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.Copy | {
"line": 206,
"column": 44
} | {
"line": 206,
"column": 55
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nX : Type u_3\nα : Type u_4\nβ : Type u_5\nγ : Type u_6\nG G₁ G₂ G₃ : SimpleGraph V\nH : SimpleGraph W\nI : SimpleGraph X\nA : SimpleGraph α\nB : SimpleGraph β\nC : SimpleGraph γ\nf : ⊤.Copy G\nv w : α\nh : G.Adj (f.toEmbedding v) (f.toEmbedding w)\n⊢ ⊤.Adj v w",
"usedCon... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Copy | {
"line": 316,
"column": 2
} | {
"line": 316,
"column": 42
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\nf : G.Copy H\nv : V\ninst✝¹ : Fintype ↑(G.neighborSet v)\ninst✝ : Fintype ↑(H.neighborSet (f v))\n⊢ G.degree v ≤ H.degree (f v)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Copy | {
"line": 327,
"column": 47
} | {
"line": 329,
"column": 23
} | [
{
"pp": "V : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\ninst✝³ : Fintype V\ninst✝² : Fintype W\ninst✝¹ : DecidableRel G.Adj\ninst✝ : DecidableRel H.Adj\nh : G ⊑ H\n⊢ G.maxDegree ≤ H.maxDegree",
"usedConstants": [
"SimpleGraph.maxDegree",
"SimpleGraph.IsContained",
"Simpl... | by
have ⟨f⟩ := h
exact f.max_degree_le | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Copy | {
"line": 511,
"column": 2
} | {
"line": 511,
"column": 38
} | [
{
"pp": "case e_s\nV : Type u_1\nW : Type u_2\nG : SimpleGraph V\nH : SimpleGraph W\ninst✝² : Fintype V\ninst✝¹ : Fintype { f // Injective ⇑f }\ninst✝ : DecidableEq G.Subgraph\n⊢ ↑{G' | Nonempty (H ≃g G'.coe)} = ↑(image Copy.toSubgraph univ)",
"usedConstants": [
"Eq.mpr",
"Set.image_univ",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 347,
"column": 8
} | {
"line": 348,
"column": 28
} | [
{
"pp": "ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : G.Subgraph\na b : V\ns : Set G.Subgraph\n⊢ ∀ {v w : V}, (∃ G' ∈ s, G'.Adj v w) → G.Adj v w",
"usedConstants": [
"SimpleGraph.Subgraph",
"SimpleGraph.Adj",
"SimpleGraph.Subgraph.adj_sub",
"Membership.mem",
... | rintro a b ⟨G', -, hab⟩
exact G'.adj_sub hab | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 347,
"column": 8
} | {
"line": 348,
"column": 28
} | [
{
"pp": "ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : G.Subgraph\na b : V\ns : Set G.Subgraph\n⊢ ∀ {v w : V}, (∃ G' ∈ s, G'.Adj v w) → G.Adj v w",
"usedConstants": [
"SimpleGraph.Subgraph",
"SimpleGraph.Adj",
"SimpleGraph.Subgraph.adj_sub",
"Membership.mem",
... | rintro a b ⟨G', -, hab⟩
exact G'.adj_sub hab | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 352,
"column": 30
} | {
"line": 352,
"column": 52
} | [
{
"pp": "ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : G.Subgraph\na✝ b✝ : V\ns : Set G.Subgraph\na b : V\nh : (fun a b ↦ ∃ G' ∈ s, G'.Adj a b) a b\n⊢ (fun a b ↦ ∃ G' ∈ s, G'.Adj a b) b a",
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleGraph.Subgraph",
"Membershi... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 487,
"column": 37
} | {
"line": 487,
"column": 48
} | [
{
"pp": "ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : G.Subgraph\na b : V\nι✝ : Type u\nκ✝ : ι✝ → Type u\nf : (a : ι✝) → κ✝ a → G.Subgraph\n⊢ (⨅ a, ⨆ b, f a b).verts = (⨆ g, ⨅ a, f a (g a)).verts",
"usedConstants": [
"Eq.mpr",
"iInf",
"congrArg",
"iSup",
"Si... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 616,
"column": 32
} | {
"line": 616,
"column": 43
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nH₁ H₂ : G.Subgraph\nh : Disjoint H₁ H₂\n⊢ H₁.edgeSet ⊓ H₂.edgeSet ≤ ⊥",
"usedConstants": [
"Eq.mpr",
"CompleteBooleanAlgebra.toCompleteDistribLattice",
"CompleteLattice.toLattice",
"OrderBot.toBot",
"PartialOrder.toPreorder",
"Preor... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 723,
"column": 66
} | {
"line": 723,
"column": 77
} | [
{
"pp": "ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : G.Subgraph\na✝ b✝ : V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nH : G.Subgraph\na b : V\n⊢ (H.verts.toFinset, fun a b ↦ decide (H.Adj a b)).2 a b = true → G.Adj a b",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 724,
"column": 19
} | {
"line": 724,
"column": 30
} | [
{
"pp": "ι : Sort u_1\nV : Type u\nW : Type v\nG : SimpleGraph V\nG₁ G₂ : G.Subgraph\na✝ b✝ : V\ninst✝² : DecidableEq V\ninst✝¹ : Fintype V\ninst✝ : DecidableRel G.Adj\nH : G.Subgraph\na b : V\n⊢ (H.verts.toFinset, fun a b ↦ decide (H.Adj a b)).2 a b = true → a ∈ (H.verts.toFinset, fun a b ↦ decide (H.Adj a b))... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 838,
"column": 54
} | {
"line": 843,
"column": 35
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nG' : G.Subgraph\nv : V\ninst✝ : Fintype ↑(G'.neighborSet v)\nhG : G'.verts.Subsingleton\n⊢ G'.degree v = 0",
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"congrArg",
"SimpleGraph.Subgraph.coe_degree",
"SimpleGraph.Subgraph.degree_of_notMem_ver... | by
by_cases hv : v ∈ G'.verts
· rw [← G'.coe_degree ⟨v, hv⟩]
have := (Set.subsingleton_coe _).mpr hG
exact G'.coe.degree_eq_zero_of_subsingleton ⟨v, hv⟩
· exact degree_of_notMem_verts hv | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 976,
"column": 28
} | {
"line": 976,
"column": 53
