Nuprl Lemma : hdf-parallel-transformation2-1
∀[L1,L2,G1,G2:Top]. ∀[m1,m2:ℕ].
(fix((λmk-hdf.(inl (λa.cbva_seq(L1[a]; λg.<mk-hdf, G1[g]>; m1))))) || fix((λmk-hdf.(inl (λa.cbva_seq(L2[a]; λg.<mk-hdf
, G2[g]
>;
m2)))))
~ fix((λmk-hdf.(inl (λa.cbva_seq(λn.if n <z m1 then L1[a] n
if n <z m1 + m2 then mk_lambdas(L2[a] (n - m1);m1)
else mk_lambdas_fun(λg1.mk_lambdas_fun(λg2.(G1[g1] + G2[g2]);m2);m1)
fi ; λg.<mk-hdf, select_fun_last(g;m1 + m2)>; (m1 + m2) + 1))))))
Proof
Definitions occuring in Statement :
hdf-parallel: X || Y,
nat: ℕ,
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
apply: f a,
fix: fix(F),
lambda: λx.A[x],
pair: <a, b>,
inl: inl x,
subtract: n - m,
add: n + m,
natural_number: $n,
sqequal: s ~ t,
bag-append: as + bs,
select_fun_last: select_fun_last(g;m),
mk_lambdas: mk_lambdas(F;m),
mk_lambdas_fun: mk_lambdas_fun(F;m),
cbva_seq: cbva_seq(L; F; m)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Latex:
\mforall{}[L1,L2,G1,G2:Top]. \mforall{}[m1,m2:\mBbbN{}].
(fix((\mlambda{}mk-hdf.(inl (\mlambda{}a.cbva\_seq(L1[a]; \mlambda{}g.<mk-hdf, G1[g]> m1)))))
|| fix((\mlambda{}mk-hdf.(inl (\mlambda{}a.cbva\_seq(L2[a]; \mlambda{}g.<mk-hdf, G2[g]> m2)))))
\msim{} fix((\mlambda{}mk-hdf.(inl (\mlambda{}a.cbva\_seq(\mlambda{}n.if n <z m1 then L1[a] n
if n <z m1 + m2 then mk\_lambdas(L2[a] (n - m1);m1)
else mk\_lambdas\_fun(\mlambda{}g1.mk\_lambdas\_fun(\mlambda{}g2.(G1[g1]
+ G2[g2]);m2);m1)
fi ; \mlambda{}g.<mk-hdf, select\_fun\_last(g;m1 + m2)> (m1 + m2)
+ 1))))))
Date html generated:
2016_05_16-AM-10_47_10
Last ObjectModification:
2015_12_28-PM-07_39_18
Theory : halting!dataflow
Home
Index