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

Goal state before:
pp: V : Type u
G : SimpleGraph V
h : G = ⊥
h' : ¬IsEmpty V
⊢ Nonempty V

Tactic:
simpa using

Goal state after:
V : Type u
G : SimpleGraph V
h : G = ⊥
h' : ¬IsEmpty V
⊢ Nonempty V
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v x : V
w : G.Walk u v
hw : w.IsTrail
hxu : x ≠ u
hxv : x ≠ v
hx : (G.neighborSet x).Subsingleton
hxw : x ∈ w.support
y : V
hxy : G.Adj x y
p : (G.deleteEdges {s(y, x)}).Walk u x
⊢ G.Adj x p.penultimate ∧ ?m.117 ∧ ¬p.penultimate = y

Tactic:
simpa using

Goal state after:
V : Type u
G : SimpleGraph V
u v x : V
w : G.Walk u v
hw : w.IsTrail
hxu : x ≠ u
hxv : x ≠ v
hx : (G.neighborSet x).Subsingleton
hxw : x ∈ w.support
y : V
hxy : G.Adj x y
p : (G.deleteEdges {s(y, x)}).Walk u x
⊢ G.Adj x p.penultimate ∧ ?m.117 ∧ ¬p.penultimate = y
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v x : V
w : G.Walk u v
hw : w.IsTrail
hxu : x ≠ u
hxv : x ≠ v
hx : (G.neighborSet x).Subsingleton
hxw : x ∈ w.support
y : V
hxy : G.Adj x y
p : (G.deleteEdges {s(y, x)}).Walk v x
⊢ G.Adj x p.penultimate ∧ ?m.183 ∧ ¬p.penultimate = y

Tactic:
simpa using

Goal state after:
V : Type u
G : SimpleGraph V
u v x : V
w : G.Walk u v
hw : w.IsTrail
hxu : x ≠ u
hxv : x ≠ v
hx : (G.neighborSet x).Subsingleton
hxw : x ∈ w.support
y : V
hxy : G.Adj x y
p : (G.deleteEdges {s(y, x)}).Walk v x
⊢ G.Adj x p.penultimate ∧ ?m.183 ∧ ¬p.penultimate = y
Module: Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex

Goal state before:
pp: V : Type u_4
W : Type u_5
inst✝¹ : Nonempty V
inst✝ : Nonempty W
⊢ (completeBipartiteGraph V W).Colorable 2

Tactic:
simpa using

Goal state after:
V : Type u_4
W : Type u_5
inst✝¹ : Nonempty V
inst✝ : Nonempty W
⊢ (completeBipartiteGraph V W).Colorable 2
Module: Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex

Goal state before:
pp: case neg
V : Type u_4
W : Type u_5
inst✝¹ : Nonempty V
inst✝ : Nonempty W
C : (completeBipartiteGraph V W).Coloring (Fin 2)
b : Fin 2
v : V
w : W
h : (completeBipartiteGraph V W).Adj (Sum.inl v) (Sum.inr w)
he : ¬C (Sum.inl v) = b
he' : ¬C (Sum.inr w) = b
⊢ ∃ a, C a = b

Tactic:
simpa using

Goal state after:
case neg
V : Type u_4
W : Type u_5
inst✝¹ : Nonempty V
inst✝ : Nonempty W
C : (completeBipartiteGraph V W).Coloring (Fin 2)
b : Fin 2
v : V
w : W
h : (completeBipartiteGraph V W).Adj (Sum.inl v) (Sum.inr w)
he : ¬C (Sum.inl v) = b
he' : ¬C (Sum.inr w) = b
⊢ (∃ a, C (Sum.inl a) = b) ∨ ∃ b_1, C (Sum.inr b_1) = b
Module: Mathlib.Combinatorics.SimpleGraph.Coloring.Vertex

Goal state before:
pp: V : Type u
G : SimpleGraph V
n : ℕ
hc : ↑G.chromaticNumber.toNat < ↑n
hne : ↑G.chromaticNumber.toNat = G.chromaticNumber
m : ℕ
hc' : G.Colorable m
this : G.Colorable G.chromaticNumber.toNat
⊢ G.chromaticNumber.toNat < n

