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

Goal state before:
pp: α : Type u_1
inst✝² : BooleanAlgebra α
inst✝¹ : Nontrivial α
inst✝ : Fintype α
s : Finset α
hs : (↑s).Intersecting
this : 2 * #s ≤ Fintype.card α
⊢ ∃ t, s ⊆ t ∧ 2 * #t = Fintype.card α ∧ (↑t).Intersecting

Tactic:
Nat.mul_comm,

Goal state after:
α : Type u_1
inst✝² : BooleanAlgebra α
inst✝¹ : Nontrivial α
inst✝ : Fintype α
s : Finset α
hs : (↑s).Intersecting
this : #s * 2 ≤ Fintype.card α
⊢ ∃ t, s ⊆ t ∧ 2 * #t = Fintype.card α ∧ (↑t).Intersecting
Module: Mathlib.Combinatorics.SetFamily.FourFunctions

Goal state before:
pp: case pos.refine_3
α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : CommSemiring β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
a : α
f₁ f₂ f₃ f₄ : Finset α → β
u : Finset α
inst✝ : ExistsAddOfLE β
hu : a ∉ u
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)
s : Finset α
hsu : s ⊆ u
t : Finset α
htu : t ⊆ u
this✝⁴ : s ⊆ insert a u
this✝³ : t ⊆ insert a u
this✝² : insert a s ⊆ insert a u
this✝¹ : insert a t ⊆ insert a u
has : a ∉ s
hat : a ∉ t
this✝ : a ∉ s ∩ t
this : a ∉ s ∪ t
h✝³ : s ∈ 𝒜
h✝² : insert a s ∈ 𝒜
h✝¹ : t ∈ ℬ
h✝ : insert a t ∈ ℬ
⊢ f₁ (insert a s) * f₂ (insert a t) ≤ f₃ (insert a s ∩ insert a t) * f₄ (insert a s ∪ insert a t)

Tactic:
· simpa [*] using h ‹insert a s ⊆ _› ‹insert a t ⊆ _›

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.LYM

Goal state before:
pp: case pos
α : Type u_2
inst✝ : DecidableEq α
k : ℕ
𝒜 : Finset (Finset α)
s : Finset α
hs : #s = k
t : Finset α
ht : t ∈ 𝒜
hst : s ⊆ t
h : s ∈ 𝒜
⊢ s ∈ 𝒜 ∧ #s = k ∨ ∃ s_1, ((∃ t ∈ 𝒜, s_1 ⊆ t) ∧ #s_1 = k + 1) ∧ ∃ a ∈ s_1, s_1.erase a = s

pp: case neg
α : Type u_2
inst✝ : DecidableEq α
k : ℕ
𝒜 : Finset (Finset α)
s : Finset α
hs : #s = k
t : Finset α
ht : t ∈ 𝒜
hst : s ⊆ t
h : s ∉ 𝒜
⊢ s ∈ 𝒜 ∧ #s = k ∨ ∃ s_1, ((∃ t ∈ 𝒜, s_1 ⊆ t) ∧ #s_1 = k + 1) ∧ ∃ a ∈ s_1, s_1.erase a = s

Tactic:
· exact Or.inl ⟨h, hs⟩

Goal state after:
case neg
α : Type u_2
inst✝ : DecidableEq α
k : ℕ
𝒜 : Finset (Finset α)
s : Finset α
hs : #s = k
t : Finset α
ht : t ∈ 𝒜
hst : s ⊆ t
h : s ∉ 𝒜
⊢ s ∈ 𝒜 ∧ #s = k ∨ ∃ s_1, ((∃ t ∈ 𝒜, s_1 ⊆ t) ∧ #s_1 = k + 1) ∧ ∃ a ∈ s_1, s_1.erase a = s
Module: Mathlib.Combinatorics.SetFamily.Shatter

Goal state before:
pp: case refine_3.refine_2
α : Type u_1
inst✝ : DecidableEq α
𝒜✝ : Finset (Finset α)
a : α
𝒜 : Finset (Finset α)
ih₀ : #(nonMemberSubfamily a 𝒜) ≤ #(nonMemberSubfamily a 𝒜).shatterer
ih₁ : #(memberSubfamily a 𝒜) ≤ #(memberSubfamily a 𝒜).shatterer
ℬ : Finset (Finset α) := image (insert a) ((memberSubfamily a 𝒜).shatterer ∩ (nonMemberSubfamily a 𝒜).shatterer)
hℬ : #ℬ = #((memberSubfamily a 𝒜).shatterer ∩ (nonMemberSubfamily a 𝒜).shatterer)
⊢ ℬ ⊆ 𝒜.shatterer

