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

Goal state before:
pp: case refine_2.refine_1
V : Type u_1
G : SimpleGraph V
n✝ : ℕ
hn : 2 < n✝
a : V
p : G.Walk a a
hp₁ : p.IsCycle
hp₂ : p.length = n✝
n : Fin n✝
⊢ ↑n.succ < p.support.length

pp: case refine_2.refine_2
V : Type u_1
G : SimpleGraph V
n : ℕ
hn : 2 < n
a : V
p : G.Walk a a
hp₁ : p.IsCycle
hp₂ : p.length = n
⊢ ∀ {a_1 b : Fin n}, (cycleGraph n).Adj a_1 b → G.Adj p.support[a_1.succ] p.support[b.succ]

pp: case refine_2.refine_3
V : Type u_1
G : SimpleGraph V
n : ℕ
hn : 2 < n
a : V
p : G.Walk a a
hp₁ : p.IsCycle
hp₂ : p.length = n
⊢ Function.Injective ⇑{ toFun := fun n_1 ↦ p.support[n_1.succ], map_rel' := ?refine_2.refine_2 }

Tactic:
·

Goal state after:
case refine_2.refine_1
V : Type u_1
G : SimpleGraph V
n✝ : ℕ
hn : 2 < n✝
a : V
p : G.Walk a a
hp₁ : p.IsCycle
hp₂ : p.length = n✝
n : Fin n✝
⊢ ↑n.succ < p.support.length
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: case refine_1.inl.inr.refine_3
V : Type u_1
G : SimpleGraph V
α : Type u_2
β : Type u_3
inst✝³ : Finite α
inst✝² : Finite β
inst✝¹ : Nonempty α
inst✝ : Nonempty β
this✝ : Fintype α
this : Fintype β
x✝ :
  ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
left right : Finset (V ⊕ V)
card_left : #left = Fintype.card α
card_right : #right = Fintype.card β
h : G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
l : V
hl : Sum.inl l ∈ left
hmem_left :
  ∀ {l' : V ⊕ V},
    l' ∈ left → ((Sum.inl l).isLeft = true → l'.isLeft = true) ∧ ((Sum.inl l).isRight = true → l'.isRight = true)
r : V
hr : Sum.inr r ∈ right
hmem_right :
  ∀ {r' : V ⊕ V},
    r' ∈ right → ((Sum.inr r).isLeft = true → r'.isLeft = true) ∧ ((Sum.inr r).isRight = true → r'.isRight = true)
i : V
hi : i ∈ ↑left.toLeft
j : V
hj : j ∈ ↑right.toRight
⊢ G.Adj i j

Tactic:
·

Goal state after:
case refine_1.inl.inr.refine_3
V : Type u_1
G : SimpleGraph V
α : Type u_2
β : Type u_3
inst✝³ : Finite α
inst✝² : Finite β
inst✝¹ : Nonempty α
inst✝ : Nonempty β
this✝ : Fintype α
this : Fintype β
x✝ :
  ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
left right : Finset (V ⊕ V)
card_left : #left = Fintype.card α
card_right : #right = Fintype.card β
h : G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
l : V
hl : Sum.inl l ∈ left
hmem_left :
  ∀ {l' : V ⊕ V},
    l' ∈ left → ((Sum.inl l).isLeft = true → l'.isLeft = true) ∧ ((Sum.inl l).isRight = true → l'.isRight = true)
r : V
hr : Sum.inr r ∈ right
hmem_right :
  ∀ {r' : V ⊕ V},
    r' ∈ right → ((Sum.inr r).isLeft = true → r'.isLeft = true) ∧ ((Sum.inr r).isRight = true → r'.isRight = true)
i : V
hi : i ∈ ↑left.toLeft
j : V
hj : j ∈ ↑right.toRight
⊢ G.Adj i j
Module: Mathlib.Combinatorics.SimpleGraph.CycleGraph

Goal state before:
pp: case refine_2.refine_2
V : Type u_1
G : SimpleGraph V
n : ℕ
hn : 2 < n
a : V
p : G.Walk a a
hp₁ : p.IsCycle
hp₂ : p.length = n
⊢ ∀ {a_1 b : Fin n}, (cycleGraph n).Adj a_1 b → G.Adj p.support[a_1.succ] p.support[b.succ]

pp: case refine_2.refine_3
V : Type u_1
G : SimpleGraph V
n : ℕ
hn : 2 < n
a : V
p : G.Walk a a
hp₁ : p.IsCycle
hp₂ : p.length = n
⊢ Function.Injective ⇑{ toFun := fun n_1 ↦ p.support[n_1.succ], map_rel' := ?refine_2.refine_2 }

