module string | startPos dict | endPos dict | nextStartPos dict | goals list | goalsAfter list | ppTac string | elaborator string | kind string |
|---|---|---|---|---|---|---|---|---|
Mathlib.Combinatorics.SimpleGraph.Walk.Traversal | {
"line": 145,
"column": 37
} | {
"line": 145,
"column": 72
} | {
"line": 145,
"column": 73
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv w : V\np : G.Walk v w\nhp : ¬p.Nil\n⊢ 0 < p.length",
"ppTerm": "?m.25",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\nv w : V\np : G.Walk v w\nhp : ¬p.Nil\n⊢ 0 < p.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 318,
"column": 58
} | {
"line": 318,
"column": 81
} | {
"line": 318,
"column": 82
} | [
{
"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\n⊢ v = v' ∧ w₁.edges = w₂.edges",
"ppTerm": "?m.351",
"assigned": false,
"usedConstants": [],
"usedFVars... | [
"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\n⊢ v = v' ∧ w₁.edges = w₂.edges"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 472,
"column": 4
} | {
"line": 472,
"column": 15
} | {
"line": 472,
"column": 16
} | [
{
"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",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"List.head",
... | [
"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 (v :: l) ⋯ ⋯).support = v :: l"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Basic | {
"line": 507,
"column": 4
} | {
"line": 507,
"column": 33
} | {
"line": 507,
"column": 34
} | [
{
"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",
"ppTerm": "?m.29",
"assigned": true,
"usedConstants": [
"List.hea... | [
"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 (d₂ :: l) ⋯ ⋯).darts = d₂ :: l"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 260,
"column": 6
} | {
"line": 261,
"column": 12
} | {
"line": 263,
"column": 0
} | [
{
"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)",
"ppTerm": "?inr",
... | [] | · 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.Subwalks | {
"line": 231,
"column": 2
} | {
"line": 238,
"column": 46
} | {
"line": 240,
"column": 0
} | [
{
"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)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Nat.recAux",
"SimpleGraph.Walk.drop_zero",
"HEq.refl",
"SimpleGraph.Walk.IsSubwalk.copy",
... | [] | 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
} | {
"line": 240,
"column": 0
} | [
{
"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)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Nat.recAux",
"SimpleGraph.Walk.drop_zero",
"HEq.refl",
"SimpleGraph.Walk.IsSubwalk.copy",
... | [] | 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
} | {
"line": 240,
"column": 0
} | [
{
"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)",
"ppTerm": "?m.23",
"assigned": true,
"usedConstants": [
"Nat.recAux",
"SimpleGraph.Walk.drop_zero",
"HEq.refl",
"SimpleGraph.Walk.IsSubwalk.copy",
... | [] | 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.Operations | {
"line": 482,
"column": 2
} | {
"line": 482,
"column": 13
} | {
"line": 482,
"column": 14
} | [
{
"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",
"ppTerm": "?m.19",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 252,
"column": 2
} | {
"line": 252,
"column": 13
} | {
"line": 252,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v w : V\np : G.Walk v w\nh : u ∈ p.support\n⊢ (p.takeUntil u h).length ≤ p.length",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v w : V\np : G.Walk v w\nh : u ∈ p.support\n⊢ (p.takeUntil u h).length ≤ p.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 258,
"column": 2
} | {
"line": 258,
"column": 13
} | {
"line": 258,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v w : V\np : G.Walk v w\nh : u ∈ p.support\n⊢ (p.dropUntil u h).length ≤ p.length",
"ppTerm": "?m.24",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\ninst✝ : DecidableEq V\nu v w : V\np : G.Walk v w\nh : u ∈ p.support\n⊢ (p.dropUntil u h).length ≤ p.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 278,
"column": 2
} | {
"line": 278,
"column": 34
} | {
"line": 278,
"column": 35
} | [
{
"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",
"ppTerm": "?m.40",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 302,
"column": 25
} | {
"line": 302,
"column": 36
} | {
"line": 302,
"column": 37
} | [
{
"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",
"ppTerm": "?m.38",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Decomp | {
"line": 355,
"column": 2
} | {
"line": 355,
"column": 13
} | {
"line": 355,
"column": 14
} | [
{
"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",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 573,