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w : V\nhvw : G.Adj v w\nu : V\nthis : w = u ↔ u = w\n⊢ u ∈ (G.subgraphOfAdj hvw).neighborSet v ↔ u ∈ {w}",
"usedConstants": [
"SimpleGraph.Subgraph.mem_neighborSet._simp_1",
"Eq.mpr",
"False",
"Sym2.Rel",
"Sym2.eq._simp_1",
"eq_fa... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Traversal | {
"line": 145,
"column": 2
} | {
"line": 145,
"column": 13
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w : V\np : G.Walk v w\nhp : ¬p.Nil\n⊢ G.Adj v p.snd",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Traversal | {
"line": 145,
"column": 37
} | {
"line": 145,
"column": 72
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w : V\np : G.Walk v w\nhp : ¬p.Nil\n⊢ 0 < p.length",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 1049,
"column": 4
} | {
"line": 1049,
"column": 15
} | [
{
"pp": "case Adj.h.h.a\nV : Type u\nG : SimpleGraph V\nG' : G.Subgraph\nG'' : G'.coe.Subgraph\nx✝¹ x✝ : ↑G'.verts\n⊢ (G'.Adj ↑x✝¹ ↑x✝ ∧ ∃ (hv : ↑x✝¹ ∈ G'.verts) (hw : ↑x✝ ∈ G'.verts), G''.Adj ⟨↑x✝¹, hv⟩ ⟨↑x✝, hw⟩) ↔ G''.Adj x✝¹ x✝",
"usedConstants": [
"Eq.mpr",
"Iff.of_eq",
"congrArg",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Subgraph | {
"line": 1204,
"column": 43
} | {
"line": 1207,
"column": 22
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nG' : G.Subgraph\n⊢ G'.IsInduced ↔ ∃ s, G' = ⊤.induce s",
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleGraph.Subgraph",
"SimpleGraph.Adj",
"Membership.mem",
"Exists",
"Eq.rec",
"id",
"SimpleGraph.Subgraph.instTop... | by
refine ⟨fun h ↦ ⟨G'.verts, h.induce_top_verts.symm⟩, fun ⟨s, h⟩ _ hu _ hv hadj ↦ ?_⟩
rw [h, (h ▸ rfl : s = G'.verts)]
exact ⟨hu, hv, hadj⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 222,
"column": 2
} | {
"line": 222,
"column": 13
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\n⊢ List.map (fun x ↦ x.toProd.2) p.darts = p.support.tail",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 312,
"column": 58
} | {
"line": 312,
"column": 78
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu✝ v✝ u c v : V\nh₁ : G.Adj u v\nw₁ : G.Walk v c\nv' : V\nh₂ : G.Adj u v'\nw₂ : G.Walk v' c\nh : (cons' u v c h₁ w₁).edges = (cons' u v' c h₂ w₂).edges\nh₃ : u ≠ v'\n⊢ v = v' ∧ w₁.edges = w₂.edges",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 260,
"column": 6
} | {
"line": 261,
"column": 12
} | [
{
"pp": "case inr\nV : Type u\nG : SimpleGraph V\nu v : V\ni : ℕ\nu✝ v✝ w✝ : V\nh : G.Adj u✝ v✝\np : G.Walk v✝ w✝\nih : p.reverse.getVert i = p.getVert (p.length - i)\nhi : ¬i < p.length\nhi' : p.length < i\n⊢ (cons ⋯ nil).getVert (i - p.length) = (cons h p).getVert (p.length + 1 - i)",
"usedConstants": [
... | · rw [Nat.eq_add_of_sub_eq (Nat.sub_pos_of_lt hi') rfl, Nat.sub_eq_zero_of_le hi']
simp | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 466,
"column": 4
} | {
"line": 466,
"column": 15
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nl✝ : List V\nhead✝ v : V\nl : List V\nhne : head✝ :: v :: l ≠ []\nhchain : List.IsChain G.Adj (head✝ :: v :: l)\n⊢ (ofSupport (head✝ :: v :: l) hne hchain).support = head✝ :: v :: l",
"usedConstants": [
"List.head",
"List.getLast",
"Eq.mpr",
"S... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 501,
"column": 4
} | {
"line": 501,
"column": 33
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nl✝ : List G.Dart\nd₁ d₂ : G.Dart\nl : List G.Dart\nhne : d₁ :: d₂ :: l ≠ []\nhchain : List.IsChain G.DartAdj (d₁ :: d₂ :: l)\n⊢ (ofDarts (d₁ :: d₂ :: l) hne hchain).darts = d₁ :: d₂ :: l",
"usedConstants": [
"List.head",
"List.getLast",
"List.IsChain... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 118,
"column": 2
} | {
"line": 119,
"column": 69
} | [
{
"pp": "α : Type u_1\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nU : Finset α\nhU : U ∈ P.parts\nV : Finset α\nhV : V ∈ P.parts\nhUV : U ≠ V\nh₂ : ¬G.IsUniform ε U V\nhX : G.nonuniformWitness ε U V ∈ P.nonunif... | grw [sum_const, smul_eq_mul, card_filter_atomise_le_two_pow (s := U) hX,
Finpartition.card_nonuniformWitnesses_le, filter_subset] <;> simp | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 468,
"column": 2
} | {
"line": 468,
"column": 13
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nd : G.Dart\nh : d ∈ p.darts\n⊢ d.toProd.2 ∈ p.support",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 559,
"column": 2
} | {
"line": 559,
"column": 78
} | [
{
"pp": "case h\nV : Type u\nG : SimpleGraph V\nu v w : V\nh : G.Adj u v\np : G.Walk v w\nn : ℕ\nhn : n ≠ 0\n⊢ ((cons h p).drop n).support = ((p.drop (n - 1)).copy ⋯ ⋯).support",
"usedConstants": [
"Iff.mpr",
"Nat.ne_zero_iff_zero_lt",
"Exists",
"Ne",
"instOfNatNat",
"ins... | obtain ⟨_, rfl⟩ := Nat.exists_add_one_eq.mpr (Nat.ne_zero_iff_zero_lt.mp hn) | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 640,
"column": 2
} | {
"line": 640,
"column": 13
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\n⊢ p.reverse.snd = p.penultimate",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Subwalks | {
"line": 231,
"column": 2
} | {
"line": 238,
"column": 46
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\nn k : ℕ\np : G.Walk u v\nh : n ≤ k\n⊢ (p.drop k).IsSubwalk (p.drop n)",
"usedConstants": [
