mathlib-initiative/mathlib-tactics huggingface dataset:
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 : Sum.inl i ∈ left
j : V
hj : j ∈ right.toRight
⊢ G.Adj i j

Tactic:
mem_toRight

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 : Sum.inl i ∈ left
j : V
hj : Sum.inr j ∈ right
⊢ G.Adj i j
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:
mem_toRight

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 : j ∈ ↑right.toLeft
⊢ G.Adj i j
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Basic

Goal state before:
pp: V : Type u_1
W : Type u_2
inst✝ : Fintype V
H : SimpleGraph W
m : ℕ
⊢ (∃ b, H.Free b ∧ m < #b.edgeFinset) ↔ ∃ G x, H.Free G ∧ m < #G.edgeFinset

Tactic:
exact ⟨fun ⟨_, h, h'⟩ ↦ ⟨_, _, h, h'⟩, fun ⟨_, _, h, h'⟩ ↦ ⟨_, h, by convert!
    h'⟩⟩

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 : V
h : G.IsTuranMaximal r
hn : ¬G.Adj s t
⊢ G.degree s = G.degree t

Tactic:
rw [IsTuranMaximal, IsExtremal] at h; contrapose! h; intro cf
  wlog hd : G.degree t < G.degree s generalizing G t s
  · replace hd : G.degree s < G.degree t := lt_of_le_of_ne (le_of_not_gt hd) h
    exact this (by rwa [adj_comm] at hn) hd.ne' cf hd
  classical
  use G.replaceVertex s t, inferInstance, cf.replaceVertex s t
  have := G.card_edgeFinset_replaceVertex_of_not_adj hn
  lia

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)
nst : s ≠ t
ntu : t ≠ u
this : ¬(G.replaceVertex s t).Adj s u
l1 : (G.replaceVertex s t).degree s = G.degree s
⊢ #((G.replaceVertex s t).neighborFinset u) = #(G.neighborFinset u) - 1

Tactic:
← card_singleton t,

Goal state after:
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)
nst : s ≠ t
ntu : t ≠ u
this : ¬(G.replaceVertex s t).Adj s u
l1 : (G.replaceVertex s t).degree s = G.degree s
⊢ #((G.replaceVertex s t).neighborFinset u) = #(G.neighborFinset u) - #{t}
Module: Mathlib.Combinatorics.SimpleGraph.Acyclic

Goal state before:
pp: case neg
V : Type u_1
G : SimpleGraph V
h : G.IsTree
inst✝² : Fintype V
inst✝¹ : Nontrivial V
inst✝ : DecidableRel G.Adj
q : ¬2 ≤ G.minDegree
⊢ G.minDegree = 1

Tactic:
have := h.preconnected.minDegree_pos_of_nontrivial
    lia

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Coloring.Constructions

Goal state before:
pp: case mpr
α : Type u_1
G : SimpleGraph α
h : ∀ (u : α) (w : G.Walk u u), ¬Odd w.length
c : G.ConnectedComponent
⊢ c.toSimpleGraph.Colorable 2

Tactic:
obtain ⟨_, hv⟩ := c.nonempty_supp

Goal state after:
case mpr
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
⊢ c.toSimpleGraph.Colorable 2
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
hab : c.toSimpleGraph.Adj a b
⊢ (completeGraph (Fin 2)).Adj (Fin.ofNat 2 (Nonempty.some ⋯).length) (Fin.ofNat 2 (Nonempty.some ⋯).length)

Tactic:
he

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
he : Fin.ofNat 2 (Nonempty.some ⋯).length = Fin.ofNat 2 (Nonempty.some ⋯).length
⊢ False
Module: Mathlib.Combinatorics.SimpleGraph.Coloring.Constructions

Goal state before:
pp: 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
he : Fin.ofNat 2 (Nonempty.some ⋯).length = Fin.ofNat 2 (Nonempty.some ⋯).length
⊢ ↑(Fin.ofNat 2 (Nonempty.some ⋯).length) = ↑(Fin.ofNat 2 (Nonempty.some ⋯).length)

Tactic:
he

Goal state after:
Module: Mathlib.Combinatorics.SimpleGraph.Coloring.EdgeLabeling

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
K : Type u_3
C : G.EdgeLabeling K
k : K
x y : V
⊢ (fromEdgeSet {e | ∃ (h : e ∈ G.edgeSet), C ⟨e, h⟩ = k}).Adj x y ↔ ∃ (H : G.Adj x y), C ⟨s(x, y), H⟩ = k

