Theorem: List.perm_lookmap
File: Mathlib/Data/List/Perm.lean

tactic: induction' p with a l₁ l₂ p IH a b l l₁ l₂ l₃ p₁ _ IH₁ IH₂
state_before: α : Type u_1
β : Type u_2
l l₁✝ l₂✝ : List α
a : α
f : α → Option α
l₁ l₂ : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁
p : l₁ ~ l₂
⊢ lookmap f l₁ ~ lookmap f l₂
state_after: case nil
α : Type u_1
β : Type u_2
l l₁✝ l₂✝ : List α
a : α
f : α → Option α
l₁ l₂ : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) []
⊢ lookmap f [] ~ lookmap f []

case cons
α : Type u_1
β : Type u_2
l l₁✝¹ l₂✝¹ : List α
a✝ : α
f : α → Option α
l₁✝ l₂✝ : List α
a : α
l₁ l₂ : List α
p : l₁ ~ l₂
IH : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁ → lookmap f l₁ ~ lookmap f l₂
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (a :: l₁)
⊢ lookmap f (a :: l₁) ~ lookmap f (a :: l₂)

case swap
α : Type u_1
β : Type u_2
l✝ l₁✝ l₂✝ : List α
a✝ : α
f : α → Option α
l₁ l₂ : List α
a b : α
l : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (b :: a :: l)
⊢ lookmap f (b :: a :: l) ~ lookmap f (a :: b :: l)

case trans
α : Type u_1
β : Type u_2
l l₁✝¹ l₂✝¹ : List α
a : α
f : α → Option α
l₁✝ l₂✝ l₁ l₂ l₃ : List α
p₁ : l₁ ~ l₂
a✝ : l₂ ~ l₃
IH₁ : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁ → lookmap f l₁ ~ lookmap f l₂
IH₂ : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₂ → lookmap f l₂ ~ lookmap f l₃
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁
⊢ lookmap f l₁ ~ lookmap f l₃

tactic: simp
state_before: case nil
α : Type u_1
β : Type u_2
l l₁✝ l₂✝ : List α
a : α
f : α → Option α
l₁ l₂ : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) []
⊢ lookmap f [] ~ lookmap f []
state_after: no goals

tactic: cases h : f a
state_before: case cons
α : Type u_1
β : Type u_2
l l₁✝¹ l₂✝¹ : List α
a✝ : α
f : α → Option α
l₁✝ l₂✝ : List α
a : α
l₁ l₂ : List α
p : l₁ ~ l₂
IH : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁ → lookmap f l₁ ~ lookmap f l₂
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (a :: l₁)
⊢ lookmap f (a :: l₁) ~ lookmap f (a :: l₂)
state_after: case cons.none
α : Type u_1
β : Type u_2
l l₁✝¹ l₂✝¹ : List α
a✝ : α
f : α → Option α
l₁✝ l₂✝ : List α
a : α
l₁ l₂ : List α
p : l₁ ~ l₂
IH : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁ → lookmap f l₁ ~ lookmap f l₂
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (a :: l₁)
h : f a = none
⊢ lookmap f (a :: l₁) ~ lookmap f (a :: l₂)

case cons.some
α : Type u_1
β : Type u_2
l l₁✝¹ l₂✝¹ : List α
a✝ : α
f : α → Option α
l₁✝ l₂✝ : List α
a : α
l₁ l₂ : List α
p : l₁ ~ l₂
IH : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁ → lookmap f l₁ ~ lookmap f l₂
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (a :: l₁)
val✝ : α
h : f a = some val✝
⊢ lookmap f (a :: l₁) ~ lookmap f (a :: l₂)

tactic: simp [h]
state_before: case cons.none
α : Type u_1
β : Type u_2
l l₁✝¹ l₂✝¹ : List α
a✝ : α
f : α → Option α
l₁✝ l₂✝ : List α
a : α
l₁ l₂ : List α
p : l₁ ~ l₂
IH : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁ → lookmap f l₁ ~ lookmap f l₂
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (a :: l₁)
h : f a = none
⊢ lookmap f (a :: l₁) ~ lookmap f (a :: l₂)
state_after: case cons.none
α : Type u_1
β : Type u_2
l l₁✝¹ l₂✝¹ : List α
a✝ : α
f : α → Option α
l₁✝ l₂✝ : List α
a : α
l₁ l₂ : List α
p : l₁ ~ l₂
IH : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁ → lookmap f l₁ ~ lookmap f l₂
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (a :: l₁)
h : f a = none
⊢ lookmap f l₁ ~ lookmap f l₂