Tactic:
·

Goal state after:
case refine_2.refine_2
V : Type u_1
G : SimpleGraph V
n : ℕ
hn : 2 < n
a : V
p : G.Walk a a
hp₁ : p.IsCycle
hp₂ : p.length = n
⊢ ∀ {a_1 b : Fin n}, (cycleGraph n).Adj a_1 b → G.Adj p.support[a_1.succ] p.support[b.succ]
Module: Mathlib.Combinatorics.SimpleGraph.CycleGraph

Goal state before:
pp: case refine_2.refine_2.inr
V : Type u_1
G : SimpleGraph V
n : ℕ
hn : 2 < n
a : V
p : G.Walk a a
hp₁ : p.IsCycle
hp₂ : p.length = n
x : ℕ
hx : x < n
y : ℕ
hy : y < n
hab : (cycleGraph n).Adj ⟨x, hx⟩ ⟨y, hy⟩
hne : x ≠ y
this :
  ∀ {V : Type u_1} {G : SimpleGraph V} {n : ℕ},
    2 < n →
      ∀ (a : V) (p : G.Walk a a),
        p.IsCycle →
          ∀ (hp₂ : p.length = n) (x : ℕ) (hx : x < n) (y : ℕ) (hy : y < n),
            (cycleGraph n).Adj ⟨x, hx⟩ ⟨y, hy⟩ → x ≠ y → x > y → G.Adj p.support[⟨x, hx⟩.succ] p.support[⟨y, hy⟩.succ]
hle : ¬x > y
⊢ G.Adj p.support[⟨x, hx⟩.succ] p.support[⟨y, hy⟩.succ]

pp: V✝ : Type u_1
G✝ : SimpleGraph V✝
V : Type u_1
G : SimpleGraph V
n : ℕ
hn : 2 < n
a : V
p : G.Walk a a
hp₁ : p.IsCycle
hp₂ : p.length = n
x : ℕ
hx : x < n
y : ℕ
hy : y < n
hab : (cycleGraph n).Adj ⟨x, hx⟩ ⟨y, hy⟩
hne : x ≠ y
hle : x > y
⊢ G.Adj p.support[⟨x, hx⟩.succ] p.support[⟨y, hy⟩.succ]

Tactic:
·

Goal state after:
case refine_2.refine_2.inr
V : Type u_1
G : SimpleGraph V
n : ℕ
hn : 2 < n
a : V
p : G.Walk a a
hp₁ : p.IsCycle
hp₂ : p.length = n
x : ℕ
hx : x < n
y : ℕ
hy : y < n
hab : (cycleGraph n).Adj ⟨x, hx⟩ ⟨y, hy⟩
hne : x ≠ y
this :
  ∀ {V : Type u_1} {G : SimpleGraph V} {n : ℕ},
    2 < n →
      ∀ (a : V) (p : G.Walk a a),
        p.IsCycle →
          ∀ (hp₂ : p.length = n) (x : ℕ) (hx : x < n) (y : ℕ) (hy : y < n),
            (cycleGraph n).Adj ⟨x, hx⟩ ⟨y, hy⟩ → x ≠ y → x > y → G.Adj p.support[⟨x, hx⟩.succ] p.support[⟨y, hy⟩.succ]
hle : ¬x > y
⊢ G.Adj p.support[⟨x, hx⟩.succ] p.support[⟨y, hy⟩.succ]
Module: Mathlib.Combinatorics.SimpleGraph.CycleGraph

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
n : ℕ
hn : 2 < n
a : V
p : G.Walk a a
hp₁ : p.IsCycle
hp₂ : p.length = n
x : ℕ
hx : x < n
y : ℕ
hy : y < n
hab : (cycleGraph n).Adj ⟨x, hx⟩ ⟨y, hy⟩
hne : x ≠ y
this :
  ∀ {V : Type u_1} {G : SimpleGraph V} {n : ℕ},
    2 < n →
      ∀ (a : V) (p : G.Walk a a),
        p.IsCycle →
          ∀ (hp₂ : p.length = n) (x : ℕ) (hx : x < n) (y : ℕ) (hy : y < n),
            (cycleGraph n).Adj ⟨x, hx⟩ ⟨y, hy⟩ → x ≠ y → x > y → G.Adj p.support[⟨x, hx⟩.succ] p.support[⟨y, hy⟩.succ]
hle : ¬x > y
⊢ y > x

