Nuprl Lemma : decidable-implies-es-interface
∀[Info,A:Type]. ∀[P:eo:EO+(Info) ─→ E ─→ A ─→ ℙ].
((∀eo:EO+(Info). ∀e:E. Dec(∃a:A. P[eo;e;a]))
⇒ (∃X:EClass(A). ∀eo:EO+(Info). ∀e:E. ((↑e ∈b X ⇐⇒ ∃a:A. P[eo;e;a]) ∧ P[eo;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-E: E,
assert: ↑b,
decidable: Dec(P),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2;s3],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
and: P ∧ Q,
function: x:A ─→ B[x],
universe: Type
Lemmas :
not_wf,
single-bag_wf,
empty-bag_wf,
equal_wf,
es-E_wf,
event-ordering+_subtype,
all_wf,
iff_wf,
assert_wf,
in-eclass_wf,
es-interface-subtype_rel2,
top_wf,
exists_wf,
eclass-val_wf,
event-ordering+_wf,
decidable_wf,
bag_size_single_lemma,
bag_only_single_lemma,
true_wf,
bag_size_empty_lemma,
false_wf
\mforall{}[Info,A:Type]. \mforall{}[P:eo:EO+(Info) {}\mrightarrow{} E {}\mrightarrow{} A {}\mrightarrow{} \mBbbP{}].
((\mforall{}eo:EO+(Info). \mforall{}e:E. Dec(\mexists{}a:A. P[eo;e;a]))
{}\mRightarrow{} (\mexists{}X:EClass(A)
\mforall{}eo:EO+(Info). \mforall{}e:E. ((\muparrow{}e \mmember{}\msubb{} X \mLeftarrow{}{}\mRightarrow{} \mexists{}a:A. P[eo;e;a]) \mwedge{} P[eo;e;X(e)] supposing \muparrow{}e \mmember{}\msubb{} X)))
Date html generated:
2015_07_17-PM-01_03_10
Last ObjectModification:
2015_01_27-PM-10_39_37
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