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

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
i j : ℕ
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
W X : Finset α
hx : ∀ x ∈ X, i ≤ #({z ∈ W | ¬G.Adj x z})
hxc : ∀ y ∈ Xᶜ, j ≤ #({z ∈ W | ¬G.Adj y z})
⊢ ∑ x ∈ X ∪ Xᶜ, #(G.neighborFinset x ∩ W) ≤ #X * (#W - i) + #Xᶜ * (#W - j)

Tactic:
sum_union disjoint_compl_right (s₁ := X),

Goal state after:
α : Type u_1
G : SimpleGraph α
i j : ℕ
inst✝² : Fintype α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
W X : Finset α
hx : ∀ x ∈ X, i ≤ #({z ∈ W | ¬G.Adj x z})
hxc : ∀ y ∈ Xᶜ, j ≤ #({z ∈ W | ¬G.Adj y z})
⊢ ∑ x ∈ X, #(G.neighborFinset x ∩ W) + ∑ x ∈ Xᶜ, #(G.neighborFinset x ∩ W) ≤ #X * (#W - i) + #Xᶜ * (#W - j)
Module: Mathlib.Combinatorics.SimpleGraph.Hall

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
inst✝ : G.LocallyFinite
p₁ p₂ : Set V
h₁ : G.IsBipartiteWith p₁ p₂
h₂ : ∀ (s : Set V), s.ncard ≤ (⋃ x ∈ s, G.neighborSet x).ncard
b : ↑p₁ → ↑p₂
hb₁ : Bijective b
hb₂ : ∀ (a : ↑p₁), G.Adj ↑a ↑(b a)
⊢ p₁ ∪ p₂ = Set.univ

Tactic:
]

Goal state after:
V : Type u_1
G : SimpleGraph V
inst✝ : G.LocallyFinite
p₁ p₂ : Set V
h₁ : G.IsBipartiteWith p₁ p₂
h₂ : ∀ (s : Set V), s.ncard ≤ (⋃ x ∈ s, G.neighborSet x).ncard
b : ↑p₁ → ↑p₂
hb₁ : Bijective b
hb₂ : ∀ (a : ↑p₁), G.Adj ↑a ↑(b a)
⊢ p₁ ∪ p₂ = Set.univ
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
⊢ ↑t / (↑r * ε) < ↑⌊↑t / (↑r * ε)⌋₊ + 1

Tactic:
]

Goal state after:
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
⊢ ↑t / (↑r * ε) < ↑⌊↑t / (↑r * ε)⌋₊ + 1
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.ErdosStoneSimonovits

Goal state before:
pp: case inr
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'
⊢ ∀ᶠ (n : ℕ) in atTop,
    ∀ {G : SimpleGraph (Fin n)} [inst : DecidableRel G.Adj],
      ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n → completeEquipartiteGraph (r + 1) t ⊑ G

Tactic:
have ⟨N', ih⟩ := eventually_atTop.mp <|
      eventually_completeEquipartiteGraph_isContained_of_minDegree hε' (r - 1) t'

Goal state after:
case inr
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
⊢ ∀ᶠ (n : ℕ) in atTop,
    ∀ {G : SimpleGraph (Fin n)} [inst : DecidableRel G.Adj],
      ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n → completeEquipartiteGraph (r + 1) t ⊑ G
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
v w : V
h : G.Adj v w
v✝ : V
hv : v✝ = v ∨ v✝ = w
⊢ ∃! w_1, (G.subgraphOfAdj h).Adj v✝ w_1

Tactic:
]

Goal state after:
V : Type u_1
G : SimpleGraph V
v w : V
h : G.Adj v w
v✝ : V
hv : v✝ = v ∨ v✝ = w
⊢ ∃! w_1, (G.subgraphOfAdj h).Adj v✝ w_1
Module: Mathlib.Combinatorics.SimpleGraph.Hamiltonian

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
G : SimpleGraph α
a b : α
p : G.Walk a b
inst✝ : Subsingleton α
v : α
⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
G : SimpleGraph α
a b : α
p : G.Walk a b
inst✝ : Fintype α
hp : p.IsHamiltonian
⊢ Fintype.card α = Fintype.card α

Tactic:
]

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
G : SimpleGraph α
β : Type u_2
H : SimpleGraph β
a b : α
p : G.Walk a b
f : G →g H
hp : p.IsHamiltonian
h : (Walk.map f p).IsPath
⊢ Set.InjOn ⇑f {v | v ∈ p.support}

Tactic:
]

