.Adj v w\nthi... | ← Cardinal.eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 552,
"column": 4
} | {
"line": 552,
"column": 12
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nv : V\ninst✝ : Finite V\nc : G.ConnectedComponent\nh : G.IsCycles\nhv : v ∈ c.supp\nw : V\nhw : w ∈ G.neighborSet v\nu : V\np : G.Walk u u\nhp : p.IsCycle ∧ s(v, w) ∈ p.edges\nhvp : v ∈ p.support\nc' : G.ConnectedComponent\nhc' : p.toSubgraph.verts = c'.supp\nthis : v ∈... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.UniversalVerts | {
"line": 54,
"column": 8
} | {
"line": 54,
"column": 22
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\ns : Set V\nh : Disjoint G.universalVerts s\nhc : s.ncard ≤ G.universalVerts.ncard\nt : Set V\nht : t ⊆ G.universalVerts ∧ t.ncard = s.ncard\n⊢ Nonempty (↑s ≃ ↑t)",
"usedConstants": [
"Eq.mpr",
"Cardinal",
"congrArg",
"Cardin... | ← Cardinal.eq, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 591,
"column": 6
} | {
"line": 591,
"column": 14
} | [
{
"pp": "case neg.inl.inr.inr\nV : Type u_1\nG G' : SimpleGraph V\nu x : V\nhalt : G.IsAlternating G'\nhnadj : ¬G'.Adj u x\nhu' : ∀ (u' : V), u' ≠ u → G.Adj x u' → G'.Adj x u'\nhx' : ∀ (x' : V), x' ≠ x → G.Adj x' u → G'.Adj x' u\nhadj : ¬G.Adj u x\nv w w' : V\nhww' : w ≠ w'\nhl : G.Adj v w\nh2 : (v = u ∧ w' = x... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 591,
"column": 6
} | {
"line": 591,
"column": 14
} | [
{
"pp": "case neg.inl.inr.inr\nV : Type u_1\nG G' : SimpleGraph V\nu x : V\nhalt : G.IsAlternating G'\nhnadj : ¬G'.Adj u x\nhu' : ∀ (u' : V), u' ≠ u → G.Adj x u' → G'.Adj x u'\nhx' : ∀ (x' : V), x' ≠ x → G.Adj x' u → G'.Adj x' u\nhadj : ¬G.Adj u x\nv w w' : V\nhww' : w ≠ w'\nhl : G.Adj v w\nh2 : (v = u ∧ w' = x... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Matching | {
"line": 591,
"column": 6
} | {
"line": 591,
"column": 14
} | [
{
"pp": "case neg.inl.inr.inr\nV : Type u_1\nG G' : SimpleGraph V\nu x : V\nhalt : G.IsAlternating G'\nhnadj : ¬G'.Adj u x\nhu' : ∀ (u' : V), u' ≠ u → G.Adj x u' → G'.Adj x u'\nhx' : ∀ (x' : V), x' ≠ x → G.Adj x' u → G'.Adj x' u\nhadj : ¬G.Adj u x\nv w w' : V\nhww' : w ≠ w'\nhl : G.Adj v w\nh2 : (v = u ∧ w' = x... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.FiveWheelLike | {
"line": 391,
"column": 61
} | {
"line": 394,
"column": 52
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nr k : ℕ\nv w₁ w₂ : α\ns t : Finset α\ninst✝² : DecidableEq α\nhw : G.IsFiveWheelLike r k v w₁ w₂ s t\nhcf : G.CliqueFree (r + 2)\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Fintype α\nhm : G.FiveWheelLikeFree r (k + 1)\nX : Finset α := {x | ∀ ⦃y : α⦄, y ∈ s ∩ t → G.Adj x y}\n... | by
simp_rw [add_assoc, add_comm k, ← add_assoc, ← Wc, add_assoc, ← two_mul, mul_add,
← hw.card_inter, card_eq_sum_ones, ← mul_assoc, mul_sum, mul_one, mul_comm 2]
gcongr with i <;> exact minDegree_le_degree .. | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 49
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\n⊢ G.IsVertexCover ∅ ↔ G = ⊥",
"usedConstants": [
"False",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"SimpleGraph.Adj",
"_private.Mathlib.Combinatorics.SimpleGraph.VertexCover.0.SimpleGraph.isVertexCover_empty._simp_1_2",
"Mem... | simp [IsVertexCover, eq_bot_iff_forall_not_adj] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 49
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\n⊢ G.IsVertexCover ∅ ↔ G = ⊥",
"usedConstants": [
