-- Mathlib/Data/QPF/Multivariate/Constructions/Fix.lean
theorem Fix.mk_dest (x : Fix F α) : Fix.mk (Fix.dest x) = x

Proof state:
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x : Fix F α
⊢ mk x.dest = x

Tactic:
change (Fix.mk ∘ Fix.dest) x = x

Result:
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x : Fix F α
⊢ (mk ∘ dest) x = x

Proof state:
case h
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x✝ : Fix F α
x : F (α ::: Fix F α)
⊢ (TypeVec.id ::: mk ∘ rec (MvFunctor.map (TypeVec.id ::: mk))) <$$> x = (TypeVec.id ::: fun x => x) <$$> x →
    mk ((TypeVec.id ::: mk ∘ rec (MvFunctor.map (TypeVec.id ::: mk))) <$$> x) = mk x

Tactic:
intro h

Result:
case h
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x✝ : Fix F α
x : F (α ::: Fix F α)
h : (TypeVec.id ::: mk ∘ rec (MvFunctor.map (TypeVec.id ::: mk))) <$$> x = (TypeVec.id ::: fun x => x) <$$> x
⊢ mk ((TypeVec.id ::: mk ∘ rec (MvFunctor.map (TypeVec.id ::: mk))) <$$> x) = mk x

Proof state:
case h
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x✝ : Fix F α
x : F (α ::: Fix F α)
h : (TypeVec.id ::: mk ∘ rec (MvFunctor.map (TypeVec.id ::: mk))) <$$> x = (TypeVec.id ::: fun x => x) <$$> x
⊢ mk ((TypeVec.id ::: mk ∘ rec (MvFunctor.map (TypeVec.id ::: mk))) <$$> x) = mk x

Tactic:
rw [h]

Result:
case h
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x✝ : Fix F α
x : F (α ::: Fix F α)
h : (TypeVec.id ::: mk ∘ rec (MvFunctor.map (TypeVec.id ::: mk))) <$$> x = (TypeVec.id ::: fun x => x) <$$> x
⊢ mk ((TypeVec.id ::: fun x => x) <$$> x) = mk x

Proof state:
case h
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x✝ : Fix F α
x : F (α ::: Fix F α)
⊢ (TypeVec.id ::: mk ∘ dest) <$$> x = (TypeVec.id ::: fun x => x) <$$> x → mk (mk x).dest = mk x

Tactic:
rw [Fix.dest, Fix.rec_eq, ← comp_map, ← appendFun_comp, id_comp]

Result:
case h
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x✝ : Fix F α
x : F (α ::: Fix F α)
⊢ (TypeVec.id ::: mk ∘ rec (MvFunctor.map (TypeVec.id ::: mk))) <$$> x = (TypeVec.id ::: fun x => x) <$$> x →
    mk ((TypeVec.id ::: mk ∘ rec (MvFunctor.map (TypeVec.id ::: mk))) <$$> x) = mk x

Proof state:
case h
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x✝ : Fix F α
x : F (α ::: Fix F α)
h : (TypeVec.id ::: mk ∘ rec (MvFunctor.map (TypeVec.id ::: mk))) <$$> x = (TypeVec.id ::: fun x => x) <$$> x
⊢ mk ((TypeVec.id ::: fun x => x) <$$> x) = mk x

Tactic:
show Fix.mk (appendFun id id <$$> x) = Fix.mk x

Result:
case h
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x✝ : Fix F α
x : F (α ::: Fix F α)
h : (TypeVec.id ::: mk ∘ rec (MvFunctor.map (TypeVec.id ::: mk))) <$$> x = (TypeVec.id ::: fun x => x) <$$> x
⊢ mk ((TypeVec.id ::: _root_.id) <$$> x) = mk x

Proof state:
case h
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x : Fix F α
⊢ ∀ (x : F (α ::: Fix F α)),
    (TypeVec.id ::: mk ∘ dest) <$$> x = (TypeVec.id ::: fun x => x) <$$> x → (mk ∘ dest) (mk x) = mk x

Tactic:
intro x

Result:
case h
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x✝ : Fix F α
x : F (α ::: Fix F α)
⊢ (TypeVec.id ::: mk ∘ dest) <$$> x = (TypeVec.id ::: fun x => x) <$$> x → (mk ∘ dest) (mk x) = mk x

Proof state:
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x : Fix F α
⊢ (mk ∘ dest) x = x

Tactic:
apply Fix.ind_rec

Result:
case h
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x : Fix F α
⊢ ∀ (x : F (α ::: Fix F α)),
    (TypeVec.id ::: mk ∘ dest) <$$> x = (TypeVec.id ::: fun x => x) <$$> x → (mk ∘ dest) (mk x) = mk x

Proof state:
case h
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x✝ : Fix F α
x : F (α ::: Fix F α)
⊢ (TypeVec.id ::: mk ∘ dest) <$$> x = (TypeVec.id ::: fun x => x) <$$> x → (mk ∘ dest) (mk x) = mk x

Tactic:
dsimp

Result:
case h
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x✝ : Fix F α
x : F (α ::: Fix F α)
⊢ (TypeVec.id ::: mk ∘ dest) <$$> x = (TypeVec.id ::: fun x => x) <$$> x → mk (mk x).dest = mk x

Proof state:
case h
n : ℕ
F : TypeVec.{u} (n + 1) → Type u
q : MvQPF F
α : TypeVec.{u} n
x✝ : Fix F α
x : F (α ::: Fix F α)
h : (TypeVec.id ::: mk ∘ rec (MvFunctor.map (TypeVec.id ::: mk))) <$$> x = (TypeVec.id ::: fun x => x) <$$> x
⊢ mk ((TypeVec.id ::: _root_.id) <$$> x) = mk x

Tactic:
rw [appendFun_id_id, MvFunctor.id_map]

Result:
no goals