Goal state after:
α : Type u_1
inst✝ : DecidableEq α
G : SimpleGraph α
β : Type u_2
H : SimpleGraph β
a b : α
p : G.Walk a b
f : G →g H
hp : p.IsHamiltonian
h : (Walk.map f p).IsPath
⊢ Set.InjOn ⇑f {v | v ∈ p.support}
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: case h.refine_2
V : Type u_1
G : SimpleGraph V
G' : G.Subgraph
M : G'.coe.Subgraph
hM : M.IsMatching
v✝ : V
hv : v✝ ∈ (Subgraph.coeSubgraph M).verts
w : ↑G'.verts
hw : (fun w ↦ M.Adj ⟨v✝, ⋯⟩ w) w ∧ ∀ (y : ↑G'.verts), (fun w ↦ M.Adj ⟨v✝, ⋯⟩ w) y → y = w
y : V
hy : (fun w ↦ (Subgraph.coeSubgraph M).Adj v✝ w) y
w✝ : v✝ ∈ G'.verts
hw' : y ∈ G'.verts
hvw : M.Adj ⟨v✝, w✝⟩ ⟨y, hw'⟩
⊢ y = ↑⟨y, hw'⟩

Tactic:
]

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

Goal state before:
pp: case cons
α : Type u_1
inst✝¹ : DecidableEq α
G : SimpleGraph α
β : Type u_2
inst✝ : DecidableEq β
H : SimpleGraph β
a : α
f : G →g H
hf : Bijective ⇑f
x v✝ : α
y : G.Adj a v✝
p : G.Walk v✝ a
hp : (cons y p).IsHamiltonianCycle
__IsCycle✝ : (Walk.map f (cons y p)).IsCycle := IsCycle.map (Bijective.injective hf) (isCycle hp)
⊢ List.count x p.support = 1

Tactic:
]

Goal state after:
case cons
α : Type u_1
inst✝¹ : DecidableEq α
G : SimpleGraph α
β : Type u_2
inst✝ : DecidableEq β
H : SimpleGraph β
a : α
f : G →g H
hf : Bijective ⇑f
x v✝ : α
y : G.Adj a v✝
p : G.Walk v✝ a
hp : (cons y p).IsHamiltonianCycle
__IsCycle✝ : (Walk.map f (cons y p)).IsCycle := IsCycle.map (Bijective.injective hf) (isCycle hp)
⊢ List.count x p.support = 1
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
⊢ 0 < n

Tactic:
]

Goal state after:
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
⊢ 0 < n
Module: Mathlib.Combinatorics.SimpleGraph.Hamiltonian

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
G : SimpleGraph α
a : α
p : G.Walk a a
inst✝ : Fintype α
hp : p.IsHamiltonianCycle
⊢ Fintype.card α = Fintype.card α

pp: α : Type u_1
inst✝¹ : DecidableEq α
G : SimpleGraph α
a : α
p : G.Walk a a
inst✝ : Fintype α
hp : p.IsHamiltonianCycle
⊢ 1 ≤ Fintype.card α

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
G : SimpleGraph α
a : α
p : G.Walk a a
inst✝ : Fintype α
hp : p.IsHamiltonianCycle
⊢ 1 ≤ Fintype.card α
Module: Mathlib.Combinatorics.SimpleGraph.Hamiltonian

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
G : SimpleGraph α
a : α
p : G.Walk a a
inst✝ : Fintype α
hp : p.IsHamiltonianCycle
⊢ Nonempty α

Tactic:
]

Goal state after:
α : Type u_1
inst✝¹ : DecidableEq α
G : SimpleGraph α
a : α
p : G.Walk a a
inst✝ : Fintype α
hp : p.IsHamiltonianCycle
⊢ Nonempty α
Module: Mathlib.Combinatorics.SimpleGraph.Hamiltonian

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
G : SimpleGraph α
a : α
p : G.Walk a a
hp : p.IsHamiltonianCycle
⊢ 1 + 1 = 2

Tactic:
]

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
G : SimpleGraph α
a b : α
p : G.Walk a a
hp : p.IsHamiltonianCycle
h : a ≠ b
⊢ 1 = 1

Tactic:
]

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

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
G : SimpleGraph α
inst✝ : Fintype α
e : Sym2 α
he : G.IsBridge e
⊢ ¬G.IsHamiltonian

Tactic:
induction e with | h u v
  have := he.nontrivial
  intro hG
  obtain ⟨p, hp⟩ := hG.exists_isHamiltonianCycle u
  refine hp.isHamiltonian_tail.isPath.isTrail.not_mem_support_of_not_reachable
    (fun huv ↦ he <| .trans ?_ huv) he (hp.isHamiltonian_tail.mem_support v)
  apply hp.isTrail.isEdgeReachable_two <;> simp