"False",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"SimpleGraph.Adj",
"_private.Mathlib.Combinatorics.SimpleGraph.VertexCover.0.SimpleGraph.isVertexCover_empty._simp_1_2",
"Mem... | simp [IsVertexCover, eq_bot_iff_forall_not_adj] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 41,
"column": 2
} | {
"line": 41,
"column": 49
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\n⊢ G.IsVertexCover ∅ ↔ G = ⊥",
"usedConstants": [
"False",
"Set.mem_empty_iff_false._simp_1",
"congrArg",
"SimpleGraph.Adj",
"_private.Mathlib.Combinatorics.SimpleGraph.VertexCover.0.SimpleGraph.isVertexCover_empty._simp_1_2",
"Mem... | simp [IsVertexCover, eq_bot_iff_forall_not_adj] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 127,
"column": 25
} | {
"line": 127,
"column": 33
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\n⊢ G = ⊥ → G.vertexCoverNum = 0",
"usedConstants": [
"SimpleGraph.vertexCoverNum",
"congrArg",
"CommSemiring.toSemiring",
"Bot.bot",
"SimpleGraph",
"ENat",
"True",
"eq_self",
"instCommSemiringENat",
"of_eq_t... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 127,
"column": 25
} | {
"line": 127,
"column": 33
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\n⊢ G = ⊥ → G.vertexCoverNum = 0",
"usedConstants": [
"SimpleGraph.vertexCoverNum",
"congrArg",
"CommSemiring.toSemiring",
"Bot.bot",
"SimpleGraph",
"ENat",
"True",
"eq_self",
"instCommSemiringENat",
"of_eq_t... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 127,
"column": 25
} | {
"line": 127,
"column": 33
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\n⊢ G = ⊥ → G.vertexCoverNum = 0",
"usedConstants": [
"SimpleGraph.vertexCoverNum",
"congrArg",
"CommSemiring.toSemiring",
"Bot.bot",
"SimpleGraph",
"ENat",
"True",
"eq_self",
"instCommSemiringENat",
"of_eq_t... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.VertexCover | {
"line": 173,
"column": 8
} | {
"line": 173,
"column": 68
} | [
{
"pp": "V : Type u_1\na✝ : Nontrivial V\nn : ℕ\nhn : ↑n ≤ ENat.card V - 1\nhh : (completeGraph V).vertexCoverNum < ↑n\nthis : ↑n - 1 ≤ ENat.card V\nt : Set V\nht₁ : t.encard = ↑(n - 1)\nht₂ : (completeGraph V).IsVertexCover t\n⊢ 1 + 1 ≤ (Set.univ \\ t).encard",
"usedConstants": [
"Eq.mpr",
"Set... | Set.encard_diff (by simp) (Set.finite_of_encard_eq_coe ht₁), | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.Tiling.Tile | {
"line": 191,
"column": 4
} | {
"line": 192,
"column": 67
} | [
{
"pp": "case refine_2.groupElts\nG : Type u_1\nX : Type u_2\nιₚ : Type u_3\ninst✝¹ : Group G\ninst✝ : MulAction G X\nps : Protoset G X ιₚ\ni₁ : ιₚ\ng₁ g₂ : G ⧸ Subgroup.map (MulAction.stabilizer G ↑(↑ps i₁)).subtype (↑ps i₁).symmetries\nx✝ :\n { index := i₁, groupElts := g₁ }.index = { index := i₁, groupElts ... | · exact heq_of_eq (Set.singleton_eq_singleton_iff.1
((Set.preimage_eq_preimage Quotient.mk''_surjective).1 hq)) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.Young.YoungDiagram | {
"line": 410,
"column": 34
} | {
"line": 410,
"column": 42
} | [
{
"pp": "case cons.zero\nhead✝ : ℕ\ntail✝ : List ℕ\ntail_ih✝ : ∀ {c : ℕ × ℕ}, c ∈ YoungDiagram.cellsOfRowLens tail✝ ↔ ∃ (h : c.1 < tail✝.length), c.2 < tail✝[c.1]\nsnd✝ : ℕ\n⊢ (0, snd✝) ∈\n {0} ×ˢ Finset.range head✝ ∪\n Finset.map ({ toFun := Nat.succ, inj' := Nat.succ_injective }.prodMap (Embedding... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Combinatorics.Young.YoungDiagram | {
"line": 410,
"column": 34
} | {
"line": 410,
"column": 42
} | [
{