tactic: exact IH (pairwise_cons.1 H).2
state_before: case cons.none
α : Type u_1
β : Type u_2
l l₁✝¹ l₂✝¹ : List α
a✝ : α
f : α → Option α
l₁✝ l₂✝ : List α
a : α
l₁ l₂ : List α
p : l₁ ~ l₂
IH : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁ → lookmap f l₁ ~ lookmap f l₂
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (a :: l₁)
h : f a = none
⊢ lookmap f l₁ ~ lookmap f l₂
state_after: no goals

tactic: simp [lookmap_cons_some _ _ h, p]
state_before: case cons.some
α : Type u_1
β : Type u_2
l l₁✝¹ l₂✝¹ : List α
a✝ : α
f : α → Option α
l₁✝ l₂✝ : List α
a : α
l₁ l₂ : List α
p : l₁ ~ l₂
IH : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁ → lookmap f l₁ ~ lookmap f l₂
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (a :: l₁)
val✝ : α
h : f a = some val✝
⊢ lookmap f (a :: l₁) ~ lookmap f (a :: l₂)
state_after: no goals

tactic: cases' h₁ : f a with c <;> cases' h₂ : f b with d
state_before: case swap
α : Type u_1
β : Type u_2
l✝ l₁✝ l₂✝ : List α
a✝ : α
f : α → Option α
l₁ l₂ : List α
a b : α
l : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (b :: a :: l)
⊢ lookmap f (b :: a :: l) ~ lookmap f (a :: b :: l)
state_after: case swap.none.none
α : Type u_1
β : Type u_2
l✝ l₁✝ l₂✝ : List α
a✝ : α
f : α → Option α
l₁ l₂ : List α
a b : α
l : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (b :: a :: l)
h₁ : f a = none
h₂ : f b = none
⊢ lookmap f (b :: a :: l) ~ lookmap f (a :: b :: l)

case swap.none.some
α : Type u_1
β : Type u_2
l✝ l₁✝ l₂✝ : List α
a✝ : α
f : α → Option α
l₁ l₂ : List α
a b : α
l : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (b :: a :: l)
h₁ : f a = none
d : α
h₂ : f b = some d
⊢ lookmap f (b :: a :: l) ~ lookmap f (a :: b :: l)

case swap.some.none
α : Type u_1
β : Type u_2
l✝ l₁✝ l₂✝ : List α
a✝ : α
f : α → Option α
l₁ l₂ : List α
a b : α
l : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (b :: a :: l)
c : α
h₁ : f a = some c
h₂ : f b = none
⊢ lookmap f (b :: a :: l) ~ lookmap f (a :: b :: l)

case swap.some.some
α : Type u_1
β : Type u_2
l✝ l₁✝ l₂✝ : List α
a✝ : α
f : α → Option α
l₁ l₂ : List α
a b : α
l : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (b :: a :: l)
c : α
h₁ : f a = some c
d : α
h₂ : f b = some d
⊢ lookmap f (b :: a :: l) ~ lookmap f (a :: b :: l)

tactic: simp [h₁, h₂]
state_before: case swap.none.none
α : Type u_1
β : Type u_2
l✝ l₁✝ l₂✝ : List α
a✝ : α
f : α → Option α
l₁ l₂ : List α
a b : α
l : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (b :: a :: l)
h₁ : f a = none
h₂ : f b = none
⊢ lookmap f (b :: a :: l) ~ lookmap f (a :: b :: l)
state_after: case swap.none.none
α : Type u_1
β : Type u_2
l✝ l₁✝ l₂✝ : List α
a✝ : α
f : α → Option α
l₁ l₂ : List α
a b : α
l : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (b :: a :: l)
h₁ : f a = none
h₂ : f b = none
⊢ b :: a :: lookmap f l ~ a :: b :: lookmap f l

tactic: apply swap
state_before: case swap.none.none
α : Type u_1
β : Type u_2
l✝ l₁✝ l₂✝ : List α
a✝ : α
f : α → Option α
l₁ l₂ : List α
a b : α
l : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (b :: a :: l)
h₁ : f a = none
h₂ : f b = none
⊢ b :: a :: lookmap f l ~ a :: b :: lookmap f l
state_after: no goals

tactic: simp [h₁, lookmap_cons_some _ _ h₂]
state_before: case swap.none.some
α : Type u_1
β : Type u_2
l✝ l₁✝ l₂✝ : List α
a✝ : α
f : α → Option α
l₁ l₂ : List α
a b : α
l : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (b :: a :: l)
h₁ : f a = none
d : α
h₂ : f b = some d
⊢ lookmap f (b :: a :: l) ~ lookmap f (a :: b :: l)
state_after: case swap.none.some
α : Type u_1
β : Type u_2
l✝ l₁✝ l₂✝ : List α
a✝ : α
f : α → Option α
l₁ l₂ : List α
a b : α
l : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (b :: a :: l)
h₁ : f a = none
d : α
h₂ : f b = some d
⊢ d :: a :: l ~ a :: d :: l