Tactic:
by lia

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

Goal state before:
pp: case inl
V✝ : Type u_1
G✝ : SimpleGraph V✝
V : Type u_1
G : SimpleGraph V
n : ℕ
hn : 2 < n
a : V
p : G.Walk a a
hp₁ : p.IsCycle
hp₂ : p.length = n
x : ℕ
hx : x < n
y : ℕ
hy : y < n
hab✝ : (cycleGraph n).Adj ⟨x, hx⟩ ⟨y, hy⟩
hne : x ≠ y
hle : x > y
hab : ↑(⟨x, hx⟩ - ⟨y, hy⟩) = 1
⊢ G.Adj p.support[⟨x, hx⟩.succ] p.support[⟨y, hy⟩.succ]

pp: case inr
V✝ : Type u_1
G✝ : SimpleGraph V✝
V : Type u_1
G : SimpleGraph V
n : ℕ
hn : 2 < n
a : V
p : G.Walk a a
hp₁ : p.IsCycle
hp₂ : p.length = n
x : ℕ
hx : x < n
y : ℕ
hy : y < n
hab✝ : (cycleGraph n).Adj ⟨x, hx⟩ ⟨y, hy⟩
hne : x ≠ y
hle : x > y
hab : ↑(⟨y, hy⟩ - ⟨x, hx⟩) = 1
⊢ G.Adj p.support[⟨x, hx⟩.succ] p.support[⟨y, hy⟩.succ]

Tactic:
·

Goal state after:
case inl
V✝ : Type u_1
G✝ : SimpleGraph V✝
V : Type u_1
G : SimpleGraph V
n : ℕ
hn : 2 < n
a : V
p : G.Walk a a
hp₁ : p.IsCycle
hp₂ : p.length = n
x : ℕ
hx : x < n
y : ℕ
hy : y < n
hab✝ : (cycleGraph n).Adj ⟨x, hx⟩ ⟨y, hy⟩
hne : x ≠ y
hle : x > y
hab : ↑(⟨x, hx⟩ - ⟨y, hy⟩) = 1
⊢ G.Adj p.support[⟨x, hx⟩.succ] p.support[⟨y, hy⟩.succ]
Module: Mathlib.Combinatorics.SimpleGraph.CycleGraph

Goal state before:
pp: case inr
V✝ : Type u_1
G✝ : SimpleGraph V✝
V : Type u_1
G : SimpleGraph V
n : ℕ
hn : 2 < n
a : V
p : G.Walk a a
hp₁ : p.IsCycle
hp₂ : p.length = n
x : ℕ
hx : x < n
y : ℕ
hy : y < n
hab✝ : (cycleGraph n).Adj ⟨x, hx⟩ ⟨y, hy⟩
hne : x ≠ y
hle : x > y
hab : ↑(⟨y, hy⟩ - ⟨x, hx⟩) = 1
⊢ G.Adj p.support[⟨x, hx⟩.succ] p.support[⟨y, hy⟩.succ]

Tactic:
·

Goal state after:
case inr
V✝ : Type u_1
G✝ : SimpleGraph V✝
V : Type u_1
G : SimpleGraph V
n : ℕ
hn : 2 < n
a : V
p : G.Walk a a
hp₁ : p.IsCycle
hp₂ : p.length = n
x : ℕ
hx : x < n
y : ℕ
hy : y < n
hab✝ : (cycleGraph n).Adj ⟨x, hx⟩ ⟨y, hy⟩
hne : x ≠ y
hle : x > y
hab : ↑(⟨y, hy⟩ - ⟨x, hx⟩) = 1
⊢ G.Adj p.support[⟨x, hx⟩.succ] p.support[⟨y, hy⟩.succ]
Module: Mathlib.Combinatorics.SimpleGraph.CycleGraph

Goal state before:
pp: case refine_2.refine_3
V : Type u_1
G : SimpleGraph V
n : ℕ
hn : 2 < n
a : V
p : G.Walk a a
hp₁ : p.IsCycle
hp₂ : p.length = n
⊢ Function.Injective ⇑{ toFun := fun n_1 ↦ p.support[n_1.succ], map_rel' := ⋯ }