Tactic:
simpa using

Goal state after:
V : Type u
G : SimpleGraph V
n : ℕ
hc : ↑G.chromaticNumber.toNat < ↑n
hne : ↑G.chromaticNumber.toNat = G.chromaticNumber
m : ℕ
hc' : G.Colorable m
this : G.Colorable G.chromaticNumber.toNat
⊢ G.chromaticNumber.toNat < n
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: V : Type u_1
v w : V
G : SimpleGraph V
s t : Set V
h : G.IsBipartiteWith s t
hv : v ∈ s
hadj : v ∈ s ∧ w ∈ t ∨ v ∈ t ∧ w ∈ s
nhv : v ∉ t
⊢ w ∈ t

Tactic:
simpa using

Goal state after:
V : Type u_1
v w : V
G : SimpleGraph V
s t : Set V
h : G.IsBipartiteWith s t
hv : v ∈ s
hadj : v ∈ s ∧ w ∈ t ∨ v ∈ t ∧ w ∈ s
nhv : v ∉ t
⊢ w ∈ t
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: V : Type u_1
v w : V
G : SimpleGraph V
s t : Set V
h : G.IsBipartiteWith s t
hw : w ∈ t
hadj : v ∈ s ∧ w ∈ t ∨ v ∈ t ∧ w ∈ s
nhw : w ∉ s
⊢ v ∈ s

Tactic:
simpa using

Goal state after:
V : Type u_1
v w : V
G : SimpleGraph V
s t : Set V
h : G.IsBipartiteWith s t
hw : w ∈ t
hadj : v ∈ s ∧ w ∈ t ∨ v ∈ t ∧ w ∈ s
nhw : w ∉ s
⊢ v ∈ s
Module: Mathlib.Combinatorics.SetFamily.FourFunctions

Goal state before:
pp: case empty.inr.inr
α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : CommSemiring β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : ExistsAddOfLE β
f₁ f₂ f₃ f₄ : Finset α → β
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ ⦃s : Finset α⦄, s ⊆ ∅ → ∀ ⦃t : Finset α⦄, t ⊆ ∅ → f₁ s * f₂ t ≤ f₃ (s ∩ t) * f₄ (s ∪ t)
⊢ (∑ s ∈ {∅}, f₁ s) * ∑ s ∈ {∅}, f₂ s ≤ (∑ s ∈ {∅} ⊼ {∅}, f₃ s) * ∑ s ∈ {∅} ⊻ {∅}, f₄ s

Tactic:
simpa using

Goal state after:
case empty.inr.inr
α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : CommSemiring β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : ExistsAddOfLE β
f₁ f₂ f₃ f₄ : Finset α → β
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ ⦃s : Finset α⦄, s ⊆ ∅ → ∀ ⦃t : Finset α⦄, t ⊆ ∅ → f₁ s * f₂ t ≤ f₃ (s ∩ t) * f₄ (s ∪ t)
⊢ f₁ ∅ * f₂ ∅ ≤ f₃ ∅ * f₄ ∅
Module: Mathlib.Combinatorics.SetFamily.FourFunctions

Goal state before:
pp: case empty.inr.inr
α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : CommSemiring β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : ExistsAddOfLE β
f₁ f₂ f₃ f₄ : Finset α → β
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ ⦃s : Finset α⦄, s ⊆ ∅ → ∀ ⦃t : Finset α⦄, t ⊆ ∅ → f₁ s * f₂ t ≤ f₃ (s ∩ t) * f₄ (s ∪ t)
⊢ (∑ s ∈ {∅}, f₁ s) * ∑ s ∈ {∅}, f₂ s ≤ (∑ s ∈ {∅} ⊼ {∅}, f₃ s) * ∑ s ∈ {∅} ⊻ {∅}, f₄ s

Tactic:
simpa using h (subset_refl ∅) subset_rfl

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
k : ℕ
u v : V
inst✝ : Fintype ↑(G.neighborSet u)
h : G.IsEdgeReachable k u v
huv : u ≠ v
hh : (G.incidenceSet u).encard < ↑k
w : (G.deleteEdges (G.incidenceSet u)).Walk u v
h✝ : w.IsPath
⊢ False

Tactic:
simpa using

Goal state after:
V : Type u_1
G : SimpleGraph V
k : ℕ
u v : V
inst✝ : Fintype ↑(G.neighborSet u)
h : G.IsEdgeReachable k u v
huv : u ≠ v
hh : (G.incidenceSet u).encard < ↑k
w : (G.deleteEdges (G.incidenceSet u)).Walk u v
h✝ : w.IsPath
⊢ False
Module: Mathlib.Combinatorics.SetFamily.FourFunctions