Tactic:
simp only [mem_edgeSet, fromEdgeSet_adj, Set.mem_ofPred_eq, Ne.eq_def]

Goal state after:
V : Type u_1
G : SimpleGraph V
K : Type u_3
C : G.EdgeLabeling K
k : K
x y : V
⊢ (∃ (h : G.Adj x y), C ⟨s(x, y), ⋯⟩ = k) ∧ ¬x = y ↔ ∃ (H : G.Adj x y), C ⟨s(x, y), H⟩ = k
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Turan

Goal state before:
pp: V : Type u_1
inst✝² : Fintype V
G : SimpleGraph V
inst✝¹ : DecidableRel G.Adj
r : ℕ
h : G.IsTuranMaximal r
inst✝ : DecidableEq V
fp : Finpartition univ := h.finpartition
⊢ fp.IsEquipartition

Tactic:
by_contra hn

Goal state after:
V : Type u_1
inst✝² : Fintype V
G : SimpleGraph V
inst✝¹ : DecidableRel G.Adj
r : ℕ
h : G.IsTuranMaximal r
inst✝ : DecidableEq V
fp : Finpartition univ := h.finpartition
hn : ¬fp.IsEquipartition
⊢ False
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Turan

Goal state before:
pp: V : Type u_1
inst✝² : Fintype V
G : SimpleGraph V
inst✝¹ : DecidableRel G.Adj
r : ℕ
h : G.IsTuranMaximal r
inst✝ : DecidableEq V
fp : Finpartition univ := h.finpartition
large : Finset V
hl : large ∈ fp.parts
small : Finset V
hs : small ∈ fp.parts
ineq : #small + 1 < #large
w : V
hw : w ∈ large
v : V
hv : v ∈ small
cf : G.CliqueFree (r + 1)
large_eq : fp.part w = large
small_eq : fp.part v = small
⊢ G.Adj v w

Tactic:
by_contra hn

Goal state after:
V : Type u_1
inst✝² : Fintype V
G : SimpleGraph V
inst✝¹ : DecidableRel G.Adj
r : ℕ
h : G.IsTuranMaximal r
inst✝ : DecidableEq V
fp : Finpartition univ := h.finpartition
large : Finset V
hl : large ∈ fp.parts
small : Finset V
hs : small ∈ fp.parts
ineq : #small + 1 < #large
w : V
hw : w ∈ large
v : V
hv : v ∈ small
cf : G.CliqueFree (r + 1)
large_eq : fp.part w = large
small_eq : fp.part v = small
hn : ¬G.Adj v w
⊢ False
Module: Mathlib.Combinatorics.SimpleGraph.Extremal.Turan

Goal state before:
pp: case inr.a
n r c : ℕ
hr : r > 0
⊢ #({i ∈ range r | c ≡ n + i [MOD r]}) ≤ 1

Tactic:
card_le_one_iff

Goal state after:
case inr.a
n r c : ℕ
hr : r > 0
⊢ ∀ {a b : ℕ}, a ∈ {i ∈ range r | c ≡ n + i [MOD r]} → b ∈ {i ∈ range r | c ≡ n + i [MOD r]} → a = b
Module: Mathlib.Combinatorics.SimpleGraph.Diam

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
h : ¬G.Connected
u : α
⊢ G.eccent u = ⊤

Tactic:
rw [connected_iff_exists_forall_reachable] at h

Goal state after:
α : Type u_1
G : SimpleGraph α
h : ¬∃ v, ∀ (w : α), G.Reachable v w
u : α
⊢ G.eccent u = ⊤
Module: Mathlib.Combinatorics.SimpleGraph.Diam

Goal state before:
pp: α : Type u_1
G : SimpleGraph α
inst✝ : Nonempty α
h : ¬G.Connected
⊢ G.ediam = ⊤

Tactic:
rw [connected_iff_exists_forall_reachable] at h

Goal state after:
α : Type u_1
G : SimpleGraph α
inst✝ : Nonempty α
h : ¬∃ v, ∀ (w : α), G.Reachable v w
⊢ G.ediam = ⊤
Module: Mathlib.Combinatorics.SimpleGraph.Ends.Properties

Goal state before:
pp: case intro
V : Type
G : SimpleGraph V
inst✝ : Finite V
s : (j : (Finset V)ᵒᵖ) → G.componentComplFunctor.obj j
property✝ : s ∈ G.end
val✝ : Fintype V
v : V
h : v ∈ ↑(s (Opposite.op Finset.univ))
⊢ False

