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

Goal state before:
pp: α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
⊢ supSum (𝒜 ∪ ℬ) + supSum (𝒜 ⊼ ℬ) = supSum 𝒜 + supSum ℬ

Tactic:
unfold supSum

Goal state after:
α : Type u_1
inst✝¹ : Fintype α
inst✝ : DecidableEq α
𝒜 ℬ : Finset (Finset α)
⊢ ∑ s, ↑(#((𝒜 ∪ ℬ).truncatedSup s)) / ((↑(card α) - ↑(#s)) * ↑((card α).choose #s)) +
      ∑ s, ↑(#((𝒜 ⊼ ℬ).truncatedSup s)) / ((↑(card α) - ↑(#s)) * ↑((card α).choose #s)) =
    ∑ s, ↑(#(𝒜.truncatedSup s)) / ((↑(card α) - ↑(#s)) * ↑((card α).choose #s)) +
      ∑ s, ↑(#(ℬ.truncatedSup s)) / ((↑(card α) - ↑(#s)) * ↑((card α).choose #s))
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
u v : Finset α
r : ℕ
huv : #u = #v
h𝒜 : Set.Sized r ↑𝒜
⊢ Set.Sized r ↑(𝓒 u v 𝒜)

Tactic:
simp_rw [Set.Sized, mem_coe, mem_compression]
  rintro s (hs | ⟨huvt, t, ht, rfl⟩)
  · exact h𝒜 hs.1
  · rw [card_compress huv, h𝒜 ht]

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: case neg
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
u v : Finset α
huv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜
𝒜' : Finset (Finset α) := 𝓒 u v 𝒜
H : ∀ s ∈ ∂ 𝒜', s ∉ ∂ 𝒜 → u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \ u ∈ ∂ 𝒜 ∧ (s ∪ v) \ u ∉ ∂ 𝒜'
s : Finset α
hs' : s ∈ ∂ 𝒜'
hs : s ∉ ∂ 𝒜
⊢ s ∈ ∂ 𝒜 ∧ compress u v s ∈ ∂ 𝒜 ∨ s ∉ ∂ 𝒜 ∧ ∃ b ∈ ∂ 𝒜, compress u v b = s

Tactic:
obtain ⟨hus, hvs, h, _⟩ := H _ hs' hs
      exact Or.inr ⟨hs, _, h, compress_of_disjoint_of_le' hvs hus⟩

Goal state after:
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
u v : Finset α
huv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜
𝒜' : Finset (Finset α) := 𝓒 u v 𝒜
s : Finset α
hs𝒜' : s ∈ ∂ 𝒜'
hs𝒜 : s ∉ ∂ 𝒜
m : ∀ y ∉ s, insert y s ∉ 𝒜
⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \ u ∈ ∂ 𝒜 ∧ (s ∪ v) \ u ∉ ∂ 𝒜'

Tactic:
obtain ⟨x, _, _⟩ := mem_shadow_iff_insert_mem.1 hs𝒜'

Goal state after:
α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
u v : Finset α
huv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜
𝒜' : Finset (Finset α) := 𝓒 u v 𝒜
s : Finset α
hs𝒜' : s ∈ ∂ 𝒜'
hs𝒜 : s ∉ ∂ 𝒜
m : ∀ y ∉ s, insert y s ∉ 𝒜
x : α
left✝ : x ∉ s
right✝ : insert x s ∈ 𝒜'
⊢ u ⊆ s ∧ Disjoint v s ∧ (s ∪ v) \ u ∈ ∂ 𝒜 ∧ (s ∪ v) \ u ∉ ∂ 𝒜'
Module: Mathlib.Combinatorics.SetFamily.Compression.UV