"Nat.recAux",
"SimpleGraph.Walk.drop_zero",
"HEq.refl",
"SimpleGraph.Walk.IsSubwalk.copy",
"SimpleGraph.Adj",
"SimpleGraph.Walk.ge... | induction k, h using Nat.le_induction with
| base => rfl
| succ k h ih =>
apply IsSubwalk.trans ?_ ih
clear h ih
induction k generalizing p u with
| zero => exact p.drop_zero ▸ (p.isSubwalk_rfl.copy rfl rfl p.getVert_zero.symm rfl).tail
| succ _ ih => cases p <;> simp [drop, ih] | _private.Lean.Elab.Tactic.Induction.0.Lean.Elab.Tactic.evalInduction | Lean.Parser.Tactic.induction |
Mathlib.Combinatorics.SimpleGraph.Walk.Subwalks | {
"line": 231,
"column": 2
} | {
"line": 238,
"column": 46
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\nn k : ℕ\np : G.Walk u v\nh : n ≤ k\n⊢ (p.drop k).IsSubwalk (p.drop n)",
"usedConstants": [
"Nat.recAux",
"SimpleGraph.Walk.drop_zero",
"HEq.refl",
"SimpleGraph.Walk.IsSubwalk.copy",
"SimpleGraph.Adj",
"SimpleGraph.Walk.ge... | induction k, h using Nat.le_induction with
| base => rfl
| succ k h ih =>
apply IsSubwalk.trans ?_ ih
clear h ih
induction k generalizing p u with
| zero => exact p.drop_zero ▸ (p.isSubwalk_rfl.copy rfl rfl p.getVert_zero.symm rfl).tail
| succ _ ih => cases p <;> simp [drop, ih] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Walk.Subwalks | {
"line": 231,
"column": 2
} | {
"line": 238,
"column": 46
} | [
{
"pp": "V : Type u_1\nG : SimpleGraph V\nu v : V\nn k : ℕ\np : G.Walk u v\nh : n ≤ k\n⊢ (p.drop k).IsSubwalk (p.drop n)",
"usedConstants": [
"Nat.recAux",
"SimpleGraph.Walk.drop_zero",
"HEq.refl",
"SimpleGraph.Walk.IsSubwalk.copy",
"SimpleGraph.Adj",
"SimpleGraph.Walk.ge... | induction k, h using Nat.le_induction with
| base => rfl
| succ k h ih =>
apply IsSubwalk.trans ?_ ih
clear h ih
induction k generalizing p u with
| zero => exact p.drop_zero ▸ (p.isSubwalk_rfl.copy rfl rfl p.getVert_zero.symm rfl).tail
| succ _ ih => cases p <;> simp [drop, ih] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 251,
"column": 2
} | {
"line": 251,
"column": 34
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w u : V\ninst✝ : DecidableEq V\np : G.Walk u v\nh : w ∈ p.support\nhsu : ¬1 ≤ (p.takeUntil w h).length\n⊢ u = w",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 275,
"column": 25
} | {
"line": 275,
"column": 36
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v w : V\np : G.Walk v w\nh : u ∈ p.support\nhuw : u ≠ w\nhl : (p.takeUntil u h).length = p.length\n⊢ u = w",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 322,
"column": 2
} | {
"line": 322,
"column": 13
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv : V\ninst✝ : DecidableEq V\nc : G.Walk v v\nu : V\nh : u ∈ c.support\n⊢ (c.rotate u h).length = c.length",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 27
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nh : p.IsTrail\n⊢ p.reverse.IsTrail",
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"_private.Mathlib.Combinatorics.SimpleGraph.Paths.0.SimpleGraph.Walk.IsTrail.reverse._simp_1_1",
"List.Nodup",
"List",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 181,
"column": 4
} | {
"line": 181,
"column": 23
} | [
{
"pp": "case a\nV : Type u\nG : SimpleGraph V\ninst✝ : Fintype ↑G.edgeSet\nu v : V\nw : G.Walk u v\nh✝ : w.IsTrail\nedges : Finset (Sym2 V) := ⋯\nthis : edges.card = w.length\ne : Sym2 V\nh : e ∈ edges\n⊢ e ∈ w.edges",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 203,
"column": 2
} | {
"line": 203,
"column": 26
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nh : p.IsPath\n⊢ p.reverse.IsPath",
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleGraph.Walk.support",
"id",
"_private.Mathlib.Combinatorics.SimpleGraph.Paths.0.SimpleGraph.Walk.IsPath.reverse._simp_1_1",
"... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 322,
"column": 13
} | {
"line": 322,
"column": 24
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu : V\np : G.Walk u u\nh : p.reverse.IsCycle\n⊢ p.IsCycle",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 398,
"column": 6
} | {
"line": 398,
"column": 55
} | [
{
"pp": "case pos\nV : Type u\nG : SimpleGraph V\nu✝ v✝ v w u : V\nh : G.Adj v w\np : G.Walk w u\nihp :\n p.IsPath → ∀ ⦃n : ℕ⦄, n ∈ {i | i ≤ p.length} → ∀ ⦃m : ℕ⦄, m ∈ {i | i ≤ p.length} → p.getVert n = p.getVert m → n = m\nhp : (cons h p).IsPath\nn : ℕ\nhn : n ≤ p.length + 1\nm : ℕ\nhm : m ≤ p.length + 1\nhnm... | simp only [hm0, Walk.getVert_cons p h hn0] at hnm | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 419,
"column": 4
} | {
"line": 419,
"column": 15
} | [
{
"pp": "case nil\nV : Type u\nG : SimpleGraph V\nu : V\ni : ℕ\nhp : Walk.nil.IsPath\nhi : i ≤ Walk.nil.length\n⊢ Walk.nil.getVert i = u ↔ i = 0",
"usedConstants": [
"Eq.mpr",
"congrArg",
"true_iff",
"id",
"instOfNatNat",
"Iff",
"SimpleGraph.Walk.nil",
"Nat",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 443,
"column": 12
} | {
"line": 443,
"column": 38
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v u✝ v✝ w✝ : V\nh : G.Adj u✝ v✝\nq : G.Walk v✝ w✝\nih : Set.InjOn q.getVert {i | i ≤ q.length} → q.IsPath\nhinj : Set.InjOn (cons h q).getVert {i | i ≤ (cons h q).length}\nn : ℕ\nhn : n ≤ q.length\nm : ℕ\nhm : m ≤ q.length\nhnm : q.getVert n = q.getVert m\n⊢ (cons h q).... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 461,