Tactic:
refine forall_mem_image.2 fun s hs ↦ mem_shatterer.2 fun t ht ↦ ?_

Goal state after:
case refine_3.refine_2
α : Type u_1
inst✝ : DecidableEq α
𝒜✝ : Finset (Finset α)
a : α
𝒜 : Finset (Finset α)
ih₀ : #(nonMemberSubfamily a 𝒜) ≤ #(nonMemberSubfamily a 𝒜).shatterer
ih₁ : #(memberSubfamily a 𝒜) ≤ #(memberSubfamily a 𝒜).shatterer
ℬ : Finset (Finset α) := image (insert a) ((memberSubfamily a 𝒜).shatterer ∩ (nonMemberSubfamily a 𝒜).shatterer)
hℬ : #ℬ = #((memberSubfamily a 𝒜).shatterer ∩ (nonMemberSubfamily a 𝒜).shatterer)
s : Finset α
hs : s ∈ (memberSubfamily a 𝒜).shatterer ∩ (nonMemberSubfamily a 𝒜).shatterer
t : Finset α
ht : t ⊆ insert a s
⊢ ∃ u ∈ 𝒜, insert a s ∩ u = t
Module: Mathlib.Data.Setoid.Partition

Goal state before:
pp: α : Type u_1
c : Set (Set α)
H : ∀ (a : α), ∃! b, b ∈ c ∧ a ∈ b
x _y : α
h : ∀ s ∈ c, x ∈ s → _y ∈ s
s : Set α
hs : s ∈ c
hy : _y ∈ s
⊢ x ∈ s

Tactic:
obtain ⟨t, ⟨ht, hx⟩, _⟩ := H x
    rwa [eq_of_mem_eqv_class H hs hy ht (h t ht hx)]

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

Goal state before:
pp: V : Type u
G : SimpleGraph V
p : Prop
u v : V
h : G.Reachable u v
hp : ∀ (a : G.Path u v), p
⊢ p

Tactic:
exact h.elim fun q => hp q.toPath

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.FourFunctions

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : CommSemiring β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
a : α
f₁ f₂ f₃ f₄ : Finset α → β
u : Finset α
inst✝ : ExistsAddOfLE β
hu : a ∉ u
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)
s : Finset α
hsu : s ⊆ u
t : Finset α
htu : t ⊆ u
this✝⁴ : s ⊆ insert a u
this✝³ : t ⊆ insert a u
this✝² : insert a s ⊆ insert a u
this✝¹ : insert a t ⊆ insert a u
has : a ∉ s
hat : a ∉ t
this✝ : a ∉ s ∩ t
this : a ∉ s ∪ t
h✝³ : s ∈ 𝒜
h✝² : insert a s ∉ 𝒜
h✝¹ : t ∉ ℬ
h✝ : insert a t ∉ ℬ
⊢ f₁ s * 0 + f₁ s * 0 ≤ collapse (𝒜 ⊼ ℬ) a f₃ (s ∩ t) * collapse (𝒜 ⊻ ℬ) a f₄ (s ∪ t)

Tactic:
· simp_rw [mul_zero, add_zero]
      exact mul_nonneg (collapse_nonneg h₃ _) <| collapse_nonneg h₄ _

Goal state after:
Module: Mathlib.Data.Setoid.Partition

Goal state before:
pp: α : Type u_2
s : Set (Set α)
h₁ : s.PairwiseDisjoint id
h₂ : ∀ (a : α), ∃ x ∈ s, a ∈ x
h₃ : ∅ ∉ s
a : α
b₁ b₂ : Set α
hb₁ : b₁ ∈ s ∧ a ∈ b₁
hb₂ : b₂ ∈ s ∧ a ∈ b₂
⊢ b₁ = b₂