tactic: apply swap
state_before: case swap.none.some
α : Type u_1
β : Type u_2
l✝ l₁✝ l₂✝ : List α
a✝ : α
f : α → Option α
l₁ l₂ : List α
a b : α
l : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (b :: a :: l)
h₁ : f a = none
d : α
h₂ : f b = some d
⊢ d :: a :: l ~ a :: d :: l
state_after: no goals

tactic: simp [lookmap_cons_some _ _ h₁, h₂]
state_before: case swap.some.none
α : Type u_1
β : Type u_2
l✝ l₁✝ l₂✝ : List α
a✝ : α
f : α → Option α
l₁ l₂ : List α
a b : α
l : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (b :: a :: l)
c : α
h₁ : f a = some c
h₂ : f b = none
⊢ lookmap f (b :: a :: l) ~ lookmap f (a :: b :: l)
state_after: case swap.some.none
α : Type u_1
β : Type u_2
l✝ l₁✝ l₂✝ : List α
a✝ : α
f : α → Option α
l₁ l₂ : List α
a b : α
l : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (b :: a :: l)
c : α
h₁ : f a = some c
h₂ : f b = none
⊢ b :: c :: l ~ c :: b :: l

tactic: apply swap
state_before: case swap.some.none
α : Type u_1
β : Type u_2
l✝ l₁✝ l₂✝ : List α
a✝ : α
f : α → Option α
l₁ l₂ : List α
a b : α
l : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (b :: a :: l)
c : α
h₁ : f a = some c
h₂ : f b = none
⊢ b :: c :: l ~ c :: b :: l
state_after: no goals

tactic: simp [lookmap_cons_some _ _ h₁, lookmap_cons_some _ _ h₂]
state_before: case swap.some.some
α : Type u_1
β : Type u_2
l✝ l₁✝ l₂✝ : List α
a✝ : α
f : α → Option α
l₁ l₂ : List α
a b : α
l : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (b :: a :: l)
c : α
h₁ : f a = some c
d : α
h₂ : f b = some d
⊢ lookmap f (b :: a :: l) ~ lookmap f (a :: b :: l)
state_after: case swap.some.some
α : Type u_1
β : Type u_2
l✝ l₁✝ l₂✝ : List α
a✝ : α
f : α → Option α
l₁ l₂ : List α
a b : α
l : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (b :: a :: l)
c : α
h₁ : f a = some c
d : α
h₂ : f b = some d
⊢ d :: a :: l ~ c :: b :: l

tactic: rcases (pairwise_cons.1 H).1 _ (mem_cons.2 (Or.inl rfl)) _ h₂ _ h₁ with ⟨rfl, rfl⟩
state_before: case swap.some.some
α : Type u_1
β : Type u_2
l✝ l₁✝ l₂✝ : List α
a✝ : α
f : α → Option α
l₁ l₂ : List α
a b : α
l : List α
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (b :: a :: l)
c : α
h₁ : f a = some c
d : α
h₂ : f b = some d
⊢ d :: a :: l ~ c :: b :: l
state_after: case swap.some.some.intro
α : Type u_1
β : Type u_2
l✝ l₁✝ l₂✝ : List α
a : α
f : α → Option α
l₁ l₂ : List α
b : α
l : List α
d : α
h₂ : f b = some d
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (b :: b :: l)
h₁ : f b = some d
⊢ d :: b :: l ~ d :: b :: l

tactic: exact Perm.refl _
state_before: case swap.some.some.intro
α : Type u_1
β : Type u_2
l✝ l₁✝ l₂✝ : List α
a : α
f : α → Option α
l₁ l₂ : List α
b : α
l : List α
d : α
h₂ : f b = some d
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) (b :: b :: l)
h₁ : f b = some d
⊢ d :: b :: l ~ d :: b :: l
state_after: no goals