Tactic:
·

Goal state after:
case refine_2.refine_3
V : Type u_1
G : SimpleGraph V
n : ℕ
hn : 2 < n
a : V
p : G.Walk a a
hp₁ : p.IsCycle
hp₂ : p.length = n
⊢ Function.Injective ⇑{ toFun := fun n_1 ↦ p.support[n_1.succ], map_rel' := ⋯ }
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: case refine_1.inr.inl
V : Type u_1
G : SimpleGraph V
α : Type u_2
β : Type u_3
inst✝³ : Finite α
inst✝² : Finite β
inst✝¹ : Nonempty α
inst✝ : Nonempty β
this✝ : Fintype α
this : Fintype β
x✝ :
  ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
left right : Finset (V ⊕ V)
card_left : #left = Fintype.card α
card_right : #right = Fintype.card β
h : G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
l : V
hl : Sum.inr l ∈ left
hmem_left :
  ∀ {l' : V ⊕ V},
    l' ∈ left → ((Sum.inr l).isLeft = true → l'.isLeft = true) ∧ ((Sum.inr l).isRight = true → l'.isRight = true)
r : V
hr : Sum.inl r ∈ right
hmem_right :
  ∀ {r' : V ⊕ V},
    r' ∈ right → ((Sum.inl r).isLeft = true → r'.isLeft = true) ∧ ((Sum.inl r).isRight = true → r'.isRight = true)
⊢ ∃ left_1 right_1, #left_1 = #left ∧ #right_1 = #right ∧ G.IsCompleteBetween ↑left_1 ↑right_1

pp: case refine_1.inr.inr
V : Type u_1
G : SimpleGraph V
α : Type u_2
β : Type u_3
inst✝³ : Finite α
inst✝² : Finite β
inst✝¹ : Nonempty α
inst✝ : Nonempty β
this✝ : Fintype α
this : Fintype β
x✝ :
  ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
left right : Finset (V ⊕ V)
card_left : #left = Fintype.card α
card_right : #right = Fintype.card β
h : G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
l : V
hl : Sum.inr l ∈ left
hmem_left :
  ∀ {l' : V ⊕ V},
    l' ∈ left → ((Sum.inr l).isLeft = true → l'.isLeft = true) ∧ ((Sum.inr l).isRight = true → l'.isRight = true)
r : V
hr : Sum.inr r ∈ right
hmem_right :
  ∀ {r' : V ⊕ V},
    r' ∈ right → ((Sum.inr r).isLeft = true → r'.isLeft = true) ∧ ((Sum.inr r).isRight = true → r'.isRight = true)
⊢ ∃ left_1 right_1, #left_1 = #left ∧ #right_1 = #right ∧ G.IsCompleteBetween ↑left_1 ↑right_1

Tactic:
·

Goal state after:
case refine_1.inr.inl
V : Type u_1
G : SimpleGraph V
α : Type u_2
β : Type u_3
inst✝³ : Finite α
inst✝² : Finite β
inst✝¹ : Nonempty α
inst✝ : Nonempty β
this✝ : Fintype α
this : Fintype β
x✝ :
  ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
left right : Finset (V ⊕ V)
card_left : #left = Fintype.card α
card_right : #right = Fintype.card β
h : G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
l : V
hl : Sum.inr l ∈ left
hmem_left :
  ∀ {l' : V ⊕ V},
    l' ∈ left → ((Sum.inr l).isLeft = true → l'.isLeft = true) ∧ ((Sum.inr l).isRight = true → l'.isRight = true)
r : V
hr : Sum.inl r ∈ right
hmem_right :
  ∀ {r' : V ⊕ V},
    r' ∈ right → ((Sum.inl r).isLeft = true → r'.isLeft = true) ∧ ((Sum.inl r).isRight = true → r'.isRight = true)
⊢ ∃ left_1 right_1, #left_1 = #left ∧ #right_1 = #right ∧ G.IsCompleteBetween ↑left_1 ↑right_1
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: case refine_1.inr.inl.refine_2
V : Type u_1
G : SimpleGraph V
α : Type u_2
β : Type u_3
inst✝³ : Finite α
inst✝² : Finite β
inst✝¹ : Nonempty α
inst✝ : Nonempty β
this✝ : Fintype α
this : Fintype β
x✝ :
  ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
left right : Finset (V ⊕ V)
card_left : #left = Fintype.card α
card_right : #right = Fintype.card β
h : G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
l : V
hl : Sum.inr l ∈ left
hmem_left :
  ∀ {l' : V ⊕ V},
    l' ∈ left → ((Sum.inr l).isLeft = true → l'.isLeft = true) ∧ ((Sum.inr l).isRight = true → l'.isRight = true)
r : V
hr : Sum.inl r ∈ right
hmem_right :
  ∀ {r' : V ⊕ V},
    r' ∈ right → ((Sum.inl r).isLeft = true → r'.isLeft = true) ∧ ((Sum.inl r).isRight = true → r'.isRight = true)
⊢ #right.toLeft = #right