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : CommSemiring β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : ExistsAddOfLE β
a : α
u : Finset α
hu : a ∉ u
f₁ f₂ f₃ f₄ : Finset α → β
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ ⦃s : Finset α⦄, s ⊆ insert a u → ∀ ⦃t : Finset α⦄, t ⊆ insert a u → f₁ s * f₂ t ≤ f₃ (s ∩ t) * f₄ (s ∪ t)
𝒜 ℬ : Finset (Finset α)
ih :
  (∑ s ∈ u.powerset, collapse 𝒜 a f₁ s) * ∑ s ∈ u.powerset, collapse ℬ a f₂ s ≤
    (∑ s ∈ u.powerset ⊼ u.powerset, collapse (𝒜 ⊼ ℬ) a f₃ s) * ∑ s ∈ u.powerset ⊻ u.powerset, collapse (𝒜 ⊻ ℬ) a f₄ s
h𝒜 : 𝒜 ⊆ (insert a u).powerset
hℬ : ℬ ⊆ (insert a u).powerset
⊢ 𝒜 ⊼ ℬ ⊆ (insert a u).powerset

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : CommSemiring β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : ExistsAddOfLE β
a : α
u : Finset α
hu : a ∉ u
f₁ f₂ f₃ f₄ : Finset α → β
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ ⦃s : Finset α⦄, s ⊆ insert a u → ∀ ⦃t : Finset α⦄, t ⊆ insert a u → f₁ s * f₂ t ≤ f₃ (s ∩ t) * f₄ (s ∪ t)
𝒜 ℬ : Finset (Finset α)
ih :
  (∑ s ∈ u.powerset, collapse 𝒜 a f₁ s) * ∑ s ∈ u.powerset, collapse ℬ a f₂ s ≤
    (∑ s ∈ u.powerset ⊼ u.powerset, collapse (𝒜 ⊼ ℬ) a f₃ s) * ∑ s ∈ u.powerset ⊻ u.powerset, collapse (𝒜 ⊻ ℬ) a f₄ s
h𝒜 : 𝒜 ⊆ (insert a u).powerset
hℬ : ℬ ⊆ (insert a u).powerset
⊢ ∀ a_1 ∈ 𝒜, ∀ b ∈ ℬ, a_1 ∩ b ⊆ insert a u
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity

Goal state before:
pp: case refine_2.inl
V : Type u_1
G : SimpleGraph V
k : ℕ
u v : V
hk : k ≠ 0
h : ∀ (e : Sym2 V), (G.deleteEdges {e}).IsEdgeReachable k u v
hs : ∅.encard < ↑(k + 1)
⊢ (G.deleteEdges ∅).Reachable u v

Tactic:
simpa using

Goal state after:
case refine_2.inl
V : Type u_1
G : SimpleGraph V
k : ℕ
u v : V
hk : k ≠ 0
h : ∀ (e : Sym2 V), (G.deleteEdges {e}).IsEdgeReachable k u v
hs : ∅.encard < ↑(k + 1)
⊢ G.Reachable u v
Module: Mathlib.Combinatorics.SetFamily.FourFunctions

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : CommSemiring β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : ExistsAddOfLE β
a : α
u : Finset α
hu : a ∉ u
f₁ f₂ f₃ f₄ : Finset α → β
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ ⦃s : Finset α⦄, s ⊆ insert a u → ∀ ⦃t : Finset α⦄, t ⊆ insert a u → f₁ s * f₂ t ≤ f₃ (s ∩ t) * f₄ (s ∪ t)
𝒜 ℬ : Finset (Finset α)
ih :
  (∑ s ∈ u.powerset, collapse 𝒜 a f₁ s) * ∑ s ∈ u.powerset, collapse ℬ a f₂ s ≤
    (∑ s ∈ u.powerset ⊼ u.powerset, collapse (𝒜 ⊼ ℬ) a f₃ s) * ∑ s ∈ u.powerset ⊻ u.powerset, collapse (𝒜 ⊻ ℬ) a f₄ s
h𝒜 : 𝒜 ⊆ (insert a u).powerset
hℬ : ℬ ⊆ (insert a u).powerset
this : 𝒜 ⊼ ℬ ⊆ (insert a u).powerset
⊢ 𝒜 ⊻ ℬ ⊆ (insert a u).powerset

