Nuprl Lemma : es-first-before
∀es:EO. ∀i:Id. ∀e':E.
∀[P:{e:E| loc(e) = i ∈ Id} ⟶ ℙ]
(∀e@i.Dec(P[e]) ⇒ ∃e<e'.e is first@ i s.t. e.P[e] ⇐⇒ ∃e<e'.P[e] supposing loc(e') = i ∈ Id)
Proof
Definitions occuring in Statement :
es-first-at: e is first@ i s.t. e.P[e],
existse-before: ∃e<e'.P[e],
alle-at: ∀e@i.P[e],
es-loc: loc(e),
es-E: E,
event_ordering: EO,
Id: Id,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
uimplies: b supposing a,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
existse-before: ∃e<e'.P[e],
es-first-at: e is first@ i s.t. e.P[e],
exists: ∃x:A. B[x],
cand: A c∧ B,
es-locl: (e <loc e'),
subtype_rel: A ⊆r B,
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
guard: {T},
not: ¬A,
false: False,
alle-at: ∀e@i.P[e],
decidable: Dec(P),
or: P ∨ Q,
es-E: E,
es-base-E: es-base-E(es),
alle-lt: ∀e<e'.P[e]
Latex:
\mforall{}es:EO. \mforall{}i:Id. \mforall{}e':E.
\mforall{}[P:\{e:E| loc(e) = i\} {}\mrightarrow{} \mBbbP{}]
(\mforall{}e@i.Dec(P[e]) {}\mRightarrow{} \mexists{}e<e'.e is first@ i s.t. e.P[e] \mLeftarrow{}{}\mRightarrow{} \mexists{}e<e'.P[e] supposing loc(e') = i)
Date html generated:
2016_05_16-AM-10_24_38
Last ObjectModification:
2015_12_28-PM-09_20_59
Theory : new!event-ordering
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