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
u v : Finset α
huv : ∀ x ∈ u, ∃ y ∈ v, IsCompressed (u.erase x) (v.erase y) 𝒜
𝒜' : Finset (Finset α) := 𝓒 u v 𝒜
s : Finset α
hs𝒜' : s ∈ ∂ 𝒜'
hs𝒜 : s ∉ ∂ 𝒜
m : ∀ y ∉ s, insert y s ∉ 𝒜
x : α
left✝ : x ∉ s
right✝ : insert x s ∈ 𝒜'
hus✝ : u ⊆ insert x s
hvs : Disjoint v (insert x s)
this✝⁴ : (insert x s ∪ v) \ u ∈ 𝒜
hsv : Disjoint s v
hvu : Disjoint v u
hxv : x ∉ v
this✝³ : v \ u = v
this✝² : x ∉ u
hus : u ⊆ s
w : α
hwB : w ∉ (s ∪ v) \ u
hw𝒜' : insert w ((s ∪ v) \ u) ∈ 𝒜'
this✝¹ : v ⊆ insert w ((s ∪ v) \ u)
hwu : w ∈ u
z : α
hz : z ∈ v
hxy : IsCompressed (u.erase w) (v.erase z) 𝒜
this✝ : insert w ((s ∪ v) \ u) ∈ 𝒜
this : (insert w ((s ∪ v) \ u) ∪ u.erase w) \ v.erase z ∈ 𝒜
⊢ insert z s = (insert w ((s ∪ v) \ u) ∪ u.erase w) \ v.erase z

Tactic:
rw [insert_union_comm, insert_erase ‹w ∈ u›,
      sdiff_union_of_subset (hus.trans subset_union_left),
      sdiff_erase (mem_union_right _ ‹z ∈ v›), union_sdiff_cancel_right hsv]

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

Goal state before:
pp: α : Type u_2
inst✝ : DecidableEq α
k : ℕ
𝒜 : Finset (Finset α)
s : Finset α
⊢ (s ∈ 𝒜 ∧ #s = k ∨ ∃ s_1 ∈ falling (k + 1) 𝒜, ∃ a ∈ s_1, s_1.erase a = s) ↔ s ∈ falling k 𝒜

Tactic:
mem_falling

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

Goal state before:
pp: α : Type u_1
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
t : Finset α
a : α
h : ∀ t ∈ 𝒜, a ∉ t
ht : 𝒜.Shatters t
⊢ a ∉ t

Tactic:
by
  obtain ⟨u, hu, htu⟩ := ht.exists_superset; exact notMem_mono htu <| h u hu

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

Goal state before:
pp: α : Type u_2
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
m n : ℕ
h𝒜 : IsAntichain (fun x1 x2 ↦ x1 ⊆ x2) ↑𝒜
s : Finset α
h₂ : s ∈ 𝒜 # m
h₁ : ∃ s_1 ∈ falling n 𝒜, ∃ a ∈ s_1, s_1.erase a = s
⊢ False

Tactic:
mem_falling

Goal state after:
α : Type u_2
inst✝ : DecidableEq α
𝒜 : Finset (Finset α)
m n : ℕ
h𝒜 : IsAntichain (fun x1 x2 ↦ x1 ⊆ x2) ↑𝒜
s : Finset α
h₂ : s ∈ 𝒜 # m
h₁ : ∃ s_1, ((∃ t ∈ 𝒜, s_1 ⊆ t) ∧ #s_1 = n) ∧ ∃ a ∈ s_1, s_1.erase a = s
⊢ False
Module: Mathlib.Combinatorics.SetFamily.KruskalKatona

Goal state before:
pp: case inr
n r : ℕ
𝒜 : Finset (Finset (Fin n))
h : Set.Sized r ↑𝒜
usable : Finset (Finset (Fin n) × Finset (Fin n)) := {t | UsefulCompression t.1 t.2 ∧ ¬IsCompressed t.1 t.2 𝒜}
husable : usable.Nonempty
⊢ ∃ ℬ, #(∂ ℬ) ≤ #(∂ 𝒜) ∧ #𝒜 = #ℬ ∧ Set.Sized r ↑ℬ ∧ ∀ (U V : Finset (Fin n)), UsefulCompression U V → IsCompressed U V ℬ

Tactic:
obtain ⟨⟨U, V⟩, hUV, t⟩ := exists_min_image usable (fun t ↦ #t.1) husable

Goal state after:
case inr
n r : ℕ
𝒜 : Finset (Finset (Fin n))
h : Set.Sized r ↑𝒜
usable : Finset (Finset (Fin n) × Finset (Fin n)) := {t | UsefulCompression t.1 t.2 ∧ ¬IsCompressed t.1 t.2 𝒜}
husable : usable.Nonempty
U V : Finset (Fin n)
hUV : (U, V) ∈ usable
t : ∀ x' ∈ usable, #(U, V).1 ≤ #x'.1
⊢ ∃ ℬ, #(∂ ℬ) ≤ #(∂ 𝒜) ∧ #𝒜 = #ℬ ∧ Set.Sized r ↑ℬ ∧ ∀ (U V : Finset (Fin n)), UsefulCompression U V → IsCompressed U V ℬ
Module: Mathlib.Combinatorics.SetFamily.KruskalKatona

