Nuprl Lemma : hdf-parallel-transformation2
∀[L,G,H,S,init,out:Base]. ∀[m:ℕ].
(inl (λa.<inr Ax , out>) || fix((λmk-hdf,s. (inl (λa.cbva_seq(L[s;a]; λg.<case H[g;s]
of inl() =>
mk-hdf S[g;s]
| inr() =>
inr Ax
, G[g]
>; m)))))
init ~ fix((λmk-hdf,s. (inl (λa.cbva_seq(λn.if (n =z m)
then case fst(s)
of inl(x) =>
mk_lambdas_fun(λg.(out + G[g]);m)
| inr(x) =>
mk_lambdas_fun(λg.G[g];m)
else L[snd(s);a] n
fi ; λg.<case H[partial_ap(g;m + 1;m);snd(s)]
of inl() =>
mk-hdf
<inr Ax
, S[partial_ap(g;m + 1;m);snd(s)]
>
| inr() =>
inr Ax
, select_fun_last(g;m)
>; m + 1)))))
<inl Ax, init>)
Proof
Definitions occuring in Statement :
hdf-parallel: X || Y,
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_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>,
decide: case b of inl(x) => s[x] | inr(y) => t[y],
inr: inr x ,
inl: inl x,
add: n + m,
natural_number: $n,
base: Base,
sqequal: s ~ t,
axiom: Ax,
bag-append: as + bs,
select_fun_last: select_fun_last(g;m),
partial_ap: partial_ap(g;n;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-int_eq,
pair-eta,
lifting-strict-decide,
empty_bag_append_lemma,
strict4-spread,
lifting-strict-callbyvalueall,
cbva_seq-spread,
cbva_seq_extend,
decide_bfalse_lemma,
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,
decidable__le,
not-le-2,
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,
decidable__lt,
cbva_seq-sqequal-n,
set_subtype_base,
int_subtype_base,
sq_stable__le,
less-iff-le,
minus-zero,
minus-minus,
le-add-cancel-alt,
le-add-cancel2,
add-mul-special,
zero-mul,
lelt_wf,
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{}[L,G,H,S,init,out:Base]. \mforall{}[m:\mBbbN{}].
(inl (\mlambda{}a.<inr Ax , out>) || fix((\mlambda{}mk-hdf,s. (inl (\mlambda{}a.cbva\_seq(L[s;a]; \mlambda{}g.<case H[g;s]
of inl() =>
mk-hdf S[g;s]
| inr() =>
inr Ax
, G[g]
> m)))))
init
\msim{} fix((\mlambda{}mk-hdf,s. (inl (\mlambda{}a.cbva\_seq(\mlambda{}n.if (n =\msubz{} m)
then case fst(s)
of inl(x) =>
mk\_lambdas\_fun(\mlambda{}g.(out + G[g]);m)
| inr(x) =>
mk\_lambdas\_fun(\mlambda{}g.G[g];m)
else L[snd(s);a] n
fi ; \mlambda{}g.<case H[partial\_ap(g;m + 1;m);snd(s)]
of inl() =>
mk-hdf <inr Ax , S[partial\_ap(g;m + 1;m);snd(s)]>
| inr() =>
inr Ax
, select\_fun\_last(g;m)
> m + 1)))))
<inl Ax, init>)
Date html generated:
2015_07_17-AM-08_09_21
Last ObjectModification:
2015_04_23-PM-10_57_37
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