Nuprl Lemma : oalist_fact
∀a:LOSet. ∀b:AbDMon. ∀ps:|oal(a;b)|. (ps = (msFor{oal_mon(a;b)} k' ∈ dom(ps). inj(k',ps[k'])) ∈ |oal(a;b)|)
Proof
Definitions occuring in Statement :
oal_inj: inj(k,v),
oal_mon: oal_mon(a;b),
lookup: as[k],
oal_dom: dom(ps),
oalist: oal(a;b),
mset_for: mset_for,
all: ∀x:A. B[x],
equal: s = t ∈ T,
abdmonoid: AbDMon,
grp_id: e,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
dset: DSet,
uall: ∀[x:A]. B[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],
implies: P ⇒ Q,
top: Top,
null_mset: 0{s},
oal_dom: dom(ps),
mk_mset: mk_mset(as),
oal_mon: oal_mon(a;b),
grp_id: e,
pi2: snd(t),
mset_inj: mset_inj{s}(x),
mset_sum: a + b,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
grp_op: *,
infix_ap: x f y,
prop: ℙ,
guard: {T},
oal_nil: 00,
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,
squash: ↓T,
grp_car: |g|,
list: T List,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
true: True,
rev_uimplies: rev_uimplies(P;Q),
mon_when: when b. p
Lemmas referenced :
lookups_same_a,
mset_for_wf,
oal_mon_wf,
oal_inj_wf,
lookup_wf,
grp_car_wf,
grp_id_wf,
set_car_wf,
oal_dom_wf,
abdmonoid_abmonoid,
oalist_wf,
abdmonoid_wf,
loset_wf,
oalist_ind_a,
equal_wf,
lookup_nil_lemma,
istype-void,
map_nil_lemma,
mset_for_null_lemma,
lookup_cons_pr_lemma,
list_ind_cons_lemma,
list_ind_nil_lemma,
map_cons_lemma,
mset_for_inj_lemma,
not_wf,
assert_wf,
before_wf,
map_wf,
set_prod_wf,
dset_of_mon_wf,
set_eq_wf,
uiff_transitivity,
equal-wf-T-base,
bool_wf,
member_wf,
eqtt_to_assert,
assert_of_dset_eq,
iff_transitivity,
bnot_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
squash_wf,
true_wf,
istype-universe,
ifthenelse_wf,
lookup_merge,
infix_ap_wf,
subtype_rel_self,
list_wf,
sd_ordered_wf,
mem_wf,
dset_of_mon_wf0,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
grp_op_wf,
lookup_oal_inj,
iff_weakening_equal,
mon_when_wf,
iabmonoid_subtype_imon,
abmonoid_subtype_iabmonoid,
subtype_rel_transitivity,
abmonoid_wf,
iabmonoid_wf,
imon_wf,
poset_sig_wf,
mset_for_functionality,
ite_rw_false,
mon_subtype_grp_sig,
dmon_subtype_mon,
abdmonoid_dmon,
dmon_wf,
mon_wf,
grp_sig_wf,
mset_mem_wf,
lookup_non_zero,
lookup_before_start,
assert_functionality_wrt_uiff,
dset_wf,
mon_ident
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
because_Cache,
hypothesis,
applyEquality,
sqequalRule,
lambdaEquality_alt,
isectElimination,
universeIsType,
independent_functionElimination,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
voidElimination,
productElimination,
equalityIsType1,
unionElimination,
equalityElimination,
baseClosed,
independent_isectElimination,
independent_pairFormation,
imageElimination,
universeEquality,
setEquality,
productEquality,
productIsType,
dependent_pairFormation_alt,
promote_hyp,
instantiate,
cumulativity,
natural_numberEquality,
imageMemberEquality
Latex:
\mforall{}a:LOSet. \mforall{}b:AbDMon. \mforall{}ps:|oal(a;b)|. (ps = (msFor\{oal\_mon(a;b)\} k' \mmember{} dom(ps). inj(k',ps[k'])))
Date html generated:
2019_10_16-PM-01_07_41
Last ObjectModification:
2018_10_08-PM-00_23_31
Theory : polynom_2
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