Theorem: Turing.TM2to1.addBottom_map
File: Mathlib/Computability/TuringMachine.lean

Proof steps:
state_before: K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
L : ListBlank ((k : K) → Option (Γ k))
⊢ default.2 = default
state_after: no goals
tactic: rfl
state_before: K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
L : ListBlank ((k : K) → Option (Γ k))
⊢ ListBlank.map { f := Prod.snd, map_pt' := ⋯ } (addBottom L) = L
state_after: K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
L : ListBlank ((k : K) → Option (Γ k))
⊢ ListBlank.cons L.head
      (ListBlank.map { f := Prod.snd, map_pt' := ⋯ } (ListBlank.map { f := Prod.mk false, map_pt' := ⋯ } L.tail)) =
    L
tactic: simp only [addBottom, ListBlank.map_cons]
state_before: K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
L : ListBlank ((k : K) → Option (Γ k))
⊢ ListBlank.cons L.head
      (ListBlank.map { f := Prod.snd, map_pt' := ⋯ } (ListBlank.map { f := Prod.mk false, map_pt' := ⋯ } L.tail)) =
    L
state_after: case h.e'_2.h.e'_4
K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
L : ListBlank ((k : K) → Option (Γ k))
⊢ ListBlank.map { f := Prod.snd, map_pt' := ⋯ } (ListBlank.map { f := Prod.mk false, map_pt' := ⋯ } L.tail) = L.tail
tactic: convert ListBlank.cons_head_tail L
state_before: case h.e'_2.h.e'_4
K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
L : ListBlank ((k : K) → Option (Γ k))
⊢ ListBlank.map { f := Prod.snd, map_pt' := ⋯ } (ListBlank.map { f := Prod.mk false, map_pt' := ⋯ } L.tail) = L.tail
state_after: case h.e'_2.h.e'_4
K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
L L' : ListBlank ((k : K) → Option (Γ k))
⊢ ListBlank.map { f := Prod.snd, map_pt' := ⋯ } (ListBlank.map { f := Prod.mk false, map_pt' := ⋯ } L') = L'
tactic: generalize ListBlank.tail L = L'
state_before: case h.e'_2.h.e'_4
K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
L L' : ListBlank ((k : K) → Option (Γ k))
⊢ ListBlank.map { f := Prod.snd, map_pt' := ⋯ } (ListBlank.map { f := Prod.mk false, map_pt' := ⋯ } L') = L'
state_after: case h.e'_2.h.e'_4
K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
L L' : ListBlank ((k : K) → Option (Γ k))
l : List ((k : K) → Option (Γ k))
⊢ ListBlank.map { f := Prod.snd, map_pt' := ⋯ } (ListBlank.map { f := Prod.mk false, map_pt' := ⋯ } (ListBlank.mk l)) =
    ListBlank.mk l
tactic: refine L'.induction_on fun l ↦ ?_
state_before: case h.e'_2.h.e'_4
K : Type u_1
inst✝² : DecidableEq K
Γ : K → Type u_2
Λ : Type u_3
inst✝¹ : Inhabited Λ
σ : Type u_4
inst✝ : Inhabited σ
L L' : ListBlank ((k : K) → Option (Γ k))
l : List ((k : K) → Option (Γ k))
⊢ ListBlank.map { f := Prod.snd, map_pt' := ⋯ } (ListBlank.map { f := Prod.mk false, map_pt' := ⋯ } (ListBlank.mk l)) =
    ListBlank.mk l
state_after: no goals
tactic: simp