"column": 2
} | {
"line": 573,
"column": 78
} | {
"line": 574,
"column": 2
} | [
{
"pp": "V : 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",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"SimpleGraph.Walk.getVert_cons",
"Nat.ne_zer... | [
"V : Type u\nG : SimpleGraph V\nu v w : V\nh : G.Adj u v\np : G.Walk v w\nw✝ : ℕ\nhn : w✝ + 1 ≠ 0\n⊢ ((cons h p).drop (w✝ + 1)).support = ((p.drop (w✝ + 1 - 1)).copy ⋯ ⋯).support"
] | 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.Paths | {
"line": 144,
"column": 2
} | {
"line": 144,
"column": 27
} | {
"line": 144,
"column": 28
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nh : p.IsTrail\n⊢ p.reverse.IsTrail",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"id",
"_private.Mathlib.Combinatorics.SimpleGraph.Paths.0.SimpleGraph.Walk.IsTrail.reverse._simp... | [
"V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nh : p.IsTrail\n⊢ p.edges.Nodup"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 184,
"column": 4
} | {
"line": 184,
"column": 23
} | {
"line": 184,
"column": 24
} | [
{
"pp": "V : 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",
"ppTerm": "?m.75",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"use... | [
"V : 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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 215,
"column": 2
} | {
"line": 215,
"column": 26
} | {
"line": 215,
"column": 27
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nh : p.IsPath\n⊢ p.reverse.IsPath",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleGraph.Walk.support",
"id",
"_private.Mathlib.Combinatorics.SimpleGraph.Paths.0.Simple... | [
"V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\nh : p.IsPath\n⊢ p.support.Nodup"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 316,
"column": 31
} | {
"line": 316,
"column": 42
} | {
"line": 316,
"column": 43
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv : V\np : G.Walk v v\nv✝ : V\nh✝¹ : G.Adj v v✝\nh✝ : G.Adj v✝ v\nhp : (cons h✝¹ (cons h✝ nil)).IsCircuit\n⊢ 3 ≤ (cons h✝¹ (cons h✝ nil)).length",
"ppTerm": "?m.50",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"congrArg",
"S... | [
"V : Type u\nG : SimpleGraph V\nv : V\np : G.Walk v v\nv✝ : V\nh✝¹ : G.Adj v v✝\nh✝ : G.Adj v✝ v\nhp : (cons h✝¹ (cons h✝ nil)).IsCircuit\n⊢ False"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 656,
"column": 2
} | {
"line": 656,
"column": 13
} | {
"line": 656,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\n⊢ p.reverse.snd = p.penultimate",
"ppTerm": "?m.15",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\nu v : V\np : G.Walk u v\n⊢ p.reverse.snd = p.penultimate"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 340,
"column": 13
} | {
"line": 340,
"column": 24
} | {
"line": 340,
"column": 25
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu : V\np : G.Walk u u\nh : p.reverse.IsCycle\n⊢ p.IsCycle",
"ppTerm": "?m.18",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\nu : V\np : G.Walk u u\nh : p.reverse.IsCycle\n⊢ p.IsCycle"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 63,
"column": 4
} | {
"line": 63,
"column": 15
} | {
"line": 63,
"column": 16
} | [
{
"pp": "case refine_2\nα : Type u_1\nG : SimpleGraph α\ns : Set α\nh : s.Pairwise G.Adj\nv : α\nhv : v ∈ s\nw : α\nhw : w ∈ s\n⊢ (induce s G).Adj ⟨v, hv⟩ ⟨w, hw⟩ ↔ ⊤.Adj ⟨v, hv⟩ ⟨w, hw⟩",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.comap_adj._simp_... | [
"case refine_2\nα : Type u_1\nG : SimpleGraph α\ns : Set α\nh : s.Pairwise G.Adj\nv : α\nhv : v ∈ s\nw : α\nhw : w ∈ s\n⊢ G.Adj v w ↔ ¬v = w"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 422,
"column": 6
} | {
"line": 422,
"column": 55
} | {
"line": 423,
"column": 6
} | [
{
"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... | [
"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\nhn0 : ¬n = 0\nh... | simp only [hm0, Walk.getVert_cons p h hn0] at hnm | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 13
} | {
"line": 184,
"column": 14
} | [
{
"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)",
"ppTerm": "?mpr",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"Finset",
"S... | [
"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⊢ G.IsClique ↑s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Walk.Operations | {
"line": 839,
"column": 2
} | {
"line": 839,
"column": 13
} | {
"line": 839,
"column": 14
} | [
{
"pp": "case cons\nV : Type u\nG : SimpleGraph V\nu v✝ : V\nh : G.Adj u v✝\np : G.Walk v✝ u\n⊢ (cons h p).support.tail ~ (cons h p).support.dropLast",
"ppTerm": "?cons",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.support",
"id",
"List.Perm",