Tactic:
apply h₁.elim hb₁.1 hb₂.1

Goal state after:
α : Type u_2
s : Set (Set α)
h₁ : s.PairwiseDisjoint id
h₂ : ∀ (a : α), ∃ x ∈ s, a ∈ x
h₃ : ∅ ∉ s
a : α
b₁ b₂ : Set α
hb₁ : b₁ ∈ s ∧ a ∈ b₁
hb₂ : b₂ ∈ s ∧ a ∈ b₂
⊢ ¬Disjoint (id b₁) (id b₂)
Module: Mathlib.Data.Setoid.Partition

Goal state before:
pp: ι : Type u_1
α : Type u_2
s : ι → Set α
hs : IndexedPartition s
β : Type u_3
f : ι → α → β
t : ι → Set β
ht : IndexedPartition t
hf : ∀ (i : ι), BijOn (f i) (s i) (t i)
g : α → β := hs.piecewise f
hg : g = hs.piecewise f
⊢ Bijective g

Tactic:
have hg_bij (i) : BijOn g (s i) (t i) := by
    refine (hf i).congr fun x hx => ?_
    rw [hg, piecewise_apply, hs.mem_iff_index_eq.mp hx]

Goal state after:
ι : Type u_1
α : Type u_2
s : ι → Set α
hs : IndexedPartition s
β : Type u_3
f : ι → α → β
t : ι → Set β
ht : IndexedPartition t
hf : ∀ (i : ι), BijOn (f i) (s i) (t i)
g : α → β := hs.piecewise f
hg : g = hs.piecewise f
hg_bij : ∀ (i : ι), BijOn g (s i) (t i)
⊢ Bijective g
Module: Mathlib.Data.Setoid.Partition

Goal state before:
pp: case refine_1
ι : Type u_1
α : Type u_2
s : ι → Set α
hs : IndexedPartition s
β : Type u_3
f : ι → α → β
t : Set β
x : α
hx : x ∈ hs.piecewise f ⁻¹' t
⊢ x ∈ ⋃ i, s i ∩ f i ⁻¹' t

Tactic:
mem_preimage,

Goal state after:
case refine_1
ι : Type u_1
α : Type u_2
s : ι → Set α
hs : IndexedPartition s
β : Type u_3
f : ι → α → β
t : Set β
x : α
hx : hs.piecewise f x ∈ t
⊢ x ∈ ⋃ i, s i ∩ f i ⁻¹' t
Module: Mathlib.Combinatorics.SetFamily.FourFunctions

Goal state before:
pp: case neg
α : Type u_1
β : Type u_2
inst✝⁴ : DecidableEq α
inst✝³ : CommSemiring β
inst✝² : LinearOrder β
inst✝¹ : IsStrictOrderedRing β
a : α
f₁ f₂ f₃ f₄ : Finset α → β
u : Finset α
inst✝ : ExistsAddOfLE β
hu : a ∉ u
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)
s : Finset α
hsu : s ⊆ u
t : Finset α
htu : t ⊆ u
this✝⁴ : s ⊆ insert a u
this✝³ : t ⊆ insert a u
this✝² : insert a s ⊆ insert a u
this✝¹ : insert a t ⊆ insert a u
has : a ∉ s
hat : a ∉ t
this✝ : a ∉ s ∩ t
this : a ∉ s ∪ t
h✝³ : s ∉ 𝒜
h✝² : insert a s ∈ 𝒜
h✝¹ : t ∉ ℬ
h✝ : insert a t ∉ ℬ
⊢ f₁ (insert a s) * 0 + f₁ (insert a s) * 0 ≤ collapse (𝒜 ⊼ ℬ) a f₃ (s ∩ t) * collapse (𝒜 ⊻ ℬ) a f₄ (s ∪ t)

Tactic:
· simp_rw [mul_zero, add_zero]
      exact mul_nonneg (collapse_nonneg h₃ _) <| collapse_nonneg h₄ _

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

Goal state before:
pp: case inr
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.isRight = true
hs₂ : s₂.isLeft = true
⊢ G.Adj (Sum.elim (Subtype.val ∘ ⇑fα) (Subtype.val ∘ ⇑fβ) s₁) (Sum.elim (Subtype.val ∘ ⇑fα) (Subtype.val ∘ ⇑fβ) s₂)

Tactic:
← Sum.inr_getRight s₁ hs₁,

Goal state after:
case inr
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.isRight = true
hs₂ : s₂.isLeft = true
⊢ G.Adj (Sum.elim (Subtype.val ∘ ⇑fα) (Subtype.val ∘ ⇑fβ) (Sum.inr (s₁.getRight 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
u v u' v' : V
w : G.Walk u v
⊢ w.toSubgraph.Adj u' v' ↔ ∃ i, s(w.getVert i, w.getVert (i + 1)) = s(u', v') ∧ i < w.length