"column": 2
} | {
"line": 461,
"column": 20
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v w : V\np : G.Walk u v\nhp : p.IsPath\nhmem : s(v, w) ∈ p.edges\n⊢ w = p.penultimate",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 131,
"column": 64
} | {
"line": 131,
"column": 75
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nthis :\n ∀ (a b : α),\n a ≠ b → {s | s ∈ G.cliqueSet 3 ∧ s(a, b) ∈ s.sym2} = {s | G.Adj a b ∧ ∃ c, G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}}\nhG :\n ∀ ⦃e : Sym2 α⦄,\n ¬e.IsDiag → {s | (∃ a b c, G.Adj a b ∧ G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}) ∧ e ∈ s.sym2}.Subsin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 132,
"column": 25
} | {
"line": 132,
"column": 36
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nthis :\n ∀ (a b : α),\n a ≠ b → {s | s ∈ G.cliqueSet 3 ∧ s(a, b) ∈ s.sym2} = {s | G.Adj a b ∧ ∃ c, G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}}\nhG :\n ∀ ⦃e : Sym2 α⦄,\n ¬e.IsDiag → {s | (∃ a b c, G.Adj a b ∧ G.Adj a c ∧ G.Adj b c ∧ s = {a, b, c}) ∧ e ∈ s.sym2}.Subsin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 157,
"column": 4
} | {
"line": 159,
"column": 11
} | [
{
"pp": "case refine_2\nα : Type u_1\nG : SimpleGraph α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : DecidableRel G.Adj\nhG : G.EdgeDisjointTriangles\ne : Sym2 α\nhe : e ∈ G.edgeFinset\n⊢ #(bipartiteBelow (fun s e ↦ e ∈ s.sym2) (G.cliqueFinset 3) e) ≤ 1",
"usedConstants": [
"Eq.mpr",
"Mu... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 166,
"column": 4
} | {
"line": 168,
"column": 11
} | [
{
"pp": "case refine_1\nα : Type u_1\nG : SimpleGraph α\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\ninst✝ : DecidableRel G.Adj\nhG : G.LocallyLinear\n⊢ ∀ a ∈ G.edgeFinset, 1 ≤ #(bipartiteAbove (fun e s ↦ e ∈ s.sym2) (G.cliqueFinset 3) a)",
"usedConstants": [
"Eq.mpr",
"MulOne.toOne",
"Sim... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 149,
"column": 2
} | {
"line": 149,
"column": 13
} | [
{
"pp": "case mpr\nα : Type u_1\nβ : Type u_2\nG : SimpleGraph α\nf : α ↪ β\ns : Finset α\nhs : G.IsClique ↑s\nht : (map f s).Nontrivial\n⊢ (SimpleGraph.map (⇑f) G).IsClique ↑(map f s)",
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"SimpleGraph.isClique_map_image_iff._simp_1",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 175,
"column": 2
} | {
"line": 175,
"column": 26
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nv w : α\ns : Set α\nhc : (G ⊔ edge v w).IsClique s\nx✝ : α\nhx : x✝ ∈ s \\ {v}\ny✝ : α\nhy : y✝ ∈ s \\ {v}\nhxy : x✝ ≠ y✝\n⊢ G.Adj x✝ y✝",
"usedConstants": [
"SimpleGraph.edge",
"SimpleGraph.Adj",
"Membership.mem",
"Set.instSingletonSet",
... | have := hc hx.1 hy.1 hxy | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 565,
"column": 38
} | {
"line": 565,
"column": 66
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\ninst✝ : DecidableEq V\nx : V\nw : G.Walk u v\nhw : w.IsTrail\nhx : x ∈ w.support\n⊢ ((w.takeUntil x hx).edges ++ (w.dropUntil x hx).edges).Nodup",
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleGraph.Walk",
"SimpleGraph.Walk.take_spec... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 254,
"column": 61
} | {
"line": 254,
"column": 72
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\nG : SimpleGraph α\nε : 𝕜\ninst✝² : Fintype α\ninst✝¹ : DecidableRel G.Adj\ninst✝ : DecidableEq α\nhG : G.EdgeDisjointTriangles\ntris_big : ε * ↑(Fintype.card α ^ 2) ≤ ↑(#(G.cliqueFinset 3))\n⊢ (↑(... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 264,
"column": 28
} | {
"line": 264,
"column": 39
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\nG : SimpleGraph α\nε : 𝕜\ninst✝² : Fintype α\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty α\nhε : G.FarFromTriangleFree ε\n⊢ ε * ↑(Fintype.card α) ^ 2 ≤ ↑(#G.edgeFinset)",
"usedConstants": [... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 267,
"column": 6
} | {
"line": 267,
"column": 36
} | [
{
"pp": "α : Type u_1\n𝕜 : Type u_3\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\nG : SimpleGraph α\nε : 𝕜\ninst✝² : Fintype α\ninst✝¹ : DecidableRel G.Adj\ninst✝ : Nonempty α\nhε : G.FarFromTriangleFree ε\n⊢ ↑((Fintype.card α).choose 2) < 2⁻¹ * ↑(Fintype.card α) ^ 2",
"use... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 832,
"column": 6
} | {
"line": 832,
"column": 17
} | [
{
"pp": "case refine_2\nV : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nv v' : V\nhvv' : G.Adj v v'\nw : G.Walk v' v\nhw : (cons hvv' w).IsCircuit\n⊢ (cons hvv' w.bypass).support.tail.Nodup",
"usedConstants": [
"SimpleGraph.Walk.support",
"SimpleGraph.Walk.bypass",
"id",
"List.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 426,
"column": 2
} | {
"line": 426,
"column": 37
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\nn : ℕ\ninst✝ : Fintype α\nhc : Nonempty ((completeGraph (Fin n)).Copy G)\n⊢ n ≤ Fintype.card α",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 344,
"column": 4
} | {
"line": 344,
"column": 45
} | [
{