"pp": "case cons.succ\nhead✝ : ℕ\ntail✝ : List ℕ\ntail_ih✝ : ∀ {c : ℕ × ℕ}, c ∈ YoungDiagram.cellsOfRowLens tail✝ ↔ ∃ (h : c.1 < tail✝.length), c.2 < tail✝[c.1]\nsnd✝ n✝ : ℕ\n⊢ (n✝ + 1, snd✝) ∈\n {0} ×ˢ Finset.range head✝ ∪\n Finset.map ({ toFun := Nat.succ, inj' := Nat.succ_injective }.prodMap (E... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Computability.Primrec.Basic | {
"line": 725,
"column": 10
} | {
"line": 725,
"column": 26
} | [
{
"pp": "this : PrimrecRel fun a b ↦ a.2 = 0 ∧ b = 0 ∨ 0 < a.2 ∧ b * a.2 ≤ a.1 ∧ a.1 < (b + 1) * a.2\na k q : ℕ\nH : ¬k = 0\n⊢ q * k ≤ a ∧ a < (q + 1) * k ↔ q = a / k",
"usedConstants": [
"Eq.mpr",
"instHDiv",
"HMul.hMul",
"congrArg",
"PartialOrder.toPreorder",
"Preorder.... | le_antisymm_iff, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Tutte | {
"line": 210,
"column": 4
} | {
"line": 210,
"column": 12
} | [
{
"pp": "case h\nV : Type u_1\nG : SimpleGraph V\ninst✝ : Finite V\nx a b c : V\nM1 : (G ⊔ edge x b).Subgraph\nM2 : (G ⊔ edge a c).Subgraph\nhxa : G.Adj x a\nhab : G.Adj a b\nhnGxb : ¬G.Adj x b\nhnGac : ¬G.Adj a c\nhnxb : x ≠ b\nhnxc : x ≠ c\nhnac : a ≠ c\nhnbc : b ≠ c\nhM1 : M1.IsPerfectMatching\nhM2 : M2.IsPe... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Computability.Primrec.Basic | {
"line": 767,
"column": 10
} | {
"line": 767,
"column": 26
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : Primcodable α\ninst✝ : Primcodable β\nn : ℕ\n⊢ encode\n (bif n.bodd then Option.map (fun b ↦ 2 * encode b + 1) (decode n.div2)\n else Option.map (fun b ↦ 2 * encode b) (decode n.div2)) =\n encode\n (match n.bodd, n.div2 with\n | false, m => Op... | cases Nat.bodd n | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | Lean.Parser.Tactic.cases |
Mathlib.Computability.Primrec.List | {
"line": 209,
"column": 24
} | {
"line": 209,
"column": 32
} | [
{
"pp": "case cons.zero.nil\nα : Type u_1\ninst✝ : Primcodable α\nF : List α → ℕ → ℕ ⊕ α :=\n fun l n ↦ List.foldl (fun s a ↦ Sum.casesOn s (fun x ↦ Nat.casesOn x (Sum.inr a) Sum.inl) Sum.inr) (Sum.inl n) l\nhF : Primrec₂ F\nthis : Primrec fun p ↦ Sum.casesOn (F p.1 p.2) (fun x ↦ none) some\na : α\n⊢ [a][0]? =... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Computability.Primrec.List | {
"line": 209,
"column": 24
} | {
"line": 209,
"column": 32
} | [
{
"pp": "case cons.zero.cons\nα : Type u_1\ninst✝ : Primcodable α\nF : List α → ℕ → ℕ ⊕ α :=\n fun l n ↦ List.foldl (fun s a ↦ Sum.casesOn s (fun x ↦ Nat.casesOn x (Sum.inr a) Sum.inl) Sum.inr) (Sum.inl n) l\nhF : Primrec₂ F\nthis : Primrec fun p ↦ Sum.casesOn (F p.1 p.2) (fun x ↦ none) some\na head✝ : α\ntail... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Computability.Partrec | {
"line": 418,
"column": 24
} | {
"line": 418,
"column": 32
} | [
{
"pp": "case some.zero\nα : Type u_1\nσ : Type u_3\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nf : α → ℕ\ng : α →. σ\nh : α → ℕ × σ →. σ\nhf : Computable f\nhg : Partrec g\nhh : Partrec₂ h\nn : ℕ\na : α\ne : decode n = Option.some a\n⊢ ((Part.map encode (Part.some 0)).bind fun n_1 ↦\n Nat.rec (Part.ma... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Computability.Partrec | {
"line": 418,
"column": 24
} | {
"line": 418,
"column": 32
} | [
{
"pp": "case some.succ\nα : Type u_1\nσ : Type u_3\ninst✝¹ : Primcodable α\ninst✝ : Primcodable σ\nf : α → ℕ\ng : α →. σ\nh : α → ℕ × σ →. σ\nhf : Computable f\nhg : Partrec g\nhh : Partrec₂ h\nn : ℕ\na : α\ne : decode n = Option.some a\nn✝ : ℕ\na✝ :\n ((Part.map encode (Part.some n✝)).bind fun n_1 ↦\n N... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Computability.Ackermann | {