"List.dropLast",
... | [
"case cons\nV : Type u\nG : SimpleGraph V\nu v✝ : V\nh : G.Adj u v✝\np : G.Walk v✝ u\n⊢ p.support ~ (u :: p.support).dropLast"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 210,
"column": 2
} | {
"line": 210,
"column": 26
} | {
"line": 211,
"column": 2
} | [
{
"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✝",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"SimpleGraph.edge",
"SimpleGraph.Adj",
"Membe... | [
"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✝\nthis : (G ⊔ edge v w).Adj x✝ y✝\n⊢ G.Adj x✝ y✝"
] | 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": 443,
"column": 4
} | {
"line": 443,
"column": 15
} | {
"line": 443,
"column": 16
} | [
{
"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",
"ppTerm": "?nil",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"true_iff",
"id",
"instOfNatNat",
"Iff",... | [
"case nil\nV : Type u\nG : SimpleGraph V\nu : V\ni : ℕ\nhp : Walk.nil.IsPath\nhi : i ≤ Walk.nil.length\n⊢ i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 467,
"column": 12
} | {
"line": 467,
"column": 38
} | {
"line": 467,
"column": 39
} | [
{
"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).... | [
"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⊢ q.getVert n = q.getVert... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 344,
"column": 4
} | {
"line": 344,
"column": 45
} | {
"line": 345,
"column": 4
} | [
{
"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 : ... | [
"case inl.refine_2\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 : U ∈ P.parts\... | · exact mod_cast G.edgeDensity_nonneg _ _ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 485,
"column": 2
} | {
"line": 485,
"column": 20
} | {
"line": 485,
"column": 21
} | [
{
"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",
"ppTerm": "?m.22",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 589,
"column": 38
} | {
"line": 589,
"column": 66
} | {
"line": 589,
"column": 67
} | [
{
"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",
"ppTerm": "?m.38",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleG... | [
"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.edges.Nodup"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 464,
"column": 2
} | {
"line": 464,
"column": 37
} | {
"line": 464,
"column": 38
} | [
{
"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",
"ppTerm": "?m.13",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\nG : SimpleGraph α\nn : ℕ\ninst✝ : Fintype α\nhc : Nonempty ((completeGraph (Fin n)).Copy G)\n⊢ n ≤ Fintype.card α"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 131,
"column": 64
} | {
"line": 131,
"column": 75
} | {
"line": 131,
"column": 76
} | [
{
"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... | [
"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}.Subsingleton\na b ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 132,
"column": 25
} | {
"line": 132,
"column": 36
} | {
"line": 132,
"column": 37
} | [
{
"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... | [
"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}.Subsingleton\na b ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 157,
"column": 4
} | {
"line": 159,
"column": 11
} | {
"line": 159,
"column": 12
} | [
{
"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",
"ppTerm": "?refine_2✝",
"assigned": true... | [
"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⊢ ∀ (a : Finset α), G.IsNClique 3 a → e ∈ a.sym2 → ∀ (b : Finset α), G.IsNClique 3 b → e ∈ b.sym2 → a = b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 166,
"column": 4
} | {
"line": 168,
"column": 11
} | {
"line": 168,
"column": 12
} | [
{
"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)",
"ppTerm": "?refine_1",
"assigned": true,
"usedConstants": [... | [
"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⊢ ∀ (x y : α), G.Adj x y → ∃ x_1, G.IsNClique 3 x_1 ∧ x ∈ x_1 ∧ y ∈ x_1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 407,
"column": 41
} | {
"line": 407,
"column": 51
} | {
"line": 407,
"column": 52
} | [
{
"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.... | [
"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.parts\nhV : ... | ← one_div, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 538,
"column": 2
} | {
"line": 539,
"column": 68
} | {
"line": 540,
"column": 2
} | [
{
"pp": "case mp\nα : Type u_1\nG : SimpleGraph α\n⊢ G.CliqueFree 2 → G = ⊥",
"ppTerm": "?mp",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"False",
"Finset.coe_singleton",
"eq_false",
"SimpleGraph.Adj.ne",
"Sym2.mk",
"congrArg",
"Finset",
"_pr... | [
"case mpr\nα : Type u_1\nG : SimpleGraph α\n⊢ G = ⊥ → G.CliqueFree 2"