Goal state before:
pp: n r : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h𝒜r : Set.Sized r ↑𝒜
h𝒞𝒜 : #𝒞 ≤ #𝒜
h𝒞 : IsInitSeg 𝒞 r
𝒜' : Finset (Finset (Fin n))
h𝒜 : 𝒜' ⊆ 𝒜
h𝒜𝒞 : #𝒜' = #𝒞
ℬ : Finset (Finset (Fin n))
hℬ𝒜 : #(∂ ℬ) ≤ #(∂ 𝒜')
h𝒜ℬ : #𝒜' = #ℬ
hℬr : Set.Sized r ↑ℬ
hℬ : ∀ (U V : Finset (Fin n)), UV.UsefulCompression U V → IsCompressed U V ℬ
hcard : #ℬ = #𝒞
⊢ ℬ = 𝒞

Tactic:
obtain h𝒞ℬ | hℬ𝒞 := h𝒞.total (UV.isInitSeg_of_compressed hℬr hℬ)

Goal state after:
case inl
n r : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h𝒜r : Set.Sized r ↑𝒜
h𝒞𝒜 : #𝒞 ≤ #𝒜
h𝒞 : IsInitSeg 𝒞 r
𝒜' : Finset (Finset (Fin n))
h𝒜 : 𝒜' ⊆ 𝒜
h𝒜𝒞 : #𝒜' = #𝒞
ℬ : Finset (Finset (Fin n))
hℬ𝒜 : #(∂ ℬ) ≤ #(∂ 𝒜')
h𝒜ℬ : #𝒜' = #ℬ
hℬr : Set.Sized r ↑ℬ
hℬ : ∀ (U V : Finset (Fin n)), UV.UsefulCompression U V → IsCompressed U V ℬ
hcard : #ℬ = #𝒞
h𝒞ℬ : 𝒞 ⊆ ℬ
⊢ ℬ = 𝒞

case inr
n r : ℕ
𝒜 𝒞 : Finset (Finset (Fin n))
h𝒜r : Set.Sized r ↑𝒜
h𝒞𝒜 : #𝒞 ≤ #𝒜
h𝒞 : IsInitSeg 𝒞 r
𝒜' : Finset (Finset (Fin n))
h𝒜 : 𝒜' ⊆ 𝒜
h𝒜𝒞 : #𝒜' = #𝒞
ℬ : Finset (Finset (Fin n))
hℬ𝒜 : #(∂ ℬ) ≤ #(∂ 𝒜')
h𝒜ℬ : #𝒜' = #ℬ
hℬr : Set.Sized r ↑ℬ
hℬ : ∀ (U V : Finset (Fin n)), UV.UsefulCompression U V → IsCompressed U V ℬ
hcard : #ℬ = #𝒞
hℬ𝒞 : ℬ ⊆ 𝒞
⊢ ℬ = 𝒞
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected

Goal state before:
pp: V : Type u
G : SimpleGraph V
v w : V
H : G.Subgraph
c : H.coe.ConnectedComponent
hw : w ∈ H.verts
hadj : H.Adj v w
w✝ : ↑H.verts
h : w✝ ∈ ↑c ∧ ↑w✝ = v
⊢ w ∈ Subtype.val '' ↑c

Tactic:
use ⟨w, hw⟩

Goal state after:
case h
V : Type u
G : SimpleGraph V
v w : V
H : G.Subgraph
c : H.coe.ConnectedComponent
hw : w ∈ H.verts
hadj : H.Adj v w
w✝ : ↑H.verts
h : w✝ ∈ ↑c ∧ ↑w✝ = v
⊢ ⟨w, hw⟩ ∈ ↑c ∧ ↑⟨w, hw⟩ = w
Module: Mathlib.Combinatorics.SetFamily.KruskalKatona

Goal state before:
pp: n r k i : ℕ
𝒜 : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ #𝒜
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
this : Set.Sized r ↑𝒞
⊢ k.choose (r - i) = #(powersetCard (r - i) range'k)

Tactic:
card_powersetCard,

Goal state after:
n r k i : ℕ
𝒜 : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ #𝒜
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
this : Set.Sized r ↑𝒞
⊢ k.choose (r - i) = (#range'k).choose (r - i)
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected

