Nuprl Lemma : hdf-state-base4-3
∀[F1,F2,F3,F4,f1,f2,f3,f4,s:Top]. ∀[hdr1,hdr2,hdr3,hdr4:Name].
(hdf-state((λx,s. f1[x;s]) o hdf-base(a.if name_eq(fst(a);hdr1) then [F1[a]] else [] fi )
|| (λx,s. f2[x;s]) o hdf-base(a.if name_eq(fst(a);hdr2) then [F2[a]] else [] fi )
|| (λx,s. f3[x;s]) o hdf-base(a.if name_eq(fst(a);hdr3) then [F3[a]] else [] fi ) || (λx,s. f4[x;s])
o hdf-base(a.if name_eq(fst(a);hdr4) then [F4[a]] else [] fi );[s])
~ fix((λmk-hdf,s. (inl (λa.cbva_seq(λn.if name_eq(fst(a);hdr1) then f1[F1[a];s]
if name_eq(fst(a);hdr2) then f2[F2[a];s]
if name_eq(fst(a);hdr3) then f3[F3[a];s]
if name_eq(fst(a);hdr4) then f4[F4[a];s]
else s
fi ; λg.<mk-hdf (g (λx.x)), g (λx.[x])>; 1)))))
s) supposing
((¬(hdr1 = hdr2 ∈ Name)) and
(¬(hdr1 = hdr3 ∈ Name)) and
(¬(hdr1 = hdr4 ∈ Name)) and
(¬(hdr2 = hdr3 ∈ Name)) and
(¬(hdr2 = hdr4 ∈ Name)) and
(¬(hdr3 = hdr4 ∈ Name)))
Proof
Definitions occuring in Statement :
hdf-parallel: X || Y,
hdf-state: hdf-state(X;bs),
hdf-compose1: f o X,
hdf-base: hdf-base(m.F[m]),
name_eq: name_eq(x;y),
name: Name,
cons: [a / b],
nil: [],
ifthenelse: if b then t else f fi ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s1;s2],
so_apply: x[s],
pi1: fst(t),
not: ¬A,
apply: f a,
fix: fix(F),
lambda: λx.A[x],
pair: <a, b>,
inl: inl x,
natural_number: $n,
sqequal: s ~ t,
equal: s = t ∈ T,
cbva_seq: cbva_seq(L; F; m)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
top: Top,
so_apply: x[s],
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
cbva_seq: cbva_seq(L; F; m),
select_fun_last: select_fun_last(g;m),
select_fun_ap: select_fun_ap(g;n;m),
mk_lambdas_fun: mk_lambdas_fun(F;m),
bag-map: bag-map(f;bs),
mk_lambdas: mk_lambdas(F;m),
all: ∀x:A. B[x],
subtract: n - m,
callbyvalueall_seq: callbyvalueall_seq(L;G;F;n;m),
le_int: i ≤z j,
lt_int: i <z j,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
btrue: tt,
bfalse: ff,
eq_int: (i =z j),
mk_lambdas-fun: mk_lambdas-fun(F;G;n;m),
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
nat_plus: ℕ+,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
nil: [],
it: ⋅,
bag-append: as + bs,
append: as @ bs,
list_ind: list_ind,
cons: [a / b],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
empty-bag: {},
subtype_rel: A ⊆r B,
name: Name,
has-valueall: has-valueall(a),
bag-null: bag-null(bs),
has-value: (a)↓,
bag-combine: ⋃x∈bs.f[x],
bag-union: bag-union(bbs),
concat: concat(ll),
reduce: reduce(f;k;as),
map: map(f;as),
null: null(as),
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Latex:
\mforall{}[F1,F2,F3,F4,f1,f2,f3,f4,s:Top]. \mforall{}[hdr1,hdr2,hdr3,hdr4:Name].
(hdf-state((\mlambda{}x,s. f1[x;s]) o hdf-base(a.if name\_eq(fst(a);hdr1) then [F1[a]] else [] fi )
|| (\mlambda{}x,s. f2[x;s]) o hdf-base(a.if name\_eq(fst(a);hdr2) then [F2[a]] else [] fi )
|| (\mlambda{}x,s. f3[x;s]) o hdf-base(a.if name\_eq(fst(a);hdr3) then [F3[a]] else [] fi )
|| (\mlambda{}x,s. f4[x;s]) o hdf-base(a.if name\_eq(fst(a);hdr4)
then [F4[a]]
else []
fi );[s]) \msim{} fix((\mlambda{}mk-hdf,s. (inl (\mlambda{}a.cbva\_seq(\mlambda{}n.if name\_eq(fst(a);hdr1)
then f1[F1[a];s]
if name\_eq(fst(a);hdr2)
then f2[F2[a];s]
if name\_eq(fst(a);hdr3)
then f3[F3[a];s]
if name\_eq(fst(a);hdr4)
then f4[F4[a];s]
else s
fi ; \mlambda{}g.<mk-hdf (g (\mlambda{}x.x))
, g (\mlambda{}x.[x])
> 1)))))
s) supposing
((\mneg{}(hdr1 = hdr2)) and
(\mneg{}(hdr1 = hdr3)) and
(\mneg{}(hdr1 = hdr4)) and
(\mneg{}(hdr2 = hdr3)) and
(\mneg{}(hdr2 = hdr4)) and
(\mneg{}(hdr3 = hdr4)))
Date html generated:
2016_05_16-AM-10_49_19
Last ObjectModification:
2016_01_18-PM-06_18_53
Theory : halting!dataflow
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