pp: case refine_1.inr.inl.refine_3
V : Type u_1
G : SimpleGraph V
α : Type u_2
β : Type u_3
inst✝³ : Finite α
inst✝² : Finite β
inst✝¹ : Nonempty α
inst✝ : Nonempty β
this✝ : Fintype α
this : Fintype β
x✝ :
  ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
left right : Finset (V ⊕ V)
card_left : #left = Fintype.card α
card_right : #right = Fintype.card β
h : G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
l : V
hl : Sum.inr l ∈ left
hmem_left :
  ∀ {l' : V ⊕ V},
    l' ∈ left → ((Sum.inr l).isLeft = true → l'.isLeft = true) ∧ ((Sum.inr l).isRight = true → l'.isRight = true)
r : V
hr : Sum.inl r ∈ right
hmem_right :
  ∀ {r' : V ⊕ V},
    r' ∈ right → ((Sum.inl r).isLeft = true → r'.isLeft = true) ∧ ((Sum.inl r).isRight = true → r'.isRight = true)
i : V
hi : i ∈ ↑left.toRight
j : V
hj : j ∈ ↑right.toLeft
⊢ G.Adj i j

Tactic:
·

Goal state after:
case refine_1.inr.inl.refine_2
V : Type u_1
G : SimpleGraph V
α : Type u_2
β : Type u_3
inst✝³ : Finite α
inst✝² : Finite β
inst✝¹ : Nonempty α
inst✝ : Nonempty β
this✝ : Fintype α
this : Fintype β
x✝ :
  ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
left right : Finset (V ⊕ V)
card_left : #left = Fintype.card α
card_right : #right = Fintype.card β
h : G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
l : V
hl : Sum.inr l ∈ left
hmem_left :
  ∀ {l' : V ⊕ V},
    l' ∈ left → ((Sum.inr l).isLeft = true → l'.isLeft = true) ∧ ((Sum.inr l).isRight = true → l'.isRight = true)
r : V
hr : Sum.inl r ∈ right
hmem_right :
  ∀ {r' : V ⊕ V},
    r' ∈ right → ((Sum.inl r).isLeft = true → r'.isLeft = true) ∧ ((Sum.inl r).isRight = true → r'.isRight = true)
⊢ #right.toLeft = #right
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: case refine_1.inr.inl.refine_3
V : Type u_1
G : SimpleGraph V
α : Type u_2
β : Type u_3
inst✝³ : Finite α
inst✝² : Finite β
inst✝¹ : Nonempty α
inst✝ : Nonempty β
this✝ : Fintype α
this : Fintype β
x✝ :
  ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
left right : Finset (V ⊕ V)
card_left : #left = Fintype.card α
card_right : #right = Fintype.card β
h : G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
l : V
hl : Sum.inr l ∈ left
hmem_left :
  ∀ {l' : V ⊕ V},
    l' ∈ left → ((Sum.inr l).isLeft = true → l'.isLeft = true) ∧ ((Sum.inr l).isRight = true → l'.isRight = true)
r : V
hr : Sum.inl r ∈ right
hmem_right :
  ∀ {r' : V ⊕ V},
    r' ∈ right → ((Sum.inl r).isLeft = true → r'.isLeft = true) ∧ ((Sum.inl r).isRight = true → r'.isRight = true)
i : V
hi : i ∈ ↑left.toRight
j : V
hj : j ∈ ↑right.toLeft
⊢ G.Adj i j

