Nuprl Lemma : oal_equal_char
∀a:LOSet. ∀b:AbDMon. ∀ps,qs:|oal(a;b)|. (ps = qs ∈ |oal(a;b)| ⇐⇒ ∀u:|a|. ((ps[u]) = (qs[u]) ∈ |b|))
Proof
Definitions occuring in Statement :
lookup: as[k],
oalist: oal(a;b),
all: ∀x:A. B[x],
iff: 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],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
dset: DSet,
prop: ℙ,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
abdmonoid: AbDMon,
dmon: DMon,
mon: Mon,
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],
squash: ↓T,
true: True,
uimplies: b supposing a,
guard: {T}
Lemmas referenced :
set_car_wf,
equal_wf,
oalist_wf,
dset_wf,
all_wf,
grp_car_wf,
lookup_wf,
grp_id_wf,
abdmonoid_wf,
loset_wf,
squash_wf,
true_wf,
list_wf,
poset_sig_wf,
iff_weakening_equal,
lookups_same_a
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
hypothesis,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
dependent_functionElimination,
applyEquality,
lambdaEquality,
sqequalRule,
because_Cache,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
productEquality,
cumulativity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDMon. \mforall{}ps,qs:|oal(a;b)|. (ps = qs \mLeftarrow{}{}\mRightarrow{} \mforall{}u:|a|. ((ps[u]) = (qs[u])))
Date html generated:
2017_10_01-AM-10_02_26
Last ObjectModification:
2017_03_03-PM-01_04_42
Theory : polynom_2
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