Nuprl Lemma : lookup_omral_scale_a
∀g:OCMon. ∀r:CDRng. ∀k,k':|g|. ∀v:|r|. ∀ps:|omral(g;r)|. (((<k,v>* ps)[k * k']) = (v * (ps[k'])) ∈ |r|)
Proof
Definitions occuring in Statement :
omral_scale: <k,v>* ps,
omralist: omral(g;r),
lookup: as[k],
infix_ap: x f y,
all: ∀x:A. B[x],
equal: s = t ∈ T,
cdrng: CDRng,
rng_times: *,
rng_zero: 0,
rng_car: |r|,
oset_of_ocmon: g↓oset,
ocmon: OCMon,
grp_op: *,
grp_car: |g|,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
cdrng: CDRng,
crng: CRng,
rng: Rng,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
and: P ∧ Q,
cand: A c∧ B,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
omon: OMon,
so_lambda: λ2x y.t[x; y],
infix_ap: x f y,
so_apply: x[s1;s2],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
implies: P ⇒ Q,
omralist: omral(g;r),
oalist: oal(a;b),
dset_set: dset_set,
mk_dset: mk_dset(T, eq),
set_car: |p|,
pi1: fst(t),
dset_list: s List,
set_prod: s × t,
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
add_grp_of_rng: r↓+gp,
grp_id: e,
pi2: snd(t),
grp_car: |g|,
dset: DSet,
omral_scale: <k,v>* ps,
ycomb: Y,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
set_eq: =b,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
squash: ↓T,
true: True,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rng_car_wf,
grp_car_wf,
cdrng_wf,
ocmon_wf,
cdrng_is_abdmonoid,
abdmonoid_dmon,
ocmon_subtype_abdmonoid,
subtype_rel_transitivity,
abdmonoid_wf,
dmon_wf,
oalist_ind,
oset_of_ocmon_wf,
ulinorder_wf,
assert_wf,
grp_le_wf,
equal_wf,
bool_wf,
grp_eq_wf,
band_wf,
set_car_wf,
omralist_wf,
dset_wf,
lookup_wf,
oset_of_ocmon_wf0,
mon_subtype_grp_sig,
dmon_subtype_mon,
mon_wf,
grp_sig_wf,
rng_zero_wf,
infix_ap_wf,
grp_op_wf,
subtype_rel_self,
omral_scale_wf,
rng_times_wf,
list_wf,
list_ind_nil_lemma,
lookup_nil_lemma,
rng_times_zero,
list_ind_cons_lemma,
lookup_cons_pr_lemma,
not_wf,
before_wf,
ocmon_subtype_omon,
map_wf,
set_prod_wf,
dset_of_mon_wf,
rng_eq_wf,
uiff_transitivity,
equal-wf-T-base,
eqtt_to_assert,
assert_of_rng_eq,
cdrng_subtype_drng,
assert_of_mon_eq,
iff_transitivity,
bnot_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
squash_wf,
true_wf,
rng_lookup_before_start,
omral_scale_wf2,
iff_weakening_equal,
rng_before_all_imp_before,
omral_scale_dom_bound,
rng_before_imp_before_all,
assert_functionality_wrt_uiff,
ocmon_cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
productElimination,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
applyEquality,
instantiate,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
dependent_set_memberEquality,
productEquality,
lambdaEquality,
because_Cache,
functionEquality,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
equalityElimination,
baseClosed,
impliesFunctionality,
imageElimination,
universeEquality,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageMemberEquality
Latex:
\mforall{}g:OCMon. \mforall{}r:CDRng. \mforall{}k,k':|g|. \mforall{}v:|r|. \mforall{}ps:|omral(g;r)|. (((<k,v>* ps)[k * k']) = (v * (ps[k'])))
Date html generated:
2018_05_22-AM-07_46_44
Last ObjectModification:
2018_05_19-AM-08_27_25
Theory : polynom_3
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