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : CommSemiring β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : ExistsAddOfLE β
a : α
u : Finset α
hu : a ∉ u
f₁ f₂ f₃ f₄ : Finset α → β
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ ⦃s : Finset α⦄, s ⊆ insert a u → ∀ ⦃t : Finset α⦄, t ⊆ insert a u → f₁ s * f₂ t ≤ f₃ (s ∩ t) * f₄ (s ∪ t)
𝒜 ℬ : Finset (Finset α)
ih :
  (∑ s ∈ u.powerset, collapse 𝒜 a f₁ s) * ∑ s ∈ u.powerset, collapse ℬ a f₂ s ≤
    (∑ s ∈ u.powerset ⊼ u.powerset, collapse (𝒜 ⊼ ℬ) a f₃ s) * ∑ s ∈ u.powerset ⊻ u.powerset, collapse (𝒜 ⊻ ℬ) a f₄ s
h𝒜 : 𝒜 ⊆ (insert a u).powerset
hℬ : ℬ ⊆ (insert a u).powerset
this : 𝒜 ⊼ ℬ ⊆ (insert a u).powerset
⊢ ∀ a_1 ∈ 𝒜, ∀ b ∈ ℬ, a_1 ∪ b ⊆ insert a u
Module: Mathlib.Combinatorics.SetFamily.FourFunctions

Goal state before:
pp: case insert
α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : CommSemiring β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : ExistsAddOfLE β
a : α
u : Finset α
hu : a ∉ u
f₁ f₂ f₃ f₄ : Finset α → β
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ ⦃s : Finset α⦄, s ⊆ insert a u → ∀ ⦃t : Finset α⦄, t ⊆ insert a u → f₁ s * f₂ t ≤ f₃ (s ∩ t) * f₄ (s ∪ t)
𝒜 ℬ : Finset (Finset α)
ih :
  (∑ s ∈ u.powerset, collapse 𝒜 a f₁ s) * ∑ s ∈ u.powerset, collapse ℬ a f₂ s ≤
    (∑ s ∈ u.powerset ⊼ u.powerset, collapse (𝒜 ⊼ ℬ) a f₃ s) * ∑ s ∈ u.powerset ⊻ u.powerset, collapse (𝒜 ⊻ ℬ) a f₄ s
h𝒜 : 𝒜 ⊆ (insert a u).powerset
hℬ : ℬ ⊆ (insert a u).powerset
this✝ : 𝒜 ⊼ ℬ ⊆ (insert a u).powerset
this : 𝒜 ⊻ ℬ ⊆ (insert a u).powerset
⊢ (∑ s ∈ 𝒜, f₁ s) * ∑ s ∈ ℬ, f₂ s ≤ (∑ s ∈ 𝒜 ⊼ ℬ, f₃ s) * ∑ s ∈ 𝒜 ⊻ ℬ, f₄ s

Tactic:
simpa using

Goal state after:
case insert
α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : CommSemiring β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
inst✝ : ExistsAddOfLE β
a : α
u : Finset α
hu : a ∉ u
f₁ f₂ f₃ f₄ : Finset α → β
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ ⦃s : Finset α⦄, s ⊆ insert a u → ∀ ⦃t : Finset α⦄, t ⊆ insert a u → f₁ s * f₂ t ≤ f₃ (s ∩ t) * f₄ (s ∪ t)
𝒜 ℬ : Finset (Finset α)
ih :
  (∑ s ∈ u.powerset, collapse 𝒜 a f₁ s) * ∑ s ∈ u.powerset, collapse ℬ a f₂ s ≤
    (∑ s ∈ u.powerset ⊼ u.powerset, collapse (𝒜 ⊼ ℬ) a f₃ s) * ∑ s ∈ u.powerset ⊻ u.powerset, collapse (𝒜 ⊻ ℬ) a f₄ s
h𝒜 : 𝒜 ⊆ (insert a u).powerset
hℬ : ℬ ⊆ (insert a u).powerset
this✝ : 𝒜 ⊼ ℬ ⊆ (insert a u).powerset
this : 𝒜 ⊻ ℬ ⊆ (insert a u).powerset
⊢ (∑ s ∈ 𝒜, f₁ s) * ∑ s ∈ ℬ, f₂ s ≤ (∑ s ∈ 𝒜 ⊼ ℬ, f₃ s) * ∑ s ∈ 𝒜 ⊻ ℬ, f₄ s
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
k : ℕ
hk : k ≠ 0
⊢ G.IsEdgeConnected (k + 1) ↔ ∀ (e : Sym2 V), (G.deleteEdges {e}).IsEdgeConnected k

