Nuprl Lemma : hdf-compose2-ap
∀[A,B,C:Type]. ∀[X:hdataflow(A;B ⟶ bag(C))]. ∀[Y:hdataflow(A;B)].
∀[a:A]. (X o Y(a) = <(fst(X(a))) o (fst(Y(a))), ⋃f∈snd(X(a)).⋃b∈snd(Y(a)).f b> ∈ (hdataflow(A;C) × bag(C)))
supposing valueall-type(C)
Proof
Definitions occuring in Statement :
hdf-compose2: X o Y,
hdf-ap: X(a),
hdataflow: hdataflow(A;B),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
apply: f a,
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: bag(T)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
hdf-ap: X(a),
hdf-compose2: X o Y,
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,
bor: p ∨bq,
ifthenelse: if b then t else f fi ,
top: Top,
subtype_rel: A ⊆r B,
ext-eq: A ≡ B,
assert: ↑b,
bfalse: ff,
false: False,
prop: ℙ,
hdf-halt: hdf-halt(),
pi1: fst(t),
pi2: snd(t),
so_lambda: λ2x.t[x],
so_apply: x[s],
callbyvalueall: callbyvalueall,
evalall: evalall(t),
bag-combine: ⋃x∈bs.f[x],
bag-union: bag-union(bbs),
concat: concat(ll),
reduce: reduce(f;k;as),
list_ind: list_ind,
bag-map: bag-map(f;bs),
map: map(f;as),
empty-bag: {},
nil: [],
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
exposed-bfalse: exposed-bfalse,
not: ¬A,
true: True,
has-value: (a)↓,
has-valueall: has-valueall(a)
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[X:hdataflow(A;B {}\mrightarrow{} bag(C))]. \mforall{}[Y:hdataflow(A;B)].
\mforall{}[a:A]. (X o Y(a) = <(fst(X(a))) o (fst(Y(a))), \mcup{}f\mmember{}snd(X(a)).\mcup{}b\mmember{}snd(Y(a)).f b>)
supposing valueall-type(C)
Date html generated:
2016_05_16-AM-10_39_31
Last ObjectModification:
2015_12_28-PM-07_45_48
Theory : halting!dataflow
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