mathlib-initiative/mathlib-tactics huggingface dataset:
Module: Mathlib.Combinatorics.SetFamily.FourFunctions

Goal state before:
pp: case inr.inr.refine_1
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
f g μ : α → β
inst✝ : Fintype α
hμ₀ : 0 ≤ μ
hf✝ : 0 ≤ f
hg✝ : 0 ≤ g
hμ : Monotone μ
hfg : ∑ a, f a = ∑ a, g a
h : ∀ (a b : α), f a * g b ≤ f (a ⊓ b) * g (a ⊔ b)
hf : 0 < f
hg : 0 < g
a b : α
⊢ g a * (μ * f) b ≤ f (a ⊓ b) * (μ * g) (a ⊔ b)

Tactic:
·

Goal state after:
case inr.inr.refine_1
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
f g μ : α → β
inst✝ : Fintype α
hμ₀ : 0 ≤ μ
hf✝ : 0 ≤ f
hg✝ : 0 ≤ g
hμ : Monotone μ
hfg : ∑ a, f a = ∑ a, g a
h : ∀ (a b : α), f a * g b ≤ f (a ⊓ b) * g (a ⊔ b)
hf : 0 < f
hg : 0 < g
a b : α
⊢ g a * (μ * f) b ≤ f (a ⊓ b) * (μ * g) (a ⊔ b)
Module: Mathlib.Combinatorics.SimpleGraph.Metric

Goal state before:
pp: case refine_1
V : Type u_1
G : SimpleGraph V
u v : V
h : 2 < G.edist u v
⊢ u ≠ v ∧ ¬G.Adj u v ∧ G.commonNeighbors u v = ∅

pp: case refine_2
V : Type u_1
G : SimpleGraph V
u v : V
h : u ≠ v ∧ ¬G.Adj u v ∧ G.commonNeighbors u v = ∅
⊢ 2 ≤ G.edist u v

pp: case refine_3
V : Type u_1
G : SimpleGraph V
u v : V
h : u ≠ v ∧ ¬G.Adj u v ∧ G.commonNeighbors u v = ∅
⊢ G.edist u v ≠ 2

Tactic:
·

Goal state after:
case refine_1
V : Type u_1
G : SimpleGraph V
u v : V
h : 2 < G.edist u v
⊢ u ≠ v ∧ ¬G.Adj u v ∧ G.commonNeighbors u v = ∅
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: case refine_1
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
⊢ ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.IsCompleteBetween ↑left ↑right

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_1
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
⊢ ∃ left right, #left = Fintype.card α ∧ #right = Fintype.card β ∧ G.IsCompleteBetween ↑left ↑right
Module: Mathlib.Combinatorics.SimpleGraph.Metric

Goal state before:
pp: case refine_2
V : Type u_1
G : SimpleGraph V
u v : V
h : u ≠ v ∧ ¬G.Adj u v ∧ G.commonNeighbors u v = ∅
⊢ 2 ≤ G.edist u v

pp: case refine_3
V : Type u_1
G : SimpleGraph V
u v : V
h : u ≠ v ∧ ¬G.Adj u v ∧ G.commonNeighbors u v = ∅
⊢ G.edist u v ≠ 2

Tactic:
·

Goal state after:
case refine_2
V : Type u_1
G : SimpleGraph V
u v : V
h : u ≠ v ∧ ¬G.Adj u v ∧ G.commonNeighbors u v = ∅
⊢ 2 ≤ G.edist u v
Module: Mathlib.Combinatorics.SimpleGraph.Metric

Goal state before:
pp: case refine_3
V : Type u_1
G : SimpleGraph V
u v : V
h : u ≠ v ∧ ¬G.Adj u v ∧ G.commonNeighbors u v = ∅
⊢ G.edist u v ≠ 2

Tactic:
·

Goal state after:
case refine_3
V : Type u_1
G : SimpleGraph V
u v : V
h : u ≠ v ∧ ¬G.Adj u v ∧ G.commonNeighbors u v = ∅
⊢ G.edist u v ≠ 2
Module: Mathlib.Combinatorics.SimpleGraph.Metric

Goal state before:
pp: case pos
V : Type u_1
G : SimpleGraph V
u v : V
G' : SimpleGraph V
h : G ≤ G'
hr : G.Reachable u v
⊢ G'.edist u v ≤ G.edist u v

pp: case neg
V : Type u_1
G : SimpleGraph V
u v : V
G' : SimpleGraph V
h : G ≤ G'
hr : ¬G.Reachable u v
⊢ G'.edist u v ≤ G.edist u v