"pp": "case inl.refine_1\nα : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhP : P.IsEquipartition\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhU : ... | · exact mod_cast G.edgeDensity_nonneg _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 500,
"column": 2
} | {
"line": 501,
"column": 68
} | [
{
"pp": "case mp\nα : Type u_1\nG : SimpleGraph α\n⊢ G.CliqueFree 2 → G = ⊥",
"usedConstants": [
"Eq.mpr",
"False",
"Finset.coe_singleton",
"eq_false",
"SimpleGraph.Adj.ne",
"Sym2.mk",
"congrArg",
"Finset",
"_private.Mathlib.Combinatorics.SimpleGraph.Cli... | · simp_rw [← edgeSet_eq_empty, Set.eq_empty_iff_forall_notMem, Sym2.forall, mem_edgeSet]
exact fun h a b hab => h _ ⟨by simpa [hab.ne], card_pair hab.ne⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 532,
"column": 6
} | {
"line": 532,
"column": 51
} | [
{
"pp": "case right\nα : Type u_1\nG : SimpleGraph α\nn : ℕ\ninst✝ : DecidableEq α\nh : Maximal (fun H ↦ H.CliqueFree (n + 1)) G\nx y : α\nhne : x ≠ y\nhn : ¬G.Adj x y\nt : Finset α\nhc : (G ⊔ edge x y).IsNClique (n + 1) t\nh1 : x ∈ t\nh2 : y ∈ t\n⊢ G.IsNClique n (insert x ((t.erase y).erase x)) ∧ G.IsNClique n... | insert_erase <| mem_erase_of_ne_of_mem hne h1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 661,
"column": 4
} | {
"line": 661,
"column": 15
} | [
{
"pp": "case inr\nα : Type u_1\nβ : Type u_2\nG : SimpleGraph α\ne : α ≃ β\nn : ℕ\nhn : n ≠ 1\n⊢ (SimpleGraph.map (⇑e) G).cliqueSet n = map e.toEmbedding '' G.cliqueSet n",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Removal | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 13
} | [
{
"pp": "n k : ℕ\nhk : 0 < k\nhn : k ≤ n\n⊢ k ≤ k * (n / k)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 761,
"column": 2
} | {
"line": 761,
"column": 37
} | [
{
"pp": "α : Type u_1\nG : SimpleGraph α\ninst✝² : Fintype α\ninst✝¹ : DecidableEq α\ninst✝ : DecidableRel G.Adj\nn : ℕ\ns : Finset α\n⊢ s ∈ G.cliqueFinset n → s ∈ powersetCard n univ",
"usedConstants": [
"Eq.mpr",
"SimpleGraph.IsNClique",
"Finset.univ",
"congrArg",
"Finset.sub... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 941,
"column": 2
} | {
"line": 941,
"column": 39
} | [
{
"pp": "α : Type u_3\nG : SimpleGraph α\ninst✝ : Finite α\nt : Finset α\ntc : Gᶜ.IsClique ↑t\n⊢ #t ≤ G.indepNum",
"usedConstants": [
"_private.Mathlib.Combinatorics.SimpleGraph.Clique.0.SimpleGraph.IsIndepSet.card_le_indepNum._simp_1_2",
"Eq.mpr",
"SimpleGraph.IsNClique",
"congrArg"... | simp_rw [indepNum, ← isNClique_compl] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 945,
"column": 2
} | {
"line": 945,
"column": 39
} | [
{
"pp": "α : Type u_3\nG : SimpleGraph α\n⊢ ∃ s, G.IsNIndepSet G.indepNum s",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Combinatorics.SimpleGraph.Clique.0.SimpleGraph.exists_isNIndepSet_indepNum._simp_1_2",
"SimpleGraph.IsNClique",
"congrArg",
"Compl.compl",
"Finset"... | simp_rw [indepNum, ← isNClique_compl] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 984,
"column": 2
} | {
"line": 984,
"column": 37
} | [
{
"pp": "α : Type u_3\nG : SimpleGraph α\ninst✝ : Finite α\nt : Finset α\ntmc : G.IsMaximumIndepSet t\n⊢ #t = G.indepNum",
"usedConstants": [
"SimpleGraph.IsMaximumIndepSet",
"congrArg",
"Compl.compl",
"Eq.mp",
"SimpleGraph",
"SimpleGraph.instCompl",
"propext",
... | rw [← isMaximumClique_compl] at tmc | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 985,
"column": 2
} | {
"line": 985,
"column": 39
} | [
{
"pp": "α : Type u_3\nG : SimpleGraph α\ninst✝ : Finite α\nt : Finset α\ntmc : Gᶜ.IsMaximumClique t\n⊢ #t = G.indepNum",
"usedConstants": [
"Eq.mpr",
"SimpleGraph.IsNClique",
"congrArg",
"Compl.compl",
"Finset",
"setOf",
"_private.Mathlib.Combinatorics.SimpleGraph.... | simp_rw [indepNum, ← isNClique_compl] | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | Mathlib.Tactic.tacticSimp_rw___ |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 407,
"column": 41
} | {
"line": 407,
"column": 51
} | [
{
"pp": "α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhε₁ : ε ≤ 1\nhU : U ∈ P.... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 449,
"column": 4
} | {
"line": 449,
"column": 15
} | [
{
"pp": "α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhε₁ : ε ≤ 1\nhU : U ∈ P.... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Removal | {
"line": 150,
"column": 18
} | {
"line": 150,
"column": 50
} | [
{
"pp": "α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nhG : G.FarFromTriangleFree ε\nh✝ : Nonempty α\nhε : 0 < ε\nl : ℕ := ⌈4 / ε⌉₊\nhl : 4 / ε ≤ ↑l\nhl' : Fintype.card α ≤ l\n⊢ 1 ≤ ↑(#(G.cliqueFinset 3))",
"usedConstants": [
"Eq.mpr",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite | {
"line": 155,
"column": 2
} | {
"line": 157,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nt : Finset (α × β × γ)\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq γ\nx y z : α ⊕ β ⊕ γ\n⊢ (graph t).Adj x y →\n (graph t).Adj x z →\n (graph t).Adj y z →\n ∃ a b c,\n {in₀ a, in₁ b, in₂ c} = {x, y, z} ∧\n ... | rintro (_ | _ | _) (_ | _ | _) (_ | _ | _) <;>
refine ⟨_, _, _, by ext; simp only [Finset.mem_insert, Finset.mem_singleton]; try tauto,
?_, ?_, ?_⟩ <;> constructor <;> assumption | Lean.Parser.Tactic.«_aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tactic_<;>__1» | Lean.Parser.Tactic.«tactic_<;>_» |