Goal state before:
pp: V : Type u
G : SimpleGraph V
v w : V
H : G.Subgraph
c : H.coe.ConnectedComponent
hv : v ∈ Subtype.val '' ↑c
hw : w ∈ H.verts
hadj : H.Adj v w
⊢ w ∈ Subtype.val '' ↑c

Tactic:
by
  obtain ⟨_, h⟩ := hv
  use ⟨w, hw⟩
  rw [← (mem_supp_iff _ _).mp h.1]
  exact ⟨connectedComponentMk_eq_of_adj <| Subgraph.Adj.coe <| h.2 ▸ hadj.symm, rfl⟩

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

Goal state before:
pp: case refine_1
V : Type u
G : SimpleGraph V
s : Set V
x✝ : Maximal (fun x ↦ (induce x G).Connected) s
hconn : (fun x ↦ (induce x G).Connected) s
h : ∀ ⦃y : Set V⦄, (fun x ↦ (induce x G).Connected) y → s ⊆ y → y ⊆ s
v : V
hv : v ∈ s
⊢ s ⊆ (G.connectedComponentMk v).supp

Tactic:
exact fun u hu ↦ ConnectedComponent.sound <|
      hconn.preconnected ⟨u, hu⟩ ⟨v, hv⟩ |>.map <| Embedding.induce s |>.toHom

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

Goal state before:
pp: case refine_1
n r k i : ℕ
𝒜 : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ #𝒜
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
this : Set.Sized r ↑𝒞
⊢ #𝒞 ≤ #𝒜

Tactic:
card_powersetCard,

Goal state after:
case refine_1
n r k i : ℕ
𝒜 : Finset (Finset (Fin n))
hir : i ≤ r
hrk : r ≤ k
hkn : k ≤ n
h₁ : Set.Sized r ↑𝒜
h₂ : k.choose r ≤ #𝒜
range'k : Finset (Fin n) := (range k).attachFin ⋯
𝒞 : Finset (Finset (Fin n)) := powersetCard r range'k
this : Set.Sized r ↑𝒞
⊢ (#range'k).choose r ≤ #𝒜
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Connected

Goal state before:
pp: case inr
V : Type u
G : SimpleGraph V
hG : G.Connected
x y : V
h : (G.deleteEdges {s(x, y)}).Reachable x y
hxy : G.Adj x y
⊢ (G.deleteEdges {s(x, y)}).Connected

Tactic:
refine (connected_iff_exists_forall_reachable _).2 ⟨x, fun w ↦ ?_⟩

Goal state after:
case inr
V : Type u
G : SimpleGraph V
hG : G.Connected
x y : V
h : (G.deleteEdges {s(x, y)}).Reachable x y
hxy : G.Adj x y
w : V
⊢ (G.deleteEdges {s(x, y)}).Reachable x w
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: V : Type u_1
v : V
G : SimpleGraph V
s t : Set V
h : G.IsBipartiteWith s t
hv : v ∈ s
⊢ G.neighborSet v = {w | w ∈ t ∧ G.Adj v w}

Tactic:
by
  ext w
  rw [mem_neighborSet, Set.mem_ofPred_eq, iff_and_self]
  exact h.mem_of_mem_adj hv

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

Goal state before:
pp: V : Type u_1
w : V
G : SimpleGraph V
s t : Finset V
inst✝¹ : Fintype ↑(G.neighborSet w)
inst✝ : DecidableRel G.Adj
h : G.IsBipartiteWith ↑s ↑t
hw : w ∈ t
⊢ {v ∈ s | G.Adj v w} = bipartiteBelow G.Adj s w

Tactic:
bipartiteBelow

Goal state after:
V : Type u_1
w : V
G : SimpleGraph V
s t : Finset V
inst✝¹ : Fintype ↑(G.neighborSet w)
inst✝ : DecidableRel G.Adj
h : G.IsBipartiteWith ↑s ↑t
hw : w ∈ t
⊢ {v ∈ s | G.Adj v w} = {a ∈ s | G.Adj a w}
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: case inl
V : Type u_1
G : SimpleGraph V
s t : Set V
h : G.IsBipartiteWith s t
v w : V
hw : G.Adj v w
hs : v ∈ s
ht : w ∈ t
⊢ (completeGraph (Fin 2)).Adj (s.indicator 1 v) (s.indicator 1 w)

