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
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
id-graph: Graph(S),
id-graph-edge: (i⟶j)∈G,
graph-rcvs: graph-rcvs(S;G;a;b;j),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
rev_implies: P ⇐ Q
Latex:
\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:
2016_05_16-AM-10_59_15
Last ObjectModification:
2015_12_29-AM-09_13_04
Theory : event-ordering
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