Def DA(l) == FinalState(DA)(Result(DA)l)

Thm* Alph,St:Type, A:Automata(Alph;St), l:Alph*. A(l)

Def Automata(Alph;States) == (StatesAlphStates)States(States)

Thm* Alph,States:Type{i}. Automata(Alph;States) Type{i'}

Def Result(DA)l == if null(l) InitialState(DA) else DA((Result(DA)tl(l)),hd(l)) fi (recursive)

Thm* Alph,St:Type, A:Automata(Alph;St), l:Alph*. (Result(A)l) St

Thm* T:Type, E:(TTProp). (EquivRel x,y:T. E(x,y)) Prop

Thm* Alph,States:Type, a:Automata(Alph;States). FinalState(a) States

Def hd(l) == Case of l; nil "?" ; h.t h

Thm* A:Type, l:A*. ||l||1 hd(l) A

Def tl(l) == Case of l; nil nil ; h.t t

Thm* A:Type, l:A*. tl(l) A*

Def a == 1of(a)

Thm* Alph,States:Type, a:Automata(Alph;States). a StatesAlphStates

Thm* Alph,States:Type, a:Automata(Alph;States). InitialState(a) States

Def null(as) == Case of as; nil true ; a.as' false

Thm* T:Type, as:T*. null(as)

Thm* null(nil)

Def Trans x,y:T. E(x;y) == a,b,c:T. E(a;b) E(b;c) E(a;c)

Thm* T:Type, E:(TTProp). Trans x,y:T. E(x,y) Prop

Def Sym x,y:T. E(x;y) == a,b:T. E(a;b) E(b;a)

Thm* T:Type, E:(TTProp). Sym x,y:T. E(x,y) Prop

Def Refl(T;x,y.E(x;y)) == a:T. E(a;a)

Thm* T:Type, E:(TTProp). Refl(T;x,y.E(x,y)) Prop

Thm* A:Type, B:(AType), p:a:AB(a). 2of(p) B(1of(p))

Thm* A:Type, B:(AType), p:a:AB(a). 1of(p) A