Nuprl Lemma : es-pplus_functionality_wrt_iff
∀es:EO. ∀e1:E. ∀e2:{e:E| loc(e) = loc(e1) ∈ Id} .
∀[p,q:{e:E| loc(e) = loc(e1) ∈ Id} ⟶ {e:E| loc(e) = loc(e1) ∈ Id} ⟶ ℙ].
((∀a,b:{e:E| loc(e) = loc(e1) ∈ Id} . ((a ∈ [e1, e2]) ⇒ (b ∈ [e1, e2]) ⇒ (p[a;b] ⇐⇒ q[a;b])))
⇒ ([e1,e2]~([a,b].p[a;b])+ ⇐⇒ [e1,e2]~([a,b].q[a;b])+))
Proof
Definitions occuring in Statement :
es-pplus: [e1,e2]~([a,b].p[a; b])+,
es-interval: [e, e'],
es-loc: loc(e),
es-E: E,
event_ordering: EO,
Id: Id,
l_member: (x ∈ l),
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
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],
member: t ∈ T,
uall: ∀[x:A]. B[x],
es-pplus: [e1,e2]~([a,b].p[a; b])+,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
so_apply: x[s]
Latex:
\mforall{}es:EO. \mforall{}e1:E. \mforall{}e2:\{e:E| loc(e) = loc(e1)\} .
\mforall{}[p,q:\{e:E| loc(e) = loc(e1)\} {}\mrightarrow{} \{e:E| loc(e) = loc(e1)\} {}\mrightarrow{} \mBbbP{}].
((\mforall{}a,b:\{e:E| loc(e) = loc(e1)\} . ((a \mmember{} [e1, e2]) {}\mRightarrow{} (b \mmember{} [e1, e2]) {}\mRightarrow{} (p[a;b] \mLeftarrow{}{}\mRightarrow{} q[a;b])))
{}\mRightarrow{} ([e1,e2]\msim{}([a,b].p[a;b])+ \mLeftarrow{}{}\mRightarrow{} [e1,e2]\msim{}([a,b].q[a;b])+))
Date html generated:
2016_05_16-AM-09_57_23
Last ObjectModification:
2015_12_28-PM-09_30_58
Theory : new!event-ordering
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