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
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
so_apply: x[s1;s2;s3],
so_apply: x[s],
eclass-val: X(e),
in-eclass: e ∈b X,
eclass: EClass(A[eo; e]),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
pi1: fst(t),
top: Top,
eq_int: (i =z j),
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
and: P ∧ Q,
cand: A c∧ B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
true: True,
uimplies: b supposing a,
bfalse: ff,
false: False,
not: ¬A,
nat: ℕ,
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla)
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:
2016_05_16-PM-11_16_33
Last ObjectModification:
2016_01_17-PM-07_12_59
Theory : event-ordering
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