Tactic:
·

Goal state after:
case pos
V : Type u_1
G : SimpleGraph V
u v : V
G' : SimpleGraph V
h : G ≤ G'
hr : G.Reachable u v
⊢ G'.edist u v ≤ G.edist u v
Module: Mathlib.Combinatorics.SimpleGraph.Metric

Goal state before:
pp: case neg
V : Type u_1
G : SimpleGraph V
u v : V
G' : SimpleGraph V
h : G ≤ G'
hr : ¬G.Reachable u v
⊢ G'.edist u v ≤ G.edist u v

Tactic:
·

Goal state after:
case neg
V : Type u_1
G : SimpleGraph V
u v : V
G' : SimpleGraph V
h : G ≤ G'
hr : ¬G.Reachable u v
⊢ G'.edist u v ≤ G.edist u v
Module: Mathlib.Combinatorics.SimpleGraph.Metric

Goal state before:
pp: case pos
V : Type u_1
G : SimpleGraph V
u v : V
h : G.Reachable u v
w : V
h' : ¬G.Reachable u w
⊢ G.dist u w ≤ G.dist u v + G.dist v w

pp: case neg
V : Type u_1
G : SimpleGraph V
u v : V
h : G.Reachable u v
w : V
h' : G.Reachable u w
⊢ G.dist u w ≤ G.dist u v + G.dist v w

Tactic:
·

Goal state after:
case pos
V : Type u_1
G : SimpleGraph V
u v : V
h : G.Reachable u v
w : V
h' : ¬G.Reachable u w
⊢ G.dist u w ≤ G.dist u v + G.dist v w
Module: Mathlib.Combinatorics.SimpleGraph.Metric

Goal state before:
pp: case pos
V : Type u_1
G : SimpleGraph V
v w : V
h : G.Reachable v w
u : V
h' : ¬G.Reachable u w
⊢ G.dist u w ≤ G.dist u v + G.dist v w

pp: case neg
V : Type u_1
G : SimpleGraph V
v w : V
h : G.Reachable v w
u : V
h' : G.Reachable u w
⊢ G.dist u w ≤ G.dist u v + G.dist v w

Tactic:
·

Goal state after:
case pos
V : Type u_1
G : SimpleGraph V
v w : V
h : G.Reachable v w
u : V
h' : ¬G.Reachable u w
⊢ G.dist u w ≤ G.dist u v + G.dist v w
Module: Mathlib.Combinatorics.SimpleGraph.Metric

Goal state before:
pp: case pos
V : Type u_1
G : SimpleGraph V
u v w : V
hadj : G.Adj v w
huw : ¬G.Reachable u w
⊢ G.dist u w = G.dist u v ∨ G.dist u w = G.dist u v + 1 ∨ G.dist u w = G.dist u v - 1

pp: case neg
V : Type u_1
G : SimpleGraph V
u v w : V
hadj : G.Adj v w
huw : G.Reachable u w
⊢ G.dist u w = G.dist u v ∨ G.dist u w = G.dist u v + 1 ∨ G.dist u w = G.dist u v - 1

Tactic:
·

Goal state after:
case pos
V : Type u_1
G : SimpleGraph V
u v w : V
hadj : G.Adj v w
huw : ¬G.Reachable u w
⊢ G.dist u w = G.dist u v ∨ G.dist u w = G.dist u v + 1 ∨ G.dist u w = G.dist u v - 1
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph

Goal state before:
pp: case inl
V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
hp : p.IsPath
i : ℕ
hl1 : p.getVert i = u
hadj : p.toSubgraph.Adj u (p.getVert (i + 1))
hi :
  (p.getVert i = u ∧ p.getVert (i + 1) = p.getVert (i + 1) ∨ p.getVert i = p.getVert (i + 1) ∧ p.getVert (i + 1) = u) ∧
    i < p.length
⊢ p.snd = p.getVert (i + 1)

pp: case inr
V : Type u
G : SimpleGraph V
u v v' : V
p : G.Walk u v
hp : p.IsPath
hadj : p.toSubgraph.Adj u v'
i : ℕ
hi : (p.getVert i = u ∧ p.getVert (i + 1) = v' ∨ p.getVert i = v' ∧ p.getVert (i + 1) = u) ∧ i < p.length
hr1 : p.getVert i = v'
hr2 : p.getVert (i + 1) = u
⊢ p.snd = v'

