Nuprl Lemma : hdf-compose0-transformation1
∀[f,L,S,G,s:Top]. ∀[m:ℕ].
(hdf-compose0(f;fix((λmk-hdf,s. (inl (λa.cbva_seq(L[s;a]; λg.<mk-hdf S[s;a;g], G[s;a;g]>; m))))) s)
~ fix((λmk-hdf,s. (inl (λa.cbva_seq(λn.if (n =z m) then mk_lambdas_fun(λg.∪b∈G[s;a;g].f b;m) else L[s;a] n fi ;
λg.<mk-hdf S[s;a;partial_ap(g;m + 1;m)], select_fun_last(g;m)>; m + 1)))))
s)
Proof
Definitions occuring in Statement :
hdf-compose0: hdf-compose0(f;X),
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s1;s2;s3],
so_apply: x[s1;s2],
apply: f a,
fix: fix(F),
lambda: λx.A[x],
pair: <a, b>,
inl: inl x,
add: n + m,
natural_number: $n,
sqequal: s ~ t,
bag-combine: ∪x∈bs.f[x],
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-decide,
has-value_wf_base,
base_wf,
lifting-strict-int_eq,
top_wf,
lifting-strict-spread,
strict4-spread,
lifting-strict-callbyvalueall,
nat_properties,
less_than_transitivity1,
less_than_irreflexivity,
ge_wf,
less_than_wf,
fun_exp0_lemma,
strictness-apply,
bottom_diverge,
decidable__le,
subtract_wf,
false_wf,
not-ge-2,
less-iff-le,
condition-implies-le,
minus-one-mul,
zero-add,
minus-add,
minus-minus,
add-associates,
add-swap,
add-commutes,
add_functionality_wrt_le,
add-zero,
le-add-cancel,
fun_exp_unroll,
le_weakening2,
le_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
le_weakening,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
nat_wf,
cbva_seq-spread,
cbva_seq_extend,
set_subtype_base,
int_subtype_base
\mforall{}[f,L,S,G,s:Top]. \mforall{}[m:\mBbbN{}].
(hdf-compose0(f;fix((\mlambda{}mk-hdf,s. (inl (\mlambda{}a.cbva\_seq(L[s;a]; \mlambda{}g.<mk-hdf S[s;a;g], G[s;a;g]> m))))) s\000C)
\msim{} fix((\mlambda{}mk-hdf,s. (inl (\mlambda{}a.cbva\_seq(\mlambda{}n.if (n =\msubz{} m)
then mk\_lambdas\_fun(\mlambda{}g.\mcup{}b\mmember{}G[s;a;g].f b;m)
else L[s;a] n
fi ; \mlambda{}g.<mk-hdf S[s;a;partial\_ap(g;m + 1;m)]
, select\_fun\_last(g;m)
> m + 1)))))
s)
Date html generated:
2015_07_17-AM-08_08_45
Last ObjectModification:
2015_01_27-PM-00_08_15
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