Mathlib.Combinatorics.SimpleGraph.Triangle.Removal | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 10
} | [
{
"pp": "α : Type u_1\ninst✝² : DecidableEq α\ninst✝¹ : Fintype α\nG : SimpleGraph α\ninst✝ : DecidableRel G.Adj\nε : ℝ\nhG : ↑(#(G.cliqueFinset 3)) < triangleRemovalBound ε * ↑(Fintype.card α) ^ 3\nh :\n ∀ G' ≤ G,\n ∀ (x : DecidableRel G'.Adj), ↑(#G.edgeFinset) - ↑(#G'.edgeFinset) < ε * ↑(Fintype.card α ^ ... | intro G' | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite | {
"line": 155,
"column": 2
} | {
"line": 157,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nt : Finset (α × β × γ)\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq γ\nx y z : α ⊕ β ⊕ γ\n⊢ (graph t).Adj x y →\n (graph t).Adj x z →\n (graph t).Adj y z →\n ∃ a b c,\n {in₀ a, in₁ b, in₂ c} = {x, y, z} ∧\n ... | rintro (_ | _ | _) (_ | _ | _) (_ | _ | _) <;>
refine ⟨_, _, _, by ext; simp only [Finset.mem_insert, Finset.mem_singleton]; try tauto,
?_, ?_, ?_⟩ <;> constructor <;> assumption | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite | {
"line": 155,
"column": 2
} | {
"line": 157,
"column": 48
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\nt : Finset (α × β × γ)\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq γ\nx y z : α ⊕ β ⊕ γ\n⊢ (graph t).Adj x y →\n (graph t).Adj x z →\n (graph t).Adj y z →\n ∃ a b c,\n {in₀ a, in₁ b, in₂ c} = {x, y, z} ∧\n ... | rintro (_ | _ | _) (_ | _ | _) (_ | _ | _) <;>
refine ⟨_, _, _, by ext; simp only [Finset.mem_insert, Finset.mem_singleton]; try tauto,
?_, ?_, ?_⟩ <;> constructor <;> assumption | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite | {
"line": 162,
"column": 42
} | {
"line": 164,
"column": 72
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\n𝕜 : Type u_4\ninst✝⁵ : Field 𝕜\ninst✝⁴ : LinearOrder 𝕜\ninst✝³ : IsStrictOrderedRing 𝕜\nt : Finset (α × β × γ)\na✝ a'✝ : α\nb✝ b'✝ : β\nc✝ c'✝ : γ\nx : α × β × γ\ninst✝² : DecidableEq α\ninst✝¹ : DecidableEq β\ninst✝ : DecidableEq γ\nx✝¹ x✝ : α × β × γ\na :... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.Corner.Roth | {
"line": 39,
"column": 97
} | {
"line": 44,
"column": 16
} | [
{
"pp": "G : Type u_1\ninst✝ : AddCommGroup G\nA : Finset (G × G)\na b c : G\n⊢ (a, b, c) ∈ triangleIndices A ↔ (a, b) ∈ A ∧ c = a + b",
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Combinatorics.Additive.Corner.Roth.0.Corners.triangleIndices",
"congrArg",
"AddCommGroup.toAddCommMo... | by
simp only [triangleIndices, Prod.ext_iff, mem_map, Embedding.coeFn_mk, Prod.exists,
eq_comm]
refine ⟨?_, fun h ↦ ⟨_, _, h.1, rfl, rfl, h.2⟩⟩
rintro ⟨_, _, h₁, rfl, rfl, h₂⟩
exact ⟨h₁, h₂⟩ | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite | {
"line": 232,
"column": 52
} | {
"line": 232,
"column": 63
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\n𝕜 : Type u_4\ninst✝⁹ : Field 𝕜\ninst✝⁸ : LinearOrder 𝕜\ninst✝⁷ : IsStrictOrderedRing 𝕜\nt : Finset (α × β × γ)\ninst✝⁶ : DecidableEq α\ninst✝⁵ : DecidableEq β\ninst✝⁴ : DecidableEq γ\ninst✝³ : Fintype α\ninst✝² : Fintype β\ninst✝¹ : Fintype γ\ninst✝ : Expli... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.Corner.Roth | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 29
} | [
{
"pp": "G : Type u_1\ninst✝² : AddCommGroup G\nA : Finset (G × G)\nε : ℝ\ninst✝¹ : Fintype G\ninst✝ : DecidableEq G\nhε : ε * ↑(Fintype.card G) ^ 2 ≤ ↑(#A)\n⊢ ε * ↑(Fintype.card G) ^ 2 ≤ ↑(#A)",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite | {
"line": 233,
"column": 38
} | {
"line": 233,
"column": 61
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\nγ : Type u_3\n𝕜 : Type u_4\ninst✝⁹ : Field 𝕜\ninst✝⁸ : LinearOrder 𝕜\ninst✝⁷ : IsStrictOrderedRing 𝕜\nt : Finset (α × β × γ)\ninst✝⁶ : DecidableEq α\ninst✝⁵ : DecidableEq β\ninst✝⁴ : DecidableEq γ\ninst✝³ : Fintype α\ninst✝² : Fintype β\ninst✝¹ : Fintype γ\ninst✝ : Expli... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 497,
"column": 8
} | {
"line": 497,
"column": 19
} | [
{
"pp": "case refine_1\nα : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhε₁ : ε ≤... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.DoublingConst | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 40
} | [
{
"pp": "G' : Type u_2\ninst✝³ : AddGroup G'\ninst✝² : DecidableEq G'\n𝕜 : Type u_3\ninst✝¹ : Semifield 𝕜\ninst✝ : CharZero 𝕜\nA B : Finset G'\n⊢ ↑(#A) * ↑σ[A, B] = ↑(#(A + B))",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"Finset.addConst"... | norm_cast; exact card_mul_addConst _ _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.DoublingConst | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 40
} | [
{
"pp": "G' : Type u_2\ninst✝³ : AddGroup G'\ninst✝² : DecidableEq G'\n𝕜 : Type u_3\ninst✝¹ : Semifield 𝕜\ninst✝ : CharZero 𝕜\nA B : Finset G'\n⊢ ↑(#A) * ↑σ[A, B] = ↑(#(A + B))",
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne",
"HMul.hMul",
"Finset.addConst"... | norm_cast; exact card_mul_addConst _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 510,
"column": 2
} | {
"line": 515,
"column": 54
} | [
{