] | · 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": 570,
"column": 6
} | {
"line": 570,
"column": 51
} | {
"line": 570,
"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... | [
"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 (t.erase y) ∧ G.IsNClique n (t.erase x)"
] | insert_erase <| mem_erase_of_ne_of_mem hne h1 | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 899,
"column": 2
} | {
"line": 899,
"column": 13
} | {
"line": 899,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\ninst✝ : DecidableEq V\np : G.Walk u v\n⊢ p.bypass.length ≤ p.length",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\nu v : V\ninst✝ : DecidableEq V\np : G.Walk u v\n⊢ p.bypass.length ≤ p.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 905,
"column": 72
} | {
"line": 905,
"column": 83
} | {
"line": 905,
"column": 84
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nu v : V\ninst✝ : DecidableEq V\np : G.Walk u v\nh : p.length ≤ p.bypass.length\n⊢ p.support.length ≤ p.bypass.support.length",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"congrArg",
"SimpleGraph.Walk.length",
"Sim... | [
"V : Type u\nG : SimpleGraph V\nu v : V\ninst✝ : DecidableEq V\np : G.Walk u v\nh : p.length ≤ p.bypass.length\n⊢ p.length ≤ p.bypass.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 956,
"column": 2
} | {
"line": 956,
"column": 13
} | {
"line": 956,
"column": 14
} | [
{
"pp": "V : Type u\nG : SimpleGraph V\nv : V\ninst✝ : DecidableEq V\nw : G.Walk v v\n⊢ w.cycleBypass.length ≤ w.length",
"ppTerm": "?m.16",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"V : Type u\nG : SimpleGraph V\nv : V\ninst✝ : DecidableEq V\nw : G.Walk v v\n⊢ w.cycleBypass.length ≤ w.length"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Paths | {
"line": 965,
"column": 6
} | {
"line": 965,
"column": 17
} | {
"line": 965,
"column": 18
} | [
{
"pp": "case refine_2\nV : Type u\nG : SimpleGraph V\nv : V\ninst✝ : DecidableEq V\nv' : V\nhvv' : G.Adj v v'\nw : G.Walk v' v\nhw : (cons hvv' w).IsCircuit\n⊢ (cons hvv' w.bypass).support.tail.Nodup",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [
"SimpleGraph.Walk.support",
... | [
"case refine_2\nV : Type u\nG : SimpleGraph V\nv : V\ninst✝ : DecidableEq V\nv' : V\nhvv' : G.Adj v v'\nw : G.Walk v' v\nhw : (cons hvv' w).IsCircuit\n⊢ w.bypass.support.Nodup"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 449,
"column": 4
} | {
"line": 449,
"column": 15
} | {
"line": 449,
"column": 16
} | [
{
"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.... | [
"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.parts\nhV : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 707,
"column": 4
} | {
"line": 707,
"column": 15
} | {
"line": 707,
"column": 16
} | [
{
"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",
"ppTerm": "?inr",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 813,
"column": 2
} | {
"line": 813,
"column": 37
} | {
"line": 813,
"column": 38
} | [
{
"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",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.IsNClique",
"Fins... | [
"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⊢ G.IsNClique n s → #s = n"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 264,
"column": 28
} | {
"line": 264,
"column": 39
} | {
"line": 264,
"column": 40
} | [
{
"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)",
"ppTerm": "?m.65",... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Basic | {
"line": 267,
"column": 6
} | {
"line": 267,
"column": 36
} | {
"line": 267,
"column": 37
} | [
{
"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",
"ppT... | [
"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) < ↑(Fintype.card α) ^ 2 / 2"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Removal | {
"line": 67,
"column": 2
} | {
"line": 67,
"column": 13
} | {
"line": 67,
"column": 14
} | [
{
"pp": "n k : ℕ\nhk : 0 < k\nhn : k ≤ n\n⊢ k ≤ k * (n / k)",
"ppTerm": "?m.87",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"n k : ℕ\nhk : 0 < k\nhn : k ≤ n\n⊢ k ≤ k * (n / k)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Clique | {
"line": 994,
"column": 2
} | {
"line": 994,
"column": 39
} | {
"line": 995,
"column": 2
} | [
{
"pp": "α : Type u_3\nG : SimpleGraph α\ninst✝ : Finite α\nt : Finset α\ntc : Gᶜ.IsClique ↑t\n⊢ #t ≤ G.indepNum",
"ppTerm": "?m.13",
"assigned": true,
"usedConstants": [
"_private.Mathlib.Combinatorics.SimpleGraph.Clique.0.SimpleGraph.IsIndepSet.card_le_indepNum._simp_1_2",
"Eq.mpr",
... | [
"α : Type u_3\nG : SimpleGraph α\ninst✝ : Finite α\nt : Finset α\ntc : Gᶜ.IsClique ↑t\n⊢ #t ≤ sSup {n | ∃ s, Gᶜ.IsNClique n s}"
