Def (l,m) == l0:(V+T)*, l1:(V+T List), l2,m1:(V+T)*. l = (l0 @ l1 @ l2) & (l1 m1) & m = (l0 @ m1 @ l2)

Thm* V,T:Type, G:Grammar(V;T). (V+T)*(V+T)*Prop

Def Grammar(V;T) == ((V+T List)(V+T)*)*V

Thm* V,T:Type{i}. Grammar(V;T) Type{i'}

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

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

Def (l1,l2) == i:||G.prod||. < l1,l2 > = G.prod[i] (V+T List)(V+T)*

Thm* V,T:Type, G:Grammar(V;T). (V+T List)(V+T)*Prop

Def T List == {l:(T*)| ||l|| > 0 }

Thm* T:Type. (T List) Type

Thm* V,T:Type, g:Grammar(V;T). g.prod ((V+T List)(V+T)*)*

Thm* A:Type, l:A*, n:. 0n n < ||l|| l[n] A

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

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

Thm* ||nil||

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

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

Thm* i,j:. i > j Prop

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

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 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 ij == j < i

Thm* i,j:. ij

Def AB == B < A

Thm* i,j:. ij Prop

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

Def A == A False

Thm* A:Prop. (A) Prop