Nuprl Lemma : previous-event-exists
∀es:EO. ∀i:Id.
∀[R:{e:E| loc(e) = i ∈ Id} ⟶ ℙ]
∀e:E
((∀e:E. Dec(R[e]) supposing loc(e) = i ∈ Id)
⇒ (∃e':E. (e' ≤loc e c∧ R[e']))
⇒ (∃e':E. (e' ≤loc e c∧ (R[e'] ∧ (∀e'':E. ((e' <loc e'') ⇒ e'' ≤loc e ⇒ (¬R[e'']))))))
supposing loc(e) = i ∈ Id)
Proof
Definitions occuring in Statement :
es-le: e ≤loc 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],
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],
member: t ∈ T,
prop: ℙ,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
uimplies: b supposing a,
so_apply: x[s],
cand: A c∧ B,
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
and: P ∧ Q,
es-locl: (e <loc e'),
wellfounded: WellFnd{i}(A;x,y.R[x; y]),
guard: {T},
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
iff: P ⇐⇒ Q,
squash: ↓T,
es-E: E,
es-base-E: es-base-E(es),
rev_implies: P ⇐ Q,
es-le: e ≤loc e'
Latex:
\mforall{}es:EO. \mforall{}i:Id.
\mforall{}[R:\{e:E| loc(e) = i\} {}\mrightarrow{} \mBbbP{}]
\mforall{}e:E
((\mforall{}e:E. Dec(R[e]) supposing loc(e) = i)
{}\mRightarrow{} (\mexists{}e':E. (e' \mleq{}loc e c\mwedge{} R[e']))
{}\mRightarrow{} (\mexists{}e':E
(e' \mleq{}loc e c\mwedge{} (R[e'] \mwedge{} (\mforall{}e'':E. ((e' <loc e'') {}\mRightarrow{} e'' \mleq{}loc e {}\mRightarrow{} (\mneg{}R[e'']))))))
supposing loc(e) = i)
Date html generated:
2016_05_16-AM-09_50_09
Last ObjectModification:
2016_01_17-PM-01_27_03
Theory : new!event-ordering
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