Tactic:
by
  grind [adj_toSubgraph_iff_mem_edges, mk_mem_edges_iff_exists]

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

Goal state before:
pp: W₁ : Type u_2
W₂ : Type u_3
⊢ (Set.range fun x ↦ s(Sum.inl x.1, Sum.inr x.2)).encard = ENat.card W₁ * ENat.card W₂

Tactic:
← ENat.card_prod,

Goal state after:
W₁ : Type u_2
W₂ : Type u_3
⊢ (Set.range fun x ↦ s(Sum.inl x.1, Sum.inr x.2)).encard = ENat.card (W₁ × W₂)
Module: Mathlib.Combinatorics.SimpleGraph.Metric

Goal state before:
pp: 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
⊢ p.tail.length < p.length

Tactic:
rw [← p.length_tail_add_one hnp]

Goal state after:
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
⊢ p.tail.length < p.tail.length + 1
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph

Goal state before:
pp: V : Type u
G : SimpleGraph V
u v v' : V
p : G.Walk u v
hp : p.IsPath
hadj : p.toSubgraph.Adj u v'
⊢ p.snd = v'

Tactic:
have ⟨i, hi⟩ := p.toSubgraph_adj_iff.mp hadj
  simp only [Sym2.eq, Sym2.rel_iff', Prod.mk.injEq, Prod.swap_prod_mk] at hi
  rcases hi.1 with ⟨hl1, rfl⟩ | ⟨hr1, hr2⟩
  · have : i = 0 := by
      apply hp.getVert_injOn (by rw [Set.mem_ofPred]; lia) (by rw [Set.mem_ofPred]; lia)
      rw [p.getVert_zero, hl1]
    simp [this]
  · have : i + 1 = 0 := by
      apply hp.getVert_injOn (by rw [Set.mem_ofPred]; lia) (by rw [Set.mem_ofPred]; lia)
      rw [p.getVert_zero, hr2]
    contradiction

Goal state after:
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:
exact ⟨h, rfl⟩

Goal state after:
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
⊢ (Walk.boxProdRight G a (cons' x y z h w)).ofBoxProdRight = cons' x y z h w

Tactic:
rw [Walk.boxProdRight, map_cons, ofBoxProdRight, Or.by_cases, dif_pos, ←
      Walk.boxProdRight]

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

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.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:
exact ⟨h, rfl⟩

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:
· apply hpc.getVert_injOn' (by rw [Set.mem_ofPred_eq]; lia)
      (by rw [Set.mem_ofPred_eq]; lia) at hl1
    simp_all

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.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:
exact Subgraph.top_induce_pair_connected_of_adj huv

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

Goal state before:
pp: V : Type u_3
W : Type u_5
G : SimpleGraph V
H : SimpleGraph W
v : V
⊢ (G ⊕g H).neighborSet (Sum.inl v) = Sum.inl '' G.neighborSet v

Tactic:
by
  ext (v' | w') <;> simp

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

Goal state before:
pp: V : Type u_3
W : Type u_5
G : SimpleGraph V
H : SimpleGraph W
w : W
⊢ (G ⊕g H).neighborSet (Sum.inr w) = Sum.inr '' H.neighborSet w

Tactic:
by
  ext (v' | w') <;> simp

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

Goal state before:
pp: n : ℕ
h : 0 < n + 3
⊢ (cycleCons n ⟨0, h⟩).length = ↑⟨0, h⟩

Tactic:
unfold cycleGraph.cycleCons

Goal state after:
n : ℕ
h : 0 < n + 3
⊢ nil.length = ↑⟨0, h⟩
Module: Mathlib.Combinatorics.SimpleGraph.CycleGraph

Goal state before:
pp: n m : ℕ
h : m + 1 < n + 3
⊢ (cycleCons n ⟨m + 1, h⟩).length = ↑⟨m + 1, h⟩

