Nuprl Lemma : member-graph-rcvs
∀S:Id List. ∀G:Graph(S). ∀a:Id ─→ Id ─→ Id. ∀b:Id. ∀j:{j:Id| (j ∈ S)} . ∀k:Knd.
((k ∈ graph-rcvs(S;G;a;b;j)) ⇐⇒ ∃i:Id. ((i ∈ S) ∧ (i─→j)∈G ∧ (k = rcv((link(a i j) from i to j),b) ∈ Knd)))
Proof
Definitions occuring in Statement :
graph-rcvs: graph-rcvs(S;G;a;b;j),
rcv: rcv(l,tg),
Knd: Knd,
mk_lnk: (link(n) from i to j),
id-graph-edge: (i─→j)∈G,
id-graph: Graph(S),
Id: Id,
l_member: (x ∈ l),
list: T List,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ─→ B[x],
equal: s = t ∈ T
Lemmas :
list-subtype,
l_member-settype,
l_member_wf,
subtype_rel_list,
Id_wf,
Knd_wf,
rcv_wf,
mk_lnk_wf,
exists_wf,
assert_wf,
deq-member_wf,
id-deq_wf,
subtype_rel-deq,
sq_stable__l_member,
decidable__equal_Id,
equal_wf,
set_wf,
assert-deq-member,
member_map_filter,
mapfilter_wf,
iff_wf,
id-graph_wf,
list_wf
\mforall{}S:Id List. \mforall{}G:Graph(S). \mforall{}a:Id {}\mrightarrow{} Id {}\mrightarrow{} Id. \mforall{}b:Id. \mforall{}j:\{j:Id| (j \mmember{} S)\} . \mforall{}k:Knd.
((k \mmember{} graph-rcvs(S;G;a;b;j))
\mLeftarrow{}{}\mRightarrow{} \mexists{}i:Id. ((i \mmember{} S) \mwedge{} (i{}\mrightarrow{}j)\mmember{}G \mwedge{} (k = rcv((link(a i j) from i to j),b))))
Date html generated:
2015_07_17-AM-09_13_36
Last ObjectModification:
2015_01_28-AM-07_58_21
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