Tactic:
replace ht : _ ∉ s := h.1.subset_compl_left ht

Goal state after:
case inl
V : Type u_1
G : SimpleGraph V
s t : Set V
h : G.IsBipartiteWith s t
v w : V
hw : G.Adj v w
hs : v ∈ s
ht : w ∉ s
⊢ (completeGraph (Fin 2)).Adj (s.indicator 1 v) (s.indicator 1 w)
Module: Mathlib.Combinatorics.SimpleGraph.Bipartite

Goal state before:
pp: case inr
V : Type u_1
G : SimpleGraph V
s t : Set V
h : G.IsBipartiteWith s t
v w : V
hw : G.Adj v w
ht : v ∈ t
hs : w ∈ s
⊢ (completeGraph (Fin 2)).Adj (s.indicator 1 v) (s.indicator 1 w)

Tactic:
replace ht : _ ∉ s := h.1.subset_compl_left ht

Goal state after:
case inr
V : Type u_1
G : SimpleGraph V
s t : Set V
h : G.IsBipartiteWith s t
v w : V
hw : G.Adj v w
hs : w ∈ s
ht : v ∉ s
⊢ (completeGraph (Fin 2)).Adj (s.indicator 1 v) (s.indicator 1 w)
Module: Mathlib.Combinatorics.SimpleGraph.Connectivity.Subgraph

Goal state before:
pp: case cons
V : Type u
G : SimpleGraph V
u v w u✝ v✝ w✝ : V
ha : G.Adj u✝ v✝
p✝ : G.Walk v✝ w✝
ih : (p✝.toSubgraph.neighborSet w).Finite
⊢ ((G.subgraphOfAdj ha).neighborSet w ∪ p✝.toSubgraph.neighborSet w).Finite

Tactic:
refine Set.Finite.union ?_ ih

Goal state after:
case cons
V : Type u
G : SimpleGraph V
u v w u✝ v✝ w✝ : V
ha : G.Adj u✝ v✝
p✝ : G.Walk v✝ w✝
ih : (p✝.toSubgraph.neighborSet w).Finite
⊢ ((G.subgraphOfAdj ha).neighborSet w).Finite
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
h_bipartite : (between (↑s) (↑s)ᶜ G).IsBipartiteWith ↑s ↑sᶜ
⊢ #(G.neighborFinset v) ≤ #((between (↑s) (↑s)ᶜ G).neighborFinset v) + #s

Tactic:
← card_union_of_disjoint (isBipartiteWith_neighborFinset_disjoint h_bipartite hv)

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
h_bipartite : (between (↑s) (↑s)ᶜ G).IsBipartiteWith ↑s ↑sᶜ
⊢ #(G.neighborFinset v) ≤ #((between (↑s) (↑s)ᶜ G).neighborFinset v ∪ 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
⊢ w ∈ (↑s)ᶜ

Tactic:
simpa using hw

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

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
h✝ : G.IsBipartite
hv : Finite V
s t : Set V
h : G.IsBipartiteWith s t
this : s.encard + t.encard ≤ ENat.card V
⊢ 4 * (s.encard * t.encard) ≤ ENat.card V ^ 2

Tactic:
ENat.card_eq_coe_natCard,

Goal state after:
V : Type u_1
G : SimpleGraph V
h✝ : G.IsBipartite
hv : Finite V
s t : Set V
h : G.IsBipartiteWith s t
this : s.encard + t.encard ≤ ↑(Nat.card V)
⊢ 4 * (s.encard * t.encard) ≤ ↑(Nat.card V) ^ 2
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 |
        match x with
        | (x, y) => G.Adj x y} =
    #G.bipartiteDoubleCover.edgeFinset

Tactic:
apply card_bij (fun (v, w) _ ↦ s(.inl v, .inr w))
    (fun _ h ↦ by simpa using h) (by grind) (fun e he ↦ ?_)

Goal state after:
V : Type u_1
G : SimpleGraph V
inst✝¹ : Fintype V
inst✝ : DecidableRel G.Adj
e : Sym2 (V ⊕ V)
he : e ∈ G.bipartiteDoubleCover.edgeFinset
⊢ ∃ a,
    ∃ (ha :
      a ∈
        {x |
          match x with
          | (x, y) => G.Adj x y}),
      (match a, ha with
        | (v, w), x => s(Sum.inl v, Sum.inr w)) =
        e
Module: Mathlib.Combinatorics.SimpleGraph.Metric