Tactic:
·

Goal state after:
case inl
V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u v
hp : p.IsPath
i : ℕ
hl1 : p.getVert i = u
hadj : p.toSubgraph.Adj u (p.getVert (i + 1))
hi :
  (p.getVert i = u ∧ p.getVert (i + 1) = p.getVert (i + 1) ∨ p.getVert i = p.getVert (i + 1) ∧ p.getVert (i + 1) = u) ∧
    i < p.length
⊢ p.snd = p.getVert (i + 1)
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph

Goal state before:
pp: case inr
V : Type u
G : SimpleGraph V
u v v' : V
p : G.Walk u v
hp : p.IsPath
hadj : p.toSubgraph.Adj u v'
i : ℕ
hi : (p.getVert i = u ∧ p.getVert (i + 1) = v' ∨ p.getVert i = v' ∧ p.getVert (i + 1) = u) ∧ i < p.length
hr1 : p.getVert i = v'
hr2 : p.getVert (i + 1) = u
⊢ p.snd = v'

Tactic:
·

Goal state after:
case inr
V : Type u
G : SimpleGraph V
u v v' : V
p : G.Walk u v
hp : p.IsPath
hadj : p.toSubgraph.Adj u v'
i : ℕ
hi : (p.getVert i = u ∧ p.getVert (i + 1) = v' ∨ p.getVert i = v' ∧ p.getVert (i + 1) = u) ∧ i < p.length
hr1 : p.getVert i = v'
hr2 : p.getVert (i + 1) = u
⊢ p.snd = v'
Module: Mathlib.Combinatorics.SimpleGraph.Metric

Goal state before:
pp: case pos
V : Type u_1
G : SimpleGraph V
v w : V
p : G.Walk v w
hp : p.length = G.dist v w
hl : 1 < G.dist v w
hnp : ¬p.Nil
this✝ : p.tail.tail.length < p.tail.length
this : p.tail.length < p.length
hv : v = p.getVert 2
⊢ G.Adj v (p.getVert 1) ∧ G.Adj (p.getVert 1) (p.getVert 2) ∧ ¬G.Adj v (p.getVert 2) ∧ v ≠ p.getVert 2

pp: case neg
V : Type u_1
G : SimpleGraph V
v w : V
p : G.Walk v w
hp : p.length = G.dist v w
hl : 1 < G.dist v w
hnp : ¬p.Nil
this✝ : p.tail.tail.length < p.tail.length
this : p.tail.length < p.length
hv : ¬v = p.getVert 2
⊢ G.Adj v (p.getVert 1) ∧ G.Adj (p.getVert 1) (p.getVert 2) ∧ ¬G.Adj v (p.getVert 2) ∧ v ≠ p.getVert 2

Tactic:
·

Goal state after:
case pos
V : Type u_1
G : SimpleGraph V
v w : V
p : G.Walk v w
hp : p.length = G.dist v w
hl : 1 < G.dist v w
hnp : ¬p.Nil
this✝ : p.tail.tail.length < p.tail.length
this : p.tail.length < p.length
hv : v = p.getVert 2
⊢ G.Adj v (p.getVert 1) ∧ G.Adj (p.getVert 1) (p.getVert 2) ∧ ¬G.Adj v (p.getVert 2) ∧ v ≠ p.getVert 2
Module: Mathlib.Combinatorics.SimpleGraph.Metric

Goal state before:
pp: case pos
V : Type u_1
G : SimpleGraph V
v w : V
p : G.Walk v w
hp : p.length = G.dist v w
hl : 1 < G.dist v w
hnp : ¬p.Nil
this✝ : p.tail.tail.length < p.tail.length
this : p.tail.length < p.length
hv : ¬v = p.getVert 2
hadj : G.Adj v (p.getVert 2)
⊢ G.Adj v (p.getVert 1) ∧ G.Adj (p.getVert 1) (p.getVert 2) ∧ ¬G.Adj v (p.getVert 2) ∧ v ≠ p.getVert 2

pp: case neg
V : Type u_1
G : SimpleGraph V
v w : V
p : G.Walk v w
hp : p.length = G.dist v w
hl : 1 < G.dist v w
hnp : ¬p.Nil
this✝ : p.tail.tail.length < p.tail.length
this : p.tail.length < p.length
hv : ¬v = p.getVert 2
hadj : ¬G.Adj v (p.getVert 2)
⊢ G.Adj v (p.getVert 1) ∧ G.Adj (p.getVert 1) (p.getVert 2) ∧ ¬G.Adj v (p.getVert 2) ∧ v ≠ p.getVert 2

