Nuprl Lemma : hdf-bind-gen-compose1-left
∀[A,B,C,U:Type]. ∀[f:B ⟶ C]. ∀[X:hdataflow(A;B)]. ∀[Y:C ⟶ hdataflow(A;U)]. ∀[hdfs:bag(hdataflow(A;U))].
(f o X (hdfs) >>= Y = X (hdfs) >>= Y o f ∈ hdataflow(A;U)) supposing (valueall-type(C) and valueall-type(U))
Proof
Definitions occuring in Statement :
hdf-bind-gen: X (hdfs) >>= Y,
hdf-compose1: f o X,
hdataflow: hdataflow(A;B),
compose: f o g,
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
bag: bag(T)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
uiff: uiff(P;Q),
and: P ∧ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
all: ∀x:A. B[x],
top: Top,
prop: ℙ,
hdf-bind-gen: X (hdfs) >>= Y,
hdf-compose1: f o X,
mk-hdf: mk-hdf(s,m.G[s; m];st.H[st];s0),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
band: p ∧b q,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
pi1: fst(t),
pi2: snd(t),
subtype_rel: A ⊆r B,
bind-nxt: bind-nxt(Y;p;a),
ext-eq: A ≡ B,
hdf-halt: hdf-halt(),
hdf-ap: X(a),
squash: ↓T,
callbyvalueall: callbyvalueall,
has-value: (a)↓,
has-valueall: has-valueall(a),
hdf-run: hdf-run(P),
evalall: evalall(t),
bag-map: bag-map(f;bs),
map: map(f;as),
list_ind: list_ind,
empty-bag: {},
nil: [],
true: True
Latex:
\mforall{}[A,B,C,U:Type]. \mforall{}[f:B {}\mrightarrow{} C]. \mforall{}[X:hdataflow(A;B)]. \mforall{}[Y:C {}\mrightarrow{} hdataflow(A;U)].
\mforall{}[hdfs:bag(hdataflow(A;U))].
(f o X (hdfs) >>= Y = X (hdfs) >>= Y o f) supposing (valueall-type(C) and valueall-type(U))
Date html generated:
2016_05_16-AM-10_43_29
Last ObjectModification:
2016_01_17-AM-11_12_35
Theory : halting!dataflow
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