] | 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": 998,
"column": 2
} | {
"line": 998,
"column": 39
} | {
"line": 999,
"column": 2
} | [
{
"pp": "α : Type u_3\nG : SimpleGraph α\n⊢ ∃ s, G.IsNIndepSet G.indepNum s",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Combinatorics.SimpleGraph.Clique.0.SimpleGraph.exists_isNIndepSet_indepNum._simp_1_2",
"SimpleGraph.IsNClique",
"co... | [
"α : Type u_3\nG : SimpleGraph α\n⊢ ∃ s, Gᶜ.IsNClique (sSup {n | ∃ s, Gᶜ.IsNClique n s}) s"
] | 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": 1037,
"column": 2
} | {
"line": 1037,
"column": 37
} | {
"line": 1038,
"column": 2
} | [
{
"pp": "α : Type u_3\nG : SimpleGraph α\ninst✝ : Finite α\nt : Finset α\ntmc : G.IsMaximumIndepSet t\n⊢ #t = G.indepNum",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"SimpleGraph.IsMaximumIndepSet",
"congrArg",
"Compl.compl",
"Eq.mp",
"SimpleGraph",
"S... | [
"α : Type u_3\nG : SimpleGraph α\ninst✝ : Finite α\nt : Finset α\ntmc : Gᶜ.IsMaximumClique t\n⊢ #t = G.indepNum"
] | 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": 1038,
"column": 2
} | {
"line": 1038,
"column": 39
} | {
"line": 1039,
"column": 2
} | [
{
"pp": "α : Type u_3\nG : SimpleGraph α\ninst✝ : Finite α\nt : Finset α\ntmc : Gᶜ.IsMaximumClique t\n⊢ #t = G.indepNum",
"ppTerm": "?m.14",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"SimpleGraph.IsNClique",
"congrArg",
"Compl.compl",
"Finset",
"setOf",
... | [
"α : Type u_3\nG : SimpleGraph α\ninst✝ : Finite α\nt : Finset α\ntmc : Gᶜ.IsMaximumClique t\n⊢ #t = sSup {n | ∃ s, Gᶜ.IsNClique n s}"
] | 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": 497,
"column": 8
} | {
"line": 497,
"column": 19
} | {
"line": 497,
"column": 20
} | [
{
"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 ≤... | [
"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 ≤ 1\nhU : U ∈... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite | {
"line": 161,
"column": 2
} | {
"line": 163,
"column": 48
} | {
"line": 165,
"column": 0
} | [
{
"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.Tripartite | {
"line": 161,
"column": 2
} | {
"line": 163,
"column": 48
} | {
"line": 165,
"column": 0
} | [
{
"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": 161,
"column": 2
} | {
"line": 163,
"column": 48
} | {
"line": 165,
"column": 0
} | [
{
"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": 168,
"column": 42
} | {
"line": 170,
"column": 72
} | {
"line": 170,
"column": 73
} | [
{
"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 :... | [
"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 : α\nb : β\nc... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Removal | {
"line": 150,
"column": 18
} | {
"line": 150,
"column": 50
} | {
"line": 150,
"column": 51
} | [
{
"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))",
"ppTerm": "?m.192",
"assigned":... | [
"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⊢ ¬G.CliqueFree 3"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Regularity.Chunk | {
"line": 510,
"column": 2
} | {
"line": 515,
"column": 54
} | {
"line": 517,
"column": 0
} | [
{
"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
} | {
"line": 517,
"column": 0
} | [
{
"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": 39,
"column": 97
} | {
"line": 44,
"column": 16
} | {
"line": 46,
"column": 0
} | [
{
"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",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"_private.Mathlib.Combinatorics.Additive.Corner.Roth.0.Corners.triangleIndices",
... | [] | 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.Removal | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 10
} | {
"line": 166,
"column": 11
} | [
{
"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 ^ ... | [
"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 ^ 2) → ¬G'.Cli... | intro G' | Lean.Elab.Tactic.evalIntro | null |
Mathlib.Combinatorics.Additive.Corner.Roth | {
"line": 62,
"column": 2
} | {
"line": 62,
"column": 29
} | {
"line": 62,
"column": 30
} | [
{
"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)",
"ppTerm": "?m.59",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
... | [
"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)"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.DoublingConst | {
"line": 176,
"column": 2
} | {
"line": 176,
"column": 40
} | {
"line": 178,
"column": 0
} | [
{
"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))",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne... | [] | 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
} | {
"line": 178,
"column": 0
} | [
{