tactic: refine (IH₁ H).trans (IH₂ ((p₁.pairwise_iff ?_).1 H))
state_before: case trans
α : Type u_1
β : Type u_2
l l₁✝¹ l₂✝¹ : List α
a : α
f : α → Option α
l₁✝ l₂✝ l₁ l₂ l₃ : List α
p₁ : l₁ ~ l₂
a✝ : l₂ ~ l₃
IH₁ : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁ → lookmap f l₁ ~ lookmap f l₂
IH₂ : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₂ → lookmap f l₂ ~ lookmap f l₃
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁
⊢ lookmap f l₁ ~ lookmap f l₃
state_after: case trans
α : Type u_1
β : Type u_2
l l₁✝¹ l₂✝¹ : List α
a : α
f : α → Option α
l₁✝ l₂✝ l₁ l₂ l₃ : List α
p₁ : l₁ ~ l₂
a✝ : l₂ ~ l₃
IH₁ : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁ → lookmap f l₁ ~ lookmap f l₂
IH₂ : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₂ → lookmap f l₂ ~ lookmap f l₃
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁
⊢ ∀ {x y : α}, (∀ c ∈ f x, ∀ d ∈ f y, x = y ∧ c = d) → ∀ c ∈ f y, ∀ d ∈ f x, y = x ∧ c = d

tactic: intro x y h c hc d hd
state_before: case trans
α : Type u_1
β : Type u_2
l l₁✝¹ l₂✝¹ : List α
a : α
f : α → Option α
l₁✝ l₂✝ l₁ l₂ l₃ : List α
p₁ : l₁ ~ l₂
a✝ : l₂ ~ l₃
IH₁ : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁ → lookmap f l₁ ~ lookmap f l₂
IH₂ : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₂ → lookmap f l₂ ~ lookmap f l₃
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁
⊢ ∀ {x y : α}, (∀ c ∈ f x, ∀ d ∈ f y, x = y ∧ c = d) → ∀ c ∈ f y, ∀ d ∈ f x, y = x ∧ c = d
state_after: case trans
α : Type u_1
β : Type u_2
l l₁✝¹ l₂✝¹ : List α
a : α
f : α → Option α
l₁✝ l₂✝ l₁ l₂ l₃ : List α
p₁ : l₁ ~ l₂
a✝ : l₂ ~ l₃
IH₁ : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁ → lookmap f l₁ ~ lookmap f l₂
IH₂ : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₂ → lookmap f l₂ ~ lookmap f l₃
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁
x y : α
h : ∀ c ∈ f x, ∀ d ∈ f y, x = y ∧ c = d
c : α
hc : c ∈ f y
d : α
hd : d ∈ f x
⊢ y = x ∧ c = d

tactic: rw [@eq_comm _ y, @eq_comm _ c]
state_before: case trans
α : Type u_1
β : Type u_2
l l₁✝¹ l₂✝¹ : List α
a : α
f : α → Option α
l₁✝ l₂✝ l₁ l₂ l₃ : List α
p₁ : l₁ ~ l₂
a✝ : l₂ ~ l₃
IH₁ : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁ → lookmap f l₁ ~ lookmap f l₂
IH₂ : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₂ → lookmap f l₂ ~ lookmap f l₃
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁
x y : α
h : ∀ c ∈ f x, ∀ d ∈ f y, x = y ∧ c = d
c : α
hc : c ∈ f y
d : α
hd : d ∈ f x
⊢ y = x ∧ c = d
state_after: case trans
α : Type u_1
β : Type u_2
l l₁✝¹ l₂✝¹ : List α
a : α
f : α → Option α
l₁✝ l₂✝ l₁ l₂ l₃ : List α
p₁ : l₁ ~ l₂
a✝ : l₂ ~ l₃
IH₁ : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁ → lookmap f l₁ ~ lookmap f l₂
IH₂ : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₂ → lookmap f l₂ ~ lookmap f l₃
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁
x y : α
h : ∀ c ∈ f x, ∀ d ∈ f y, x = y ∧ c = d
c : α
hc : c ∈ f y
d : α
hd : d ∈ f x
⊢ x = y ∧ d = c

tactic: apply h d hd c hc
state_before: case trans
α : Type u_1
β : Type u_2
l l₁✝¹ l₂✝¹ : List α
a : α
f : α → Option α
l₁✝ l₂✝ l₁ l₂ l₃ : List α
p₁ : l₁ ~ l₂
a✝ : l₂ ~ l₃
IH₁ : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁ → lookmap f l₁ ~ lookmap f l₂
IH₂ : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₂ → lookmap f l₂ ~ lookmap f l₃
H : Pairwise (fun a b => ∀ c ∈ f a, ∀ d ∈ f b, a = b ∧ c = d) l₁
x y : α
h : ∀ c ∈ f x, ∀ d ∈ f y, x = y ∧ c = d
c : α
hc : c ∈ f y
d : α
hd : d ∈ f x
⊢ x = y ∧ d = c
state_after: no goals