Nuprl Lemma : es-interface-from-decidable
∀[Info:Type]. ∀[A:es:EO+(Info) ─→ e:E ─→ Type]. ∀[R:es:EO+(Info) ─→ e:E ─→ A[es;e] ─→ ℙ].
((∀es:EO+(Info). ∀e:E. Dec(∃a:A[es;e]. R[es;e;a]))
⇒ (∃X:EClass(A[es;e]). ∀es:EO+(Info). ∀e:E. ((↑e ∈b X ⇐⇒ ∃a:A[es;e]. R[es;e;a]) ∧ R[es;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],
so_apply: x[s1;s2],
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 :
all_wf,
event-ordering+_wf,
es-E_wf,
event-ordering+_subtype,
decidable_wf,
exists_wf,
single-bag_wf,
empty-bag_wf,
equal_wf,
bag_size_single_lemma,
bag_only_single_lemma,
true_wf,
bag_size_empty_lemma,
false_wf,
iff_wf,
assert_wf,
eq_int_wf,
bag-size_wf,
nat_wf,
bag-only_wf2,
assert_of_eq_int,
single-valued-bag-if-le1,
le_weakening,
decidable__lt,
le_antisymmetry_iff,
add_functionality_wrt_le,
add-zero,
le-add-cancel
Latex:
\mforall{}[Info:Type]. \mforall{}[A:es:EO+(Info) {}\mrightarrow{} e:E {}\mrightarrow{} Type]. \mforall{}[R:es:EO+(Info) {}\mrightarrow{} e:E {}\mrightarrow{} A[es;e] {}\mrightarrow{} \mBbbP{}].
((\mforall{}es:EO+(Info). \mforall{}e:E. Dec(\mexists{}a:A[es;e]. R[es;e;a]))
{}\mRightarrow{} (\mexists{}X:EClass(A[es;e])
\mforall{}es:EO+(Info). \mforall{}e:E.
((\muparrow{}e \mmember{}\msubb{} X \mLeftarrow{}{}\mRightarrow{} \mexists{}a:A[es;e]. R[es;e;a]) \mwedge{} R[es;e;X(e)] supposing \muparrow{}e \mmember{}\msubb{} X)))
Date html generated:
2015_07_20-PM-03_55_30
Last ObjectModification:
2015_01_27-PM-09_59_43
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