"line": 71,
"column": 46
} | {
"line": 71,
"column": 57
} | [
{
"pp": "n : ℕ\n⊢ ack 0 n = n + 1",
"usedConstants": [
"Eq.mpr",
"ack",
"congrArg",
"id",
"instOfNatNat",
"instHAdd",
"ack.eq_1",
"HAdd.hAdd",
"Nat",
"instAddNat",
"Eq.refl",
"OfNat.ofNat",
"Eq"
]
}
] | by rw [ack] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.Ackermann | {
"line": 74,
"column": 59
} | {
"line": 74,
"column": 70
} | [
{
"pp": "m : ℕ\n⊢ ack (m + 1) 0 = ack m 1",
"usedConstants": [
"Eq.mpr",
"ack",
"congrArg",
"id",
"instOfNatNat",
"instHAdd",
"HAdd.hAdd",
"Nat",
"instAddNat",
"Eq.refl",
"OfNat.ofNat",
"Nat.succ",
"Eq",
"ack.eq_2"
]
}... | by rw [ack] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.Ackermann | {
"line": 77,
"column": 81
} | {
"line": 77,
"column": 92
} | [
{
"pp": "m n : ℕ\n⊢ ack (m + 1) (n + 1) = ack m (ack (m + 1) n)",
"usedConstants": [
"Eq.mpr",
"ack",
"congrArg",
"id",
"instOfNatNat",
"ack.eq_3",
"instHAdd",
"HAdd.hAdd",
"Nat",
"instAddNat",
"Eq.refl",
"OfNat.ofNat",
"Nat.succ"... | by rw [ack] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.Ackermann | {
"line": 97,
"column": 65
} | {
"line": 97,
"column": 86
} | [
{
"pp": "case succ\nn : ℕ\nIH : ack 3 n = 2 ^ (n + 3) - 3\n⊢ 2 * 2 ^ (n + 3) + 3 - (3 + 3) = 2 * 2 ^ (n + 3) - 3",
"usedConstants": [
"instPowNat",
"Eq.mpr",
"HMul.hMul",
"CommSemiring.toNonUnitalCommSemiring",
"congrArg",
"Nat.instMonoid",
"HSub.hSub",
"id",
... | Nat.add_sub_add_right | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 85,
"column": 2
} | {
"line": 86,
"column": 37
} | [
{
"pp": "f : ℝ → ℝ\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nhf : GrowsPolynomially f\n⊢ ∃ c > 0, ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (b * ↑n) ↑n, c * f ↑n ≤ f u",
"usedConstants": [
"Real.instIsOrderedRing",
"Real.partialOrder",
"Real.instLE",
"Real",
"Real.instArchimedean",
"HMul.hM... | obtain ⟨c, hc_mem, hc⟩ := hf.eventually_atTop_ge hb
exact ⟨c, hc_mem, hc.natCast_atTop⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 85,
"column": 2
} | {
"line": 86,
"column": 37
} | [
{
"pp": "f : ℝ → ℝ\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nhf : GrowsPolynomially f\n⊢ ∃ c > 0, ∀ᶠ (n : ℕ) in atTop, ∀ u ∈ Set.Icc (b * ↑n) ↑n, c * f ↑n ≤ f u",
"usedConstants": [
"Real.instIsOrderedRing",
"Real.partialOrder",
"Real.instLE",
"Real",
"Real.instArchimedean",
"HMul.hM... | obtain ⟨c, hc_mem, hc⟩ := hf.eventually_atTop_ge hb
exact ⟨c, hc_mem, hc.natCast_atTop⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 104,
"column": 6
} | {
"line": 104,
"column": 74
} | [
{
"pp": "case zero\nf : ℝ → ℝ\nhf✝ : GrowsPolynomially f\nhf' : ∀ (a : ℝ), ∃ b ≥ a, f b = 0\nc₁ : ℝ\nhc₁_mem : c₁ > 0\nc₂ : ℝ\nhc₂_mem : c₂ > 0\nhf : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (1 / 2 * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)\nx : ℝ\nhx_pos : 0 < x\nx₀ : ℝ\nhx₀_ge : x₀ ≥ max x 1\nhx₀ : f x₀ = 0\nx₀_po... | simp only [hx₀, mul_zero, Set.Icc_self, Set.mem_singleton_iff] at hx | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 284,
"column": 8
} | {
"line": 286,
"column": 53
} | [
{
"pp": "case inl.inr\nf g : ℝ → ℝ\nhf : GrowsPolynomially f\nhg : GrowsPolynomially g\nthis : GrowsPolynomially fun x ↦ |f x| * |g x|\nhf' : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x\nhg' : ∀ᶠ (x : ℝ) in atTop, g x ≤ 0\n⊢ GrowsPolynomially fun x ↦ f x * g x",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