Tactic:
exact Set.disjoint_iff.mp (s _).disjoint_right
        ⟨by simp only [Finset.coe_univ, Set.mem_univ], h⟩

Goal state after:
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:
rw [← add_rotate, ← add_rotate _ _ (r.choose 2)]

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 =
    r.choose 2 * 2 * (n' / r) + n' % r * (n' / r + 1) * (r - 1) + r.choose 2
Module: Mathlib.Combinatorics.SimpleGraph.Ends.Defs

Goal state before:
pp: V : Type u
G : SimpleGraph V
s : (j : (Finset V)ᵒᵖ) → G.componentComplFunctor.obj j
sec : s ∈ G.end
K L : (Finset V)ᵒᵖ
h : L ⟶ K
v : V
vnL : v ∉ Opposite.unop L
hs : s L = G.componentComplMk vnL
⊢ s K = G.componentComplMk ⋯

Tactic:
rw [← sec h, hs]
  apply ComponentCompl.hom_mk _ (le_of_op_hom h)

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

Goal state before:
pp: case hn
W : Type u_1
H : SimpleGraph W
n : ℕ
hn : n ≥ 2
G : SimpleGraph (Fin (n + 1))
inst✝ : DecidableRel G.Adj
h : H.Free G
e : Sym2 (Fin (n + 1))
⊢ e ∈ G.edgeFinset → n - 1 ≤ ↑(#(bipartiteBelow (fun v e ↦ v ∉ e) univ e))

Tactic:
he

Goal state after:
case hn
W : Type u_1
H : SimpleGraph W
n : ℕ
hn : n ≥ 2
G : SimpleGraph (Fin (n + 1))
inst✝ : DecidableRel G.Adj
h : H.Free G
e : Sym2 (Fin (n + 1))
he : e ∈ G.edgeFinset
⊢ n - 1 ≤ ↑(#(bipartiteBelow (fun v e ↦ v ∉ e) univ e))
Module: Mathlib.Combinatorics.SimpleGraph.Girth

Goal state before:
pp: α : Type u_1
h : 3 ≤ ENat.card α
s : Finset α
hcard : 3 = s.card
x y z : α
hxy : x ≠ y
hxz : x ≠ z
hyz : y ≠ z
w : ⊤.Walk x x := Walk.cons hxy (Walk.cons hyz (Walk.cons ⋯ Walk.nil))
hw : w = Walk.cons hxy (Walk.cons hyz (Walk.cons ⋯ Walk.nil))
this : w.IsCycle
⊢ ↑w.length ≤ 3

Tactic:
simp [hw]

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

Goal state before:
pp: case refine_2
V : Type u_1
G : SimpleGraph V
M : G.Subgraph
u v : V
h : M.IsMatching
huv : M.Adj u v
w : ↑M.verts
hw : w ∈ {⟨u, ⋯⟩, ⟨v, ⋯⟩}
⊢ w ∈ h.toEdge ⁻¹' {⟨s(u, v), huv⟩}

Tactic:
rcases hw with rfl | rfl

Goal state after:
case refine_2.inl
V : Type u_1
G : SimpleGraph V
M : G.Subgraph
u v : V
h : M.IsMatching
huv : M.Adj u v
⊢ ⟨u, ⋯⟩ ∈ h.toEdge ⁻¹' {⟨s(u, v), huv⟩}

case refine_2.inr
V : Type u_1
G : SimpleGraph V
M : G.Subgraph
u v : V
h : M.IsMatching
huv : M.Adj u v
⊢ ⟨v, ⋯⟩ ∈ h.toEdge ⁻¹' {⟨s(u, v), huv⟩}
Module: Mathlib.Combinatorics.SimpleGraph.Hamiltonian

Goal state before:
pp: α : Type u_1
inst✝¹ : DecidableEq α
G : SimpleGraph α
a : α
p : G.Walk a a
inst✝ : Fintype α
⊢ p.IsHamiltonianCycle ↔ p.IsCycle ∧ p.length = Fintype.card α

Tactic:
refine ⟨fun h ↦ ⟨h.isCycle, h.length_eq⟩, fun ⟨h₁, h₂⟩ ↦ ⟨h₁, ?_⟩⟩
  refine isHamiltonian_iff_isPath_and_length_eq.mpr ⟨h₁.isPath_tail, ?_⟩
  grind [length_tail_add_one, IsCycle.not_nil]

