- wo_subset
-
|- !P Q. P wo_subset Q = (!x. P x ==> Q x)
- wo_Union
-
|- !P. wo_Union P = (\x. ?p. P p /\ p x)
- wo_fl
-
|- !l x. wo_fl l x = (?y. l (x,y) \/ l (y,x))
- wo_poset
-
|- !l.
wo_poset l =
(!x. wo_fl l x ==> l (x,x)) /\
(!x y z. l (x,y) /\ l (y,z) ==> l (x,z)) /\
(!x y. l (x,y) /\ l (y,x) ==> (x = y))
- wo_chain
-
|- !l P. wo_chain l P = (!x y. P x /\ P y ==> l (x,y) \/ l (y,x))
- wo_woset
-
|- !l.
wo_woset l =
(!x. wo_fl l x ==> l (x,x)) /\
(!x y z. l (x,y) /\ l (y,z) ==> l (x,z)) /\
(!x y. l (x,y) /\ l (y,x) ==> (x = y)) /\
(!x y. wo_fl l x /\ wo_fl l y ==> l (x,y) \/ l (y,x)) /\
(!P.
(!x. P x ==> wo_fl l x) /\ (?x. P x) ==>
(?y. P y /\ (!z. P z ==> l (y,z))))
- wo_inseg
-
|- !l m.
wo_inseg (l,m) =
wo_woset l /\ wo_woset m /\ (!x y. l (x,y) = wo_fl l y /\ m (x,y))
- wo_less
-
|- !l x y. wo_less l (x,y) = l (x,y) /\ ~(x = y)
- SUBSET_REFL
-
|- !P. P wo_subset P
- SUBSET_ANTISYM
-
|- !P Q. P wo_subset Q /\ Q wo_subset P ==> (P = Q)
- SUBSET_TRANS
-
|- !P Q R. P wo_subset Q /\ Q wo_subset R ==> P wo_subset R
- POSET_FLEQ
-
|- !l. wo_poset l ==> (!x. wo_fl l x = l (x,x))
- CHAIN_SUBSET
-
|- !l P Q. wo_chain l P /\ Q wo_subset P ==> wo_chain l Q
- WOSET_POSET
-
|- !l. wo_woset l ==> wo_poset l
- WOSET_FLEQ
-
|- !l. wo_woset l ==> (!x. wo_fl l x = l (x,x))
- WOSET_TRANS_LESS
-
|- !l. wo_woset l ==> (!x y z. wo_less l (x,y) /\ l (y,z) ==> wo_less l (x,z))
- WOSET
-
|- !l.
wo_woset l =
(!x y. l (x,y) /\ l (y,x) ==> (x = y)) /\
(!P.
(!x. P x ==> wo_fl l x) /\ (?x. P x) ==>
(?y. P y /\ (!z. P z ==> l (y,z))))
- CPO_FIX
-
|- !l.
wo_poset l /\
(!A.
wo_chain l A ==>
(?m.
wo_fl l m /\
(!x. A x ==> l (x,m)) /\
(!m'. wo_fl l m' /\ (!x. A x ==> l (x,m')) ==> l (m,m')))) /\
(!x. wo_fl l x ==> l (x,f x)) ==>
(?y. wo_fl l y /\ (f y = y))
- POSET_ORD
-
|- wo_poset (\(U,V). P wo_subset U /\ U wo_subset V /\ wo_chain l V)
- KL
-
|- !l.
wo_poset l ==>
(!Q.
wo_chain l Q ==>
(?P.
(wo_chain l P /\ Q wo_subset P) /\
(!R. wo_chain l R /\ P wo_subset R ==> (R = P))))
- HP
-
|- !l.
wo_poset l ==>
(?P. wo_chain l P /\ (!Q. wo_chain l Q /\ P wo_subset Q ==> (Q = P)))
- ZL
-
|- !l.
wo_poset l /\
(!P. wo_chain l P ==> (?y. wo_fl l y /\ (!x. P x ==> l (x,y)))) ==>
(?y. wo_fl l y /\ (!x. l (y,x) ==> (y = x)))
- INSEG_WOSET
-
|- !l m. wo_inseg (l,m) ==> wo_woset l /\ wo_woset m
- INSEG_FL
-
|- !l. wo_fl wo_inseg l = wo_woset l
- INSEG_SUBSET
-
|- !l m. wo_inseg (l,m) ==> (!x y. l (x,y) ==> m (x,y))
- INSEG_POSET
-
|- wo_poset wo_inseg
- FL_UNION
-
|- !P x. wo_fl (wo_Union P) x = (?l. P l /\ wo_fl l x)
- INSEG_WOSET_UNION
-
|- !P. wo_chain wo_inseg P ==> wo_woset (wo_Union P)
- WOSET_MAXIMAL
-
|- ?y. wo_fl wo_inseg y /\ (!x. wo_inseg (y,x) ==> (y = x))
- WO_EXFL
-
|- !x.
wo_fl (\(x,y). l (x,y) \/ (wo_fl l x \/ (x = m)) /\ (y = m)) x =
wo_fl l x \/ (x = m)
- WO_TYPE
-
|- ?l. wo_woset l /\ (!x. wo_fl l x)
- WO_RESTFL
-
|- !l.
wo_woset l ==>
(!P. wo_fl (\(x,y). P x /\ P y /\ l (x,y)) x = P x /\ wo_fl l x)
- WO
-
|- !P. ?l. wo_woset l /\ (wo_fl l = P)
- WO_INDUCT
-
|- !P l.
wo_woset l /\
(!x. wo_fl l x /\ (!y. wo_less l (y,x) ==> P y) ==> P x) ==>
(!x. wo_fl l x ==> P x)
- AGREE_LEMMA
-
|- !l h ms m n f g z.
wo_woset l /\
(!x. wo_fl l (ms x)) /\
(!f f' x.
(!y. wo_less l (ms y,ms x) ==> (f y = f' y)) ==> (h f x = h f' x)) /\
(!x. l (ms x,m) ==> (f x = h f x)) /\
(!x. l (ms x,n) ==> (g x = h g x)) /\
l (ms z,m) /\
l (ms z,n) ==>
(f z = g z)
- WO_RECURSE_LOCAL
-
|- !l h ms.
wo_woset l /\
(!x. wo_fl l (ms x)) /\
(!f f' x.
(!y. wo_less l (ms y,ms x) ==> (f y = f' y)) ==> (h f x = h f' x)) ==>
(!n. ?f. !x'. l (ms x',n) ==> (f x' = h f x'))
- WO_RECURSE_EXISTS
-
|- !l h ms.
wo_woset l /\
(!x. wo_fl l (ms x)) /\
(!f f' x.
(!y. wo_less l (ms y,ms x) ==> (f y = f' y)) ==> (h f x = h f' x)) ==>
(?f. !x. f x = h f x)
- WO_RECURSE
-
|- !l h ms.
wo_woset l /\
(!x. wo_fl l (ms x)) /\
(!f g x.
(!y. wo_less l (ms y,ms x) ==> (f y = g y)) ==> (h f x = h g x)) ==>
(?!f. !x. f x = h f x)
- FL_NUM
-
|- !n. wo_fl (\(m,n). m <= n) n
- WOSET_NUM
-
|- wo_woset (\(m,n). m <= n)
- WO_RECURSE_NUM
-
|- !h ms.
(!f g x. (!y. ms y < ms x ==> (f y = g y)) ==> (h f x = h g x)) ==>
(?!f. !x. f x = h f x)