Tactic:
by
  simp [IsEdgeConnected, isEdgeReachable_add_one hk, forall_comm (α := Sym2 _)]

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

Goal state before:
pp: V : Type u_1
v w : V
G : SimpleGraph V
s t : Set V
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
left right : Finset V
card_left : #left = Fintype.card α
card_right : #right = Fintype.card β
h : G.IsCompleteBetween ↑left ↑right
this✝ : Nonempty (α ↪ ↥left)
fα : α ↪ ↥left := Classical.arbitrary (α ↪ ↥left)
this : Nonempty (β ↪ ↥right)
fβ : β ↪ ↥right := Classical.arbitrary (β ↪ ↥right)
s₁ s₂ : α ⊕ β
p₁ : β
p₂ : α
⊢ Sum.elim (Subtype.val ∘ ⇑fα) (Subtype.val ∘ ⇑fβ) (Sum.inr p₁) =
      Sum.elim (Subtype.val ∘ ⇑fα) (Subtype.val ∘ ⇑fβ) (Sum.inl p₂) →
    Sum.inr p₁ = Sum.inl p₂

Tactic:
simpa using

Goal state after:
V : Type u_1
v w : V
G : SimpleGraph V
s t : Set V
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
left right : Finset V
card_left : #left = Fintype.card α
card_right : #right = Fintype.card β
h : G.IsCompleteBetween ↑left ↑right
this✝ : Nonempty (α ↪ ↥left)
fα : α ↪ ↥left := Classical.arbitrary (α ↪ ↥left)
this : Nonempty (β ↪ ↥right)
fβ : β ↪ ↥right := Classical.arbitrary (β ↪ ↥right)
s₁ s₂ : α ⊕ β
p₁ : β
p₂ : α
⊢ ¬↑(fβ p₁) = ↑(fα p₂)
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: V : Type u_1
v w : V
G : SimpleGraph V
s t : Set V
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
left right : Finset V
card_left : #left = Fintype.card α
card_right : #right = Fintype.card β
h : G.IsCompleteBetween ↑left ↑right
this✝ : Nonempty (α ↪ ↥left)
fα : α ↪ ↥left := Classical.arbitrary (α ↪ ↥left)
this : Nonempty (β ↪ ↥right)
fβ : β ↪ ↥right := Classical.arbitrary (β ↪ ↥right)
s₁ s₂ : α ⊕ β
p₁ : α
p₂ : β
⊢ Sum.elim (Subtype.val ∘ ⇑fα) (Subtype.val ∘ ⇑fβ) (Sum.inl p₁) =
      Sum.elim (Subtype.val ∘ ⇑fα) (Subtype.val ∘ ⇑fβ) (Sum.inr p₂) →
    Sum.inl p₁ = Sum.inr p₂

Tactic:
simpa using

Goal state after:
V : Type u_1
v w : V
G : SimpleGraph V
s t : Set V
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
left right : Finset V
card_left : #left = Fintype.card α
card_right : #right = Fintype.card β
h : G.IsCompleteBetween ↑left ↑right
this✝ : Nonempty (α ↪ ↥left)
fα : α ↪ ↥left := Classical.arbitrary (α ↪ ↥left)
this : Nonempty (β ↪ ↥right)
fβ : β ↪ ↥right := Classical.arbitrary (β ↪ ↥right)
s₁ s₂ : α ⊕ β
p₁ : α
p₂ : β
⊢ ¬↑(fα p₁) = ↑(fβ p₂)
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity

Goal state before:
pp: case pos
V : Type u_1
G : SimpleGraph V
u v : V
hne : u ≠ v
h : G.IsEdgeReachable 2 u v
w : G.Walk u v
hw : w.IsPath
this✝ : G.Adj u w.snd
hs : {s(u, w.snd)}.encard < ↑2
hh : s(u, w.snd) ∈ w.tail.edges
this : u = w.getVert 2
⊢ False

Tactic:
simpa using

Goal state after:
case pos
V : Type u_1
G : SimpleGraph V
u v : V
hne : u ≠ v
h : G.IsEdgeReachable 2 u v
w : G.Walk u v
hw : w.IsPath
this✝ : G.Adj u w.snd
hs : {s(u, w.snd)}.encard < ↑2
hh : s(u, w.snd) ∈ w.tail.edges
this : u = w.getVert 2
⊢ False
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
u v x : V
w : G.Walk u v
hw : w.IsTrail
huv : G.IsEdgeReachable 2 u v
huy : ¬G.IsEdgeReachable 2 u x
⊢ ?m.25