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

Goal state before:
pp: case neg
V : Type u_1
G G' : SimpleGraph V
M : G.Subgraph
M' : G'.Subgraph
hM : M.IsPerfectMatching
hM' : M'.IsPerfectMatching
v w : V
hw : (fun w ↦ M.Adj v w) w ∧ ∀ (y : V), (fun w ↦ M.Adj v w) y → y = w
w' : V
hw' : (fun w ↦ M'.Adj v w) w' ∧ ∀ (y : V), (fun w ↦ M'.Adj v w) y → y = w'
hww' : ¬w = w'
⊢ ∃ x y, ¬x = y ∧ (M.spanningCoe \ M'.spanningCoe ⊔ M'.spanningCoe \ M.spanningCoe).neighborSet v = {x, y}

Tactic:
use w, w'

Goal state after:
case h
V : Type u_1
G G' : SimpleGraph V
M : G.Subgraph
M' : G'.Subgraph
hM : M.IsPerfectMatching
hM' : M'.IsPerfectMatching
v w : V
hw : (fun w ↦ M.Adj v w) w ∧ ∀ (y : V), (fun w ↦ M.Adj v w) y → y = w
w' : V
hw' : (fun w ↦ M'.Adj v w) w' ∧ ∀ (y : V), (fun w ↦ M'.Adj v w) y → y = w'
hww' : ¬w = w'
⊢ ¬w = w' ∧ (M.spanningCoe \ M'.spanningCoe ⊔ M'.spanningCoe \ M.spanningCoe).neighborSet v = {w, w'}
Module: Mathlib.Combinatorics.SimpleGraph.UniversalVerts

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
s : Set V
h : Disjoint G.universalVerts s
hc : s.ncard ≤ G.universalVerts.ncard
t : Set V
ht : t ⊆ G.universalVerts ∧ t.ncard = s.ncard
f : ↑s ≃ ↑t
hd : Disjoint t s := Set.disjoint_of_subset_left ht.left h
⊢ ∃ M, M.verts = s ∪ t ∧ M.IsMatching

Tactic:
have hadj (v : s) : G.Adj v (f v) := ht.1 (f v).2 (hd.ne_of_mem (f v).2 v.2) |>.symm

Goal state after:
V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
s : Set V
h : Disjoint G.universalVerts s
hc : s.ncard ≤ G.universalVerts.ncard
t : Set V
ht : t ⊆ G.universalVerts ∧ t.ncard = s.ncard
f : ↑s ≃ ↑t
hd : Disjoint t s := Set.disjoint_of_subset_left ht.left h
hadj : ∀ (v : ↑s), G.Adj ↑v ↑(f v)
⊢ ∃ M, M.verts = s ∪ t ∧ M.IsMatching
Module: Mathlib.Combinatorics.SimpleGraph.VertexCover

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
c : Set V
hi : G.IsIndepSet cᶜ
v w : V
hadj : G.Adj v w
hh : v ∉ c ∧ w ∉ c
⊢ False

Tactic:
exact hi hh.1 hh.2 (Adj.ne hadj) hadj

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

Goal state before:
pp: V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
h : ¬G.IsTutteViolator G.universalVerts
h' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp
hrep :
  ConnectedComponent.Represents (Quot.out '' G.deleteUniversalVerts.coe.oddComponents)
    G.deleteUniversalVerts.coe.oddComponents
t : Set V
ht : t ⊆ G.universalVerts
M1 : G.Subgraph
hM1 : M1.verts = Subtype.val '' Quot.out '' G.deleteUniversalVerts.coe.oddComponents ∪ t ∧ M1.IsMatching
complMatch : G.deleteUniversalVerts.coe.ConnectedComponent → G.Subgraph
hcomplMatch_compl :
  ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), (complMatch K).verts = Subtype.val '' K.supp \ M1.verts
hcomplMatch_match : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), (complMatch K).IsMatching
M2 : G.Subgraph := ⨆ K, complMatch K
i j : G.deleteUniversalVerts.coe.ConnectedComponent
hij : i ≠ j
⊢ Disjoint (Subtype.val '' i.supp \ M1.verts) (Subtype.val '' j.supp \ M1.verts)

Tactic:
exact Set.disjoint_of_subset Set.sdiff_subset Set.sdiff_subset <|
      Set.disjoint_image_of_injective Subtype.val_injective <|
        SimpleGraph.pairwise_disjoint_supp_connectedComponent _ hij

