Nuprl Lemma : hdf-until-ap
∀[A,B,C:Type]. ∀[X:hdataflow(A;B)]. ∀[Y:hdataflow(A;C)]. ∀[a:A].
(hdf-until(X;Y)(a)
= <if bag-null(snd(Y(a))) then hdf-until(fst(X(a));fst(Y(a))) else hdf-halt() fi , snd(X(a))>
∈ (hdataflow(A;B) × bag(B)))
Proof
Definitions occuring in Statement :
hdf-until: hdf-until(X;Y),
hdf-halt: hdf-halt(),
hdf-ap: X(a),
hdataflow: hdataflow(A;B),
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
pi1: fst(t),
pi2: snd(t),
pair: <a, b>,
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T,
bag-null: bag-null(bs),
bag: bag(T)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
hdf-ap: X(a),
hdf-until: hdf-until(X;Y),
mk-hdf: mk-hdf(s,m.G[s; m];st.H[st];s0),
pi1: fst(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
top: Top,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
ext-eq: A ≡ B,
assert: ↑b,
bfalse: ff,
false: False,
prop: ℙ,
hdf-halt: hdf-halt(),
pi2: snd(t),
hdf-halted: hdf-halted(P),
isr: isr(x),
hdf-run: hdf-run(P),
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
not: ¬A,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[X:hdataflow(A;B)]. \mforall{}[Y:hdataflow(A;C)]. \mforall{}[a:A].
(hdf-until(X;Y)(a)
= <if bag-null(snd(Y(a))) then hdf-until(fst(X(a));fst(Y(a))) else hdf-halt() fi , snd(X(a))>)
Date html generated:
2016_05_16-AM-10_41_06
Last ObjectModification:
2016_01_17-AM-11_12_38
Theory : halting!dataflow
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