"pp": "α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhU : U ∈ P.parts\nhV : V... | apply (edgeDensity_chunk_aux (hP := hP) hPα hPε hU hV).trans
have key : (16 : ℝ) ^ #P.parts = #((chunk hP G ε hU).parts ×ˢ (chunk hP G ε hV).parts) := by
rw [card_product, cast_mul, card_chunk (m_pos hPα).ne', card_chunk (m_pos hPα).ne', ←
cast_mul, ← mul_pow]; norm_cast
simp_rw [key]
convert! sum_div_c... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 510,
"column": 2
} | {
"line": 515,
"column": 54
} | [
{
"pp": "α : Type u_1\ninst✝³ : Fintype α\ninst✝² : DecidableEq α\nP : Finpartition univ\nhP : P.IsEquipartition\nG : SimpleGraph α\ninst✝¹ : DecidableRel G.Adj\nε : ℝ\nU V : Finset α\ninst✝ : Nonempty α\nhPα : #P.parts * 16 ^ #P.parts ≤ Fintype.card α\nhPε : 100 ≤ 4 ^ #P.parts * ε ^ 5\nhU : U ∈ P.parts\nhV : V... | apply (edgeDensity_chunk_aux (hP := hP) hPα hPε hU hV).trans
have key : (16 : ℝ) ^ #P.parts = #((chunk hP G ε hU).parts ×ˢ (chunk hP G ε hV).parts) := by
rw [card_product, cast_mul, card_chunk (m_pos hPα).ne', card_chunk (m_pos hPα).ne', ←
cast_mul, ← mul_pow]; norm_cast
simp_rw [key]
convert! sum_div_c... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.Additive.Corner.Roth | {
"line": 153,
"column": 4
} | {
"line": 154,
"column": 11
} | [
{
"pp": "G : Type u_1\ninst✝¹ : AddCommGroup G\ninst✝ : Fintype G\nε : ℝ\nhε : 0 < ε\nhG : cornersTheoremBound ε ≤ Fintype.card G\nA : Finset G\nhAε : ε * ↑(Fintype.card G) ≤ ↑(#A)\nhA : ThreeAPFree ↑A\nB : Finset (G × G) :=\n {x |\n match x with\n | (x, y) => y - x ∈ A}\nthis : ε * ↑(Fintype.card G) ^ 2... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.Corner.Roth | {
"line": 158,
"column": 20
} | {
"line": 158,
"column": 31
} | [
{
"pp": "G : Type u_1\ninst✝¹ : AddCommGroup G\ninst✝ : Fintype G\nε : ℝ\nhε : 0 < ε\nhG : cornersTheoremBound ε ≤ Fintype.card G\nA : Finset G\nhAε : ε * ↑(Fintype.card G) ≤ ↑(#A)\nhA : ThreeAPFree ↑A\nB : Finset (G × G) :=\n {x |\n match x with\n | (x, y) => y - x ∈ A}\nthis✝ : ε * ↑(Fintype.card G) ^ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.Corner.Roth | {
"line": 179,
"column": 8
} | {
"line": 179,
"column": 49
} | [
{
"pp": "n : ℕ\nε : ℝ\nhε : 0 < ε\nhG : cornersTheoremBound (ε / 3) ≤ n\nA : Finset ℕ\nhAn : ↑A ⊆ Set.Iio n\nhAε : ε * ↑n ≤ ↑(#A)\nhA : ThreeAPFree (Fin.val '' Nat.cast '' ↑A)\nthis✝ : ↑A = Fin.val '' Nat.cast '' ↑A\nthis : IsAddFreimanIso 2 (Set.Iio ↑n) (Set.Iio n) Fin.val\nx : ℕ\nhx : x ∈ Set.Iio n\n⊢ x < n",... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.ChevalleyWarning | {
"line": 132,
"column": 8
} | {
"line": 132,
"column": 55
} | [
{
"pp": "K : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁵ : Fintype K\ninst✝⁴ : Field K\ninst✝³ : Fintype σ\ninst✝² : DecidableEq σ\ninst✝¹ : DecidableEq K\np : ℕ\ninst✝ : CharP K p\ns : Finset ι\nf : ι → MvPolynomial σ K\nh : ∑ i ∈ s, (f i).totalDegree < Fintype.card σ\nhq : 0 < q - 1\nS : Finset (σ → K) := {x... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.ChevalleyWarning | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 13
} | [
{
"pp": "K : Type u_1\nσ : Type u_2\nι : Type u_3\ninst✝⁶ : Fintype K\ninst✝⁵ : Field K\ninst✝⁴ : Fintype σ\ninst✝³ : DecidableEq σ\ninst✝² : DecidableEq K\np : ℕ\ninst✝¹ : CharP K p\ninst✝ : Fintype ι\nf : ι → MvPolynomial σ K\nh : ∑ i, (f i).totalDegree < Fintype.card σ\n⊢ p ∣ Fintype.card { x // ∀ (i : ι), (... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.ChevalleyWarning | {
"line": 194,
"column": 2
} | {
"line": 194,
"column": 37
} | [
{
"pp": "K : Type u_1\nσ : Type u_2\ninst✝⁵ : Fintype K\ninst✝⁴ : Field K\ninst✝³ : Fintype σ\ninst✝² : DecidableEq σ\ninst✝¹ : DecidableEq K\np : ℕ\ninst✝ : CharP K p\nf₁ f₂ : MvPolynomial σ K\nh : f₁.totalDegree + f₂.totalDegree < Fintype.card σ\nF : Bool → MvPolynomial σ K := fun b ↦ bif b then f₂ else f₁\nt... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.Randomisation | {
"line": 53,
"column": 31
} | {
"line": 53,
"column": 61
} | [
{
"pp": "G : Type u_1\ninst✝¹ : Fintype G\ninst✝ : AddCommGroup G\nc : AddChar G ℂ → ℝ\nd : AddChar G ℂ → ℂ\nhcd : AddDissociated {ψ | d ψ ≠ 0}\nt : Finset (AddChar G ℂ)\nht : t ≠ ∅\nu : Finset (AddChar G ℂ)\nx✝ : u ∈ t.powerset\nh : (∀ a ∈ u, d a ≠ 0) ∧ ∀ a ∈ t \\ u, (starRingEnd ℂ) (d a) ≠ 0\n⊢ t \\ u ∈ {t | ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.SubsetSum | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 33
} | [
{
"pp": "M : Type u_1\ninst✝³ : DecidableEq M\ninst✝² : AddCommMonoid M\nA : Finset M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedCancelAddMonoid M\nA_nonneg : ∀ x ∈ A, 0 ≤ x\n⊢ ∀ x ∈ A.subsetSum, 0 ≤ x",
"usedConstants": [
"Eq.mpr",
"Finset.subsetSum",
"Finset",
"AddMonoid.toAddZeroCl... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.SubsetSum | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 60
} | [
{
"pp": "M : Type u_1\ninst✝³ : DecidableEq M\ninst✝² : AddCommMonoid M\nA : Finset M\na : M\ninst✝¹ : LinearOrder M\ninst✝ : IsOrderedCancelAddMonoid M\nhA : ∀ x ∈ A, 0 < x\nhAa : ∀ x ∈ A, x < a\nha : 0 < a\nthis : ∀ x ∈ A.subsetSum, 0 ≤ x\n⊢ Disjoint (insert 0 A) (a +ᵥ A.subsetSum)",
"usedConstants": [
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.Energy | {
"line": 147,
"column": 8