Tactic:
unfold cycleGraph.cycleCons

Goal state after:
n m : ℕ
h : m + 1 < n + 3
⊢ (have hadj := ⋯;
      cons hadj (cycleCons n ⟨m, ⋯⟩)).length =
    ↑⟨m + 1, h⟩
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
x : ℕ
hx : x < n
y : ℕ
hy : y < n
⊢ (cycleGraph n).Adj ⟨x, hx⟩ ⟨y, hy⟩ → G.Adj p.support[⟨x, hx⟩.succ] p.support[⟨y, hy⟩.succ]

Tactic:
hab

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
x : ℕ
hx : x < n
y : ℕ
hy : y < n
hab : (cycleGraph n).Adj ⟨x, hx⟩ ⟨y, hy⟩
⊢ G.Adj p.support[⟨x, hx⟩.succ] p.support[⟨y, hy⟩.succ]
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
⊢ Disjoint {v | v.isLeft = true} {w | w.isRight = true}

Tactic:
simp [Set.disjoint_iff_forall_ne]

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

Goal state before:
pp: α : Type u_1
V : Type u_2
G : SimpleGraph V
inst✝⁴ : DecidableRel G.Adj
inst✝³ : DecidableEq V
inst✝² : AddZeroClass α
inst✝¹ : One α
H : SimpleGraph V
inst✝ : DecidableRel H.Adj
h : IsCompl G H
⊢ adjMatrix α G + adjMatrix α H = adjMatrix α G + adjMatrix α Gᶜ

Tactic:
have := h.compl_eq

Goal state after:
α : Type u_1
V : Type u_2
G : SimpleGraph V
inst✝⁴ : DecidableRel G.Adj
inst✝³ : DecidableEq V
inst✝² : AddZeroClass α
inst✝¹ : One α
H : SimpleGraph V
inst✝ : DecidableRel H.Adj
h : IsCompl G H
this : Gᶜ = H
⊢ adjMatrix α G + adjMatrix α H = adjMatrix α G + adjMatrix α Gᶜ
Module: Mathlib.Combinatorics.SimpleGraph.Coloring.Constructions

Goal state before:
pp: case map_rel'
α : Type u_1
G : SimpleGraph α
h : ∀ (u : α) (w : G.Walk u u), ¬Odd w.length
c : G.ConnectedComponent
w✝ : α
hv : w✝ ∈ c.supp
a b : ↥c
⊢ c.toSimpleGraph.Adj a b →
    (completeGraph (Fin 2)).Adj (Fin.ofNat 2 (Nonempty.some ⋯).length) (Fin.ofNat 2 (Nonempty.some ⋯).length)

Tactic:
hab

Goal state after:
case map_rel'
α : Type u_1
G : SimpleGraph α
h : ∀ (u : α) (w : G.Walk u u), ¬Odd w.length
c : G.ConnectedComponent
w✝ : α
hv : w✝ ∈ c.supp
a b : ↥c
hab : c.toSimpleGraph.Adj a b
⊢ (completeGraph (Fin 2)).Adj (Fin.ofNat 2 (Nonempty.some ⋯).length) (Fin.ofNat 2 (Nonempty.some ⋯).length)
Module: Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite

Goal state before:
pp: case inj'
α : Type u
G✝ : SimpleGraph α
s : Set α
V : Type u_1
G : SimpleGraph V
r t : ℕ
K : G.CompleteEquipartiteSubgraph r t
ht : ¬t = 0
this✝ : Nonempty (Fin r ↪ ↥K.parts)
fᵣ : Fin r ↪ ↥K.parts := Classical.arbitrary (Fin r ↪ ↥K.parts)
this : ∀ (p : ↥K.parts), Nonempty (Fin t ↪ ↥↑p)
fₜ : (p : ↥K.parts) → Fin t ↪ ↥↑p := fun p ↦ Classical.arbitrary (Fin t ↪ ↥↑p)
i₁ : Fin r
j₁ : Fin t
i₂ : Fin r
j₂ : Fin t
heq :
  (fun x ↦
        match x with
        | (i, j) => ↑((fₜ (fᵣ i)) j))
      (i₁, j₁) =
    (fun x ↦
        match x with
        | (i, j) => ↑((fₜ (fᵣ i)) j))
      (i₂, j₂)
⊢ i₁ = i₂ ∧ j₁ = j₂