Tactic:
·

Goal state after:
case pos
V : Type u_1
G : SimpleGraph V
v w : V
p : G.Walk v w
hp : p.length = G.dist v w
hl : 1 < G.dist v w
hnp : ¬p.Nil
this✝ : p.tail.tail.length < p.tail.length
this : p.tail.length < p.length
hv : ¬v = p.getVert 2
hadj : G.Adj v (p.getVert 2)
⊢ G.Adj v (p.getVert 1) ∧ G.Adj (p.getVert 1) (p.getVert 2) ∧ ¬G.Adj v (p.getVert 2) ∧ v ≠ p.getVert 2
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph

Goal state before:
pp: V : Type u
G : SimpleGraph V
u : V
i : ℕ
p : G.Walk u u
hpc : p.IsCycle
h : i ≠ 0
h' : i < p.length
⊢ i - 1 < p.length

Tactic:
by lia

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

Goal state before:
pp: case inl
V : Type u
G : SimpleGraph V
u : V
i : ℕ
p : G.Walk u u
hpc : p.IsCycle
v : V
h : ¬i = 0
h' : i < p.length
hadj1 :
  ∃ i_1,
    (p.getVert i_1 = p.getVert i ∧ p.getVert (i_1 + 1) = p.getVert (i - 1) ∨
        p.getVert i_1 = p.getVert (i - 1) ∧ p.getVert (i_1 + 1) = p.getVert i) ∧
      i_1 < p.length
i' : ℕ
right✝ : i' < p.length
hl1 : p.getVert i' = p.getVert i
hl2 : p.getVert (i' + 1) = v
⊢ v = p.getVert (i - 1) ∨ v = p.getVert (i + 1)

pp: case inr
V : Type u
G : SimpleGraph V
u : V
i : ℕ
p : G.Walk u u
hpc : p.IsCycle
v : V
h : ¬i = 0
h' : i < p.length
hadj1 :
  ∃ i_1,
    (p.getVert i_1 = p.getVert i ∧ p.getVert (i_1 + 1) = p.getVert (i - 1) ∨
        p.getVert i_1 = p.getVert (i - 1) ∧ p.getVert (i_1 + 1) = p.getVert i) ∧
      i_1 < p.length
i' : ℕ
right✝ : i' < p.length
hr1 : p.getVert i' = v
hr2 : p.getVert (i' + 1) = p.getVert i
⊢ v = p.getVert (i - 1) ∨ v = p.getVert (i + 1)

Tactic:
·

Goal state after:
case inl
V : Type u
G : SimpleGraph V
u : V
i : ℕ
p : G.Walk u u
hpc : p.IsCycle
v : V
h : ¬i = 0
h' : i < p.length
hadj1 :
  ∃ i_1,
    (p.getVert i_1 = p.getVert i ∧ p.getVert (i_1 + 1) = p.getVert (i - 1) ∨
        p.getVert i_1 = p.getVert (i - 1) ∧ p.getVert (i_1 + 1) = p.getVert i) ∧
      i_1 < p.length
i' : ℕ
right✝ : i' < p.length
hl1 : p.getVert i' = p.getVert i
hl2 : p.getVert (i' + 1) = v
⊢ v = p.getVert (i - 1) ∨ v = p.getVert (i + 1)
Module: Mathlib.Combinatorics.SimpleGraph.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
G : SimpleGraph α
H : SimpleGraph β
inst✝¹ : DecidableEq β
inst✝ : DecidableRel G.Adj
a₁ a₂ : α
b : β
x z y : α
h : G.Adj x y
w : G.Walk y z
⊢ cons ⋯ (Walk.boxProdLeft H b w).ofBoxProdLeft = cons' x y z h w

pp: case hc
α : Type u_1
β : Type u_2
G : SimpleGraph α
H : SimpleGraph β
inst✝¹ : DecidableEq β
inst✝ : DecidableRel G.Adj
a₁ a₂ : α
b : β
x z y : α
h : G.Adj x y
w : G.Walk y z
⊢ G.Adj (x, b).1 ((G.boxProdLeft H b).toHom y).1 ∧ (x, b).2 = ((G.boxProdLeft H b).toHom y).2