} | {
"line": 147,
"column": 19
} | [
{
"pp": "α : Type u_1\ninst✝¹ : DecidableEq α\ninst✝ : Mul α\ns t u : Finset α\n⊢ (∑ c ∈ u, #({xy ∈ s ×ˢ t | xy.1 * xy.2 = c})) ^ 2 ≤ #u * ∑ c ∈ u, #({xy ∈ s ×ˢ t | xy.1 * xy.2 = c}) ^ 2",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 85,
"column": 6
} | {
"line": 85,
"column": 76
} | [
{
"pp": "case refine_2.refine_2\nι : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\ns : Finset ι\na : ι → ZMod p\nhs : #s = 2 * p - 1\nthis : NeZero p\nN : ℕ := Fintype.card { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 }\nzero_sol : { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 } := ⟨0, ⋯⟩\nhN₀ : 0... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 90,
"column": 4
} | {
"line": 90,
"column": 62
} | [
{
"pp": "case refine_3\nι : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\ns : Finset ι\na : ι → ZMod p\nhs : #s = 2 * p - 1\nthis : NeZero p\nN : ℕ := Fintype.card { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 }\nzero_sol : { x // (eval x) (f₁ s a) = 0 ∧ (eval x) (f₂ s a) = 0 } := ⟨0, ⋯⟩\nhN₀ : 0 < N\nhs'... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 98,
"column": 2
} | {
"line": 99,
"column": 9
} | [
{
"pp": "ι : Type u_1\np : ℕ\ninst✝ : Fact (Nat.Prime p)\ns : Finset ι\na : ι → ℤ\nhs : #s = 2 * p - 1\n⊢ ∃ t ⊆ s, #t = p ∧ ↑p ∣ ∑ i ∈ t, a i",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 119,
"column": 11
} | {
"line": 119,
"column": 22
} | [
{
"pp": "case one\nι : Type u_1\ns : Finset ι\na : ι → ℤ\nhs : 2 * 1 - 1 ≤ #s\n⊢ ∃ t ⊆ s, #t = 1 ∧ ↑1 ∣ ∑ i ∈ t, a i",
"usedConstants": [
"Eq.mpr",
"Int.instAddCommMonoid",
"MulOne.toOne",
"Dvd.dvd",
"and_true",
"Monoid.toMulOneClass",
"congrArg",
"Finset",
... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 135,
"column": 6
} | {
"line": 145,
"column": 29
} | [
{
"pp": "m : ℕ\nhm : 2 ≤ m\nihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i\nn : ℕ\nhn : 2 ≤ n\nihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i\nι : Type u_1\ns : Finset ι\na : ι → ℤ\nhs : 2 * (m * n)... | obtain ⟨𝒜, h𝒜card, h𝒜disj, h𝒜⟩ := this _ le_rfl
-- By induction hypothesis on `m`, find a subfamily `ℬ` of size `m` such that the sum over
-- `t ∈ ℬ` of `(∑ i ∈ t, a i) / n` is divisible by `m`.
obtain ⟨ℬ, hℬ𝒜, hℬcard, hℬ⟩ := ihm (fun t ↦ (∑ i ∈ t, a i) / n) h𝒜card.ge
-- We are done.
... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 135,
"column": 6
} | {
"line": 145,
"column": 29
} | [
{
"pp": "m : ℕ\nhm : 2 ≤ m\nihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i\nn : ℕ\nhn : 2 ≤ n\nihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i\nι : Type u_1\ns : Finset ι\na : ι → ℤ\nhs : 2 * (m * n)... | obtain ⟨𝒜, h𝒜card, h𝒜disj, h𝒜⟩ := this _ le_rfl
-- By induction hypothesis on `m`, find a subfamily `ℬ` of size `m` such that the sum over
-- `t ∈ ℬ` of `(∑ i ∈ t, a i) / n` is divisible by `m`.
obtain ⟨ℬ, hℬ𝒜, hℬcard, hℬ⟩ := ihm (fun t ↦ (∑ i ∈ t, a i) / n) h𝒜card.ge
-- We are done.
... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Data.Nat.BitIndices | {
"line": 132,
"column": 28
} | {
"line": 132,
"column": 39
} | [
{
"pp": "a n : ℕ\nha : a ∈ n.bitIndices\n⊢ n.testBit a = true",
"usedConstants": []
}
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Compactness | {
"line": 71,
"column": 2
} | {
"line": 73,
"column": 54
} | [
{
"pp": "α : Type u_1\nβ : α → Type u_2\ninst✝ : ∀ (a : α), Finite (β a)\ng : Finset α → (a : α) → β a\ninstTop : (a : α) → TopologicalSpace (β a) := fun a ↦ ⊥\ninstDiscr : ∀ (a : α), DiscreteTopology (β a)\ne : Finset α → Set ((a : α) → β a) := fun s ↦ {f | ∃ t, s ⊆ t ∧ ∀ x ∈ s, f x = g t x}\nthis : ∀ (s : Fin... | have he' (s : Finset α) : IsClosed (e s) := by
rw [← this]
exact (isClosed_discrete _).preimage (by fun_prop) | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 171,
"column": 8
} | {
"line": 171,
"column": 45
} | [
{
"pp": "m : ℕ\nhm : 2 ≤ m\nihm : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * m - 1 ≤ #s → ∃ t ⊆ s, #t = m ∧ ↑m ∣ ∑ i ∈ t, a i\nn : ℕ\nhn : 2 ≤ n\nihn : ∀ {ι : Type u_1} {s : Finset ι} (a : ι → ℤ), 2 * n - 1 ≤ #s → ∃ t ⊆ s, #t = n ∧ ↑n ∣ ∑ i ∈ t, a i\nι : Type u_1\ns : Finset ι\na : ι → ℤ\nhs : 2 * (m * n)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Compactness | {
"line": 79,
"column": 2
} | {
"line": 79,
"column": 13
} | [
{
"pp": "α : Type u_1\nβ : α → Type u_2\ninst✝ : ∀ (a : α), Finite (β a)\ng : Finset α → (a : α) → β a\ninstTop : (a : α) → TopologicalSpace (β a) := fun a ↦ ⊥\ninstDiscr : ∀ (a : α), DiscreteTopology (β a)\ne : Finset α → Set ((a : α) → β a) := fun s ↦ {f | ∃ t, s ⊆ t ∧ ∀ x ∈ s, f x = g t x}\nthis : ∀ (s : Fin... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Compactness | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 18
} | [
{
"pp": "α : Type u_1\nβ : α → Type u_2\ninst✝ : ∀ (a : α), Finite (β a)\ng : (s : Finset α) → (a : ↥s) → β ↑a\nthis : ∀ (a : α), Nonempty (β a)\ng' : Finset α → (a : α) → β a := fun s a ↦ if ha : a ∈ s then g s ⟨a, ha⟩ else Classical.arbitrary (β a)\nhg : ∀ (s : Finset α) (x : ↥s), g s x = g' s ↑x\n⊢ ∃ χ, ∀ (s... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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