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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