Tactic:
simpa using

Goal state after:
V : Type u_1
G : SimpleGraph V
u v x : V
w : G.Walk u v
hw : w.IsTrail
huv : G.IsEdgeReachable 2 u v
huy : ¬G.IsEdgeReachable 2 u x
⊢ ?m.25
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.EdgeConnectivity

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
u v x : V
w : G.Walk u v
hw : w.IsTrail
huv : G.IsEdgeReachable 2 u v
huy : ¬G.IsEdgeReachable 2 u x
e : Sym2 V
he : ¬(G.deleteEdges {e}).Reachable u x
he' : ¬(G.deleteEdges {e}).Reachable v x
hy : x ∈ w.support
⊢ e ∈ (w.dropUntil x hy).edges

Tactic:
simpa using

Goal state after:
V : Type u_1
G : SimpleGraph V
u v x : V
w : G.Walk u v
hw : w.IsTrail
huv : G.IsEdgeReachable 2 u v
huy : ¬G.IsEdgeReachable 2 u x
e : Sym2 V
he : ¬(G.deleteEdges {e}).Reachable u x
he' : ¬(G.deleteEdges {e}).Reachable v x
hy : x ∈ w.support
⊢ e ∈ (w.dropUntil x hy).edges
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v : V
huv : G.Adj u v
⊢ (⊤.induce {u, v}).Connected

Tactic:
rw [← subgraphOfAdj_eq_induce huv]

Goal state after:
V : Type u
G : SimpleGraph V
u v : V
huv : G.Adj u v
⊢ (G.subgraphOfAdj huv).Connected
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: case inl
V : Type u_1
v w : V
G : SimpleGraph V
s t : Set V
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
left right : Finset V
card_left : #left = Fintype.card α
card_right : #right = Fintype.card β
h : G.IsCompleteBetween ↑left ↑right
this✝ : Nonempty (α ↪ ↥left)
fα : α ↪ ↥left := Classical.arbitrary (α ↪ ↥left)
this : Nonempty (β ↪ ↥right)
fβ : β ↪ ↥right := Classical.arbitrary (β ↪ ↥right)
f : α ⊕ β ↪ V := { toFun := Sum.elim (Subtype.val ∘ ⇑fα) (Subtype.val ∘ ⇑fβ), inj' := ⋯ }
s₁ s₂ : α ⊕ β
hs₁ : s₁.isLeft = true
hs₂ : s₂.isRight = true
⊢ G.Adj (Sum.elim (Subtype.val ∘ ⇑fα) (Subtype.val ∘ ⇑fβ) s₁) (Sum.elim (Subtype.val ∘ ⇑fα) (Subtype.val ∘ ⇑fβ) s₂)

Tactic:
← Sum.inl_getLeft s₁ hs₁,

Goal state after:
case inl
V : Type u_1
v w : V
G : SimpleGraph V
s t : Set V
α : Type u_2
β : Type u_3
inst✝¹ : Fintype α
inst✝ : Fintype β
left right : Finset V
card_left : #left = Fintype.card α
card_right : #right = Fintype.card β
h : G.IsCompleteBetween ↑left ↑right
this✝ : Nonempty (α ↪ ↥left)
fα : α ↪ ↥left := Classical.arbitrary (α ↪ ↥left)
this : Nonempty (β ↪ ↥right)
fβ : β ↪ ↥right := Classical.arbitrary (β ↪ ↥right)
f : α ⊕ β ↪ V := { toFun := Sum.elim (Subtype.val ∘ ⇑fα) (Subtype.val ∘ ⇑fβ), inj' := ⋯ }
s₁ s₂ : α ⊕ β
hs₁ : s₁.isLeft = true
hs₂ : s₂.isRight = true
⊢ G.Adj (Sum.elim (Subtype.val ∘ ⇑fα) (Subtype.val ∘ ⇑fβ) (Sum.inl (s₁.getLeft hs₁)))
    (Sum.elim (Subtype.val ∘ ⇑fα) (Subtype.val ∘ ⇑fβ) s₂)
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph

Goal state before:
pp: V : Type u
G : SimpleGraph V
H : G.Subgraph
h : H.Preconnected
v : ↑H.verts
inst✝ : Fintype ↑(H.neighborSet ↑v)
hv : H.degree ↑v = 0
hn : H.verts.Nontrivial
this : Nontrivial ↑H.verts
⊢ False

