Nuprl Lemma : hdf-parallel-transformation1-2-1
∀[L1,L2,G1,G2,S1,S2,init1,init2:Base]. ∀[m1,m2:ℕ].
(fix((λmk-hdf,s. (inl (λa.cbva_seq(L1[s;a]; λg.<mk-hdf S1[g;s], G1[g]>; m1))))) init1
|| fix((λmk-hdf,s. (inl (λa.cbva_seq(L2[s;a]; λg.<mk-hdf S2[g;s], G2[g]>; m2))))) init2
~ fix((λmk-hdf,s. (inl (λa.cbva_seq(λn.if n <z m1 then L1[fst(s);a] n
if n <z m1 + m2 then mk_lambdas(L2[snd(s);a] (n - m1);m1)
else mk_lambdas_fun(λg1.mk_lambdas_fun(λg2.(G1[g1] + G2[g2]);m2);m1)
fi ; λg.<mk-hdf
<S1[partial_ap_gen(g;(m1 + m2) + 1;0;m1);fst(s)]
, S2[partial_ap_gen(g;(m1 + m2) + 1;m1;m2);snd(s)]
>
, select_fun_last(g;m1 + m2)
>; (m1 + m2) + 1)))))
<init1, init2>)
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],
so_apply: x[s1;s2],
so_apply: x[s],
pi1: fst(t),
pi2: snd(t),
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,
base: Base,
sqequal: s ~ t,
bag-append: as + bs,
select_fun_last: select_fun_last(g;m),
partial_ap_gen: partial_ap_gen(g;n;s;m),
mk_lambdas: mk_lambdas(F;m),
mk_lambdas_fun: mk_lambdas_fun(F;m),
cbva_seq: cbva_seq(L; F; m)
Lemmas :
lifting-strict-spread,
top_wf,
has-value_wf_base,
base_wf,
base_sq,
lifting-strict-less,
decide_bfalse_lemma,
pair-eta,
lifting-strict-decide,
strict4-spread,
lifting-strict-callbyvalueall,
less_than_transitivity1,
less_than_irreflexivity,
int_seg_wf,
decidable__equal_int,
subtype_rel-int_seg,
false_wf,
le_weakening,
subtract_wf,
int_seg_properties,
le_wf,
cbva_seq-spread,
cbva_seq_extend,
cbva_seq-combine2,
decidable__le,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
lifting-strict-int_eq,
decidable__lt,
general_add_assoc,
cbva_seq-list-case2,
cbva_seq-sqequal-n,
set_subtype_base,
int_subtype_base,
partial_ap_of_partial_ap_gen1,
partial_ap_is_gen,
less-iff-le,
minus-zero,
minus-minus,
le-add-cancel2,
add-mul-special,
zero-mul,
le-add-cancel-alt,
lelt_wf,
select_fun_last_partial_ap_gen1,
subtype_base_sq,
not-equal-2,
nat_wf,
all_wf,
sqequal_n_wf,
int_seg_subtype-nat,
set_wf,
less_than_wf,
primrec-wf2
\mforall{}[L1,L2,G1,G2,S1,S2,init1,init2:Base]. \mforall{}[m1,m2:\mBbbN{}].
(fix((\mlambda{}mk-hdf,s. (inl (\mlambda{}a.cbva\_seq(L1[s;a]; \mlambda{}g.<mk-hdf S1[g;s], G1[g]> m1))))) init1
|| fix((\mlambda{}mk-hdf,s. (inl (\mlambda{}a.cbva\_seq(L2[s;a]; \mlambda{}g.<mk-hdf S2[g;s], G2[g]> m2))))) init2
\msim{} fix((\mlambda{}mk-hdf,s.
(inl (\mlambda{}a.cbva\_seq(\mlambda{}n.if n <z m1 then L1[fst(s);a] n
if n <z m1 + m2 then mk\_lambdas(L2[snd(s);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
<S1[partial\_ap\_gen(g;(m1 + m2) + 1;0;m1);fst(s)]
, S2[partial\_ap\_gen(g;(m1 + m2) + 1;m1;m2);snd(s)]
>
, select\_fun\_last(g;m1 + m2)
> (m1 + m2) + 1)))))
<init1, init2>)
Date html generated:
2015_07_17-AM-08_09_18
Last ObjectModification:
2015_04_23-PM-10_57_50
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