-- Mathlib/Computability/PostTuringMachine.lean
theorem Machine.map_step {S : Set Λ} (f₂₁ : Function.RightInverse f₁ f₂)
    (g₂₁ : ∀ q ∈ S, g₂ (g₁ q) = q) :
    ∀ c : Cfg Γ Λ,
      c.q ∈ S → (step M c).map (Cfg.map f₁ g₁) = step (M.map f₁ f₂ g₁ g₂) (Cfg.map f₁ g₁ c)
  | ⟨q, T⟩, h => by
    unfold step Machine.map Cfg.map
    simp only [Turing.Tape.map_fst, g₂₁ q h, f₂₁ _]
    rcases M q T.1 with (_ | ⟨q', d | a⟩); · rfl
    · simp only [step, Cfg.map, Option.map_some', Tape.map_move f₁]
      rfl
    · simp only [step, Cfg.map, Option.map_some', Tape.map_write]
      rfl

Proof state:
Γ : Type u_1
inst✝³ : Inhabited Γ
Γ' : Type u_2
inst✝² : Inhabited Γ'
Λ : Type u_3
inst✝¹ : Inhabited Λ
Λ' : Type u_4
inst✝ : Inhabited Λ'
M : Machine Γ Λ
f₁ : PointedMap Γ Γ'
f₂ : PointedMap Γ' Γ
g₁ : Λ → Λ'
g₂ : Λ' → Λ
S : Set Λ
f₂₁ : Function.RightInverse f₁.f f₂.f
g₂₁ : ∀ q ∈ S, g₂ (g₁ q) = q
q : Λ
T : Tape Γ
h : { q := q, Tape := T }.q ∈ S
⊢ Option.map
      (fun x =>
        match x with
        | { q := q, Tape := T } => { q := g₁ q, Tape := Tape.map f₁ T })
      (match { q := q, Tape := T } with
      | { q := q, Tape := T } =>
        Option.map
          (fun x =>
            match x with
            | (q', a) =>
              { q := q',
                Tape :=
                  match a with
                  | Stmt.move d => Tape.move d T
                  | Stmt.write a => Tape.write a T })
          (M q T.head)) =
    match
      match { q := q, Tape := T } with
      | { q := q, Tape := T } => { q := g₁ q, Tape := Tape.map f₁ T } with
    | { q := q, Tape := T } =>
      Option.map
        (fun x =>
          match x with
          | (q', a) =>
            { q := q',
              Tape :=
                match a with
                | Stmt.move d => Tape.move d T
                | Stmt.write a => Tape.write a T })
        (match q, T.head with
        | q, l => Option.map (Prod.map g₁ (Stmt.map f₁)) (M (g₂ q) (f₂.f l)))

Tactic:
simp only [Turing.Tape.map_fst, g₂₁ q h, f₂₁ _]

Result:
Γ : Type u_1
inst✝³ : Inhabited Γ
Γ' : Type u_2
inst✝² : Inhabited Γ'
Λ : Type u_3
inst✝¹ : Inhabited Λ
Λ' : Type u_4
inst✝ : Inhabited Λ'
M : Machine Γ Λ
f₁ : PointedMap Γ Γ'
f₂ : PointedMap Γ' Γ
g₁ : Λ → Λ'
g₂ : Λ' → Λ
S : Set Λ
f₂₁ : Function.RightInverse f₁.f f₂.f
g₂₁ : ∀ q ∈ S, g₂ (g₁ q) = q
q : Λ
T : Tape Γ
h : { q := q, Tape := T }.q ∈ S
⊢ Option.map (fun x => { q := g₁ x.q, Tape := Tape.map f₁ x.Tape })
      (Option.map
        (fun x =>
          { q := x.1,
            Tape :=
              match x.2 with
              | Stmt.move d => Tape.move d T
              | Stmt.write a => Tape.write a T })
        (M q T.head)) =
    Option.map
      (fun x =>
        { q := x.1,
          Tape :=
            match x.2 with
            | Stmt.move d => Tape.move d (Tape.map f₁ T)
            | Stmt.write a => Tape.write a (Tape.map f₁ T) })
      (Option.map (Prod.map g₁ (Stmt.map f₁)) (M q T.head))

Proof state:
Γ : Type u_1
inst✝³ : Inhabited Γ
Γ' : Type u_2
inst✝² : Inhabited Γ'
Λ : Type u_3
inst✝¹ : Inhabited Λ
Λ' : Type u_4
inst✝ : Inhabited Λ'
M : Machine Γ Λ
f₁ : PointedMap Γ Γ'
f₂ : PointedMap Γ' Γ
g₁ : Λ → Λ'
g₂ : Λ' → Λ
S : Set Λ
f₂₁ : Function.RightInverse f₁.f f₂.f
g₂₁ : ∀ q ∈ S, g₂ (g₁ q) = q
q : Λ
T : Tape Γ
h : { q := q, Tape := T }.q ∈ S
⊢ Option.map (fun x => { q := g₁ x.q, Tape := Tape.map f₁ x.Tape })
      (Option.map
        (fun x =>
          { q := x.1,
            Tape :=
              match x.2 with
              | Stmt.move d => Tape.move d T
              | Stmt.write a => Tape.write a T })
        (M q T.head)) =
    Option.map
      (fun x =>
        { q := x.1,
          Tape :=
            match x.2 with
            | Stmt.move d => Tape.move d (Tape.map f₁ T)
            | Stmt.write a => Tape.write a (Tape.map f₁ T) })
      (Option.map (Prod.map g₁ (Stmt.map f₁)) (M q T.head))

Tactic:
rcases M q T.1 with (_ | ⟨q', d | a⟩)