"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))",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"NonAssocSemiring.toAddCommMonoidWithOne... | [] | norm_cast; exact card_mul_addConst _ _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite | {
"line": 238,
"column": 52
} | {
"line": 238,
"column": 63
} | {
"line": 238,
"column": 64
} | [
{
"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... | [
"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✝ : ExplicitDisjoint ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.SimpleGraph.Triangle.Tripartite | {
"line": 239,
"column": 38
} | {
"line": 239,
"column": 61
} | {
"line": 239,
"column": 62
} | [
{
"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... | [
"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✝ : ExplicitDisjoint ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.Energy | {
"line": 147,
"column": 8
} | {
"line": 147,
"column": 19
} | {
"line": 147,
"column": 20
} | [
{
"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",
"ppTerm": "?m.141",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.Corner.Roth | {
"line": 153,
"column": 4
} | {
"line": 154,
"column": 11
} | {
"line": 154,
"column": 12
} | [
{
"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... | [
"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 ≤ ↑(#B)\n⊢ ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.Corner.Roth | {
"line": 158,
"column": 20
} | {
"line": 158,
"column": 31
} | {
"line": 158,
"column": 32
} | [
{
"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) ^ ... | [
"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 ≤ ↑(#B)\nx... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.Corner.Roth | {
"line": 179,
"column": 8
} | {
"line": 179,
"column": 49
} | {
"line": 179,
"column": 50
} | [
{
"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",... | [
"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.Combinatorics.Additive.SubsetSum | {
"line": 68,
"column": 2
} | {
"line": 68,
"column": 33
} | {
"line": 68,
"column": 34
} | [
{
"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",
"ppTerm": "?m.22",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Finset.subsetSum",... | [
"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⊢ ∀ a ⊆ A, 0 ≤ ∑ b ∈ a, b"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.SubsetSum | {
"line": 78,
"column": 4
} | {
"line": 78,
"column": 60
} | {
"line": 79,
"column": 6
} | [
{
"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)",
"ppTerm": "?m.71",
... | [
"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⊢ (∀ x ∈ A.subsetSum, ¬a + x = 0) ∧ ∀ a_1 ∈ A, ∀ x ∈ A.subsetSum, ¬a + x = a_1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.ChevalleyWarning | {
"line": 132,
"column": 8
} | {
"line": 132,
"column": 55
} | {
"line": 132,
"column": 56
} | [
{
"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... | [
"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 | ∀ i ∈ s, ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 84,
"column": 6
} | {
"line": 84,
"column": 76
} | {
"line": 84,
"column": 77
} | [
{
"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... | [
"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 < N\nhs' : ... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 89,
"column": 4
} | {
"line": 89,
"column": 62
} | {
"line": 89,
"column": 63
} | [
{
"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'... | [
"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' : 2 * p - 1... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 97,
"column": 2
} | {
"line": 98,
"column": 9
} | {
"line": 98,
"column": 10
} | [
{
"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",
"ppTerm": "?m.31",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"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"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 118,
"column": 11
} | {
"line": 118,
"column": 22
} | {
"line": 118,
"column": 23
} | [
{
"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",
"ppTerm": "?one",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Int.instAddCommMonoid",
"MulOne.toOne",
"Dvd.dvd",
"and_true",
"Monoid.toMulOne... | [
"case one\nι : Type u_1\ns : Finset ι\na : ι → ℤ\nhs : 2 * 1 - 1 ≤ #s\n⊢ ∃ t ⊆ s, #t = 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Data.Nat.BitIndices | {
"line": 132,
"column": 28
} | {
"line": 132,
"column": 39
} | {
"line": 132,
"column": 40
} | [
{
"pp": "a n : ℕ\nha : a ∈ n.bitIndices\n⊢ n.testBit a = true",
"ppTerm": "?m.14",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"a n : ℕ\nha : a ∈ n.bitIndices\n⊢ n.testBit a = true"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.FieldTheory.ChevalleyWarning | {
"line": 169,
"column": 2
} | {
"line": 169,
"column": 13
} | {
"line": 169,
"column": 14
} | [
{
"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), (... | [
"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), (eval x) (f i... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Compactness | {
"line": 71,
"column": 2
} | {
"line": 73,
"column": 54
} | {
"line": 74,
"column": 2
} | [
{
"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... | [
"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 : Finset α), s.re... | 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.Compactness | {
"line": 95,
"column": 2
} | {
"line": 95,
"column": 18
} | {
"line": 95,
"column": 19
} | [
{
"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... | [
"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 : Finset α)... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 134,
"column": 6
} | {
"line": 144,
"column": 29
} | {
"line": 146,
"column": 4
} | [
{
"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": 134,
"column": 6
} | {
"line": 144,
"column": 29
} | {
"line": 146,
"column": 4
} | [
{
"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.Combinatorics.Colex | {
"line": 225,
"column": 2
} | {
"line": 225,
"column": 13
} | {
"line": 225,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝¹ : PartialOrder α\ns t : Finset α\ninst✝ : DecidableEq α\n⊢ toColex (s \\ t) ≤ toColex (t \\ s) ↔ toColex s ≤ toColex t",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝¹ : PartialOrder α\ns t : Finset α\ninst✝ : DecidableEq α\n⊢ toColex (s \\ t) ≤ toColex (t \\ s) ↔ toColex s ≤ toColex t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Colex | {
"line": 229,
"column": 2
} | {
"line": 229,
"column": 13
} | {
"line": 229,
"column": 14
} | [
{
"pp": "α : Type u_1\ninst✝¹ : PartialOrder α\ns t : Finset α\ninst✝ : DecidableEq α\n⊢ toColex (s \\ t) < toColex (t \\ s) ↔ toColex s < toColex t",
"ppTerm": "?m.17",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"α : Type u_1\ninst✝¹ : PartialOrder α\ns t : Finset α\ninst✝ : DecidableEq α\n⊢ toColex (s \\ t) < toColex (t \\ s) ↔ toColex s < toColex t"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 170,
"column": 8
} | {
"line": 170,
"column": 45
} | {
"line": 170,
"column": 46
} | [
{
"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)... | [
"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) - 1 ≤ #s\nk... | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Colex | {
"line": 305,
"column": 6
} | {
"line": 305,
"column": 40
} | {
"line": 305,
"column": 41
} | [
{
"pp": "α : Type u_1\ninst✝ : LinearOrder α\ns t : Finset α\nh : toColex s ≤ toColex t\nhst : s ≠ t\nm : α := (s ∆ t).max' ⋯\nhmt : m ∉ t\n⊢ m ∈ s",
"ppTerm": "?m.63",
"assigned": true,
"usedConstants": [
"Finset",
"Membership.mem",
"id",
"Finset.instSetLike",
"SetLike... | [
"α : Type u_1\ninst✝ : LinearOrder α\ns t : Finset α\nh : toColex s ≤ toColex t\nhst : s ≠ t\nm : α := (s ∆ t).max' ⋯\nhmt : m ∉ t\n⊢ (s ∆ t).max' ⋯ ∈ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Colex | {
"line": 311,
"column": 4
} | {
"line": 311,
"column": 37
} | {
"line": 311,
"column": 38
} | [
{
"pp": "case refine_2\nα : Type u_1\ninst✝ : LinearOrder α\ns t : Finset α\nh : ∀ (hst : s ≠ t), (s ∆ t).max' ⋯ ∈ t\na : α\nhas : a ∈ ofColex (toColex s)\nhat : a ∉ ofColex (toColex t)\nhst : s ≠ t\n⊢ (s ∆ t).max' ⋯ ∉ ofColex (toColex s)",
"ppTerm": "?refine_2",
"assigned": true,
"usedConstants": [... | [
"case refine_2\nα : Type u_1\ninst✝ : LinearOrder α\ns t : Finset α\nh : ∀ (hst : s ≠ t), (s ∆ t).max' ⋯ ∈ t\na : α\nhas : a ∈ ofColex (toColex s)\nhat : a ∉ ofColex (toColex t)\nhst : s ≠ t\n⊢ (s ∆ t).max' ⋯ ∉ s"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 184,
"column": 2
} | {
"line": 184,
"column": 51
} | {
"line": 184,
"column": 52
} | [
{
"pp": "ι : Type u_1\nn : ℕ\ns : Finset ι\na : ι → ZMod n\nhs : 2 * n - 1 ≤ #s\n⊢ ∃ t ⊆ s, #t = n ∧ ∑ i ∈ t, a i = 0",
"ppTerm": "?m.30",
"assigned": false,
"usedConstants": [],
"usedFVars": [],
"usedGoals": []
}
] | [
"ι : Type u_1\nn : ℕ\ns : Finset ι\na : ι → ZMod n\nhs : 2 * n - 1 ≤ #s\n⊢ ∃ t ⊆ s, #t = n ∧ ∑ i ∈ t, a i = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 192,
"column": 83
} | {
"line": 192,
"column": 94
} | {
"line": 192,
"column": 95
} | [
{
"pp": "n : ℕ\ns : Multiset ℤ\nhs : 2 * n - 1 ≤ s.card\n⊢ 2 * ?m.31 - 1 ≤ #s.toEnumFinset",
"ppTerm": "?m.36",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instOrderedSub",
"HMul.hMul",
"congrArg",
"HSub.hSub",
"Int.instDecidableEq",
"id",
"ins... | [
