Nuprl Lemma : es-interface-le-pred-bool
∀[Info:Type]
∀P:es:EO+(Info) ⟶ E ⟶ 𝔹
∃X:EClass({e:E| ↑(P es e)} )
∀es:EO+(Info). ∀e:E.
((↑e ∈b X ⇐⇒ ∃a:E. (es-p-le-pred(es;λe.(↑(P es e))) e a))
∧ es-p-le-pred(es;λe.(↑(P es e))) e X(e) supposing ↑e ∈b X)
Proof
Definitions occuring in Statement :
eclass-val: X(e),
in-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
event-ordering+: EO+(Info),
es-p-le-pred: es-p-le-pred(es;P),
es-E: E,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
prop: ℙ,
and: P ∧ Q,
es-p-le-pred: es-p-le-pred(es;P),
iff: P ⇐⇒ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
top: Top,
eclass: EClass(A[eo; e]),
eclass-val: X(e),
in-eclass: e ∈b X
Latex:
\mforall{}[Info:Type]
\mforall{}P:es:EO+(Info) {}\mrightarrow{} E {}\mrightarrow{} \mBbbB{}
\mexists{}X:EClass(\{e:E| \muparrow{}(P es e)\} )
\mforall{}es:EO+(Info). \mforall{}e:E.
((\muparrow{}e \mmember{}\msubb{} X \mLeftarrow{}{}\mRightarrow{} \mexists{}a:E. (es-p-le-pred(es;\mlambda{}e.(\muparrow{}(P es e))) e a))
\mwedge{} es-p-le-pred(es;\mlambda{}e.(\muparrow{}(P es e))) e X(e) supposing \muparrow{}e \mmember{}\msubb{} X)
Date html generated:
2016_05_16-PM-11_17_30
Last ObjectModification:
2015_12_29-AM-10_30_42
Theory : event-ordering
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