Tactic:
·

Goal state after:
case refine_1.inr.inl.refine_3
V : Type u_1
G : SimpleGraph V
α : Type u_2
β : Type u_3
inst✝³ : Finite α
inst✝² : Finite β
inst✝¹ : Nonempty α
inst✝ : Nonempty β
this✝ : Fintype α
this : Fintype β
x✝ :
  ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
left right : Finset (V ⊕ V)
card_left : #left = Fintype.card α
card_right : #right = Fintype.card β
h : G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
l : V
hl : Sum.inr l ∈ left
hmem_left :
  ∀ {l' : V ⊕ V},
    l' ∈ left → ((Sum.inr l).isLeft = true → l'.isLeft = true) ∧ ((Sum.inr l).isRight = true → l'.isRight = true)
r : V
hr : Sum.inl r ∈ right
hmem_right :
  ∀ {r' : V ⊕ V},
    r' ∈ right → ((Sum.inl r).isLeft = true → r'.isLeft = true) ∧ ((Sum.inl r).isRight = true → r'.isRight = true)
i : V
hi : i ∈ ↑left.toRight
j : V
hj : j ∈ ↑right.toLeft
⊢ G.Adj i j
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: case refine_1.inr.inr
V : Type u_1
G : SimpleGraph V
α : Type u_2
β : Type u_3
inst✝³ : Finite α
inst✝² : Finite β
inst✝¹ : Nonempty α
inst✝ : Nonempty β
this✝ : Fintype α
this : Fintype β
x✝ :
  ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
left right : Finset (V ⊕ V)
card_left : #left = Fintype.card α
card_right : #right = Fintype.card β
h : G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
l : V
hl : Sum.inr l ∈ left
hmem_left :
  ∀ {l' : V ⊕ V},
    l' ∈ left → ((Sum.inr l).isLeft = true → l'.isLeft = true) ∧ ((Sum.inr l).isRight = true → l'.isRight = true)
r : V
hr : Sum.inr r ∈ right
hmem_right :
  ∀ {r' : V ⊕ V},
    r' ∈ right → ((Sum.inr r).isLeft = true → r'.isLeft = true) ∧ ((Sum.inr r).isRight = true → r'.isRight = true)
⊢ ∃ left_1 right_1, #left_1 = #left ∧ #right_1 = #right ∧ G.IsCompleteBetween ↑left_1 ↑right_1

Tactic:
·

Goal state after:
case refine_1.inr.inr
V : Type u_1
G : SimpleGraph V
α : Type u_2
β : Type u_3
inst✝³ : Finite α
inst✝² : Finite β
inst✝¹ : Nonempty α
inst✝ : Nonempty β
this✝ : Fintype α
this : Fintype β
x✝ :
  ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
left right : Finset (V ⊕ V)
card_left : #left = Fintype.card α
card_right : #right = Fintype.card β
h : G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
l : V
hl : Sum.inr l ∈ left
hmem_left :
  ∀ {l' : V ⊕ V},
    l' ∈ left → ((Sum.inr l).isLeft = true → l'.isLeft = true) ∧ ((Sum.inr l).isRight = true → l'.isRight = true)
r : V
hr : Sum.inr r ∈ right
hmem_right :
  ∀ {r' : V ⊕ V},
    r' ∈ right → ((Sum.inr r).isLeft = true → r'.isLeft = true) ∧ ((Sum.inr r).isRight = true → r'.isRight = true)
⊢ ∃ left_1 right_1, #left_1 = #left ∧ #right_1 = #right ∧ G.IsCompleteBetween ↑left_1 ↑right_1
Module: Mathlib.Combinatorics.SimpleGraph.Cayley

Goal state before:
pp: case mp
M : Type u_1
s : Set M
inst✝ : Mul M
G : SimpleGraph M
⊢ (∀ (v w : M), ¬v = w → ∀ x ∈ s, v * x = w ∨ v = w * x → G.Adj v w) → ∀ g ∈ s, ∀ (a : M), ¬a * g = a → G.Adj (a * g) a

pp: case mpr
M : Type u_1
s : Set M
inst✝ : Mul M
G : SimpleGraph M
⊢ (∀ g ∈ s, ∀ (a : M), ¬a * g = a → G.Adj (a * g) a) → ∀ (v w : M), ¬v = w → ∀ x ∈ s, v * x = w ∨ v = w * x → G.Adj v w

Tactic:
·

Goal state after:
case mp
M : Type u_1
s : Set M
inst✝ : Mul M
G : SimpleGraph M
⊢ (∀ (v w : M), ¬v = w → ∀ x ∈ s, v * x = w ∨ v = w * x → G.Adj v w) → ∀ g ∈ s, ∀ (a : M), ¬a * g = a → G.Adj (a * g) a
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: case refine_2
V : Type u_1
G : SimpleGraph V
α : Type u_2
β : Type u_3
inst✝³ : Finite α
inst✝² : Finite β
inst✝¹ : Nonempty α
inst✝ : Nonempty β
this✝ : Fintype α
this : Fintype β
x✝ : ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.IsCompleteBetween ↑left ↑right
left right : Finset V
card_left : #left = Fintype.card α
card_right : #right = Fintype.card β
h : G.IsCompleteBetween ↑left ↑right
⊢ ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right