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

Goal state before:
pp: R : Type u_1
α : Type u_2
G : SimpleGraph α
inst✝² : MulZeroOneClass R
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
a b : α
e : Sym2 α
hab : a ≠ b
h : ¬G.Adj a b
⊢ 0 = 0

pp: case h
R : Type u_1
α : Type u_2
G : SimpleGraph α
inst✝² : MulZeroOneClass R
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
a b : α
e : Sym2 α
hab : a ≠ b
h : ¬G.Adj a b
⊢ e ∉ G.incidenceSet a ∩ G.incidenceSet b

Tactic:
]

Goal state after:
case h
R : Type u_1
α : Type u_2
G : SimpleGraph α
inst✝² : MulZeroOneClass R
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
a b : α
e : Sym2 α
hab : a ≠ b
h : ¬G.Adj a b
⊢ e ∉ G.incidenceSet a ∩ G.incidenceSet b
Module: Mathlib.Combinatorics.SimpleGraph.IncMatrix

Goal state before:
pp: case h
R : Type u_1
α : Type u_2
G : SimpleGraph α
inst✝² : MulZeroOneClass R
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
a b : α
e : Sym2 α
hab : a ≠ b
h : ¬G.Adj a b
⊢ e ∉ ∅

Tactic:
]

Goal state after:
case h
R : Type u_1
α : Type u_2
G : SimpleGraph α
inst✝² : MulZeroOneClass R
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
a b : α
e : Sym2 α
hab : a ≠ b
h : ¬G.Adj a b
⊢ e ∉ ∅
Module: Mathlib.Combinatorics.SimpleGraph.IncMatrix

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

Tactic:
]

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

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

Tactic:
]

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

Goal state before:
pp: case h.refl
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
v : V
hv : v ∈ M.verts
v' : V
hv' : (fun w ↦ M.Adj v w) v' ∧ ∀ (y : V), (fun w ↦ M.Adj v w) y → y = v'
w' : V
hw : M.Adj v w'
hw' : f v = f v
⊢ f v' = f v'

Tactic:
]

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 = ∑ e, incMatrix R G a e

Tactic:
]

Goal state after:
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 = ∑ e, incMatrix R G a e
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:
refine Set.ext fun v ↦ .trans ⟨fun hv ↦ ?_, fun ⟨e, he, hv⟩ ↦ ?_⟩ Set.mem_iUnion₂.symm

Goal state after:
case refine_1
V : Type u_1
G : SimpleGraph V
M : G.Subgraph
h : M.IsMatching
v : V
hv : v ∈ M.verts
⊢ ∃ i, ∃ (_ : i ∈ M.edgeSet), v ∈ ↑i

case refine_2
V : Type u_1
G : SimpleGraph V
M : G.Subgraph
h : M.IsMatching
v : V
x✝ : ∃ i, ∃ (_ : i ∈ M.edgeSet), v ∈ ↑i
e : Sym2 V
he : e ∈ M.edgeSet
hv : v ∈ ↑e
⊢ v ∈ M.verts
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
M₁ : G.Subgraph
h₁ : M₁ ∈ Set.ofPred IsMatching
M₂ : G.Subgraph
h₂ : M₂ ∈ Set.ofPred IsMatching
h : M₁.edgeSet = M₂.edgeSet
⊢ ⋃ e ∈ M₂.edgeSet, ↑e = ⋃ e ∈ M₂.edgeSet, ↑e

Tactic:
]

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
a b : α
h : G.Adj a b
⊢ ∑ a_1, incMatrix R G a_1 s(a, b) = 2

Tactic:
]

Goal state after:
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
a b : α
h : G.Adj a b
⊢ ∑ a_1, incMatrix R G a_1 s(a, b) = 2
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
M : G.Subgraph
inst✝ : Fintype ↑M.verts
h : ∀ v ∈ M.verts, M.degree v = 1
⊢ Even M.verts.toFinset.card

Tactic:
]

Goal state after:
V : Type u_1
G : SimpleGraph V
M : G.Subgraph
inst✝ : Fintype ↑M.verts
h : ∀ v ∈ M.verts, M.degree v = 1
⊢ Even M.verts.toFinset.card
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
a b : α
h : G.Adj a b
⊢ ∑ a_1, incMatrix R G a_1 s(a, b) = ↑(#{a, b})

Tactic:
]

Goal state after:
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
a b : α
h : G.Adj a b
⊢ ∑ a_1, incMatrix R G a_1 s(a, b) = ↑(#{a, b})
Module: Mathlib.Combinatorics.SimpleGraph.Matching

