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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