mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Combinatorics.SimpleGraph.IncMatrix

Goal state before:
pp: R : Type u_1
α : Type u_2
G : SimpleGraph α
inst✝³ : MulZeroOneClass R
inst✝² : DecidableEq α
inst✝¹ : DecidableRel G.Adj
a : α
e : Sym2 α
inst✝ : Nontrivial R
⊢ incMatrix R G a e = 0 ↔ e ∉ G.incidenceSet a

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.IncMatrix

Goal state before:
pp: R : Type u_1
α : Type u_2
G : SimpleGraph α
inst✝³ : MulZeroOneClass R
inst✝² : DecidableEq α
inst✝¹ : DecidableRel G.Adj
a : α
e : Sym2 α
inst✝ : Nontrivial R
⊢ incMatrix R G a e = 1 ↔ e ∈ G.incidenceSet a

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.FiveWheelLike

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
r k : ℕ
x v w₁ w₂ : α
s t : Finset α
inst✝¹ : DecidableEq α
hw : G.IsFiveWheelLike r k v w₁ w₂ s t
hcf : G.CliqueFree (r + 2)
inst✝ : DecidableRel G.Adj
hW : ∀ ⦃y : α⦄, y ∈ s ∩ t → G.Adj x y
h2 : #({z ∈ {v} ∪ ({w₁} ∪ ({w₂} ∪ (s ∪ t))) | ¬G.Adj x z}) ≤ 2
a b c d : α
ha : a ∈ insert w₁ s
haj : ¬G.Adj x a
hb : b ∈ insert w₂ t
hbj : ¬G.Adj x b
hc : c ∈ insert v s
hcj : ¬G.Adj x c
hd : d ∈ insert v t
hdj : ¬G.Adj x d
hab : a ≠ b
had : a ≠ d
hbc : b ≠ c
hat : a ∉ t
hbs : b ∉ s
W : Finset α := {v} ∪ ({w₁} ∪ ({w₂} ∪ (s ∪ t)))
hf : c = a → d ≠ b
h : ¬a = c
⊢ a ∈ {z ∈ {v} ∪ ({w₁} ∪ ({w₂} ∪ (s ∪ t))) | ¬G.Adj x z} ∧
    b ∈ {z ∈ {v} ∪ ({w₁} ∪ ({w₂} ∪ (s ∪ t))) | ¬G.Adj x z} ∧
      c ∈ {z ∈ {v} ∪ ({w₁} ∪ ({w₂} ∪ (s ∪ t))) | ¬G.Adj x z} ∧ a ≠ b ∧ a ≠ c ∧ b ≠ c

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.IncMatrix

Goal state before:
pp: R : Type u_1
α : Type u_2
G : SimpleGraph α
inst✝⁴ : NonAssocSemiring R
inst✝³ : DecidableEq α
inst✝² : DecidableRel G.Adj
a : α
inst✝¹ : Fintype (Sym2 α)
inst✝ : Fintype ↑(G.neighborSet a)
⊢ ∑ e, incMatrix R G a e = ↑(G.degree a)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
W : Type u_2
G : SimpleGraph V
G' : SimpleGraph W
M : G.Subgraph
f : G →g G'
hf : Injective ⇑f
hM : M.IsMatching
⊢ (Subgraph.map f M).IsMatching

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
W : Type u_2
G : SimpleGraph V
G' : SimpleGraph W
M : G.Subgraph
f : G ≃g G'
h : (Subgraph.map f.toHom M).IsMatching
⊢ M.IsMatching

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.IncMatrix

Goal state before:
pp: R : Type u_1
α : Type u_2
G : SimpleGraph α
inst✝⁴ : NonAssocSemiring R
inst✝³ : DecidableEq α
inst✝² : DecidableRel G.Adj
a : α
inst✝¹ : Fintype (Sym2 α)
inst✝ : Fintype ↑(G.neighborSet a)
⊢ (incMatrix R G * (incMatrix R G)ᵀ) a a = ↑(G.degree a)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits

Goal state before:
pp: ε : ℝ
hε : 0 < ε
r t : ℕ
hr_pos : 0 < r
ht_pos : 0 < t
ε' : ℝ := 1 / (↑(r - 1) * ↑r) + ε
hε' : 0 < ε'
t' : ℕ := ⌊↑t / (↑r * ε)⌋₊ + 1
ht_lt_rt'ε : ↑t < ↑r * ↑t' * ε
ht'_pos : 0 < t'
N' : ℕ
ih :
  ∀ (b : ℕ),
    N' ≤ b →
      ∀ {G : SimpleGraph (Fin b)} [inst : DecidableRel G.Adj],
        ↑G.minDegree ≥ (1 - 1 / ↑(r - 1) + ε') * ↑b → completeEquipartiteGraph (r - 1 + 1) t' ⊑ G
N : ℕ := max (max 1 N') ⌈(↑(t'.choose t) ^ r * ↑t + ↑r * ↑t') * (↑t' - ↑t) / (↑r * ↑t' * ε - ↑t)⌉₊
n : ℕ
hn : N ≤ n
G : SimpleGraph (Fin n)
x✝ : DecidableRel G.Adj
hδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n
this : Nonempty (Fin n)
h1_le_rε : 1 ≤ ↑r * ε
⊢ 0 < ↑r

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
M : G.Subgraph
h : M.IsMatching
⊢ M.verts = ⋃ e ∈ M.edgeSet, ↑e

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
⊢ Set.InjOn edgeSet (Set.ofPred IsMatching)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.FiveWheelLike

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
r k : ℕ
x v w₁ w₂ : α
s t : Finset α
inst✝¹ : DecidableEq α
hw : G.IsFiveWheelLike r k v w₁ w₂ s t
hcf : G.CliqueFree (r + 2)
inst✝ : DecidableRel G.Adj
hW : ∀ ⦃y : α⦄, y ∈ s ∩ t → G.Adj x y
h2 : #({z ∈ {v} ∪ ({w₁} ∪ ({w₂} ∪ (s ∪ t))) | ¬G.Adj x z}) ≤ 2
a b c d : α
ha : a ∈ insert w₁ s
haj : ¬G.Adj x a
hb : b ∈ insert w₂ t
hbj : ¬G.Adj x b
hc : c ∈ insert v s
hcj : ¬G.Adj x c
hd : d ∈ insert v t
hdj : ¬G.Adj x d
hab : a ≠ b
had : a ≠ d
hbc : b ≠ c
hat : a ∉ t
hbs : b ∉ s
W : Finset α := {v} ∪ ({w₁} ∪ ({w₂} ∪ (s ∪ t)))
⊢ c = a ∧ d = b

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
M : G.Subgraph
h : M.IsMatching
⊢ M.support = M.verts

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
M : G.Subgraph
inst✝ : (v : V) → Fintype ↑(M.neighborSet v)
⊢ M.IsMatching ↔ ∀ v ∈ M.verts, M.degree v = 1

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
M : G.Subgraph
inst✝ : Fintype ↑M.verts
h : M.IsMatching
⊢ Even M.verts.toFinset.card

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.IncMatrix

Goal state before:
pp: R : Type u_1
α : Type u_2
G : SimpleGraph α
inst✝³ : NonAssocSemiring R
inst✝² : DecidableEq α
inst✝¹ : DecidableRel G.Adj
e : Sym2 α
inst✝ : Fintype α
⊢ e ∈ G.edgeSet → ∑ a, incMatrix R G a e = 2

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
M : G.Subgraph
⊢ M.IsPerfectMatching ↔ ∀ (v : V), ∃! w, M.Adj v w

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
M : G.Subgraph
inst✝ : (v : V) → Fintype ↑(M.neighborSet v)
⊢ M.IsPerfectMatching ↔ ∀ (v : V), M.degree v = 1

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
M : G.Subgraph
inst✝ : Fintype V
h : M.IsPerfectMatching
⊢ Even (Fintype.card V)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
M : G.Subgraph
h : M.IsMatching
c : G.ConnectedComponent
⊢ (M.induce (M.verts ∩ c.supp)).IsMatching

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
M : G.Subgraph
h : M.IsPerfectMatching
c : G.ConnectedComponent
⊢ (M.induce c.supp).IsMatching

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G G' : SimpleGraph V
M : G.Subgraph
h : M.spanningCoe ≤ G'
⊢ (toSubgraph M.spanningCoe h).IsPerfectMatching ↔ M.IsPerfectMatching