Tactic:
simpa using

Goal state after:
V : Type u
G : SimpleGraph V
H : G.Subgraph
h : H.Preconnected
v : ↑H.verts
inst✝ : Fintype ↑(H.neighborSet ↑v)
hv : H.degree ↑v = 0
hn : H.verts.Nontrivial
this : Nontrivial ↑H.verts
⊢ False
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph

Goal state before:
pp: V : Type u
G : SimpleGraph V
H : G.Subgraph
hc : H.Connected
h : ∀ v ∈ H.verts, ∀ (w : V), G.Adj v w → H.Adj v w
v : V
hv : v ∈ H.verts
w : V
hw : w ∈ H.verts
⊢ G.Reachable w v

Tactic:
simpa using

Goal state after:
V : Type u
G : SimpleGraph V
H : G.Subgraph
hc : H.Connected
h : ∀ v ∈ H.verts, ∀ (w : V), G.Adj v w → H.Adj v w
v : V
hv : v ∈ H.verts
w : V
hw : w ∈ H.verts
⊢ G.Reachable w v
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v : V
w : G.Walk u v
h : ¬w.Nil
⊢ w.toSubgraph.Adj u w.snd

Tactic:
simpa using

Goal state after:
V : Type u
G : SimpleGraph V
u v : V
w : G.Walk u v
h : ¬w.Nil
⊢ w.toSubgraph.Adj u w.snd
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v : V
w : G.Walk u v
h : 0 < w.length
⊢ w.toSubgraph.Adj w.penultimate v

Tactic:
simpa using

Goal state after:
V : Type u
G : SimpleGraph V
u v : V
w : G.Walk u v
h : 0 < w.length
⊢ w.toSubgraph.Adj w.penultimate v
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: V : Type u_1
v : V
G : SimpleGraph V
inst✝² : DecidableEq V
inst✝¹ : Fintype V
s : Finset V
inst✝ : DecidableRel G.Adj
hv : v ∈ s
⊢ G.neighborFinset v ⊆ (between (↑s) (↑s)ᶜ G).neighborFinset v ∪ s

Tactic:
simpa using

Goal state after:
V : Type u_1
v : V
G : SimpleGraph V
inst✝² : DecidableEq V
inst✝¹ : Fintype V
s : Finset V
inst✝ : DecidableRel G.Adj
hv : v ∈ s
⊢ G.neighborSet v ⊆ (between (↑s) (↑s)ᶜ G).neighborSet v ∪ ↑s
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: V : Type u_1
v : V
G : SimpleGraph V
inst✝² : DecidableEq V
inst✝¹ : Fintype V
s : Finset V
inst✝ : DecidableRel G.Adj
hv : v ∈ s
⊢ (between (↑s) (↑s)ᶜ G).IsBipartiteWith ↑s ↑sᶜ

Tactic:
simpa using

Goal state after:
V : Type u_1
v : V
G : SimpleGraph V
inst✝² : DecidableEq V
inst✝¹ : Fintype V
s : Finset V
inst✝ : DecidableRel G.Adj
hv : v ∈ s
⊢ (between (↑s) (↑s)ᶜ G).IsBipartiteWith (↑s) (↑s)ᶜ
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: V : Type u_1
w : V
G : SimpleGraph V
inst✝² : DecidableEq V
inst✝¹ : Fintype V
s : Finset V
inst✝ : DecidableRel G.Adj
hw : w ∈ sᶜ
⊢ G.neighborFinset w ⊆ (between (↑s) (↑s)ᶜ G).neighborFinset w ∪ sᶜ

Tactic:
simpa using

Goal state after:
V : Type u_1
w : V
G : SimpleGraph V
inst✝² : DecidableEq V
inst✝¹ : Fintype V
s : Finset V
inst✝ : DecidableRel G.Adj
hw : w ∈ sᶜ
⊢ G.neighborSet w ⊆ (between (↑s) (↑s)ᶜ G).neighborSet w ∪ (↑s)ᶜ
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: V : Type u_1
w : V
G : SimpleGraph V
inst✝² : DecidableEq V
inst✝¹ : Fintype V
s : Finset V
inst✝ : DecidableRel G.Adj
hw : w ∈ sᶜ
⊢ w ∈ (↑s)ᶜ

