Nuprl Lemma : hdf-bind-gen-ap
∀[A,B,C:Type]. ∀[X:hdataflow(A;B)]. ∀[Y:B ─→ hdataflow(A;C)]. ∀[hdfs:bag(hdataflow(A;C))]. ∀[a:A].
X (hdfs) >>= Y(a) ~ <fst(X(a)) ([y∈bag-map(λP.(fst(P(a)));hdfs + bag-map(Y;snd(X(a))))|¬bhdf-halted(y)]) >>= Y
, ∪p∈bag-map(λP.P(a);hdfs + bag-map(Y;snd(X(a)))).snd(p)
>
supposing valueall-type(C)
Proof
Definitions occuring in Statement :
hdf-bind-gen: X (hdfs) >>= Y,
hdf-halted: hdf-halted(P),
hdf-ap: X(a),
hdataflow: hdataflow(A;B),
valueall-type: valueall-type(T),
bnot: ¬bb,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
lambda: λx.A[x],
function: x:A ─→ B[x],
pair: <a, b>,
universe: Type,
sqequal: s ~ t,
bag-combine: ∪x∈bs.f[x],
bag-filter: [x∈b|p[x]],
bag-append: as + bs,
bag-map: bag-map(f;bs),
bag: bag(T)
Lemmas :
hdf-halted_wf,
bool_wf,
eqtt_to_assert,
assert-bag-null,
hdf_ap_halt_lemma,
hdf-halted-is-inr,
bag_map_empty_lemma,
equal-empty-bag,
empty_bag_append_lemma,
bag_filter_empty_lemma,
bag_combine_empty_lemma,
hdf-bind-gen-halt-left,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
hdf-ap-run,
hdataflow-ext,
bag_wf,
unit_wf2,
hdf_halted_inl_red_lemma,
false_wf,
hdf_halted_halt_red_lemma,
bag-map_wf,
hdataflow_wf,
hdf-ap_wf,
bag-append_wf,
empty-bag_wf,
valueall-type-has-valueall,
bag-valueall-type,
product-valueall-type,
hdataflow-valueall-type,
evalall-reduce,
assert_wf,
bnot_wf,
set-valueall-type,
bag-filter_wf,
bag-combine_wf,
bag-map-map,
true_wf,
hdf-ap-inl,
not_wf,
valueall-type_wf
\mforall{}[A,B,C:Type]. \mforall{}[X:hdataflow(A;B)]. \mforall{}[Y:B {}\mrightarrow{} hdataflow(A;C)]. \mforall{}[hdfs:bag(hdataflow(A;C))]. \mforall{}[a:A].
X (hdfs) >>= Y(a) \msim{} <fst(X(a)) ([y\mmember{}bag-map(\mlambda{}P.(fst(P(a)));hdfs + bag-map(Y;snd(X(a))))|
\mneg{}\msubb{}hdf-halted(y)]) >>= Y
, \mcup{}p\mmember{}bag-map(\mlambda{}P.P(a);hdfs + bag-map(Y;snd(X(a)))).snd(p)
>
supposing valueall-type(C)
Date html generated:
2015_07_17-AM-08_07_02
Last ObjectModification:
2015_01_27-PM-00_06_54
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