Tactic:
·

Goal state after:
case refine_2
V : Type u_1
G : SimpleGraph V
α : Type u_2
β : Type u_3
inst✝³ : Finite α
inst✝² : Finite β
inst✝¹ : Nonempty α
inst✝ : Nonempty β
this✝ : Fintype α
this : Fintype β
x✝ : ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.IsCompleteBetween ↑left ↑right
left right : Finset V
card_left : #left = Fintype.card α
card_right : #right = Fintype.card β
h : G.IsCompleteBetween ↑left ↑right
⊢ ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.bipartiteDoubleCover.IsCompleteBetween ↑left ↑right
Module: Mathlib.Combinatorics.SimpleGraph.Cayley

Goal state before:
pp: case mpr
M : Type u_1
s : Set M
inst✝ : Mul M
G : SimpleGraph M
⊢ (∀ g ∈ s, ∀ (a : M), ¬a * g = a → G.Adj (a * g) a) → ∀ (v w : M), ¬v = w → ∀ x ∈ s, v * x = w ∨ v = w * x → G.Adj v w

Tactic:
·

Goal state after:
case mpr
M : Type u_1
s : Set M
inst✝ : Mul M
G : SimpleGraph M
⊢ (∀ g ∈ s, ∀ (a : M), ¬a * g = a → G.Adj (a * g) a) → ∀ (v w : M), ¬v = w → ∀ x ∈ s, v * x = w ∨ v = w * x → G.Adj v w
Module: Mathlib.Combinatorics.SimpleGraph.Cayley

Goal state before:
pp: case mpr.inl
M : Type u_1
s : Set M
inst✝ : Mul M
G : SimpleGraph M
h : ∀ g ∈ s, ∀ (a : M), ¬a * g = a → G.Adj (a * g) a
v g : M
hg : g ∈ s
hvw : ¬v = v * g
⊢ G.Adj v (v * g)

pp: case mpr.inr
M : Type u_1
s : Set M
inst✝ : Mul M
G : SimpleGraph M
h : ∀ g ∈ s, ∀ (a : M), ¬a * g = a → G.Adj (a * g) a
w g : M
hg : g ∈ s
hvw : ¬w * g = w
⊢ G.Adj (w * g) w

Tactic:
·

Goal state after:
case mpr.inl
M : Type u_1
s : Set M
inst✝ : Mul M
G : SimpleGraph M
h : ∀ g ∈ s, ∀ (a : M), ¬a * g = a → G.Adj (a * g) a
v g : M
hg : g ∈ s
hvw : ¬v = v * g
⊢ G.Adj v (v * g)
Module: Mathlib.Combinatorics.SimpleGraph.Cayley

Goal state before:
pp: case mpr.inr
M : Type u_1
s : Set M
inst✝ : Mul M
G : SimpleGraph M
h : ∀ g ∈ s, ∀ (a : M), ¬a * g = a → G.Adj (a * g) a
w g : M
hg : g ∈ s
hvw : ¬w * g = w
⊢ G.Adj (w * g) w

Tactic:
·

Goal state after:
case mpr.inr
M : Type u_1
s : Set M
inst✝ : Mul M
G : SimpleGraph M
h : ∀ g ∈ s, ∀ (a : M), ¬a * g = a → G.Adj (a * g) a
w g : M
hg : g ∈ s
hvw : ¬w * g = w
⊢ G.Adj (w * g) w
Module: Mathlib.Combinatorics.SimpleGraph.Circulant

Goal state before:
pp: case mp
G : Type u_1
inst✝ : AddGroup G
s : Set G
u v : G
h : u ≠ v
⊢ u - v ∈ s ∨ v - u ∈ s → u - v ∈ s \ {0} ∨ v - u ∈ s \ {0}

pp: case mpr
G : Type u_1
inst✝ : AddGroup G
s : Set G
u v : G
h : u ≠ v
⊢ u - v ∈ s \ {0} ∨ v - u ∈ s \ {0} → u - v ∈ s ∨ v - u ∈ s

Tactic:
·

Goal state after:
case mp
G : Type u_1
inst✝ : AddGroup G
s : Set G
u v : G
h : u ≠ v
⊢ u - v ∈ s ∨ v - u ∈ s → u - v ∈ s \ {0} ∨ v - u ∈ s \ {0}
Module: Mathlib.Combinatorics.SimpleGraph.Circulant