Goal state before:
pp: case inr.inr
V : Type u_1
G : SimpleGraph V
u v w : V
huv : G.edist u v ≠ ⊤
hvw : G.edist v w ≠ ⊤
p : G.Walk u v
hp : ↑p.length = G.edist u v
q : G.Walk v w
hq : ↑q.length = G.edist v w
⊢ G.edist u w ≤ ↑(p.length + q.length)

Tactic:
← Walk.length_append

Goal state after:
case inr.inr
V : Type u_1
G : SimpleGraph V
u v w : V
huv : G.edist u v ≠ ⊤
hvw : G.edist v w ≠ ⊤
p : G.Walk u v
hp : ↑p.length = G.edist u v
q : G.Walk v w
hq : ↑q.length = G.edist v w
⊢ G.edist u w ≤ ↑(p.append q).length
Module: Mathlib.Combinatorics.SimpleGraph.Metric

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
u v w : V
hconn : G.Connected
p : G.Walk u v
hp : p.length = G.dist u v
q : G.Walk v w
hq : q.length = G.dist v w
⊢ G.dist u w ≤ p.length + q.length

Tactic:
← Walk.length_append

Goal state after:
V : Type u_1
G : SimpleGraph V
u v w : V
hconn : G.Connected
p : G.Walk u v
hp : p.length = G.dist u v
q : G.Walk v w
hq : q.length = G.dist v w
⊢ G.dist u w ≤ (p.append q).length
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:
· 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]

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.Prod

Goal state before:
pp: α : Type u_1
β : Type u_2
G : SimpleGraph α
H : SimpleGraph β
x : α × β
⊢ (G □ H).neighborSet x = G.neighborSet x.1 ×ˢ {x.2} ∪ {x.1} ×ˢ H.neighborSet x.2

Tactic:
ext ⟨a', b'⟩
  simp only [mem_neighborSet, Set.mem_union, boxProd_adj, Set.mem_prod, Set.mem_singleton_iff]
  simp only [eq_comm, and_comm]

Goal state after:
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.map (G.boxProdLeft H b).toHom w)).ofBoxProdLeft = cons' x y z h w

Tactic:
ofBoxProdLeft,

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
⊢ (Or.by_cases ⋯ (fun hG ↦ cons ⋯ (Walk.map (G.boxProdLeft H b).toHom w).ofBoxProdLeft) fun hH ↦
      ⋯ ▸ (Walk.map (G.boxProdLeft H b).toHom w).ofBoxProdLeft) =
    cons' x y z h w
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 : Sum.inr i ∈ left
j : V
hj : j ∈ ↑right.toLeft
⊢ G.Adj i j

Tactic:
rw [mem_coe, mem_toLeft] at hj

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 : Sum.inr i ∈ left
j : V
hj : Sum.inl j ∈ right
⊢ G.Adj i j
Module: Mathlib.Combinatorics.SimpleGraph.Acyclic

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
inst✝¹ : Fintype V
inst✝ : Fintype ↑G.edgeSet
hG : G.IsTree
this✝ : Nonempty V
inhabited_h : Inhabited V
this : {default}ᶜ.card + 1 = Fintype.card V
f : (x : V) → G.Walk x default
hf : ∀ (x : V), (fun p ↦ p.IsPath) (f x)
hf' : ∀ (x : V) (y : G.Walk x default), (fun p ↦ p.IsPath) y → y = f x
w : V
hw : w ∈ {default}ᶜ
⊢ w ≠ default

Tactic:
simpa using hw

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

Goal state before:
pp: α : Type u_1
V : Type u_2
inst✝³ : DecidableEq α
inst✝² : DecidableEq V
inst✝¹ : Zero α
inst✝ : One α
A B : Matrix V V α
hA : A.IsAdjMatrix
hB : B.IsAdjMatrix
h : A.compl = B.compl
i j : V
⊢ A i j = B i j

Tactic:
#adaptation_note /-- Before https://github.com/leanprover/lean4/pull/13166
    (replacing grind's canonicalizer with a type-directed normalizer), `grind` closed this goal
    without the `simp`. This is probably a problem at Mathlib's end rather than `grind`'s,
    as we are relying on seeing through the definition of `Matrix`, and `of`.
    The original proof was: `grind [of, congr($h i j), compl, IsAdjMatrix]` -/