... | have hmain : (fun x => f x * g x) =ᶠ[atTop] fun x => -|f x| * |g x| := by
filter_upwards [hf', hg'] with x hx₁ hx₂
simp [abs_of_nonneg hx₁, abs_of_nonpos hx₂] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Computability.AkraBazzi.AkraBazzi | {
"line": 185,
"column": 4
} | {
"line": 185,
"column": 56
} | [
{
"pp": "case inl\np : ℝ\nhp : p = 0\nh₁ : (fun x ↦ ‖deriv (fun z ↦ z ^ p * (1 - ε z)) x‖) =ᶠ[atTop] fun z ↦ z⁻¹ / log z ^ 2\n⊢ GrowsPolynomially fun x ↦ ‖deriv (fun z ↦ z ^ p * (1 - ε z)) x‖",
"usedConstants": [
"Norm.norm",
"Real.instPow",
"Real",
"instHDiv",
"Semiring.toModu... | refine GrowsPolynomially.congr_of_eventuallyEq h₁ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Computability.AkraBazzi.AkraBazzi | {
"line": 205,
"column": 4
} | {
"line": 205,
"column": 56
} | [
{
"pp": "case inl\np : ℝ\nhp : p = 0\nh₁ : (fun x ↦ ‖deriv (fun z ↦ z ^ p * (1 + ε z)) x‖) =ᶠ[atTop] fun z ↦ z⁻¹ / log z ^ 2\n⊢ GrowsPolynomially fun x ↦ ‖deriv (fun z ↦ z ^ p * (1 + ε z)) x‖",
"usedConstants": [
"Norm.norm",
"Real.instPow",
"Real",
"instHDiv",
"Semiring.toModu... | refine GrowsPolynomially.congr_of_eventuallyEq h₁ ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 250,
"column": 84
} | {
"line": 250,
"column": 92
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\nn : ℕ\nh_ind : ∀ m < n, 0 < T m\nhn : R.n₀ ≤ n\nthis : ∀ x ≥ 0, 0 ≤ g x\n⊢ 0 ≤ g ↑n",
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 250,
"column": 84
} | {
"line": 250,
"column": 92
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\nn : ℕ\nh_ind : ∀ m < n, 0 < T m\nhn : R.n₀ ≤ n\nthis : ∀ x ≥ 0, 0 ≤ g x\n⊢ 0 ≤ g ↑n",
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 250,
"column": 84
} | {
"line": 250,
"column": 92
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\nn : ℕ\nh_ind : ∀ m < n, 0 < T m\nhn : R.n₀ ≤ n\nthis : ∀ x ≥ 0, 0 ≤ g x\n⊢ 0 ≤ g ↑n",
"usedConstants": [
"Real.instIsOrderedRing",
"NonAssocSemiring... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 274,
"column": 2
} | {
"line": 324,
"column": 44
} | [
{
"pp": "f g : ℝ → ℝ\nhf : GrowsPolynomially f\nhg : GrowsPolynomially g\n⊢ GrowsPolynomially fun x ↦ f x * g x",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Eq.mpr",
"NegZeroClass.toNeg",
"mul_no... | suffices GrowsPolynomially fun x => |f x| * |g x| by
cases eventually_atTop_nonneg_or_nonpos hf with
| inl hf' =>
cases eventually_atTop_nonneg_or_nonpos hg with
| inl hg' =>
have hmain : (fun x => f x * g x) =ᶠ[atTop] fun x => |f x| * |g x| := by
filter_upwards [hf', hg'] with x h... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 274,
"column": 2
} | {
"line": 324,
"column": 44
} | [
{
"pp": "f g : ℝ → ℝ\nhf : GrowsPolynomially f\nhg : GrowsPolynomially g\n⊢ GrowsPolynomially fun x ↦ f x * g x",
"usedConstants": [
"Mathlib.Tactic.Ring.Common.mul_pf_left",
"AddGroup.toSubtractionMonoid",
"Real.instIsOrderedRing",
"Eq.mpr",
"NegZeroClass.toNeg",
"mul_no... | suffices GrowsPolynomially fun x => |f x| * |g x| by
cases eventually_atTop_nonneg_or_nonpos hf with
| inl hf' =>
cases eventually_atTop_nonneg_or_nonpos hg with
| inl hg' =>
have hmain : (fun x => f x * g x) =ᶠ[atTop] fun x => |f x| * |g x| := by
filter_upwards [hf', hg'] with x h... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 393,
"column": 56
} | {
"line": 393,
"column": 74
} | [
{
"pp": "case h\nx : ℝ\nhx : 1 < x\n⊢ deriv (fun x ↦ 1 + ε x) x = deriv ε x",
