Def ActionSet(T) == car:TypeTcarcar

Thm* T:Type{i}. ActionSet(T) Type{i'}

Def (Ln) == if n=0 nil else (Ln-1) @ L fi (recursive)

Thm* Alph:Type, L:Alph*, n:. (Ln) Alph*

Def as @ bs == Case of as; nil bs ; a.as' a.(as' @ bs) (recursive)

Thm* T:Type, as,bs:T*. (as @ bs) T*

Thm* T:Type, a:ActionSet(T). a.car Type

Def {i..j} == {k:| i k < j }

Thm* m,n:. {m..n} Type

Thm* A,B:Type, f:(AB), g:(BA). InvFuns(A; B; f; g) Prop

Def ||as|| == Case of as; nil 0 ; a.as' ||as'||+1 (recursive)

Thm* A:Type, l:A*. ||l||

Thm* ||nil||

Def (S:Ls) == if null(L) s else S.act(hd(L),(S:tl(L)s)) fi (recursive)

Thm* Alph:Type, S:ActionSet(Alph), L:Alph*, s:S.car. (S:Ls) S.car

Def == {i:| 0i }

Thm* Type

Def == {i:| 0 < i }

Thm* Type

Def as[m..n] == firstn(n-m;nth_tl(m;as))

Thm* T:Type, as:T*, m,n:. (as[m..n]) T*

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

Thm* A:Type. Id AA

Thm* A,B,C:Type, f:(BC), g:(AB). f o g AC

Def i=j == if i=j true ; false fi

Thm* i,j:. i=j

Def nth_tl(n;as) == if n0 as else nth_tl(n-1;tl(as)) fi (recursive)

Thm* A:Type, as:A*, i:. nth_tl(i;as) A*

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

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

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

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

Thm* T:Type, a:ActionSet(T). a.act Ta.cara.car

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

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

Thm* null(nil)

Def AB == B < A

Thm* i,j:. ij Prop

Def firstn(n;as) == Case of as; nil nil ; a.as' if 0 < n a.firstn(n-1;as') else nil fi (recursive)

Thm* A:Type, as:A*, n:. firstn(n;as) A*

Thm* A:Type. Id AA

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

Def A == A False

Thm* A:Prop. (A) Prop

Def ij == j < i

Thm* i,j:. ij

Def i < j == if i < j true ; false fi

Thm* i,j:. i < j

Def b == if b false else true fi

Thm* b:. b