Goal state after:
α : Type u_1
V : Type u_2
inst✝³ : DecidableEq α
inst✝² : DecidableEq V
inst✝¹ : Zero α
inst✝ : One α
A B : Matrix V V α
hA : A.IsAdjMatrix
hB : B.IsAdjMatrix
h : A.compl = B.compl
i j : V
⊢ A i j = B i j
Module: Mathlib.Combinatorics.SimpleGraph.AdjMatrix

Goal state before:
pp: α : Type u_1
V : Type u_2
G : SimpleGraph V
inst✝² : DecidableRel G.Adj
inst✝¹ : Fintype V
inst✝ : NonAssocSemiring α
v : V
vec : V → α
⊢ (adjMatrix α G *ᵥ vec) v = ∑ u ∈ G.neighborFinset v, vec u

Tactic:
by
  rw [mulVec, adjMatrix_dotProduct]

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

Goal state before:
pp: V : Type u_1
G F : SimpleGraph V
h : Maximal (fun H ↦ H ≤ G ∧ H.IsAcyclic) F
⊢ F.Reachable = G.Reachable

Tactic:
by
  ext u v
  refine ⟨.mono h.prop.left, fun ⟨p⟩ ↦ ?_⟩
  by_contra
  let s : F.ConnectedComponent := .mk _ u
  have : v ∉ s := this ∘ s.reachable_of_mem_supp rfl
  have : ∃ d ∈ p.darts, d.fst ∈ s ∧ d.snd ∉ s := p.exists_boundary_dart s rfl this
  rcases this with ⟨⟨⟨u', v'⟩, huv⟩, _, hu, hv⟩
  have : ¬F.Reachable v' u' := mt ConnectedComponent.sound <| s.mem_supp_iff u' |>.mp hu ▸ hv
  suffices F ⊔ edge v' u' ≤ F by grind [Adj.reachable, sup_le_iff, le_iff_adj]
  refine h.le_of_ge ⟨?_, h.prop.right.sup_edge_of_not_reachable this⟩ le_sup_left
  grind [Maximal, sup_le, le_iff_adj, huv.symm]

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Turan

Goal state before:
pp: V : Type u_1
inst✝¹ : Fintype V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
r : ℕ
s t u : V
hts : ¬G.Adj t s
hsu : ¬G.Adj s u
hst : ¬G.Adj s t
dst : G.degree s = G.degree t
dsu : G.degree s = G.degree u
h : G.Adj t u
cf : G.CliqueFree (r + 1)
⊢ ∃ G' inst, G'.CliqueFree (r + 1) ∧ #G.edgeFinset < #G'.edgeFinset

Tactic:
use (G.replaceVertex s t).replaceVertex s u, inferInstance,
    (cf.replaceVertex s t).replaceVertex s u

Goal state after:
case right
V : Type u_1
inst✝¹ : Fintype V
G : SimpleGraph V
inst✝ : DecidableRel G.Adj
r : ℕ
s t u : V
hts : ¬G.Adj t s
hsu : ¬G.Adj s u
hst : ¬G.Adj s t
dst : G.degree s = G.degree t
dsu : G.degree s = G.degree u
h : G.Adj t u
cf : G.CliqueFree (r + 1)
⊢ #G.edgeFinset < #((G.replaceVertex s t).replaceVertex s u).edgeFinset
Module: Mathlib.Combinatorics.SimpleGraph.Coloring.EdgeLabeling

Goal state before:
pp: V : Type u_1
V' : Type u_2
G : SimpleGraph V
G' : SimpleGraph V'
K : Type u_3
K' : Type u_4
C : G.EdgeLabeling K
f : (x y : V) → G.Adj x y → K
f_symm : ∀ (x y : V) (H : G.Adj x y), f y x ⋯ = f x y H
e : Sym2 V
he : e ∈ G.edgeSet
⊢ K

Tactic:
revert he

Goal state after:
V : Type u_1
V' : Type u_2
G : SimpleGraph V
G' : SimpleGraph V'
K : Type u_3
K' : Type u_4
C : G.EdgeLabeling K
f : (x y : V) → G.Adj x y → K
f_symm : ∀ (x y : V) (H : G.Adj x y), f y x ⋯ = f x y H
e : Sym2 V
⊢ e ∈ G.edgeSet → K