Tactic:
·

Goal state after:
α : Type u_1
β : Type u_2
G : SimpleGraph α
H : SimpleGraph β
inst✝¹ : DecidableEq β
inst✝ : DecidableRel G.Adj
a₁ a₂ : α
b : β
x z y : α
h : G.Adj x y
w : G.Walk y z
⊢ cons ⋯ (Walk.boxProdLeft H b w).ofBoxProdLeft = cons' x y z h w
Module: Mathlib.Combinatorics.SimpleGraph.Prod

Goal state before:
pp: case hc
α : Type u_1
β : Type u_2
G : SimpleGraph α
H : SimpleGraph β
inst✝¹ : DecidableEq β
inst✝ : DecidableRel G.Adj
a₁ a₂ : α
b : β
x z y : α
h : G.Adj x y
w : G.Walk y z
⊢ G.Adj (x, b).1 ((G.boxProdLeft H b).toHom y).1 ∧ (x, b).2 = ((G.boxProdLeft H b).toHom y).2

Tactic:
·

Goal state after:
case hc
α : Type u_1
β : Type u_2
G : SimpleGraph α
H : SimpleGraph β
inst✝¹ : DecidableEq β
inst✝ : DecidableRel G.Adj
a₁ a₂ : α
b : β
x z y : α
h : G.Adj x y
w : G.Walk y z
⊢ G.Adj (x, b).1 ((G.boxProdLeft H b).toHom y).1 ∧ (x, b).2 = ((G.boxProdLeft H b).toHom y).2
Module: Mathlib.Combinatorics.SimpleGraph.Prod

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
a b₁ b₂ x z y : α
h : G.Adj x y
w : G.Walk y z
⊢ cons ⋯ (Walk.boxProdRight G a w).ofBoxProdRight = cons' x y z h w

pp: case hc
α : Type u_1
G : SimpleGraph α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
a b₁ b₂ x z y : α
h : G.Adj x y
w : G.Walk y z
⊢ G.Adj (a, x).2 ((G.boxProdRight G a).toHom y).2 ∧ (a, x).1 = ((G.boxProdRight G a).toHom y).1

Tactic:
·

Goal state after:
α : Type u_1
G : SimpleGraph α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
a b₁ b₂ x z y : α
h : G.Adj x y
w : G.Walk y z
⊢ cons ⋯ (Walk.boxProdRight G a w).ofBoxProdRight = cons' x y z h w
Module: Mathlib.Combinatorics.SimpleGraph.Prod

Goal state before:
pp: case hc
α : Type u_1
G : SimpleGraph α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
a b₁ b₂ x z y : α
h : G.Adj x y
w : G.Walk y z
⊢ G.Adj (a, x).2 ((G.boxProdRight G a).toHom y).2 ∧ (a, x).1 = ((G.boxProdRight G a).toHom y).1

Tactic:
·

Goal state after:
case hc
α : Type u_1
G : SimpleGraph α
inst✝¹ : DecidableEq α
inst✝ : DecidableRel G.Adj
a b₁ b₂ x z y : α
h : G.Adj x y
w : G.Walk y z
⊢ G.Adj (a, x).2 ((G.boxProdRight G a).toHom y).2 ∧ (a, x).1 = ((G.boxProdRight G a).toHom y).1
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph

Goal state before:
pp: case inr
V : Type u
G : SimpleGraph V
u : V
i : ℕ
p : G.Walk u u
hpc : p.IsCycle
v : V
h : ¬i = 0
h' : i < p.length
hadj1 :
  ∃ i_1,
    (p.getVert i_1 = p.getVert i ∧ p.getVert (i_1 + 1) = p.getVert (i - 1) ∨
        p.getVert i_1 = p.getVert (i - 1) ∧ p.getVert (i_1 + 1) = p.getVert i) ∧
      i_1 < p.length
i' : ℕ
right✝ : i' < p.length
hr1 : p.getVert i' = v
hr2 : p.getVert (i' + 1) = p.getVert i
⊢ v = p.getVert (i - 1) ∨ v = p.getVert (i + 1)

Tactic:
·

Goal state after:
case inr
V : Type u
G : SimpleGraph V
u : V
i : ℕ
p : G.Walk u u
hpc : p.IsCycle
v : V
h : ¬i = 0
h' : i < p.length
hadj1 :
  ∃ i_1,
    (p.getVert i_1 = p.getVert i ∧ p.getVert (i_1 + 1) = p.getVert (i - 1) ∨
        p.getVert i_1 = p.getVert (i - 1) ∧ p.getVert (i_1 + 1) = p.getVert i) ∧
      i_1 < p.length
i' : ℕ
right✝ : i' < p.length
hr1 : p.getVert i' = v
hr2 : p.getVert (i' + 1) = p.getVert i
⊢ v = p.getVert (i - 1) ∨ v = p.getVert (i + 1)
Module: Mathlib.Combinatorics.SimpleGraph.Prod