"usedConstants": [
"Eq.mpr",
"Real",
"Semiring.toModule",
"Real.denselyNormedField",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"deriv",
"NormedSpace.toModule",
"PseudoMetricS... | rw [deriv_fun_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 409,
"column": 56
} | {
"line": 409,
"column": 74
} | [
{
"pp": "case h\nx : ℝ\nhx : 1 < x\n⊢ deriv (fun x ↦ 1 + ε x) x = deriv ε x",
"usedConstants": [
"Eq.mpr",
"Real",
"Semiring.toModule",
"Real.denselyNormedField",
"congrArg",
"AddCommGroup.toAddCommMonoid",
"deriv",
"NormedSpace.toModule",
"PseudoMetricS... | rw [deriv_fun_add] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 452,
"column": 55
} | {
"line": 452,
"column": 63
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\ni : α\nn : ℕ\nhn : 1 < n\nhn' : ∀ (i : α), 0 < log (b i * ↑n)\n⊢ 1 < ↑n",
"usedConstants": [
"Real.partialOrder",
"Real",
"Preorder.toLT",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 452,
"column": 55
} | {
"line": 452,
"column": 63
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\ni : α\nn : ℕ\nhn : 1 < n\nhn' : ∀ (i : α), 0 < log (b i * ↑n)\n⊢ 1 < ↑n",
"usedConstants": [
"Real.partialOrder",
"Real",
"Preorder.toLT",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 452,
"column": 55
} | {
"line": 452,
"column": 63
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\ni : α\nn : ℕ\nhn : 1 < n\nhn' : ∀ (i : α), 0 < log (b i * ↑n)\n⊢ 1 < ↑n",
"usedConstants": [
"Real.partialOrder",
"Real",
"Preorder.toLT",
... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.PartrecCode | {
"line": 554,
"column": 4
} | {
"line": 563,
"column": 37
} | [
{
"pp": "case refine_2\nf : ℕ →. ℕ\n⊢ (∃ c, c.eval = f) → Nat.Partrec f",
"usedConstants": [
"Nat.Partrec.comp",
"Nat.Partrec",
"PFun",
"Nat.Partrec.prec",
"Nat.Partrec.zero",
"Exists",
"Nat.Partrec.Code",
"Nat.Partrec.rfind'",
"Nat.Partrec.right",
... | rintro ⟨c, rfl⟩
induction c with
| zero => exact Nat.Partrec.zero
| succ => exact Nat.Partrec.succ
| left => exact Nat.Partrec.left
| right => exact Nat.Partrec.right
| pair cf cg pf pg => exact pf.pair pg
| comp cf cg pf pg => exact pf.comp pg
| prec cf cg pf pg => exact pf.prec pg
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.PartrecCode | {
"line": 554,
"column": 4
} | {
"line": 563,
"column": 37
} | [
{
"pp": "case refine_2\nf : ℕ →. ℕ\n⊢ (∃ c, c.eval = f) → Nat.Partrec f",
"usedConstants": [
"Nat.Partrec.comp",
"Nat.Partrec",
"PFun",
"Nat.Partrec.prec",
"Nat.Partrec.zero",
"Exists",
"Nat.Partrec.Code",
"Nat.Partrec.rfind'",
"Nat.Partrec.right",
... | rintro ⟨c, rfl⟩
induction c with
| zero => exact Nat.Partrec.zero
| succ => exact Nat.Partrec.succ
| left => exact Nat.Partrec.left
| right => exact Nat.Partrec.right
| pair cf cg pf pg => exact pf.pair pg
| comp cf cg pf pg => exact pf.comp pg
| prec cf cg pf pg => exact pf.prec pg
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 458,
"column": 38
} | {
"line": 458,
"column": 46
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\ni : α\nn : ℕ\nhn : n ≠ 0\nthis : 0 < b i\n⊢ ↑n ≠ 0",
"usedConstants": [
"False",
"Real",
"eq_false",
"Real.instZero",
"Real.instRC... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 458,
"column": 38
} | {
"line": 458,
"column": 46
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\ni : α\nn : ℕ\nhn : n ≠ 0\nthis : 0 < b i\n⊢ ↑n ≠ 0",
"usedConstants": [
"False",
"Real",
"eq_false",
"Real.instZero",