Tactic:
contrapose! heq with hne

Goal state after:
case inj'
α : Type u
G✝ : SimpleGraph α
s : Set α
V : Type u_1
G : SimpleGraph V
r t : ℕ
K : G.CompleteEquipartiteSubgraph r t
ht : ¬t = 0
this✝ : Nonempty (Fin r ↪ ↥K.parts)
fᵣ : Fin r ↪ ↥K.parts := Classical.arbitrary (Fin r ↪ ↥K.parts)
this : ∀ (p : ↥K.parts), Nonempty (Fin t ↪ ↥↑p)
fₜ : (p : ↥K.parts) → Fin t ↪ ↥↑p := fun p ↦ Classical.arbitrary (Fin t ↪ ↥↑p)
i₁ : Fin r
j₁ : Fin t
i₂ : Fin r
j₂ : Fin t
hne : i₁ = i₂ → j₁ ≠ j₂
⊢ ↑((fₜ (fᵣ i₁)) j₁) ≠ ↑((fₜ (fᵣ i₂)) j₂)
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Turan

Goal state before:
pp: n r : ℕ
⊢ ∑ v, ((∑ x ∈ range n, if ↑v % r ≠ x % r then 1 else 0) + ∑ x ∈ range r, if ↑v % r ≠ (n + x) % r then 1 else 0) =
    (∑ x, ∑ i, if ↑x % r ≠ ↑i % r then 1 else 0) + 2 * (n * (r - 1)) + 2 * r.choose 2

Tactic:
rw [sum_add_distrib,
    Fin.sum_univ_eq_sum_range fun v ↦ ∑ w ∈ range n, if v % r ≠ w % r then 1 else 0,
    Fin.sum_univ_eq_sum_range fun v ↦ ∑ w ∈ range r, if v % r ≠ (n + w) % r then 1 else 0,
    sum_range_add, sum_range_add, add_assoc, add_assoc]

Goal state after:
n r : ℕ
⊢ (∑ x ∈ range n, ∑ w ∈ range n, if x % r ≠ w % r then 1 else 0) +
      ((∑ x ∈ range r, ∑ w ∈ range n, if (n + x) % r ≠ w % r then 1 else 0) +
        ((∑ x ∈ range n, ∑ w ∈ range r, if x % r ≠ (n + w) % r then 1 else 0) +
          ∑ x ∈ range r, ∑ w ∈ range r, if (n + x) % r ≠ (n + w) % r then 1 else 0)) =
    (∑ x, ∑ i, if ↑x % r ≠ ↑i % r then 1 else 0) + (2 * (n * (r - 1)) + 2 * r.choose 2)
Module: Mathlib.Combinatorics.SimpleGraph.CompleteMultipartite

Goal state before:
pp: α : Type u
G✝ : SimpleGraph α
s : Set α
V : Type u_1
G : SimpleGraph V
r t : ℕ
K : G.CompleteEquipartiteSubgraph r t
ht : ¬t = 0
this✝ : Nonempty (Fin r ↪ ↥K.parts)
fᵣ : Fin r ↪ ↥K.parts := Classical.arbitrary (Fin r ↪ ↥K.parts)
this : ∀ (p : ↥K.parts), Nonempty (Fin t ↪ ↥↑p)
fₜ : (p : ↥K.parts) → Fin t ↪ ↥↑p := fun p ↦ Classical.arbitrary (Fin t ↪ ↥↑p)
⊢ Fin r × Fin t ↪ V

Tactic:
use fun (i, j) ↦ fₜ (fᵣ i) j
      intro (i₁, j₁) (i₂, j₂) heq
      rw [Prod.mk.injEq]
      contrapose! heq with hne
      rcases eq_or_ne i₁ i₂ with heq | hne
      · rw [heq, ← Subtype.ext_iff.ne]
        exact (fₜ _).injective.ne (hne heq)
      · refine (K.isCompleteBetween (fᵣ _).prop (fᵣ _).prop ?_ (fₜ _ _).prop (fₜ _ _).prop).ne
        exact Subtype.ext_iff.ne.mp <| fᵣ.injective.ne hne

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

Goal state before:
pp: case neg
α : Type u_1
G : SimpleGraph α
inst✝ : Nonempty α
h : ¬G.diam = 0
⊢ ∃ u v, G.dist u v = G.diam

Tactic:
obtain ⟨u, v, huv⟩ := exists_edist_eq_ediam_of_ne_top <| ediam_ne_top_of_diam_ne_zero h

