Nuprl Lemma : lookup_non_zero
∀a:LOSet. ∀b:AbDMon. ∀k:|a|. ∀ps:|oal(a;b)|. (¬((ps[k]) = e ∈ |b|) ⇐⇒ ↑(k ∈b dom(ps)))
Proof
Definitions occuring in Statement :
lookup: as[k],
oal_dom: dom(ps),
oalist: oal(a;b),
mset_mem: mset_mem,
assert: ↑b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
equal: s = t ∈ T,
abdmonoid: AbDMon,
grp_id: e,
grp_car: |g|,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
abdmonoid: AbDMon,
dmon: DMon,
mon: Mon,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
dset: DSet,
subtype_rel: A ⊆r B,
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,
dset_of_mon: g↓set,
rev_implies: P ⇐ Q,
not: ¬A,
false: False,
decidable: Dec(P),
or: P ∨ Q,
oal_dom: dom(ps),
mset_mem: mset_mem,
mk_mset: mk_mset(as),
so_lambda: λ2x.t[x],
pi2: snd(t),
so_apply: x[s],
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
set_eq: =b,
infix_ap: x f y,
uiff: uiff(P;Q),
uimplies: b supposing a,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
guard: {T}
Lemmas referenced :
not_wf,
equal_wf,
grp_car_wf,
lookup_wf,
grp_id_wf,
set_car_wf,
oalist_wf,
assert_wf,
mset_mem_wf,
oal_dom_wf,
abdmonoid_abmonoid,
abdmonoid_wf,
loset_wf,
decidable__assert,
dset_wf,
lookup_fails,
list_induction,
mem_wf,
dset_of_mon_wf,
map_wf,
dset_of_mon_wf0,
list_wf,
map_nil_lemma,
lookup_nil_lemma,
mem_nil_lemma,
map_cons_lemma,
mem_cons_lemma,
false_wf,
lookup_cons_pr_lemma,
iff_transitivity,
bor_wf,
infix_ap_wf,
bool_wf,
grp_eq_wf,
or_wf,
iff_weakening_uiff,
assert_of_bor,
pi2_wf,
assert_of_mon_eq,
set_eq_wf,
pi1_wf,
assert_of_dset_eq,
cons_wf,
uiff_transitivity,
equal-wf-T-base,
eqtt_to_assert,
bnot_wf,
eqff_to_assert,
assert_of_bnot,
ifthenelse_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
applyEquality,
lambdaEquality,
sqequalRule,
independent_functionElimination,
voidElimination,
unionElimination,
productElimination,
productEquality,
functionEquality,
isect_memberEquality,
voidEquality,
addLevel,
impliesFunctionality,
independent_pairEquality,
orFunctionality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
impliesLevelFunctionality,
equalityElimination,
baseClosed,
inlFormation,
inrFormation
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDMon. \mforall{}k:|a|. \mforall{}ps:|oal(a;b)|. (\mneg{}((ps[k]) = e) \mLeftarrow{}{}\mRightarrow{} \muparrow{}(k \mmember{}\msubb{} dom(ps)))
Date html generated:
2017_10_01-AM-10_02_09
Last ObjectModification:
2017_03_03-PM-01_05_07
Theory : polynom_2
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