"Real.instRC... | simp_all | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 458,
"column": 38
} | {
"line": 458,
"column": 46
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\ni : α\nn : ℕ\nhn : n ≠ 0\nthis : 0 < b i\n⊢ ↑n ≠ 0",
"usedConstants": [
"False",
"Real",
"eq_false",
"Real.instZero",
"Real.instRC... | simp_all | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.AkraBazzi.SumTransform | {
"line": 540,
"column": 28
} | {
"line": 540,
"column": 87
} | [
{
"pp": "α : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\n⊢ ∃ a, ?m.28 a = ?m.29",
"usedConstants": [
"Eq.mpr",
"Real.instPow",
"Set.mem_range",
"Real",
"HMul.hMul",
"Finset.univ",
... | by rw [← Set.mem_range]; exact R.one_mem_range_sumCoeffsExp | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Computability.AkraBazzi.AkraBazzi | {
"line": 505,
"column": 8
} | {
"line": 505,
"column": 72
} | [
{
"pp": "case bc.h.hbc.ha\nα : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := ⋯\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), c₁ * g ↑n ≤ sumTransform (p a b) g (r i n... | refine add_nonneg zero_le_one <| Finset.sum_nonneg fun j _ => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Computability.PartrecCode | {
"line": 974,
"column": 8
} | {
"line": 974,
"column": 38
} | [
{
"pp": "case succ.rfind'\nx✝ : Unit\np n : ℕ\nthis : List.range p = List.range (Nat.pair (unpair p).1 (encode (ofNat Code (unpair p).2)))\nk' : ℕ\nk : ℕ := k' + 1\nnk : n ≤ k'\ncf : Code\nhg :\n ∀ {k' : ℕ} {c' : Code} {n : ℕ},\n Nat.pair k' (encode c') < Nat.pair k (encode cf.rfind') →\n lup\n ... | rcases evaln k cf n with - | x | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.Computability.Language | {
"line": 281,
"column": 2
} | {
"line": 281,
"column": 59
} | [
{
"pp": "α : Type u_1\nl : Language α\n⊢ 1 + l∗ * l = l∗",
"usedConstants": [
"Eq.mpr",
"Language.instOne",
"HMul.hMul",
"Language.instAdd",
"congrArg",
"KStar.kstar",
"Language.one_add_self_mul_kstar_eq_kstar",
"id",
"Language.instKStar",
"instHAd... | rw [mul_self_kstar_comm, one_add_self_mul_kstar_eq_kstar] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Computability.Language | {
"line": 281,
"column": 2
} | {
"line": 281,
"column": 59
} | [
{
"pp": "α : Type u_1\nl : Language α\n⊢ 1 + l∗ * l = l∗",
"usedConstants": [
"Eq.mpr",
"Language.instOne",
"HMul.hMul",
"Language.instAdd",
"congrArg",
"KStar.kstar",
"Language.one_add_self_mul_kstar_eq_kstar",
"id",
"Language.instKStar",
"instHAd... | rw [mul_self_kstar_comm, one_add_self_mul_kstar_eq_kstar] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Computability.Language | {
"line": 281,
"column": 2
} | {
"line": 281,
"column": 59
} | [
{
"pp": "α : Type u_1\nl : Language α\n⊢ 1 + l∗ * l = l∗",
"usedConstants": [
"Eq.mpr",
"Language.instOne",
"HMul.hMul",
"Language.instAdd",
"congrArg",
"KStar.kstar",
"Language.one_add_self_mul_kstar_eq_kstar",
"id",
"Language.instKStar",
"instHAd... | rw [mul_self_kstar_comm, one_add_self_mul_kstar_eq_kstar] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Computability.ContextFreeGrammar | {
"line": 106,
"column": 2
} | {
"line": 106,
"column": 10
} | [
{
"pp": "case h\nT : Type u_1\nN : Type u_2\nr : ContextFreeRule T N\nu v p x y : List (Symbol T N)\nhxy : u = x ++ [Symbol.nonterminal r.input] ++ y ∧ v = x ++ r.output ++ y\n⊢ p ++ u = p ++ x ++ [Symbol.nonterminal r.input] ++ y ∧ p ++ v = p ++ x ++ r.output ++ y",
"usedConstants": [
"Symbol",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Computability.ContextFreeGrammar | {
