Nuprl Lemma : es-first-at-implies
∀es:EO. ∀i:Id.
∀[P,Q:{e:E| loc(e) = i ∈ Id} ⟶ ℙ].
((∀e:{e:E| loc(e) = i ∈ Id} . Dec(Q[e]))
⇒ (∀e:{e:E| loc(e) = i ∈ Id} . (P[e] ⇒ ∃e':E. ((e' <loc e) ∧ P[e']) supposing ¬Q[e]))
⇒ (∀e:E. {e is first@ i s.t. e.P[e] ⇒ Q[e]}))
Proof
Definitions occuring in Statement :
es-first-at: e is first@ i s.t. 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],
prop: ℙ,
guard: {T},
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 :
guard: {T},
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
es-first-at: e is first@ i s.t. e.P[e],
and: P ∧ Q,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
exists: ∃x:A. B[x],
alle-lt: ∀e<e'.P[e],
not: ¬A,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
es-locl: (e <loc e')
Latex:
\mforall{}es:EO. \mforall{}i:Id.
\mforall{}[P,Q:\{e:E| loc(e) = i\} {}\mrightarrow{} \mBbbP{}].
((\mforall{}e:\{e:E| loc(e) = i\} . Dec(Q[e]))
{}\mRightarrow{} (\mforall{}e:\{e:E| loc(e) = i\} . (P[e] {}\mRightarrow{} \mexists{}e':E. ((e' <loc e) \mwedge{} P[e']) supposing \mneg{}Q[e]))
{}\mRightarrow{} (\mforall{}e:E. \{e is first@ i s.t. e.P[e] {}\mRightarrow{} Q[e]\}))
Date html generated:
2016_05_16-AM-09_49_26
Last ObjectModification:
2015_12_28-PM-09_35_32
Theory : new!event-ordering
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