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

Goal state before:
pp: case refine_4
ε : ℝ
hε : 0 < ε
r t : ℕ
hr_pos : 0 < r
ht_pos : 0 < t
ε' : ℝ := 1 / (↑(r - 1) * ↑r) + ε
hε' : 0 < ε'
t' : ℕ := ⌊↑t / (↑r * ε)⌋₊ + 1
ht_lt_rt'ε : ↑t < ↑r * ↑t' * ε
ht'_pos : 0 < t'
N' : ℕ
N : ℕ := max (max 1 N') ⌈(↑(t'.choose t) ^ r * ↑t + ↑r * ↑t') * (↑t' - ↑t) / (↑r * ↑t' * ε - ↑t)⌉₊
n : ℕ
hn : N ≤ n
G : SimpleGraph (Fin n)
x✝ : DecidableRel G.Adj
hδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n
this : Nonempty (Fin n)
hrε_lt_1 : ↑r * ε < 1
ht_lt_t' : t < t'
ih : completeEquipartiteGraph r t' ⊑ G
K : G.CompleteEquipartiteSubgraph r t'
y : (a : Finset (Fin n)) → a ∈ K.parts → Finset (Fin n)
hy✝ : y ∈ K.parts.pi fun x ↦ powersetCard t x
hy : ∀ (a : Finset (Fin n)) (h : a ∈ K.parts), y a h ∈ powersetCard t a
ht_le_card_filter : t ≤ #{w | ErdosStone.filter.pi K w = ⟨y, hy✝⟩}
s : Finset ↥(ErdosStone.filter K t)
hs_subset : s ⊆ {w | ErdosStone.filter.pi K w = ⟨y, hy✝⟩}
hcards : #s = t
p₁✝ p₂✝ : Finset (Fin n)
v₁ v₂ : Fin n
p₁ : Finset (Fin n)
hp₁ : p₁ ∈ K.parts
hv₁ : v₁ ∈ ↑(y p₁ ⋯)
hyp₁ : y p₁ ⋯ = p₁✝
hy₁' : y p₁ hp₁ ⊆ p₁ ∧ #(y p₁ hp₁) = t
p₂ : Finset (Fin n)
hp₂ : p₂ ∈ K.parts
hne : y p₁ ⋯ ≠ y p₂ ⋯
hv₂ : v₂ ∈ ↑(y p₂ ⋯)
hyp₂ : y p₂ ⋯ = p₂✝
hy₂' : y p₂ hp₂ ⊆ p₂ ∧ #(y p₂ hp₂) = t
⊢ G.Adj v₁ v₂

Tactic:
refine K.isCompleteBetween hp₁ hp₂ ?_ (hy₁'.left hv₁) (hy₂'.left hv₂)

Goal state after:
case refine_4
ε : ℝ
hε : 0 < ε
r t : ℕ
hr_pos : 0 < r
ht_pos : 0 < t
ε' : ℝ := 1 / (↑(r - 1) * ↑r) + ε
hε' : 0 < ε'
t' : ℕ := ⌊↑t / (↑r * ε)⌋₊ + 1
ht_lt_rt'ε : ↑t < ↑r * ↑t' * ε
ht'_pos : 0 < t'
N' : ℕ
N : ℕ := max (max 1 N') ⌈(↑(t'.choose t) ^ r * ↑t + ↑r * ↑t') * (↑t' - ↑t) / (↑r * ↑t' * ε - ↑t)⌉₊
n : ℕ
hn : N ≤ n
G : SimpleGraph (Fin n)
x✝ : DecidableRel G.Adj
hδ : ↑G.minDegree ≥ (1 - 1 / ↑r + ε) * ↑n
this : Nonempty (Fin n)
hrε_lt_1 : ↑r * ε < 1
ht_lt_t' : t < t'
ih : completeEquipartiteGraph r t' ⊑ G
K : G.CompleteEquipartiteSubgraph r t'
y : (a : Finset (Fin n)) → a ∈ K.parts → Finset (Fin n)
hy✝ : y ∈ K.parts.pi fun x ↦ powersetCard t x
hy : ∀ (a : Finset (Fin n)) (h : a ∈ K.parts), y a h ∈ powersetCard t a
ht_le_card_filter : t ≤ #{w | ErdosStone.filter.pi K w = ⟨y, hy✝⟩}
s : Finset ↥(ErdosStone.filter K t)
hs_subset : s ⊆ {w | ErdosStone.filter.pi K w = ⟨y, hy✝⟩}
hcards : #s = t
p₁✝ p₂✝ : Finset (Fin n)
v₁ v₂ : Fin n
p₁ : Finset (Fin n)
hp₁ : p₁ ∈ K.parts
hv₁ : v₁ ∈ ↑(y p₁ ⋯)
hyp₁ : y p₁ ⋯ = p₁✝
hy₁' : y p₁ hp₁ ⊆ p₁ ∧ #(y p₁ hp₁) = t
p₂ : Finset (Fin n)
hp₂ : p₂ ∈ K.parts
hne : y p₁ ⋯ ≠ y p₂ ⋯
hv₂ : v₂ ∈ ↑(y p₂ ⋯)
hyp₂ : y p₂ ⋯ = p₂✝
hy₂' : y p₂ hp₂ ⊆ p₂ ∧ #(y p₂ hp₂) = t
⊢ p₁ ≠ p₂
Module: Mathlib.Combinatorics.SimpleGraph.Tutte