Goal state before:
pp: case mpr
G : Type u_1
inst✝ : AddGroup G
s : Set G
u v : G
h : u ≠ v
⊢ u - v ∈ s \ {0} ∨ v - u ∈ s \ {0} → u - v ∈ s ∨ v - u ∈ s

Tactic:
·

Goal state after:
case mpr
G : Type u_1
inst✝ : AddGroup G
s : Set G
u v : G
h : u ≠ v
⊢ u - v ∈ s \ {0} ∨ v - u ∈ s \ {0} → u - v ∈ s ∨ v - u ∈ s
Module: Mathlib.Combinatorics.SimpleGraph.Circulant

Goal state before:
pp: case zero
⊢ cycleGraph (0 + 1) = circulantGraph {1}

pp: case succ
n✝ : ℕ
⊢ cycleGraph (n✝ + 1 + 1) = circulantGraph {1}

Tactic:
·

Goal state after:
case zero
⊢ cycleGraph (0 + 1) = circulantGraph {1}
Module: Mathlib.Combinatorics.SimpleGraph.Circulant

Goal state before:
pp: case succ
n✝ : ℕ
⊢ cycleGraph (n✝ + 1 + 1) = circulantGraph {1}

Tactic:
·

Goal state after:
case succ
n✝ : ℕ
⊢ cycleGraph (n✝ + 1 + 1) = circulantGraph {1}
Module: Mathlib.Combinatorics.SimpleGraph.Acyclic

Goal state before:
pp: case inl
V : Type u_1
h_acyc : ∀ H ∈ ∅, H.IsAcyclic
h_dir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) ∅
⊢ (sSup ∅).IsAcyclic

pp: case inr
V : Type u_1
Hs : Set (SimpleGraph V)
h_acyc : ∀ H ∈ Hs, H.IsAcyclic
h_dir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) Hs
hnemp : Hs.Nonempty
⊢ (sSup Hs).IsAcyclic

Tactic:
·

Goal state after:
case inl
V : Type u_1
h_acyc : ∀ H ∈ ∅, H.IsAcyclic
h_dir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) ∅
⊢ (sSup ∅).IsAcyclic
Module: Mathlib.Combinatorics.SimpleGraph.Acyclic

Goal state before:
pp: case inr
V : Type u_1
Hs : Set (SimpleGraph V)
h_acyc : ∀ H ∈ Hs, H.IsAcyclic
h_dir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) Hs
hnemp : Hs.Nonempty
⊢ (sSup Hs).IsAcyclic

Tactic:
·

Goal state after:
case inr
V : Type u_1
Hs : Set (SimpleGraph V)
h_acyc : ∀ H ∈ Hs, H.IsAcyclic
h_dir : DirectedOn (fun x1 x2 ↦ x1 ≤ x2) Hs
hnemp : Hs.Nonempty
⊢ (sSup Hs).IsAcyclic
Module: Mathlib.Combinatorics.SimpleGraph.Acyclic

Goal state before:
pp: case refine_1
V : Type u_1
G H : SimpleGraph V
hHG : H ≤ G
hH : H.IsAcyclic
c : Set (SimpleGraph V)
hcs : c ⊆ {H | H ≤ G ∧ H.IsAcyclic}
hc : IsChain (fun x1 x2 ↦ x1 ≤ x2) c
y : SimpleGraph V
hy : y ∈ c
⊢ sSup c ≤ G

pp: case refine_2
V : Type u_1
G H : SimpleGraph V
hHG : H ≤ G
hH : H.IsAcyclic
c : Set (SimpleGraph V)
hcs : c ⊆ {H | H ≤ G ∧ H.IsAcyclic}
hc : IsChain (fun x1 x2 ↦ x1 ≤ x2) c
y : SimpleGraph V
hy : y ∈ c
⊢ (sSup c).IsAcyclic

Tactic:
·

Goal state after:
case refine_1
V : Type u_1
G H : SimpleGraph V
hHG : H ≤ G
hH : H.IsAcyclic
c : Set (SimpleGraph V)
hcs : c ⊆ {H | H ≤ G ∧ H.IsAcyclic}
hc : IsChain (fun x1 x2 ↦ x1 ≤ x2) c
y : SimpleGraph V
hy : y ∈ c
⊢ sSup c ≤ G