Nuprl Lemma : hdf-compose1-transformation1-2
∀[f,L,G:Base]. ∀[m:ℕ].
(f o fix((λmk-hdf.(inl (λa.cbva_seq(L[a]; λg.<mk-hdf, G[a;g]>; m)))))
~ fix((λmk-hdf.(inl (λa.cbva_seq(λn.if (n =z m) then mk_lambdas_fun(λg.bag-map(f;G[a;g]);m) else L[a] n fi ;
λg.<mk-hdf, select_fun_ap(g;m + 1;m)>; m + 1))))))
Proof
Definitions occuring in Statement :
hdf-compose1: f o X,
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],
apply: f a,
fix: fix(F),
lambda: λx.A[x],
pair: <a, b>,
inl: inl x,
add: n + m,
natural_number: $n,
base: Base,
sqequal: s ~ t,
bag-map: bag-map(f;bs),
select_fun_ap: select_fun_ap(g;n;m),
mk_lambdas_fun: mk_lambdas_fun(F;m),
cbva_seq: cbva_seq(L; F; m)
Lemmas :
lifting-strict-decide,
top_wf,
has-value_wf_base,
base_wf,
lifting-strict-int_eq,
lifting-strict-spread,
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,
decidable__lt,
cbva_seq-sqequal-n,
set_subtype_base,
int_subtype_base,
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,
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{}[f,L,G:Base]. \mforall{}[m:\mBbbN{}].
(f o fix((\mlambda{}mk-hdf.(inl (\mlambda{}a.cbva\_seq(L[a]; \mlambda{}g.<mk-hdf, G[a;g]> m)))))
\msim{} fix((\mlambda{}mk-hdf.(inl (\mlambda{}a.cbva\_seq(\mlambda{}n.if (n =\msubz{} m)
then mk\_lambdas\_fun(\mlambda{}g.bag-map(f;G[a;g]);m)
else L[a] n
fi ; \mlambda{}g.<mk-hdf, select\_fun\_ap(g;m + 1;m)> m + 1))))))
Date html generated:
2015_07_17-AM-08_08_37
Last ObjectModification:
2015_01_27-PM-00_07_37
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