%% Results on (potentially) infinite lists
Kind nat type.
Type z nat.
Type s nat -> nat.
CoDefine colist : olist -> prop by
colist nil ;
colist (X :: L) := colist L.
CoDefine eqi : olist -> olist -> prop by
eqi nil nil ;
eqi (X :: L) (X :: R) := eqi L R.
Define tk : nat -> olist -> olist -> prop by
tk z L nil ;
tk N nil nil ;
tk (s N) (X :: L) (X :: R) := tk N L R.
Define eqf : olist -> olist -> prop by
eqf L R := (forall N S, tk N L S -> tk N R S) /\
(forall N S, tk N R S -> tk N L S).
%% First we show eqi implies eqf
Theorem eqi_take : forall L R S N,
eqi L R -> tk N L S -> tk N R S.
induction on 2. intros. case H2.
search.
case H1. search.
case H1. apply IH to H4 H3. search.
Theorem eqi_sym : forall L R,
eqi L R -> eqi R L.
coinduction. intros. case H1.
search.
apply CH to H2. search.
Theorem eqi_eqf : forall L R,
eqi L R -> eqf L R.
intros. unfold.
intros. apply eqi_take to H1 H2. search.
intros. apply eqi_sym to H1. apply eqi_take to H3 H2. search.
%% Second we show eqf implies eqi
Theorem eqf_nil : forall L,
eqf nil L -> L = nil.
intros. case H1.
apply H2 to _ with N = s z. case H4. search.
Theorem eqf_cons : forall X L R,
eqf (X :: L) R -> exists S, R = X :: S.
intros. case H1. apply H2 to _ with N = s z.
case H4. search.
Theorem eqf_sym : forall L R,
eqf L R -> eqf R L.
intros. case H1. search.
Theorem eqf_downward : forall X L R,
eqf (X :: L) (X :: R) -> eqf L R.
intros. case H1. unfold.
intros. assert tk (s N) (X :: L) (X :: S). apply H2 to H5.
case H6. search.
intros. assert tk (s N) (X :: R) (X :: S). apply H3 to H5.
case H6. search.
Theorem eqf_eqi : forall L R,
colist L -> eqf L R -> eqi L R.
coinduction. intros. case H1.
apply eqf_nil to H2. search.
apply eqf_cons to H2. apply eqf_downward to H2.
apply CH to H3 H4. search.
% We define co-inductive append and show it is associative
CoDefine coapp : olist -> olist -> olist -> prop by
coapp (X :: A) B (X :: C) := coapp A B C ;
coapp nil (X :: B) (X :: C) := coapp nil B C ;
coapp nil nil nil.
Theorem coapp_assoc : forall A B C AB ABC BC,
coapp A B AB -> coapp AB C ABC -> coapp B C BC -> coapp A BC ABC.
coinduction. intros. case H1.
case H2. apply CH to H4 H5 H3. search.
case H2. case H3. apply CH to H4 H5 H6. search.
case H2.
case H3. apply CH to _ H4 H5. search.
case H3. search.