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.FiveWheelLike

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
r k : ℕ
x v w₁ w₂ : α
s t : Finset α
inst✝¹ : DecidableEq α
hw : G.IsFiveWheelLike r k v w₁ w₂ s t
hcf : G.CliqueFree (r + 2)
inst✝ : DecidableRel G.Adj
hW : ∀ ⦃y : α⦄, y ∈ s ∩ t → G.Adj x y
h2 : #({z ∈ {v} ∪ ({w₁} ∪ ({w₂} ∪ (s ∪ t))) | ¬G.Adj x z}) ≤ 2
a b c d : α
haj : ¬G.Adj x a
hbj : ¬G.Adj x b
hab : a ≠ b
hat : a ∉ t
hbs : b ∉ s
W : Finset α := {v} ∪ ({w₁} ∪ ({w₂} ∪ (s ∪ t)))
hcj : ¬G.Adj x a
hbc : b ≠ a
hca : True
hdj : ¬G.Adj x b
had : a ≠ b
hdb : True
ha : a = w₁ ∨ a ∈ s
hb : b = w₂ ∨ b ∈ t
hc : a = v ∨ a ∈ s
hd : b = v ∨ b ∈ t
⊢ a ∈ s ∧ b ∈ t ∧ a ≠ v ∧ b ≠ v ∧ a ≠ w₂ ∧ b ≠ w₁

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
u : Set V
hc : G.IsClique u
hu : u.Finite
⊢ (∃ M, M.verts = u ∧ M.IsMatching) → Even u.ncard

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits

Goal state before:
pp: ε : ℝ
hε : 0 < ε
r t : ℕ
hr_pos : 0 < r
ht_pos : 0 < t
ε' : ℝ := 1 / (↑(r - 1) * ↑r) + ε
hε' : 0 < ε'
t' : ℕ := ⌊↑t / (↑r * ε)⌋₊ + 1
ht_lt_rt'ε : ↑t < ↑r * ↑t' * ε
ht'_pos : 0 < t'
N' : ℕ
ih :
  ∀ (b : ℕ),
    N' ≤ b →
      ∀ {G : SimpleGraph (Fin b)} [inst : DecidableRel G.Adj],
        ↑G.minDegree ≥ (1 - 1 / ↑(r - 1) + ε') * ↑b → completeEquipartiteGraph (r - 1 + 1) t' ⊑ G
N : ℕ := max (max 1 N') ⌈(↑(t'.choose t) ^ r * ↑t + ↑r * ↑t') * (↑t' - ↑t) / (↑r * ↑t' * ε - ↑t)⌉₊
n : ℕ
hn : N ≤ n
G : SimpleGraph (Fin n)
x✝ : DecidableRel G.Adj
hδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n
this : Nonempty (Fin n)
⊢ ↑r * ε < 1

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
t u : Set V
hc : G.IsClique (t ∪ u)
hu : (t ∪ u).Finite
h : Even (t ∪ u).ncard
hd : Disjoint t u
hcard : t.ncard = u.ncard
⊢ Nonempty (↑t ≃ ↑u)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
t u : Set V
hc : G.IsClique (t ∪ u)
hu : (t ∪ u).Finite
h : Even (t ∪ u).ncard
hd : Disjoint t u
hcard : t.ncard = u.ncard
f : ↑t ≃ ↑u
v : ↑t
⊢ ↑v ∈ t ∪ u

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.IncMatrix

Goal state before:
pp: R : Type u_1
α : Type u_2
G : SimpleGraph α
inst✝⁴ : NonAssocSemiring R
inst✝³ : DecidableEq α
inst✝² : DecidableRel G.Adj
e : Sym2 α
inst✝¹ : Fintype α
inst✝ : Decidable (e ∈ G.edgeSet)
⊢ ((incMatrix R G)ᵀ * incMatrix R G) e e = if e ∈ G.edgeSet then 2 else 0

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
t u : Set V
hc : G.IsClique (t ∪ u)
hu : (t ∪ u).Finite
h : Even (t ∪ u).ncard
hd : Disjoint t u
hcard : t.ncard = u.ncard
f : ↑t ≃ ↑u
v : ↑t
⊢ ↑(f v) ∈ t ∪ u