Goal state before:
pp: case pos
V : Type u_1
G : SimpleGraph V
inst✝ : Finite V
h✝ : ¬G.IsTutteViolator G.universalVerts
h' : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), G.deleteUniversalVerts.coe.IsClique K.supp
hrep :
  ConnectedComponent.Represents (Quot.out '' G.deleteUniversalVerts.coe.oddComponents)
    G.deleteUniversalVerts.coe.oddComponents
t : Set V
ht : t ⊆ G.universalVerts
M1 : G.Subgraph
hM1 : M1.verts = Subtype.val '' Quot.out '' G.deleteUniversalVerts.coe.oddComponents ∪ t ∧ M1.IsMatching
complMatch : G.deleteUniversalVerts.coe.ConnectedComponent → G.Subgraph
hcomplMatch_compl :
  ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), (complMatch K).verts = Subtype.val '' K.supp \ M1.verts
hcomplMatch_match : ∀ (K : G.deleteUniversalVerts.coe.ConnectedComponent), (complMatch K).IsMatching
M2 : G.Subgraph := ⨆ K, complMatch K
hM2 : M2.IsMatching
disjointM12 : Disjoint M1.support M2.support
v : V
hv : v ∈ Set.univ \ G.universalVerts
h : v ∈ M1.verts
⊢ v ∈ M1.verts ∪ M2.verts

Tactic:
exact M1.verts.mem_union_left _ h

Goal state after:
Module: Mathlib.Computability.Primrec.Basic

Goal state before:
pp: α : Type u_1
β : Type u_2
σ : Type u_3
inst✝² : Primcodable α
inst✝¹ : Primcodable β
inst✝ : Primcodable σ
f : α → β → σ
a : Option α
b : Option β
⊢ Option.map (fun p ↦ f p.1 p.2) (a.bind fun a ↦ Option.map (Prod.mk a) b) = a.bind fun a ↦ Option.map (f a) b

Tactic:
cases a <;> cases b <;> rfl

Goal state after:
Module: Mathlib.Computability.Partrec

Goal state before:
pp: case succ
p : ℕ →. Bool
H : ∃ n, true ∈ p n ∧ ∀ k < n, (p k).Dom
n : ℕ
pn : true ∈ p n ∧ ∀ k < n, (p k).Dom
m : ℕ
IH : ∀ (k : ℕ), n ≤ k + m → Acc (lbp p) k
k : ℕ
kn : n ≤ k + (m + 1)
a : ∀ k_1 ≤ k, false ∈ p k_1
⊢ Acc (lbp p) (k + 1)

Tactic:
exact IH _ (by rw [Nat.add_right_comm]; exact kn)

Goal state after:
Module: Mathlib.Computability.Partrec

Goal state before:
pp: case succ.refine_1
α : Type u_1
σ : Type u_2
inst✝¹ : Primcodable α
inst✝ : Primcodable σ
f : α → ℕ
g : α → σ
h : α → ℕ →. σ
hf : Computable f
hg : Computable g
hh : Partrec₂ h
a : α
n : ℕ
b : σ
H : b ∈ Nat.rec (Part.some (g a)) (fun y IH ↦ IH.bind fun i ↦ h a (n + 1 - 1)) (n + 1)
⊢ b ∈ Nat.rec (Part.some (g a)) (fun n n_ih ↦ h a n) (n + 1)

Tactic:
rcases mem_bind_iff.1 H with ⟨c, _, h₂⟩
        exact h₂

Goal state after:
Module: Mathlib.Computability.AkraBazzi.SumTransform

Goal state before:
pp: x : ℝ
hx : rexp 1 ≤ x
⊢ 1 < x

Tactic:
calc 1 = exp 0 := by simp
         _ < exp 1 := by simp
         _ ≤ x := hx

Goal state after: