Nuprl Lemma : hdf-bind-ap
∀[A,B,C:Type]. ∀[X:hdataflow(A;B)]. ∀[Y:B ─→ hdataflow(A;C)]. ∀[a:A].
X >>= Y(a)
= <fst(X(a)) ([y∈bag-map(λx.(fst(Y x(a)));snd(X(a)))|¬bhdf-halted(y)]) >>= Y, ∪p∈bag-map(λx.Y x(a);snd(X(a))).snd(p)>
∈ (hdataflow(A;C) × bag(C))
supposing valueall-type(C)
Proof
Definitions occuring in Statement :
hdf-bind-gen: X (hdfs) >>= Y,
hdf-bind: X >>= 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),
apply: f a,
lambda: λx.A[x],
function: x:A ─→ B[x],
pair: <a, b>,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T,
bag-combine: ∪x∈bs.f[x],
bag-filter: [x∈b|p[x]],
bag-map: bag-map(f;bs),
bag: bag(T)
Lemmas :
hdf-bind-as-gen,
hdf-bind-gen-ap,
empty-bag_wf,
hdataflow_wf,
empty_bag_append_lemma,
valueall-type_wf,
hdf-bind-gen_wf,
hdf-ap_wf,
bag_wf,
bag-filter_wf,
bnot_wf,
hdf-halted_wf,
bag-map_wf,
subtype_rel_bag,
assert_wf,
bag-combine_wf,
bag-map-map
\mforall{}[A,B,C:Type]. \mforall{}[X:hdataflow(A;B)]. \mforall{}[Y:B {}\mrightarrow{} hdataflow(A;C)]. \mforall{}[a:A].
X >>= Y(a)
= <fst(X(a)) ([y\mmember{}bag-map(\mlambda{}x.(fst(Y x(a)));snd(X(a)))|\mneg{}\msubb{}hdf-halted(y)]) >>= Y
, \mcup{}p\mmember{}bag-map(\mlambda{}x.Y x(a);snd(X(a))).snd(p)
>
supposing valueall-type(C)
Date html generated:
2015_07_17-AM-08_07_06
Last ObjectModification:
2015_01_27-PM-00_06_21
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