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
t u : Set V
hc : G.IsClique (t ∪ u)
hu : (t ∪ u).Finite
h : Even (t ∪ u).ncard
hd : Disjoint t u
hcard : t.ncard = u.ncard
f : ↑t ≃ ↑u
v : ↑t
⊢ ↑v ∈ t

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
t u : Set V
hc : G.IsClique (t ∪ u)
hu : (t ∪ u).Finite
h : Even (t ∪ u).ncard
hd : Disjoint t u
hcard : t.ncard = u.ncard
f : ↑t ≃ ↑u
v : ↑t
⊢ ↑(f v) ∈ u

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
u : Set V
hc : G.IsClique u
hu : u.Finite
⊢ Even u.ncard ↔ ∃ M, M.verts = u ∧ M.IsMatching

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.IncMatrix

Goal state before:
pp: R : Type u_1
α : Type u_2
G : SimpleGraph α
inst✝³ : Fintype (Sym2 α)
inst✝² : DecidableEq α
inst✝¹ : DecidableRel G.Adj
inst✝ : Semiring R
a b : α
h : G.Adj a b
⊢ (incMatrix R G * (incMatrix R G)ᵀ) a b = 1

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
M : G.Subgraph
inst✝² : Fintype V
inst✝¹ : DecidableEq V
inst✝ : DecidableRel G.Adj
c : G.ConnectedComponent
hM : M.IsPerfectMatching
⊢ Even (Fintype.card ↑c.supp)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.IncMatrix

Goal state before:
pp: R : Type u_1
α : Type u_2
G : SimpleGraph α
inst✝⁴ : Fintype (Sym2 α)
inst✝³ : DecidableEq α
inst✝² : DecidableRel G.Adj
inst✝¹ : Semiring R
inst✝ : G.LocallyFinite
⊢ incMatrix R G * (incMatrix R G)ᵀ = of fun a b ↦ if a = b then ↑(G.degree a) else if G.Adj a b then 1 else 0

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix

Goal state before:
pp: α : Type u_1
V : Type u_2
inst✝¹ : NonAssocSemiring α
inst✝ : StarRing α
A : Matrix V V α
h : A.IsAdjMatrix
⊢ A.IsHermitian

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix

Goal state before:
pp: V : Type u_1
inst✝² : Fintype V
G : SimpleGraph V
inst✝¹ : DecidableRel G.Adj
R : Type u_3
inst✝ : AddCommMonoidWithOne R
i : V
⊢ ↑(G.degree i) = ∑ j, if G.Adj i j then 1 else 0

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix

Goal state before:
pp: V : Type u_1
R : Type u_2
inst✝⁴ : Fintype V
G : SimpleGraph V
inst✝³ : DecidableRel G.Adj
inst✝² : DecidableEq V
inst✝¹ : NonAssocSemiring R
inst✝ : StarRing R
⊢ ∀ (i : V), IsSelfAdjoint ↑(G.degree i)

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix

Goal state before:
pp: V : Type u_1
R : Type u_2
inst✝³ : Fintype V
G : SimpleGraph V
inst✝² : DecidableRel G.Adj
inst✝¹ : DecidableEq V
inst✝ : NonAssocSemiring R
v : V
vec : V → R
⊢ (degMatrix R G *ᵥ vec) v = ↑(G.degree v) * vec v

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits

Goal state before:
pp: ε : ℝ
hε : 0 < ε
r t : ℕ
hr_pos : 0 < r
ht_pos : 0 < t
ε' : ℝ := 1 / (↑(r - 1) * ↑r) + ε
hε' : 0 < ε'
t' : ℕ := ⌊↑t / (↑r * ε)⌋₊ + 1
ht_lt_rt'ε : ↑t < ↑r * ↑t' * ε
ht'_pos : 0 < t'
N' : ℕ
ih :
  ∀ (b : ℕ),
    N' ≤ b →
      ∀ {G : SimpleGraph (Fin b)} [inst : DecidableRel G.Adj],
        ↑G.minDegree ≥ (1 - 1 / ↑(r - 1) + ε') * ↑b → completeEquipartiteGraph (r - 1 + 1) t' ⊑ G
N : ℕ := max (max 1 N') ⌈(↑(t'.choose t) ^ r * ↑t + ↑r * ↑t') * (↑t' - ↑t) / (↑r * ↑t' * ε - ↑t)⌉₊
n : ℕ
hn : N ≤ n
G : SimpleGraph (Fin n)
x✝ : DecidableRel G.Adj
hδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n
this : Nonempty (Fin n)
hrε_lt_1 : ↑r * ε < 1
⊢ t < t'

