Nuprl Lemma : hdf-union-ap
∀[A,B,C:Type]. ∀[X:hdataflow(A;B)]. ∀[Y:hdataflow(A;C)]. ∀[a:A].
(X + Y(a)
= <fst(X(a)) + fst(Y(a)), bag-map(λx.(inl x);snd(X(a))) + bag-map(λx.(inr x );snd(Y(a)))>
∈ (hdataflow(A;B + C) × bag(B + C))) supposing
(valueall-type(B) and
valueall-type(C))
Proof
Definitions occuring in Statement :
hdf-union: X + 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),
lambda: λx.A[x],
pair: <a, b>,
product: x:A × B[x],
inr: inr x ,
inl: inl x,
union: left + right,
universe: Type,
equal: s = t ∈ T,
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-ap: X(a),
hdf-union: X + 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,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
hdf-halt: hdf-halt(),
top: Top,
pi1: fst(t),
pi2: snd(t),
exposed-bfalse: exposed-bfalse,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
bag-map: bag-map(f;bs),
map: map(f;as),
list_ind: list_ind,
empty-bag: {},
nil: [],
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
subtype_rel: A ⊆r B,
ext-eq: A ≡ B,
not: ¬A,
true: True,
has-value: (a)↓,
has-valueall: has-valueall(a),
so_lambda: λ2x.t[x],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
hdf-halted: hdf-halted(P),
isr: isr(x)
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[X:hdataflow(A;B)]. \mforall{}[Y:hdataflow(A;C)]. \mforall{}[a:A].
(X + Y(a)
= <fst(X(a)) + fst(Y(a))
, bag-map(\mlambda{}x.(inl x);snd(X(a))) + bag-map(\mlambda{}x.(inr x );snd(Y(a)))
>) supposing
(valueall-type(B) and
valueall-type(C))
Date html generated:
2016_05_16-AM-10_42_20
Last ObjectModification:
2015_12_28-PM-07_45_52
Theory : halting!dataflow
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