"line": 113,
"column": 2
} | {
"line": 113,
"column": 15
} | [
{
"pp": "T : Type u_1\nN : Type u_2\nr : ContextFreeRule T N\nu v p x y : List (Symbol T N)\nhxy : u = x ++ [Symbol.nonterminal r.input] ++ y ∧ v = x ++ r.output ++ y\n⊢ ∃ p_1 q, u ++ p = p_1 ++ [Symbol.nonterminal r.input] ++ q ∧ v ++ p = p_1 ++ r.output ++ q",
"usedConstants": [
"Symbol",
"Exi... | use x, y ++ p | Mathlib.Tactic._aux_Mathlib_Tactic_Use___elabRules_Mathlib_Tactic_useSyntax_1 | Mathlib.Tactic.useSyntax |
Mathlib.Computability.ContextFreeGrammar | {
"line": 114,
"column": 2
} | {
"line": 114,
"column": 10
} | [
{
"pp": "case h\nT : Type u_1\nN : Type u_2\nr : ContextFreeRule T N\nu v p x y : List (Symbol T N)\nhxy : u = x ++ [Symbol.nonterminal r.input] ++ y ∧ v = x ++ r.output ++ y\n⊢ u ++ p = x ++ [Symbol.nonterminal r.input] ++ (y ++ p) ∧ v ++ p = x ++ r.output ++ (y ++ p)",
"usedConstants": [
"Symbol",
... | simp_all | Lean.Elab.Tactic.evalSimpAll | Lean.Parser.Tactic.simpAll |
Mathlib.Computability.AkraBazzi.AkraBazzi | {
"line": 612,
"column": 8
} | {
"line": 612,
"column": 72
} | [
{
"pp": "case bc.h.hbc.ha\nα : Type u_1\ninst✝¹ : Fintype α\nT : ℕ → ℝ\ng : ℝ → ℝ\na b : α → ℝ\nr : α → ℕ → ℕ\ninst✝ : Nonempty α\nR : AkraBazziRecurrence T g a b r\nb' : ℝ := ⋯\nhb_pos : 0 < b'\nc₁ : ℝ\nhc₁ : c₁ > 0\nh_sumTransform_aux : ∀ᶠ (n : ℕ) in atTop, ∀ (i : α), sumTransform (p a b) g (r i n) n ≤ c₁ * g... | refine add_nonneg zero_le_one <| Finset.sum_nonneg fun j _ => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Computability.AkraBazzi.GrowsPolynomially | {
"line": 537,
"column": 2
} | {
"line": 537,
"column": 25
} | [
{
"pp": "f : ℝ → ℝ\np : ℝ\nhf : GrowsPolynomially f\nhf_nonneg : ∀ᶠ (x : ℝ) in atTop, 0 ≤ f x\nb : ℝ\nhb : b ∈ Set.Ioo 0 1\nc₁ : ℝ\nhc₁_mem : 0 < c₁\nc₂ : ℝ\nhc₂_mem : c₂ > 0\nhfnew : ∀ᶠ (x : ℝ) in atTop, ∀ u ∈ Set.Icc (b * x) x, f u ∈ Set.Icc (c₁ * f x) (c₂ * f x)\nhc₁p : 0 < c₁ ^ p\nhc₂p : 0 < c₂ ^ p\nx✝ : 0 ... | cases le_or_gt 0 p with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalCases | null |
Mathlib.Data.Nat.Bitwise | {
"line": 221,
"column": 2
} | {
"line": 221,
"column": 54
} | [
{
"pp": "case h.h\nf : Bool → Bool → Bool\nm n : ℕ\n⊢ bitwise (swap f) m n = bitwise f n m",
"usedConstants": [
"Nat.bit",
"Eq.mpr",
"congrArg",
"Function.swap",
"id",
"Nat.binaryRec'",
"instDecidableEqBool",
"instOfNatNat",
"Bool.true",
"Nat.bitwi... | induction m using Nat.binaryRec' generalizing n with | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | null |
Mathlib.Data.Nat.Bitwise | {
"line": 244,
"column": 2
} | {
"line": 244,
"column": 37
} | [
{
"pp": "n i : ℕ\n⊢ n &&& 2 ^ i = (n.testBit i).toNat * 2 ^ i",
"usedConstants": [
"HMul.hMul",
"Nat.instAndOp",
"Nat.instMonoid",
"Bool.toNat",
"instMulNat",
"instOfNatNat",
"Nat.eq_of_testBit_eq",
"Monoid.toPow",
"HPow.hPow",
"Nat",
"Nat.te... | refine eq_of_testBit_eq fun j => ?_ | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Data.List.ReduceOption | {
"line": 95,
"column": 11
} | {
"line": 95,
"column": 19
} | [
{
"pp": "case h\nα : Type u_1\nl : List (Option α)\nl' : List α\na : α\nl₁ w✝ : List (Option α)\nh : l = l₁ ++ w✝\nhl₁ : l₁.reduceOption = l'\nm n : ℕ\nhl₂ : w✝ = replicate m none ++ some a :: replicate n none\n⊢ True ∧ True ∧ True",
"usedConstants": [
"congrArg",
"and_self",
"And",
... | and_self | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
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