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix

Goal state before:
pp: V : Type u_1
R : Type u_2
inst✝³ : Fintype V
G : SimpleGraph V
inst✝² : DecidableRel G.Adj
inst✝¹ : DecidableEq V
inst✝ : NonAssocRing R
v : V
vec : V → R
⊢ (lapMatrix R G *ᵥ vec) v = ↑(G.degree v) * vec v - ∑ u ∈ G.neighborFinset v, vec u

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix

Goal state before:
pp: V : Type u_1
R : Type u_2
inst✝³ : Fintype V
G : SimpleGraph V
inst✝² : DecidableRel G.Adj
inst✝¹ : DecidableEq V
inst✝ : NonAssocRing R
⊢ (lapMatrix R G *ᵥ fun x ↦ 1) = 0

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix

Goal state before:
pp: V : Type u_1
R : Type u_2
inst✝³ : Fintype V
G : SimpleGraph V
inst✝² : DecidableRel G.Adj
inst✝¹ : DecidableEq V
inst✝ : CommSemiring R
x : V → R
⊢ x ⬝ᵥ degMatrix R G *ᵥ x = ∑ i, ↑(G.degree i) * x i * x i

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
M : G.Subgraph
inst✝ : Finite V
u : Set V
hM : M.IsPerfectMatching
c : ↑(⊤.deleteVerts u).coe.oddComponents
h : ∀ w ∈ u, ∀ (v : ↑(⊤.deleteVerts u).verts), M.Adj (↑v) w → v ∉ (↑c).supp
w v' : V
hv' : v' ∈ (⊤.deleteVerts u).verts
hv : ⟨v', hv'⟩ ∈ (↑c).supp
hw : (fun w ↦ M.Adj (↑⟨v', hv'⟩) w) w ∧ ∀ (y : V), (fun w ↦ M.Adj (↑⟨v', hv'⟩) w) y → y = w
hwnu : w ∉ u
⊢ w ∈ ⊤.verts

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix

Goal state before:
pp: V : Type u_1
R : Type u_2
inst✝⁴ : Fintype V
G : SimpleGraph V
inst✝³ : DecidableRel G.Adj
inst✝² : DecidableEq V
inst✝¹ : Field R
inst✝ : CharZero R
x : V → R
⊢ (∑ i, ↑(G.degree i) * x i * x i - ∑ i, ∑ j, if G.Adj i j then x i * x j else 0) =
    (∑ i, ∑ j, if G.Adj i j then (x i - x j) ^ 2 else 0) / 2

Tactic:
← sum_sub_distrib,

Goal state after:
V : Type u_1
R : Type u_2
inst✝⁴ : Fintype V
G : SimpleGraph V
inst✝³ : DecidableRel G.Adj
inst✝² : DecidableEq V
inst✝¹ : Field R
inst✝ : CharZero R
x : V → R
⊢ ∑ x_1, (↑(G.degree x_1) * x x_1 * x x_1 - ∑ j, if G.Adj x_1 j then x x_1 * x j else 0) =
    (∑ i, ∑ j, if G.Adj i j then (x i - x j) ^ 2 else 0) / 2
Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix

Goal state before:
pp: V : Type u_1
R : Type u_2
inst✝⁴ : Fintype V
G : SimpleGraph V
inst✝³ : DecidableRel G.Adj
inst✝² : DecidableEq V
inst✝¹ : Field R
inst✝ : CharZero R
x : V → R
⊢ ∑ x_1, ((∑ x_2, if G.Adj x_1 x_2 then x x_1 * x x_1 else 0) - ∑ j, if G.Adj x_1 j then x x_1 * x j else 0) =
    (∑ i, ∑ j, if G.Adj i j then (x i - x j) ^ 2 else 0) / 2

