Nuprl Lemma : hdf-union-eq-disju
∀[A,B,C:Type]. ∀[X:hdataflow(A;B)]. ∀[Y:hdataflow(A;C)].
(X + Y = (λx.(inl x)) o X || (λx.(inr x )) o Y ∈ hdataflow(A;B + C)) supposing
(valueall-type(B) and
valueall-type(C))
Proof
Definitions occuring in Statement :
hdf-union: X + Y,
hdf-parallel: X || Y,
hdf-compose1: f o X,
hdataflow: hdataflow(A;B),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
lambda: λx.A[x],
inr: inr x ,
inl: inl x,
union: left + right,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
uiff: uiff(P;Q),
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
nat: ℕ,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
or: P ∨ Q,
cons: [a / b],
colength: colength(L),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
guard: {T},
decidable: Dec(P),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
hdf-compose1: f o X,
hdf-parallel: X || Y,
hdf-union: X + Y,
mk-hdf: mk-hdf(s,m.G[s; m];st.H[st];s0),
bool: 𝔹,
unit: Unit,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
hdf-ap: X(a),
hdf-halt: hdf-halt(),
exposed-bfalse: exposed-bfalse,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
empty-bag: {},
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
true: True,
pi1: fst(t),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
ext-eq: A ≡ B,
has-value: (a)↓,
has-valueall: has-valueall(a),
pi2: snd(t),
bag-map: bag-map(f;bs),
map: map(f;as),
list_ind: list_ind,
bag-append: as + bs,
append: as @ bs
Latex:
\mforall{}[A,B,C:Type]. \mforall{}[X:hdataflow(A;B)]. \mforall{}[Y:hdataflow(A;C)].
(X + Y = (\mlambda{}x.(inl x)) o X || (\mlambda{}x.(inr x )) o Y) supposing
(valueall-type(B) and
valueall-type(C))
Date html generated:
2016_05_16-AM-10_42_27
Last ObjectModification:
2016_01_17-AM-11_13_16
Theory : halting!dataflow
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