Result:
case none
Γ : Type u_1
inst✝³ : Inhabited Γ
Γ' : Type u_2
inst✝² : Inhabited Γ'
Λ : Type u_3
inst✝¹ : Inhabited Λ
Λ' : Type u_4
inst✝ : Inhabited Λ'
M : Machine Γ Λ
f₁ : PointedMap Γ Γ'
f₂ : PointedMap Γ' Γ
g₁ : Λ → Λ'
g₂ : Λ' → Λ
S : Set Λ
f₂₁ : Function.RightInverse f₁.f f₂.f
g₂₁ : ∀ q ∈ S, g₂ (g₁ q) = q
q : Λ
T : Tape Γ
h : { q := q, Tape := T }.q ∈ S
⊢ Option.map (fun x => { q := g₁ x.q, Tape := Tape.map f₁ x.Tape })
      (Option.map
        (fun x =>
          { q := x.1,
            Tape :=
              match x.2 with
              | Stmt.move d => Tape.move d T
              | Stmt.write a => Tape.write a T })
        none) =
    Option.map
      (fun x =>
        { q := x.1,
          Tape :=
            match x.2 with
            | Stmt.move d => Tape.move d (Tape.map f₁ T)
            | Stmt.write a => Tape.write a (Tape.map f₁ T) })
      (Option.map (Prod.map g₁ (Stmt.map f₁)) none)

case some.mk.move
Γ : Type u_1
inst✝³ : Inhabited Γ
Γ' : Type u_2
inst✝² : Inhabited Γ'
Λ : Type u_3
inst✝¹ : Inhabited Λ
Λ' : Type u_4
inst✝ : Inhabited Λ'
M : Machine Γ Λ
f₁ : PointedMap Γ Γ'
f₂ : PointedMap Γ' Γ
g₁ : Λ → Λ'
g₂ : Λ' → Λ
S : Set Λ
f₂₁ : Function.RightInverse f₁.f f₂.f
g₂₁ : ∀ q ∈ S, g₂ (g₁ q) = q
q : Λ
T : Tape Γ
h : { q := q, Tape := T }.q ∈ S
q' : Λ
d : Dir
⊢ Option.map (fun x => { q := g₁ x.q, Tape := Tape.map f₁ x.Tape })
      (Option.map
        (fun x =>
          { q := x.1,
            Tape :=
              match x.2 with
              | Stmt.move d => Tape.move d T
              | Stmt.write a => Tape.write a T })
        (some (q', Stmt.move d))) =
    Option.map
      (fun x =>
        { q := x.1,
          Tape :=
            match x.2 with
            | Stmt.move d => Tape.move d (Tape.map f₁ T)
            | Stmt.write a => Tape.write a (Tape.map f₁ T) })
      (Option.map (Prod.map g₁ (Stmt.map f₁)) (some (q', Stmt.move d)))

case some.mk.write
Γ : Type u_1
inst✝³ : Inhabited Γ
Γ' : Type u_2
inst✝² : Inhabited Γ'
Λ : Type u_3
inst✝¹ : Inhabited Λ
Λ' : Type u_4
inst✝ : Inhabited Λ'
M : Machine Γ Λ
f₁ : PointedMap Γ Γ'
f₂ : PointedMap Γ' Γ
g₁ : Λ → Λ'
g₂ : Λ' → Λ
S : Set Λ
f₂₁ : Function.RightInverse f₁.f f₂.f
g₂₁ : ∀ q ∈ S, g₂ (g₁ q) = q
q : Λ
T : Tape Γ
h : { q := q, Tape := T }.q ∈ S
q' : Λ
a : Γ
⊢ Option.map (fun x => { q := g₁ x.q, Tape := Tape.map f₁ x.Tape })
      (Option.map
        (fun x =>
          { q := x.1,
            Tape :=
              match x.2 with
              | Stmt.move d => Tape.move d T
              | Stmt.write a => Tape.write a T })
        (some (q', Stmt.write a))) =
    Option.map
      (fun x =>
        { q := x.1,
          Tape :=
            match x.2 with
            | Stmt.move d => Tape.move d (Tape.map f₁ T)
            | Stmt.write a => Tape.write a (Tape.map f₁ T) })
      (Option.map (Prod.map g₁ (Stmt.map f₁)) (some (q', Stmt.write a)))

Proof state:
case some.mk.write
Γ : Type u_1
inst✝³ : Inhabited Γ
Γ' : Type u_2
inst✝² : Inhabited Γ'
Λ : Type u_3
inst✝¹ : Inhabited Λ
Λ' : Type u_4
inst✝ : Inhabited Λ'
M : Machine Γ Λ
f₁ : PointedMap Γ Γ'
f₂ : PointedMap Γ' Γ
g₁ : Λ → Λ'
g₂ : Λ' → Λ
S : Set Λ
f₂₁ : Function.RightInverse f₁.f f₂.f
g₂₁ : ∀ q ∈ S, g₂ (g₁ q) = q
q : Λ
T : Tape Γ
h : { q := q, Tape := T }.q ∈ S
q' : Λ
a : Γ
⊢ some { q := g₁ q', Tape := Tape.write (f₁.f a) (Tape.map f₁ T) } =
    some
      { q := (Prod.map g₁ (Stmt.map f₁) (q', Stmt.write a)).1,
        Tape :=
          match (Prod.map g₁ (Stmt.map f₁) (q', Stmt.write a)).2 with
          | Stmt.move d => Tape.move d (Tape.map f₁ T)
          | Stmt.write a => Tape.write a (Tape.map f₁ T) }

Tactic:
rfl

Result:
no goals