Tactic:
← sum_sub_distrib,

Goal state after:
V : Type u_1
R : Type u_2
inst✝⁴ : Fintype V
G : SimpleGraph V
inst✝³ : DecidableRel G.Adj
inst✝² : DecidableEq V
inst✝¹ : Field R
inst✝ : CharZero R
x : V → R
⊢ ∑ x_1, ∑ x_2, ((if G.Adj x_1 x_2 then x x_1 * x x_1 else 0) - if G.Adj x_1 x_2 then x x_1 * x x_2 else 0) =
    (∑ i, ∑ j, if G.Adj i j then (x i - x j) ^ 2 else 0) / 2
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
M : G.Subgraph
inst✝ : Finite V
u : Set V
hM : M.IsPerfectMatching
c : ↑(⊤.deleteVerts u).coe.oddComponents
h : ∀ w ∈ u, ∀ (v : ↑(⊤.deleteVerts u).verts), M.Adj (↑v) w → v ∉ (↑c).supp
⊢ (M.induce (Subtype.val '' (↑c).supp)).IsMatching

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
M : G.Subgraph
inst✝ : Finite V
u : Set V
hM : M.IsPerfectMatching
c : ↑(⊤.deleteVerts u).coe.oddComponents
⊢ ∃ w ∈ u, ∃ v, M.Adj (↑v) w ∧ v ∈ (↑c).supp

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix

Goal state before:
pp: V : Type u_1
R : Type u_2
inst✝⁴ : Fintype V
G : SimpleGraph V
inst✝³ : DecidableRel G.Adj
inst✝² : DecidableEq V
inst✝¹ : Field R
inst✝ : CharZero R
x : V → R
⊢ (((toLinearMap₂' R) (lapMatrix R G)) x) x = (∑ i, ∑ j, if G.Adj i j then (x i - x j) ^ 2 else 0) / 2

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G G' : SimpleGraph V
h : G ≤ G'
hmf : G'.IsMatchingFree
⊢ G.IsMatchingFree

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
h : G.IsMatchingFree
⊢ ∃ Gmax, G ≤ Gmax ∧ Gmax.IsMatchingFree ∧ ∀ G' > Gmax, ∃ M, M.IsPerfectMatching

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix

Goal state before:
pp: V : Type u_1
R : Type u_2
inst✝⁷ : Fintype V
G : SimpleGraph V
inst✝⁶ : DecidableRel G.Adj
inst✝⁵ : DecidableEq V
inst✝⁴ : Field R
inst✝³ : LinearOrder R
inst✝² : IsStrictOrderedRing R
inst✝¹ : StarRing R
inst✝ : TrivialStar R
⊢ (lapMatrix R G).PosSemidef

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
v w : V
h : G.IsCycles
hadj : w ∈ G.neighborSet v
this : (G.neighborSet v).ncard = 2
⊢ 1 < (G.neighborSet v).ncard

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix

Goal state before:
pp: V : Type u_1
R : Type u_2
inst✝⁵ : Fintype V
G : SimpleGraph V
inst✝⁴ : DecidableRel G.Adj
inst✝³ : DecidableEq V
inst✝² : Field R
inst✝¹ : LinearOrder R
inst✝ : IsStrictOrderedRing R
x : V → R
⊢ (((toLinearMap₂' R) (lapMatrix R G)) x) x = 0 ↔ ∀ (i j : V), G.Adj i j → x i = x j

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
v w : V
h : G.IsCycles
hadj : G.Adj v w
⊢ ∃ w', w ≠ w' ∧ G.Adj v w'

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.FiveWheelLike

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
r k : ℕ
x v w₁ w₂ : α
s t : Finset α
inst✝¹ : DecidableEq α
hw : G.IsFiveWheelLike r k v w₁ w₂ s t
hcf : G.CliqueFree (r + 2)
inst✝ : DecidableRel G.Adj
hW : ∀ ⦃y : α⦄, y ∈ s ∩ t → G.Adj x y
h2 : #({z ∈ {v} ∪ ({w₁} ∪ ({w₂} ∪ (s ∪ t))) | ¬G.Adj x z}) ≤ 2
a b c d : α
haj : ¬G.Adj x a
hbj : ¬G.Adj x b
hab : a ≠ b
hat : a ∉ t
hbs : b ∉ s
W : Finset α := {v} ∪ ({w₁} ∪ ({w₂} ∪ (s ∪ t)))
hcj : ¬G.Adj x a
hbc : b ≠ a
hca : True
hdj : ¬G.Adj x b
had : a ≠ b
hdb : True
ha : a = w₁ ∨ a ∈ s
hb : b = w₂ ∨ b ∈ t
hc : a = v ∨ a ∈ s
hd : b = v ∨ b ∈ t
has : a ∈ s
hbt : b ∈ t
hav : a ≠ v
hbv : b ≠ v
haw : a ≠ w₂
hbw : b ≠ w₁
⊢ v ≠ x ∧ w₁ ≠ x ∧ w₂ ≠ x

