Nuprl Lemma : oal_lk_wf
∀s:LOSet. ∀g:AbDMon. ∀ps:|oal(s;g)|.  ((¬(ps = 00 ∈ |oal(s;g)|)) 
⇒ (lk(ps) ∈ |s|))
Proof
Definitions occuring in Statement : 
oal_lk: lk(ps)
, 
oal_nil: 00
, 
oalist: oal(a;b)
, 
all: ∀x:A. B[x]
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
member: t ∈ T
, 
equal: s = t ∈ T
, 
abdmonoid: AbDMon
, 
loset: LOSet
, 
set_car: |p|
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
member: t ∈ T
, 
oal_lk: lk(ps)
, 
uall: ∀[x:A]. B[x]
, 
uimplies: b supposing a
, 
oalist: oal(a;b)
, 
dset_set: dset_set, 
mk_dset: mk_dset(T, eq)
, 
set_car: |p|
, 
pi1: fst(t)
, 
dset_list: s List
, 
loset: LOSet
, 
poset: POSet{i}
, 
qoset: QOSet
, 
abdmonoid: AbDMon
, 
subtype_rel: A ⊆r B
, 
or: P ∨ Q
, 
set_prod: s × t
, 
dset_of_mon: g↓set
, 
top: Top
, 
assert: ↑b
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
bfalse: ff
, 
false: False
, 
and: P ∧ Q
, 
not: ¬A
, 
nil: []
, 
it: ⋅
, 
oal_nil: 00
, 
cons: [a / b]
, 
guard: {T}
, 
nat: ℕ
, 
ge: i ≥ j 
, 
decidable: Dec(P)
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
prop: ℙ
, 
uiff: uiff(P;Q)
, 
sq_stable: SqStable(P)
, 
squash: ↓T
, 
subtract: n - m
, 
le: A ≤ B
, 
less_than': less_than'(a;b)
, 
true: True
, 
dset: DSet
, 
so_lambda: λ2x.t[x]
, 
dmon: DMon
, 
mon: Mon
, 
so_apply: x[s]
Lemmas referenced : 
hd_wf, 
set_car_wf, 
set_prod_wf, 
dset_of_mon_wf, 
list-cases, 
length_of_nil_lemma, 
map_nil_lemma, 
sd_ordered_nil_lemma, 
mem_nil_lemma, 
oal_nil_wf, 
product_subtype_list, 
length_of_cons_lemma, 
length_wf_nat, 
nat_wf, 
decidable__le, 
false_wf, 
not-ge-2, 
sq_stable__le, 
condition-implies-le, 
minus-add, 
minus-one-mul, 
add-swap, 
minus-one-mul-top, 
add-associates, 
add-commutes, 
add_functionality_wrt_le, 
add-zero, 
le-add-cancel2, 
equal_wf, 
pi1_wf, 
dset_of_mon_wf0, 
not_wf, 
oalist_wf, 
dset_wf, 
abdmonoid_wf, 
loset_wf
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation, 
cut, 
sqequalRule, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
because_Cache, 
independent_isectElimination, 
setElimination, 
rename, 
hypothesisEquality, 
hypothesis, 
applyEquality, 
dependent_functionElimination, 
unionElimination, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
productElimination, 
independent_functionElimination, 
lambdaEquality, 
promote_hyp, 
hypothesis_subsumption, 
natural_numberEquality, 
addEquality, 
independent_pairFormation, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
minusEquality, 
equalityTransitivity, 
equalitySymmetry, 
independent_pairEquality
Latex:
\mforall{}s:LOSet.  \mforall{}g:AbDMon.  \mforall{}ps:|oal(s;g)|.    ((\mneg{}(ps  =  00))  {}\mRightarrow{}  (lk(ps)  \mmember{}  |s|))
Date html generated:
2017_10_01-AM-10_03_27
Last ObjectModification:
2017_03_03-PM-01_06_43
Theory : polynom_2
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