Nuprl Lemma : es-first-at-since-iff
∀es:EO. ∀i:Id. ∀e:E.
∀[P,R:{e:E| loc(e) = i ∈ Id} ⟶ ℙ].
((∀e:E. Dec(R[e]) supposing loc(e) = i ∈ Id)
⇒ (es-first-at-since(es;i;e;e.R[e];e.P[e])
⇐⇒ (loc(e) = i ∈ Id)
c∧ ((P[e] ∧ (¬R[e]))
∧ ((∃e'':E. ((e'' <loc e) c∧ P[e'']))
⇒ (∃e':E
((e' <loc e)
c∧ (R[e'] ∧ (∀e'':E. ((e' <loc e'') ⇒ (e'' <loc e) ⇒ ((¬P[e'']) ∧ (¬R[e''])))))))))))
Proof
Definitions occuring in Statement :
es-first-at-since: es-first-at-since(es;i;e;e.R[e];e.P[e]),
es-locl: (e <loc 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],
cand: A c∧ B,
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
implies: P ⇒ Q,
and: 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,
iff: P ⇐⇒ Q,
and: P ∧ Q,
cand: A c∧ B,
es-first-at-since: es-first-at-since(es;i;e;e.R[e];e.P[e]),
not: ¬A,
false: False,
member: t ∈ T,
prop: ℙ,
so_apply: x[s],
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
es-locl: (e <loc e'),
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
alle-lt: ∀e<e'.P[e],
guard: {T},
es-le: e ≤loc e' ,
or: P ∨ Q,
squash: ↓T
Latex:
\mforall{}es:EO. \mforall{}i:Id. \mforall{}e:E.
\mforall{}[P,R:\{e:E| loc(e) = i\} {}\mrightarrow{} \mBbbP{}].
((\mforall{}e:E. Dec(R[e]) supposing loc(e) = i)
{}\mRightarrow{} (es-first-at-since(es;i;e;e.R[e];e.P[e])
\mLeftarrow{}{}\mRightarrow{} (loc(e) = i)
c\mwedge{} ((P[e] \mwedge{} (\mneg{}R[e]))
\mwedge{} ((\mexists{}e'':E. ((e'' <loc e) c\mwedge{} P[e'']))
{}\mRightarrow{} (\mexists{}e':E
((e' <loc e)
c\mwedge{} (R[e']
\mwedge{} (\mforall{}e'':E
((e' <loc e'') {}\mRightarrow{} (e'' <loc e) {}\mRightarrow{} ((\mneg{}P[e'']) \mwedge{} (\mneg{}R[e''])))))))))))
Date html generated:
2016_05_16-AM-10_25_02
Last ObjectModification:
2016_01_17-PM-01_27_28
Theory : new!event-ordering
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