Goal state after:
case neg
α : Type u_1
G : SimpleGraph α
inst✝ : Nonempty α
h : ¬G.diam = 0
u v : α
huv : G.edist u v = G.ediam
⊢ ∃ u v, G.dist u v = G.diam
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Turan

Goal state before:
pp: case inr.inr.e_a.e_a
n r : ℕ
hr : r > 0
ring₁ : ∀ (n : ℕ), (n ^ 2 - (n % r) ^ 2) * (r - 1) / (2 * r) = n % r * (n / r) * (r - 1) + r * (r - 1) * (n / r) ^ 2 / 2
h : r ≤ n
n' : ℕ := n - r
n'r : n = n' + r
rd : 2 ∣ r * (r - 1)
ring₂ :
  n' % r * (n' / r + 1) * (r - 1) + r.choose 2 * (n' / r + 1) ^ 2 =
    n' % r * (n' / r + 1) * (r - 1) + r.choose 2 + r.choose 2 * 2 * (n' / r) + r.choose 2 * (n' / r) ^ 2
⊢ n' * (r - 1) + r.choose 2 + n' % r * (n' / r) * (r - 1) =
    n' % r * (n' / r + 1) * (r - 1) + r.choose 2 + r.choose 2 * 2 * (n' / r)

Tactic:
← add_rotate,

Goal state after:
case inr.inr.e_a.e_a
n r : ℕ
hr : r > 0
ring₁ : ∀ (n : ℕ), (n ^ 2 - (n % r) ^ 2) * (r - 1) / (2 * r) = n % r * (n / r) * (r - 1) + r * (r - 1) * (n / r) ^ 2 / 2
h : r ≤ n
n' : ℕ := n - r
n'r : n = n' + r
rd : 2 ∣ r * (r - 1)
ring₂ :
  n' % r * (n' / r + 1) * (r - 1) + r.choose 2 * (n' / r + 1) ^ 2 =
    n' % r * (n' / r + 1) * (r - 1) + r.choose 2 + r.choose 2 * 2 * (n' / r) + r.choose 2 * (n' / r) ^ 2
⊢ n' % r * (n' / r) * (r - 1) + n' * (r - 1) + r.choose 2 =
    n' % r * (n' / r + 1) * (r - 1) + r.choose 2 + r.choose 2 * 2 * (n' / r)
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Turan

Goal state before:
pp: case inr.inr.e_a.e_a.e_a
n r : ℕ
hr : r > 0
ring₁ : ∀ (n : ℕ), (n ^ 2 - (n % r) ^ 2) * (r - 1) / (2 * r) = n % r * (n / r) * (r - 1) + r * (r - 1) * (n / r) ^ 2 / 2
h : r ≤ n
n' : ℕ := n - r
n'r : n = n' + r
rd : 2 ∣ r * (r - 1)
ring₂ :
  n' % r * (n' / r + 1) * (r - 1) + r.choose 2 * (n' / r + 1) ^ 2 =
    n' % r * (n' / r + 1) * (r - 1) + r.choose 2 + r.choose 2 * 2 * (n' / r) + r.choose 2 * (n' / r) ^ 2
⊢ n' % r * (n' / r) * (r - 1) + n' * (r - 1) = r * (r - 1) * (n' / r) + n' % r * (n' / r) * (r - 1) + n' % r * (r - 1)

Tactic:
← add_rotate,

Goal state after:
case inr.inr.e_a.e_a.e_a
n r : ℕ
hr : r > 0
ring₁ : ∀ (n : ℕ), (n ^ 2 - (n % r) ^ 2) * (r - 1) / (2 * r) = n % r * (n / r) * (r - 1) + r * (r - 1) * (n / r) ^ 2 / 2
h : r ≤ n
n' : ℕ := n - r
n'r : n = n' + r
rd : 2 ∣ r * (r - 1)
ring₂ :
  n' % r * (n' / r + 1) * (r - 1) + r.choose 2 * (n' / r + 1) ^ 2 =
    n' % r * (n' / r + 1) * (r - 1) + r.choose 2 + r.choose 2 * 2 * (n' / r) + r.choose 2 * (n' / r) ^ 2
⊢ n' % r * (n' / r) * (r - 1) + n' * (r - 1) = n' % r * (r - 1) + r * (r - 1) * (n' / r) + n' % r * (n' / r) * (r - 1)