Goal state before:
pp: case inl
α : Type u_1
β : Type u_2
G : SimpleGraph α
H : SimpleGraph β
a₁ a₂✝ : α
b₁ b₂✝ : β
inst✝³ : DecidableEq α
inst✝² : DecidableEq β
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
w : (G □ H).Walk (a₁, b₁) (a₂✝, b₂✝)
b₂ : β
a₂ : α
c : α × β
x : (G □ H).Adj (a₁, b₁) c
w' : (G □ H).Walk c (a₂, b₂)
h₁ : G.Adj a₁ c.1 ∧ b₁ = c.2
⊢ w'.ofBoxProdLeft.length + w'.ofBoxProdRight.length + 1 =
    (Or.by_cases x (fun hG ↦ cons ⋯ w'.ofBoxProdLeft) fun hH ↦ ⋯ ▸ w'.ofBoxProdLeft).length +
      (⋯.by_cases (fun hH ↦ cons ⋯ w'.ofBoxProdRight) fun hG ↦ ⋯ ▸ w'.ofBoxProdRight).length

pp: case inr
α : Type u_1
β : Type u_2
G : SimpleGraph α
H : SimpleGraph β
a₁ a₂✝ : α
b₁ b₂✝ : β
inst✝³ : DecidableEq α
inst✝² : DecidableEq β
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
w : (G □ H).Walk (a₁, b₁) (a₂✝, b₂✝)
b₂ : β
a₂ : α
c : α × β
x : (G □ H).Adj (a₁, b₁) c
w' : (G □ H).Walk c (a₂, b₂)
h₂ : H.Adj b₁ c.2 ∧ a₁ = c.1
⊢ w'.ofBoxProdLeft.length + w'.ofBoxProdRight.length + 1 =
    (Or.by_cases x (fun hG ↦ cons ⋯ w'.ofBoxProdLeft) fun hH ↦ ⋯ ▸ w'.ofBoxProdLeft).length +
      (⋯.by_cases (fun hH ↦ cons ⋯ w'.ofBoxProdRight) fun hG ↦ ⋯ ▸ w'.ofBoxProdRight).length

Tactic:
·

Goal state after:
case inl
α : Type u_1
β : Type u_2
G : SimpleGraph α
H : SimpleGraph β
a₁ a₂✝ : α
b₁ b₂✝ : β
inst✝³ : DecidableEq α
inst✝² : DecidableEq β
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
w : (G □ H).Walk (a₁, b₁) (a₂✝, b₂✝)
b₂ : β
a₂ : α
c : α × β
x : (G □ H).Adj (a₁, b₁) c
w' : (G □ H).Walk c (a₂, b₂)
h₁ : G.Adj a₁ c.1 ∧ b₁ = c.2
⊢ w'.ofBoxProdLeft.length + w'.ofBoxProdRight.length + 1 =
    (Or.by_cases x (fun hG ↦ cons ⋯ w'.ofBoxProdLeft) fun hH ↦ ⋯ ▸ w'.ofBoxProdLeft).length +
      (⋯.by_cases (fun hH ↦ cons ⋯ w'.ofBoxProdRight) fun hG ↦ ⋯ ▸ w'.ofBoxProdRight).length
Module: Mathlib.Combinatorics.SimpleGraph.Prod

Goal state before:
pp: case inr
α : Type u_1
β : Type u_2
G : SimpleGraph α
H : SimpleGraph β
a₁ a₂✝ : α
b₁ b₂✝ : β
inst✝³ : DecidableEq α
inst✝² : DecidableEq β
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
w : (G □ H).Walk (a₁, b₁) (a₂✝, b₂✝)
b₂ : β
a₂ : α
c : α × β
x : (G □ H).Adj (a₁, b₁) c
w' : (G □ H).Walk c (a₂, b₂)
h₂ : H.Adj b₁ c.2 ∧ a₁ = c.1
⊢ w'.ofBoxProdLeft.length + w'.ofBoxProdRight.length + 1 =
    (Or.by_cases x (fun hG ↦ cons ⋯ w'.ofBoxProdLeft) fun hH ↦ ⋯ ▸ w'.ofBoxProdLeft).length +
      (⋯.by_cases (fun hH ↦ cons ⋯ w'.ofBoxProdRight) fun hG ↦ ⋯ ▸ w'.ofBoxProdRight).length

