induction' w with x xs ih
α qs : Type
M : DFA α qs
w : Word { x // x ∈ (DFA.toTotalDFA M).Z }
⊢ ↑(TotalDFA.del_star' (DFA.toTotalDFA M) ({ val := none, property := ⋯ }, w)) = none
case nil
α qs : Type
M : DFA α qs
⊢ ↑(TotalDFA.del_star' (DFA.toTotalDFA M) ({ val := none, property := ⋯ }, [])) = none

case cons
α qs : Type
M : DFA α qs
x : { x // x ∈ (DFA.toTotalDFA M).Z }
xs : List { x // x ∈ (DFA.toTotalDFA M).Z }
ih : ↑(TotalDFA.del_star' (DFA.toTotalDFA M) ({ val := none, property := ⋯ }, xs)) = none
⊢ ↑(TotalDFA.del_star' (DFA.toTotalDFA M) ({ val := none, property := ⋯ }, x :: xs)) = none