Nuprl Lemma : lookup_oal_lk
∀s:LOSet. ∀g:AbDMon. ∀ps:|oal(s;g)|. ((¬(ps = 00 ∈ |oal(s;g)|)) ⇒ ((ps[lk(ps)]) = lv(ps) ∈ |g|))
Proof
Definitions occuring in Statement :
oal_lv: lv(ps),
oal_lk: lk(ps),
lookup: as[k],
oal_nil: 00,
oalist: oal(a;b),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
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],
member: t ∈ T,
so_lambda: λ2x.t[x],
implies: P ⇒ Q,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
dset: DSet,
abdmonoid: AbDMon,
dmon: DMon,
mon: Mon,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
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,
so_apply: x[s],
guard: {T},
oal_nil: 00,
false: False,
not: ¬A,
top: Top,
oal_lk: lk(ps),
oal_lv: lv(ps),
pi2: snd(t),
oal_cons_pr: oal_cons_pr(x;y;ws),
infix_ap: x f y,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
oalist_cases_a,
not_wf,
equal_wf,
set_car_wf,
oalist_wf,
dset_wf,
oal_nil_wf,
grp_car_wf,
lookup_wf,
grp_id_wf,
oal_lk_wf,
oal_lv_wf,
abdmonoid_wf,
loset_wf,
lookup_cons_pr_lemma,
reduce_hd_cons_lemma,
oal_cons_pr_wf,
assert_wf,
before_wf,
map_wf,
set_prod_wf,
dset_of_mon_wf,
pi1_wf_top,
set_eq_wf,
bool_wf,
equal-wf-T-base,
member_wf,
bnot_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_dset_eq,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
functionEquality,
isectElimination,
hypothesis,
applyEquality,
setElimination,
rename,
because_Cache,
independent_functionElimination,
voidElimination,
isect_memberEquality,
voidEquality,
productElimination,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
unionElimination,
equalityElimination,
independent_isectElimination,
independent_pairFormation,
impliesFunctionality
Latex:
\mforall{}s:LOSet. \mforall{}g:AbDMon. \mforall{}ps:|oal(s;g)|. ((\mneg{}(ps = 00)) {}\mRightarrow{} ((ps[lk(ps)]) = lv(ps)))
Date html generated:
2019_10_16-PM-01_08_18
Last ObjectModification:
2018_08_22-AM-11_39_53
Theory : polynom_2
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