"n : ℕ\ns : Multiset ℤ\nhs : 2 * n - 1 ≤ s.card\n⊢ 2 * ?m.31 ≤ s.card + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 193,
"column": 78
} | {
"line": 193,
"column": 89
} | {
"line": 193,
"column": 90
} | [
{
"pp": "n : ℕ\ns : Multiset ℤ\nhs : 2 * n - 1 ≤ s.card\nt : Finset (ℤ × ℕ)\nhts : t ⊆ s.toEnumFinset\nht : #t = n ∧ ↑n ∣ ∑ i ∈ t, i.1\n⊢ (Multiset.map Prod.fst t.val).card = n ∧ ↑n ∣ (Multiset.map Prod.fst t.val).sum",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Multiset.sum",
... | [
"n : ℕ\ns : Multiset ℤ\nhs : 2 * n - 1 ≤ s.card\nt : Finset (ℤ × ℕ)\nhts : t ⊆ s.toEnumFinset\nht : #t = n ∧ ↑n ∣ ∑ i ∈ t, i.1\n⊢ #t = n ∧ ↑n ∣ ∑ a ∈ t, a.1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 201,
"column": 84
} | {
"line": 201,
"column": 95
} | {
"line": 201,
"column": 96
} | [
{
"pp": "n : ℕ\ns : Multiset (ZMod n)\nhs : 2 * n - 1 ≤ s.card\n⊢ 2 * n - 1 ≤ #s.toEnumFinset",
"ppTerm": "?m.34",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Nat.instOrderedSub",
"HMul.hMul",
"congrArg",
"ZMod.decidableEq",
"HSub.hSub",
"id",
"ins... | [
"n : ℕ\ns : Multiset (ZMod n)\nhs : 2 * n - 1 ≤ s.card\n⊢ 2 * n ≤ s.card + 1"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Additive.ErdosGinzburgZiv | {
"line": 202,
"column": 78
} | {
"line": 202,
"column": 89
} | {
"line": 202,
"column": 90
} | [
{
"pp": "n : ℕ\ns : Multiset (ZMod n)\nhs : 2 * n - 1 ≤ s.card\nt : Finset (ZMod n × ℕ)\nhts : t ⊆ s.toEnumFinset\nht : #t = n ∧ ∑ i ∈ t, i.1 = 0\n⊢ (Multiset.map Prod.fst t.val).card = n ∧ (Multiset.map Prod.fst t.val).sum = 0",
"ppTerm": "?m.75",
"assigned": true,
"usedConstants": [
"Multise... | [
"n : ℕ\ns : Multiset (ZMod n)\nhs : 2 * n - 1 ≤ s.card\nt : Finset (ZMod n × ℕ)\nhts : t ⊆ s.toEnumFinset\nht : #t = n ∧ ∑ i ∈ t, i.1 = 0\n⊢ #t = n ∧ ∑ a ∈ t, a.1 = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.Combinatorics.Colex | {
"line": 356,
"column": 4
} | {
"line": 356,
"column": 68
} | {
"line": 358,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : LinearOrder α\ns : Finset α\na : α\nha : a ∈ s\nhst : toColex s ≤ toColex s\nhcard : #s ≤ #s\nht : s.Nonempty\nm : α := s.min' ht\n⊢ toColex (s.erase a) ≤ toColex (s.erase m)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Finset.min'",
... | [] | exact (erase_le_erase ha <| min'_mem _ _).2 <| min'_le _ _ <| ha | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Combinatorics.Colex | {
"line": 356,
"column": 4
} | {
"line": 356,
"column": 68
} | {
"line": 358,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : LinearOrder α\ns : Finset α\na : α\nha : a ∈ s\nhst : toColex s ≤ toColex s\nhcard : #s ≤ #s\nht : s.Nonempty\nm : α := s.min' ht\n⊢ toColex (s.erase a) ≤ toColex (s.erase m)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Finset.min'",
... | [] | exact (erase_le_erase ha <| min'_mem _ _).2 <| min'_le _ _ <| ha | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Combinatorics.Colex | {
"line": 356,
"column": 4
} | {
"line": 356,
"column": 68
} | {
"line": 358,
"column": 2
} | [
{
"pp": "case inl\nα : Type u_1\ninst✝ : LinearOrder α\ns : Finset α\na : α\nha : a ∈ s\nhst : toColex s ≤ toColex s\nhcard : #s ≤ #s\nht : s.Nonempty\nm : α := s.min' ht\n⊢ toColex (s.erase a) ≤ toColex (s.erase m)",
"ppTerm": "?inl",
"assigned": true,
"usedConstants": [
"Finset.min'",
... | [] | exact (erase_le_erase ha <| min'_mem _ _).2 <| min'_le _ _ <| ha | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.Projectivization.Basic | {
"line": 244,
"column": 4
} | {
"line": 244,
"column": 19
} | {
"line": 244,
"column": 20
} | [
{
"pp": "case pos\nK : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nD D' : ℙ K V\nh : D = D'\n⊢ LinearIndepOn K id {D.rep, D'.rep}",
"ppTerm": "?pos✝",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"instIsTorsionFreeOfIsDomainOfNoZeroSMul... | [
"case pos\nK : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nD D' : ℙ K V\nh : D = D'\n⊢ ¬D'.rep = 0"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
Mathlib.LinearAlgebra.Projectivization.Basic | {
"line": 247,
"column": 4
} | {
"line": 247,
"column": 15
} | {
"line": 247,
"column": 16
} | [
{
"pp": "case neg\nK : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nD D' : ℙ K V\nh : LinearIndepOn K id (Set.range ![D'.rep, D.rep])\n⊢ LinearIndepOn K id {D.rep, D'.rep}",
"ppTerm": "?neg✝",
"assigned": false,
"usedConstants": [],
"usedFVars": [... | [
"case neg\nK : Type u_1\nV : Type u_2\ninst✝² : DivisionRing K\ninst✝¹ : AddCommGroup V\ninst✝ : Module K V\nD D' : ℙ K V\nh : LinearIndepOn K id (Set.range ![D'.rep, D.rep])\n⊢ LinearIndepOn K id {D.rep, D'.rep}"
] | simpa using | Lean.Elab.Tactic.Simpa.evalSimpa | null |
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