Nuprl Lemma : lookup_oal_eq_id
∀a:LOSet. ∀b:AbDMon. ∀k:|a|. ∀ps:|oal(a;b)|. ((¬↑(k ∈b dom(ps))) ⇒ ((ps[k]) = e ∈ |b|))
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],
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],
implies: P ⇒ Q,
member: t ∈ T,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
uall: ∀[x:A]. B[x],
abdmonoid: AbDMon,
dmon: DMon,
mon: Mon,
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,
dset: DSet,
prop: ℙ,
oal_dom: dom(ps),
mset_mem: mset_mem,
mk_mset: mk_mset(as)
Lemmas referenced :
lookup_fails,
grp_car_wf,
grp_id_wf,
set_car_wf,
oalist_wf,
dset_wf,
not_wf,
assert_wf,
mset_mem_wf,
oal_dom_wf,
abdmonoid_abmonoid,
abdmonoid_wf,
loset_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
isectElimination,
hypothesis,
applyEquality,
lambdaEquality,
sqequalRule,
independent_functionElimination,
because_Cache
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDMon. \mforall{}k:|a|. \mforall{}ps:|oal(a;b)|. ((\mneg{}\muparrow{}(k \mmember{}\msubb{} dom(ps))) {}\mRightarrow{} ((ps[k]) = e))
Date html generated:
2016_05_16-AM-08_16_53
Last ObjectModification:
2015_12_28-PM-06_27_32
Theory : polynom_2
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