Tactic:
·

Goal state after:
case inr
α : Type u_1
β : Type u_2
G : SimpleGraph α
H : SimpleGraph β
a₁ a₂✝ : α
b₁ b₂✝ : β
inst✝³ : DecidableEq α
inst✝² : DecidableEq β
inst✝¹ : DecidableRel G.Adj
inst✝ : DecidableRel H.Adj
w : (G □ H).Walk (a₁, b₁) (a₂✝, b₂✝)
b₂ : β
a₂ : α
c : α × β
x : (G □ H).Adj (a₁, b₁) c
w' : (G □ H).Walk c (a₂, b₂)
h₂ : H.Adj b₁ c.2 ∧ a₁ = c.1
⊢ w'.ofBoxProdLeft.length + w'.ofBoxProdRight.length + 1 =
    (Or.by_cases x (fun hG ↦ cons ⋯ w'.ofBoxProdLeft) fun hH ↦ ⋯ ▸ w'.ofBoxProdLeft).length +
      (⋯.by_cases (fun hH ↦ cons ⋯ w'.ofBoxProdRight) fun hG ↦ ⋯ ▸ w'.ofBoxProdRight).length
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph

Goal state before:
pp: case pos
V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u u
hpc : p.IsCycle
i : ℕ
hi : p.getVert i = v ∧ i ≤ p.length
he : i = 0 ∨ i = p.length
⊢ (p.toSubgraph.neighborSet v).ncard = 2

pp: case neg
V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u u
hpc : p.IsCycle
i : ℕ
hi : p.getVert i = v ∧ i ≤ p.length
he : i ≠ 0 ∧ i ≠ p.length
⊢ (p.toSubgraph.neighborSet v).ncard = 2

Tactic:
·

Goal state after:
case pos
V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u u
hpc : p.IsCycle
i : ℕ
hi : p.getVert i = v ∧ i ≤ p.length
he : i = 0 ∨ i = p.length
⊢ (p.toSubgraph.neighborSet v).ncard = 2
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u u
hpc : p.IsCycle
i : ℕ
hi : p.getVert i = v ∧ i ≤ p.length
he : i ≠ 0 ∧ i ≠ p.length
⊢ i < p.length

Tactic:
by lia

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v : V
p : G.Walk u u
hpc : p.IsCycle
i : ℕ
hi : p.getVert i = v ∧ i ≤ p.length
he : i ≠ 0 ∧ i ≠ p.length
⊢ i ≤ p.length

Tactic:
by lia

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

Goal state before:
pp: case inl
V : Type u
G : SimpleGraph V
v w : V
p : G.Walk v v
h : p.IsCycle
hadj : p.toSubgraph.Adj v w
hl : w = p.snd
⊢ ∃ p', p'.IsCycle ∧ p'.snd = w ∧ p'.toSubgraph.verts = p.toSubgraph.verts

pp: case inr
V : Type u
G : SimpleGraph V
v w : V
p : G.Walk v v
h : p.IsCycle
hadj : p.toSubgraph.Adj v w
hr : w = p.penultimate
⊢ ∃ p', p'.IsCycle ∧ p'.snd = w ∧ p'.toSubgraph.verts = p.toSubgraph.verts

Tactic:
·

Goal state after:
case inl
V : Type u
G : SimpleGraph V
v w : V
p : G.Walk v v
h : p.IsCycle
hadj : p.toSubgraph.Adj v w
hl : w = p.snd
⊢ ∃ p', p'.IsCycle ∧ p'.snd = w ∧ p'.toSubgraph.verts = p.toSubgraph.verts
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph

Goal state before:
pp: case inr
V : Type u
G : SimpleGraph V
v w : V
p : G.Walk v v
h : p.IsCycle
hadj : p.toSubgraph.Adj v w
hr : w = p.penultimate
⊢ ∃ p', p'.IsCycle ∧ p'.snd = w ∧ p'.toSubgraph.verts = p.toSubgraph.verts

Tactic:
·

Goal state after:
case inr
V : Type u
G : SimpleGraph V
v w : V
p : G.Walk v v
h : p.IsCycle
hadj : p.toSubgraph.Adj v w
hr : w = p.penultimate
⊢ ∃ p', p'.IsCycle ∧ p'.snd = w ∧ p'.toSubgraph.verts = p.toSubgraph.verts
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph

Goal state before:
pp: case h.inl
V : Type u
G : SimpleGraph V
inst✝ : DecidableEq V
u : V
n : ℕ
ih :
  ∀ m < n,
    ∀ {v : V} {p : G.Walk u v} (s : Finset V),
      {x ∈ s | x ∈ p.support}.Nonempty →
        p.length + #s = m → ∃ x ∈ s, ∃ (h : x ∈ p.support), {t ∈ s.erase x | t ∈ (p.takeUntil x ⋯).support} ⊆ ∅
v : V
p : G.Walk u v
s : Finset V
hp : p.length + #s = n
x : V
hxs : x ∈ s
hx : x ∈ p.support
h : {t ∈ s.erase x | t ∈ (p.takeUntil x hx).support} = ∅
⊢ ∃ x ∈ s, ∃ (h : x ∈ p.support), {t ∈ s.erase x | t ∈ (p.takeUntil x ⋯).support} ⊆ ∅

pp: case h.inr
V : Type u
G : SimpleGraph V
inst✝ : DecidableEq V
u : V
n : ℕ
ih :
  ∀ m < n,
    ∀ {v : V} {p : G.Walk u v} (s : Finset V),
      {x ∈ s | x ∈ p.support}.Nonempty →
        p.length + #s = m → ∃ x ∈ s, ∃ (h : x ∈ p.support), {t ∈ s.erase x | t ∈ (p.takeUntil x ⋯).support} ⊆ ∅
v : V
p : G.Walk u v
s : Finset V
hp : p.length + #s = n
x : V
hxs : x ∈ s
hx : x ∈ p.support
h : {t ∈ s.erase x | t ∈ (p.takeUntil x hx).support}.Nonempty
⊢ ∃ x ∈ s, ∃ (h : x ∈ p.support), {t ∈ s.erase x | t ∈ (p.takeUntil x ⋯).support} ⊆ ∅

Tactic:
·

Goal state after:
case h.inl
V : Type u
G : SimpleGraph V
inst✝ : DecidableEq V
u : V
n : ℕ
ih :
  ∀ m < n,
    ∀ {v : V} {p : G.Walk u v} (s : Finset V),
      {x ∈ s | x ∈ p.support}.Nonempty →
        p.length + #s = m → ∃ x ∈ s, ∃ (h : x ∈ p.support), {t ∈ s.erase x | t ∈ (p.takeUntil x ⋯).support} ⊆ ∅
v : V
p : G.Walk u v
s : Finset V
hp : p.length + #s = n
x : V
hxs : x ∈ s
hx : x ∈ p.support
h : {t ∈ s.erase x | t ∈ (p.takeUntil x hx).support} = ∅
⊢ ∃ x ∈ s, ∃ (h : x ∈ p.support), {t ∈ s.erase x | t ∈ (p.takeUntil x ⋯).support} ⊆ ∅
Module: Mathlib.Combinatorics.SimpleGraph.Prod

Goal state before:
pp: case mp
α : Type u_1
β : Type u_2
G : SimpleGraph α
H : SimpleGraph β
x y : α × β
⊢ (G □ H).Reachable x y → G.Reachable x.1 y.1 ∧ H.Reachable x.2 y.2

pp: case mpr
α : Type u_1
β : Type u_2
G : SimpleGraph α
H : SimpleGraph β
x y : α × β
⊢ G.Reachable x.1 y.1 ∧ H.Reachable x.2 y.2 → (G □ H).Reachable x y

Tactic:
·

Goal state after:
case mp
α : Type u_1
β : Type u_2
G : SimpleGraph α
H : SimpleGraph β
x y : α × β
⊢ (G □ H).Reachable x y → G.Reachable x.1 y.1 ∧ H.Reachable x.2 y.2
Module: Mathlib.Combinatorics.SimpleGraph.Prod

Goal state before:
pp: case mp
α : Type u_1
β : Type u_2
G : SimpleGraph α
H : SimpleGraph β
x y : α × β
w : (G □ H).Walk x y
⊢ G.Reachable x.1 y.1 ∧ H.Reachable x.2 y.2

Tactic:
exact ⟨⟨w.ofBoxProdLeft⟩, ⟨w.ofBoxProdRight⟩⟩

Goal state after: