- re_Union
-
|- !S'. re_Union S' = (\x. ?s. S' s /\ s x)
- re_union
-
|- !P Q. P re_union Q = (\x. P x \/ Q x)
- re_intersect
-
|- !P Q. P re_intersect Q = (\x. P x /\ Q x)
- re_null
-
|- re_null = (\x. F)
- re_universe
-
|- re_universe = (\x. T)
- re_subset
-
|- !P Q. P re_subset Q = (!x. P x ==> Q x)
- re_compl
-
|- !S'. re_compl S' = (\x. ~(S' x))
- istopology
-
|- !L.
istopology L =
L re_null /\
L re_universe /\
(!a b. L a /\ L b ==> L (a re_intersect b)) /\
(!P. P re_subset L ==> L (re_Union P))
- topology_TY_DEF
-
|- ?rep. TYPE_DEFINITION istopology rep
- topology_tybij
-
|- (!a. topology (open a) = a) /\ (!r. istopology r = open (topology r) = r)
- neigh
-
|- !top N x. neigh top (N,x) = (?P. open top P /\ P re_subset N /\ P x)
- closed
-
|- !L S'. closed L S' = open L (re_compl S')
- limpt
-
|- !top x S'.
limpt top x S' = (!N. neigh top (N,x) ==> (?y. ~(x = y) /\ S' y /\ N y))
- ismet
-
|- !m.
ismet m =
(!x y. (m (x,y) = & 0) = x = y) /\
(!x y z. m (y,z) |<=| m (x,y) |+| m (x,z))
- metric_TY_DEF
-
|- ?rep. TYPE_DEFINITION ismet rep
- metric_tybij
-
|- (!a. metric (dist a) = a) /\ (!r. ismet r = dist (metric r) = r)
- mtop
-
|- !m.
mtop m =
topology
(\S'.
!x. S' x ==> (?e. & 0 |<| e /\ (!y. dist m (x,y) |<| e ==> S' y)))
- ball
-
|- !m x e. B m (x,e) = (\y. dist m (x,y) |<| e)
- mr1
-
|- mr1 = metric (\(x,y). abs (y |-| x))
- SUBSET_REFL
-
|- !S'. S' re_subset S'
- COMPL_MEM
-
|- !S' x. S' x = ~(re_compl S' x)
- SUBSET_ANTISYM
-
|- !P Q. P re_subset Q /\ Q re_subset P = P = Q
- SUBSET_TRANS
-
|- !P Q R. P re_subset Q /\ Q re_subset R ==> P re_subset R
- TOPOLOGY
-
|- !L.
open L re_null /\
open L re_universe /\
(!x y. open L x /\ open L y ==> open L (x re_intersect y)) /\
(!P. P re_subset open L ==> open L (re_Union P))
- TOPOLOGY_UNION
-
|- !L P. P re_subset open L ==> open L (re_Union P)
- OPEN_OWN_NEIGH
-
|- !S' top x. open top S' /\ S' x ==> neigh top (S',x)
- OPEN_UNOPEN
-
|- !S' top. open top S' = re_Union (\P. open top P /\ P re_subset S') = S'
- OPEN_SUBOPEN
-
|- !S' top.
open top S' = (!x. S' x ==> (?P. P x /\ open top P /\ P re_subset S'))
- OPEN_NEIGH
-
|- !S' top.
open top S' = (!x. S' x ==> (?N. neigh top (N,x) /\ N re_subset S'))
- CLOSED_LIMPT
-
|- !top S'. closed top S' = (!x. limpt top x S' ==> S' x)
- METRIC_ISMET
-
|- !m. ismet (dist m)
- METRIC_ZERO
-
|- !m x y. (dist m (x,y) = & 0) = x = y
- METRIC_SAME
-
|- !m x. dist m (x,x) = & 0
- METRIC_POS
-
|- !m x y. & 0 |<=| dist m (x,y)
- METRIC_SYM
-
|- !m x y. dist m (x,y) = dist m (y,x)
- METRIC_TRIANGLE
-
|- !m x y z. dist m (x,z) |<=| dist m (x,y) |+| dist m (y,z)
- METRIC_NZ
-
|- !m x y. ~(x = y) ==> & 0 |<| dist m (x,y)
- mtop_istopology
-
|- !m.
istopology
(\S'.
!x. S' x ==> (?e. & 0 |<| e /\ (!y. dist m (x,y) |<| e ==> S' y)))
- MTOP_OPEN
-
|- !S' m.
open (mtop m) S' =
(!x. S' x ==> (?e. & 0 |<| e /\ (!y. dist m (x,y) |<| e ==> S' y)))
- BALL_OPEN
-
|- !m x e. & 0 |<| e ==> open (mtop m) (B m (x,e))
- BALL_NEIGH
-
|- !m x e. & 0 |<| e ==> neigh (mtop m) (B m (x,e),x)
- MTOP_LIMPT
-
|- !m x S'.
limpt (mtop m) x S' =
(!e. & 0 |<| e ==> (?y. ~(x = y) /\ S' y /\ dist m (x,y) |<| e))
- ISMET_R1
-
|- ismet (\(x,y). abs (y |-| x))
- MR1_DEF
-
|- !x y. dist mr1 (x,y) = abs (y |-| x)
- MR1_ADD
-
|- !x d. dist mr1 (x,x |+| d) = abs d
- MR1_SUB
-
|- !x d. dist mr1 (x,x |-| d) = abs d
- MR1_ADD_POS
-
|- !x d. & 0 |<=| d ==> (dist mr1 (x,x |+| d) = d)
- MR1_SUB_LE
-
|- !x d. & 0 |<=| d ==> (dist mr1 (x,x |-| d) = d)
- MR1_ADD_LT
-
|- !x d. & 0 |<| d ==> (dist mr1 (x,x |+| d) = d)
- MR1_SUB_LT
-
|- !x d. & 0 |<| d ==> (dist mr1 (x,x |-| d) = d)
- MR1_BETWEEN1
-
|- !x y z. x |<| z /\ dist mr1 (x,y) |<| z |-| x ==> y |<| z
- MR1_LIMPT
-
|- !x. limpt (mtop mr1) x re_universe