Goal state before:
pp: case h
V : Type u_1
G : SimpleGraph V
M : G.Subgraph
inst✝ : Fintype ↑M.verts
h : ∀ v ∈ M.verts, M.degree v = 1
⊢ M.verts.toFinset.card = ∑ v, M.coe.degree v

Tactic:
]

Goal state after:
case h
V : Type u_1
G : SimpleGraph V
M : G.Subgraph
inst✝ : Fintype ↑M.verts
h : ∀ v ∈ M.verts, M.degree v = 1
⊢ M.verts.toFinset.card = ∑ v, M.coe.degree v
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 ≤ 1 - 1 / ↑r + ε
⊢ ↑n ≤ ↑G.minDegree

Tactic:
]

Goal state after:
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 ≤ 1 - 1 / ↑r + ε
⊢ ↑n ≤ ↑G.minDegree
Module: Mathlib.Combinatorics.SimpleGraph.IncMatrix

Goal state before:
pp: case pos
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)
v w : α
h : s(v, w) ∈ G.edgeSet
⊢ ↑(#{x | s(v, w) ∈ G.incidenceSet x}) = ↑(#{(v, w).1, (v, w).2})

Tactic:
]

Goal state after:
case pos
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)
v w : α
h : s(v, w) ∈ G.edgeSet
⊢ ↑(#{x | s(v, w) ∈ G.incidenceSet x}) = ↑(#{(v, w).1, (v, w).2})
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 b, G.IsFiveWheelLike r (k + 1) v w₁ w₂ (insert x (s.erase a)) (insert x (t.erase b))

Tactic:
have ⟨has, hbt, hav, hbv, haw, hbw⟩ : a ∈ s ∧ b ∈ t ∧ a ≠ v ∧ b ≠ v ∧ a ≠ w₂ ∧ b ≠ w₁ := by grind

Goal state after:
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
⊢ ∃ a b, G.IsFiveWheelLike r (k + 1) v w₁ w₂ (insert x (s.erase a)) (insert x (t.erase b))
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
⊢ ↑t.ncard = ↑u.ncard

Tactic:
]

Goal state after:
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
⊢ ↑t.ncard = ↑u.ncard
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
⊢ {x | x ∈ G.incidenceSet a ∩ G.incidenceSet b} = {s(a, b)}

Tactic:
]

Goal state after:
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
⊢ {x | x ∈ G.incidenceSet a ∩ G.incidenceSet b} = {s(a, b)}
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 < ↑t' * (↑r * 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:
]

Goal state after:
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 < ↑t' * (↑r * 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'
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
⊢ ↑(#(filter (G.Adj i) univ)) = ↑(#{x | G.Adj i x})

Tactic:
]

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
⊢ ↑(G.degree v) * vec v = ↑(G.degree v) * vec v

Tactic:
]

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
i : V
⊢ ↑(G.degree i) * 1 - ∑ u ∈ G.neighborFinset i, 1 = 0 i

Tactic:
]

Goal state after:
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
i : V
⊢ ↑(G.degree i) * 1 - ∑ u ∈ G.neighborFinset i, 1 = 0 i
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 - x x_1 * x x_2 else 0) +
        ∑ x_1, ∑ x_2, if G.Adj x_1 x_2 then x x_1 * x x_1 - x x_1 * x x_2 else 0) /
      2 =
    (∑ i, ∑ j, if G.Adj i j then (x i - x j) ^ 2 else 0) / 2

Tactic:
]

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 - x x_1 * x x_2 else 0) +
        ∑ x_1, ∑ x_2, if G.Adj x_1 x_2 then x x_1 * x x_1 - x x_1 * x x_2 else 0) /
      2 =
    (∑ 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: case refine_1
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 = lapMatrix R G

Tactic:
]

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

Goal state before:
pp: case refine_2
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
x : V → R
⊢ 0 ≤ (∑ i, ∑ j, if G.Adj i j then (x i - x j) ^ 2 else 0) / 2

Tactic:
]

Goal state after:
case refine_2
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
x : V → R
⊢ 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
inst✝² : Fintype V
G : SimpleGraph V
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableEq V
x : V → ℝ
⊢ (∀ (i j : V), G.Adj i j → x i = x j) ↔ ∀ (i j : V), G.Adj i j → x i = x j

Tactic:
]

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
⊢ (∀ (i j : V), G.Adj i j → x i = x j) ↔ ∀ (i j : V), G.Reachable i j → x i = x j

Tactic:
]

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