Tactic:
simpa using

Goal state after:
V : Type u_1
w : V
G : SimpleGraph V
inst✝² : DecidableEq V
inst✝¹ : Fintype V
s : Finset V
inst✝ : DecidableRel G.Adj
hw : w ∈ sᶜ
⊢ w ∉ s
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: V : Type u_1
w : V
G : SimpleGraph V
inst✝² : DecidableEq V
inst✝¹ : Fintype V
s : Finset V
inst✝ : DecidableRel G.Adj
hw : w ∈ sᶜ
⊢ (between (↑s) (↑s)ᶜ G).IsBipartiteWith ↑s ↑sᶜ

Tactic:
simpa using

Goal state after:
V : Type u_1
w : V
G : SimpleGraph V
inst✝² : DecidableEq V
inst✝¹ : Fintype V
s : Finset V
inst✝ : DecidableRel G.Adj
hw : w ∈ sᶜ
⊢ (between (↑s) (↑s)ᶜ G).IsBipartiteWith (↑s) (↑s)ᶜ
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
inst✝¹ : Fintype V
inst✝ : DecidableRel G.Adj
x✝ : V × V
h :
  x✝ ∈
    {x |
      match x with
      | (x, y) => G.Adj x y}
⊢ (match x✝, h with
    | (v, w), x => s(Sum.inl v, Sum.inr w)) ∈
    G.bipartiteDoubleCover.edgeFinset

Tactic:
simpa using

Goal state after:
V : Type u_1
G : SimpleGraph V
inst✝¹ : Fintype V
inst✝ : DecidableRel G.Adj
x✝ : V × V
h :
  x✝ ∈
    {x |
      match x with
      | (x, y) => G.Adj x y}
⊢ G.Adj x✝.1 x✝.2
Module: Mathlib.Combinatorics.SetFamily.FourFunctions

Goal state before:
pp: α : Type u_1
inst✝¹ : DistribLattice α
inst✝ : DecidableEq α
s t : Finset α
⊢ #s * #t ≤ #(s ⊼ t) * #(s ⊻ t)

Tactic:
simpa using

Goal state after:
α : Type u_1
inst✝¹ : DistribLattice α
inst✝ : DecidableEq α
s t : Finset α
⊢ #s * #t ≤ #(s ⊼ t) * #(s ⊻ t)
Module: Mathlib.Combinatorics.SetFamily.FourFunctions

Goal state before:
pp: α : Type u_1
β : Type u_2
inst✝⁵ : DistribLattice α
inst✝⁴ : CommSemiring β
inst✝³ : LinearOrder β
inst✝² : IsStrictOrderedRing β
inst✝¹ : ExistsAddOfLE β
f₁ f₂ f₃ f₄ : α → β
inst✝ : Fintype α
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ (a b : α), f₁ a * f₂ b ≤ f₃ (a ⊓ b) * f₄ (a ⊔ b)
⊢ (∑ a, f₁ a) * ∑ a, f₂ a ≤ (∑ a, f₃ a) * ∑ a, f₄ a

Tactic:
simpa using

Goal state after:
α : Type u_1
β : Type u_2
inst✝⁵ : DistribLattice α
inst✝⁴ : CommSemiring β
inst✝³ : LinearOrder β
inst✝² : IsStrictOrderedRing β
inst✝¹ : ExistsAddOfLE β
f₁ f₂ f₃ f₄ : α → β
inst✝ : Fintype α
h₁ : 0 ≤ f₁
h₂ : 0 ≤ f₂
h₃ : 0 ≤ f₃
h₄ : 0 ≤ f₄
h : ∀ (a b : α), f₁ a * f₂ b ≤ f₃ (a ⊓ b) * f₄ (a ⊔ b)
⊢ (∑ a, f₁ a) * ∑ a, f₂ a ≤ (∑ a, f₃ a) * ∑ a, f₄ a
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
hp : p.IsPath
hnp : ¬p.Nil
⊢ p.toSubgraph.neighborSet v = {p.penultimate}

Tactic:
simpa using

Goal state after:
V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
hp : p.IsPath
hnp : ¬p.Nil
⊢ p.toSubgraph.neighborSet v = {p.penultimate}
Module: Mathlib.Combinatorics.SimpleGraph.Metric

Goal state before:
pp: case inl
V : Type u_1
G : SimpleGraph V
u v w : V
huv : G.edist u v = ⊤
⊢ G.edist u w ≤ G.edist u v + G.edist v w

Tactic:
simp [huv]

Goal state after: