Nuprl Lemma : lookups_same
∀a:LOSet. ∀b:AbDMon. ∀ps,qs:|oal(a;b)|. ((∀u:|a|. ((ps[u]) = (qs[u]) ∈ |b|)) ⇒ (ps = qs ∈ ((|a| × |b|) List)))
Proof
Definitions occuring in Statement :
lookup: as[k],
oalist: oal(a;b),
list: T List,
all: ∀x:A. B[x],
implies: P ⇒ Q,
product: x:A × B[x],
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],
member: t ∈ T,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
prop: ℙ,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
dset: DSet,
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],
guard: {T},
top: Top,
infix_ap: x f y,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
or: P ∨ Q,
squash: ↓T,
true: True,
decidable: Dec(P)
Lemmas referenced :
oalist_ind,
all_wf,
set_car_wf,
oalist_wf,
equal_wf,
grp_car_wf,
lookup_wf,
grp_id_wf,
list_wf,
nil_wf,
cons_wf,
not_wf,
assert_wf,
before_wf,
map_wf,
set_prod_wf,
dset_of_mon_wf,
abdmonoid_wf,
loset_wf,
oalist_cases,
equal-wf-base-T,
dset_wf,
lookup_nil_lemma,
lookup_cons_pr_lemma,
set_eq_wf,
bool_wf,
uiff_transitivity,
equal-wf-T-base,
eqtt_to_assert,
assert_of_dset_eq,
iff_transitivity,
bnot_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
loset_trichot,
squash_wf,
true_wf,
set_lt_transitivity_2,
set_leq_weakening_eq,
set_lt_irreflexivity,
iff_weakening_equal,
lookup_before_start,
before_trans,
decidable__dset_eq,
poset_sig_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
sqequalRule,
lambdaEquality,
isectElimination,
hypothesis,
applyEquality,
because_Cache,
functionEquality,
setElimination,
rename,
productEquality,
independent_functionElimination,
independent_pairEquality,
productElimination,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
independent_pairFormation,
impliesFunctionality,
imageElimination,
universeEquality,
natural_numberEquality,
imageMemberEquality,
cumulativity
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDMon. \mforall{}ps,qs:|oal(a;b)|. ((\mforall{}u:|a|. ((ps[u]) = (qs[u]))) {}\mRightarrow{} (ps = qs))
Date html generated:
2017_10_01-AM-10_02_23
Last ObjectModification:
2017_03_03-PM-01_05_24
Theory : polynom_2
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