Nuprl Lemma : lookup_omral_scale_b
∀g:OCMon. ∀r:CDRng. ∀k,k':|g|. ∀v:|r|. ∀ps:(|g| × |r|) List.
((¬(∃d:|g|. ((↑(d ∈b dom(ps))) ∧ ((k * d) = k' ∈ |g|)))) ⇒ (((<k,v>* ps)[k']) = 0 ∈ |r|))
Proof
Definitions occuring in Statement :
omral_scale: <k,v>* ps,
omral_dom: dom(ps),
lookup: as[k],
mset_mem: mset_mem,
list: T List,
assert: ↑b,
infix_ap: x f y,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
and: P ∧ Q,
product: x:A × B[x],
equal: s = t ∈ T,
cdrng: CDRng,
rng_zero: 0,
rng_car: |r|,
oset_of_ocmon: g↓oset,
ocmon: OCMon,
grp_op: *,
grp_car: |g|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
cdrng: CDRng,
crng: CRng,
rng: Rng,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
and: P ∧ Q,
omon: OMon,
so_lambda: λ2x y.t[x; y],
infix_ap: x f y,
so_apply: x[s1;s2],
guard: {T},
uimplies: b supposing a,
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
set_car: |p|,
pi1: fst(t),
so_apply: x[s],
grp_car: |g|,
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],
pi2: snd(t),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
set_eq: =b,
cand: A c∧ B,
omral_dom: dom(ps),
mset_inj: mset_inj{s}(x),
oal_dom: dom(ps),
mset_sum: a + b,
mk_mset: mk_mset(as),
append: as @ bs,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
omon_inc,
ocmon_subtype_omon,
abdmonoid_dmon,
list_induction,
grp_car_wf,
rng_car_wf,
not_wf,
exists_wf,
assert_wf,
mset_mem_wf,
oset_of_ocmon_wf,
ulinorder_wf,
grp_le_wf,
equal_wf,
bool_wf,
grp_eq_wf,
band_wf,
qoset_subtype_dset,
poset_subtype_qoset,
loset_subtype_poset,
subtype_rel_transitivity,
loset_wf,
poset_wf,
qoset_wf,
dset_wf,
omral_dom_wf,
grp_op_wf,
lookup_wf,
oset_of_ocmon_wf0,
rng_zero_wf,
omral_scale_wf,
subtype_rel_self,
list_wf,
set_car_wf,
nil_wf,
cons_wf,
cdrng_wf,
ocmon_wf,
list_ind_nil_lemma,
lookup_nil_lemma,
list_ind_cons_lemma,
rng_eq_wf,
rng_times_wf,
eqtt_to_assert,
assert_of_rng_eq,
cdrng_subtype_drng,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
lookup_cons_pr_lemma,
map_cons_lemma,
mset_mem_inj_sum_lemma,
iff_transitivity,
bor_wf,
or_wf,
iff_weakening_uiff,
assert_of_bor,
assert_of_mon_eq,
uiff_transitivity,
equal-wf-T-base,
bnot_wf,
assert_of_bnot,
member_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
hypothesis,
sqequalRule,
promote_hyp,
isectElimination,
productEquality,
setElimination,
rename,
lambdaEquality,
functionEquality,
dependent_set_memberEquality,
productElimination,
because_Cache,
instantiate,
independent_isectElimination,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
cumulativity,
independent_pairFormation,
orFunctionality,
inrFormation,
independent_pairEquality,
baseClosed,
impliesFunctionality,
inlFormation
Latex:
\mforall{}g:OCMon. \mforall{}r:CDRng. \mforall{}k,k':|g|. \mforall{}v:|r|. \mforall{}ps:(|g| \mtimes{} |r|) List.
((\mneg{}(\mexists{}d:|g|. ((\muparrow{}(d \mmember{}\msubb{} dom(ps))) \mwedge{} ((k * d) = k')))) {}\mRightarrow{} (((<k,v>* ps)[k']) = 0))
Date html generated:
2018_05_22-AM-07_46_50
Last ObjectModification:
2018_05_19-AM-08_27_40
Theory : polynom_3
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