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix

Goal state before:
pp: V : Type u_1
inst✝² : Fintype V
G : SimpleGraph V
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableEq V
x : V → ℝ
⊢ lapMatrix ℝ G *ᵥ x = 0 ↔ ∀ (i j : V), G.Adj i j → x i = x j

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
v w : V
h : G.IsCycles
hadj : G.Adj v w
⊢ ∃! w', w ≠ w' ∧ G.Adj v w'

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
c : G.ConnectedComponent
h : G.IsCycles
⊢ c.toSimpleGraph.spanningCoe.IsCycles

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
u : V
p : G.Walk u u
hpc : p.IsCycle
⊢ p.toSubgraph.spanningCoe.IsCycles

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
u v : V
p : G.Walk u v
hp : p.IsPath
h : u ≠ v
hs : s(v, u) ∉ p.edges
⊢ (completeGraph V).Adj v u

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix

Goal state before:
pp: V : Type u_1
inst✝² : Fintype V
G : SimpleGraph V
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableEq V
x : V → ℝ
⊢ (((toLinearMap₂' ℝ) (lapMatrix ℝ G)) x) x = 0 ↔ ∀ (i j : V), G.Reachable i j → x i = x j

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
u v : V
p : G.Walk u v
hp : p.IsPath
h : u ≠ v
hs : s(v, u) ∉ p.edges
c : (completeGraph V).Walk v v := cons ⋯ (Walk.mapLe ⋯ p)
⊢ p.toSubgraph.spanningCoe ⊔ edge v u = c.toSubgraph.spanningCoe

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
u v : V
p : G.Walk u v
hp : p.IsPath
h : u ≠ v
hs : s(v, u) ∉ p.edges
c : (completeGraph V).Walk v v := cons ⋯ (Walk.mapLe ⋯ p)
this : p.toSubgraph.spanningCoe ⊔ edge v u = c.toSubgraph.spanningCoe
⊢ c.IsCycle

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix

Goal state before:
pp: V : Type u_1
inst✝² : Fintype V
G : SimpleGraph V
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableEq V
x : V → ℝ
⊢ lapMatrix ℝ G *ᵥ x = 0 ↔ ∀ (i j : V), G.Reachable i j → x i = x j

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
u v : V
p : G.Walk u v
hp : p.IsPath
h : u ≠ v
hs : s(v, u) ∉ p.edges
⊢ (p.toSubgraph.spanningCoe ⊔ edge v u).IsCycles

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
v : V
inst✝ : G.LocallyFinite
u : V
p : G.Walk u u
hp : p.IsCycle
hcyc : G.IsCycles
hv : v ∈ p.toSubgraph.verts
w : V
⊢ ?m.46 ∈ p.support

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
v : V
inst✝ : G.LocallyFinite
u : V
p : G.Walk u u
hp : p.IsCycle
hcyc : G.IsCycles
hv : v ∈ p.toSubgraph.verts
w : V
this : (p.toSubgraph.neighborSet v).ncard = 2
⊢ (p.toSubgraph.neighborSet v).ncard ≠ 0

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
v : V
inst✝ : G.LocallyFinite
u : V
p : G.Walk u u
hp : p.IsCycle
hcyc : G.IsCycles
hv : v ∈ p.toSubgraph.verts
⊢ ∀ (w : V), p.toSubgraph.Adj v w ↔ G.Adj v w

Tactic:
by

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.LapMatrix

Goal state before:
pp: V : Type u_1
inst✝² : Fintype V
G : SimpleGraph V
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableEq V
h : Nonempty V
⊢ (lapMatrix ℝ G).det = 0

Tactic:
by

Goal state after: