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)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
hdf-bind-gen: X (hdfs) >>= Y,
hdf-ap: X(a),
bind-nxt: bind-nxt(Y;p;a),
mk-hdf: mk-hdf(s,m.G[s; m];st.H[st];s0),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
top: Top,
hdf-halt: hdf-halt(),
pi1: fst(t),
pi2: snd(t),
so_lambda: λ2x.t[x],
so_apply: x[s],
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
subtype_rel: A ⊆r B,
ext-eq: A ≡ B,
squash: ↓T,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
compose: f o g,
true: True
Latex:
\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:
2016_05_16-AM-10_43_07
Last ObjectModification:
2016_01_17-AM-11_12_11
Theory : halting!dataflow
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