Nuprl Lemma : oal_umap_char
∀s:LOSet. ∀g:AbDMon. ∀h:AbMon. ∀f:|s| ⟶ MonHom(g,h).
(λps:|oal(s;g)|. msFor{h} k ∈ dom(ps)
(f k (ps[k]))) = !v:|oal(s;g)| ⟶ |h|
(IsMonHom{oal_mon(s;g),h}(v)
∧ (∀j:|s|. ((f j) = (v o (λw.inj(j,w))) ∈ (|g| ⟶ |h|))))
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,
compose: f o g,
tlambda: λx:T. b[x],
uni_sat: a = !x:T. Q[x],
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
equal: s = t ∈ T,
monoid_hom: MonHom(M1,M2),
monoid_hom_p: IsMonHom{M1,M2}(f),
abdmonoid: AbDMon,
abmonoid: AbMon,
grp_id: e,
grp_car: |g|,
loset: LOSet,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
dset: DSet,
abdmonoid: AbDMon,
dmon: DMon,
mon: Mon,
abmonoid: AbMon,
monoid_hom_p: IsMonHom{M1,M2}(f),
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
uni_sat: a = !x:T. Q[x],
and: P ∧ Q,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
guard: {T},
uimplies: b supposing a,
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,
oal_mon: oal_mon(a;b),
grp_car: |g|,
prop: ℙ,
monoid_hom: MonHom(M1,M2),
tlambda: λx:T. b[x],
grp_op: *,
pi2: snd(t),
grp_id: e,
infix_ap: x f y,
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
band: p ∧b q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bor: p ∨bq,
bnot: ¬bb,
assert: ↑b,
false: False,
rev_uimplies: rev_uimplies(P;Q),
oal_nil: 00,
null_mset: 0{s},
oal_dom: dom(ps),
mk_mset: mk_mset(as),
top: Top,
compose: f o g,
finite_set: FiniteSet{s},
not: ¬A,
mon_when: when b. p
Lemmas referenced :
set_car_wf,
monoid_hom_wf,
abmonoid_wf,
abdmonoid_wf,
loset_wf,
monoid_hom_properties,
monoid_hom_p_wf,
oal_mon_wf,
mon_subtype_grp_sig,
dmon_subtype_mon,
abdmonoid_dmon,
subtype_rel_transitivity,
dmon_wf,
mon_wf,
grp_sig_wf,
grp_car_wf,
compose_wf,
oalist_wf,
oal_inj_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
mset_for_dom_shift,
abmonoid_subtype_iabmonoid,
lookup_wf,
grp_id_wf,
oal_merge_wf,
oal_dom_wf,
abdmonoid_abmonoid,
mset_union_wf,
oal_dom_merge,
assert_wf,
mset_mem_wf,
mset_diff_wf,
grp_op_wf,
subtype_rel_self,
iff_weakening_equal,
lookup_oal_eq_id,
oal_merge_wf2,
assert_functionality_wrt_uiff,
bor_wf,
eqtt_to_assert,
assert_of_bor,
bnot_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
bfalse_wf,
mset_mem_diff,
mset_union_wf_f,
oal_dom_wf2,
band_wf,
fset_mem_union,
iff_transitivity,
or_wf,
not_wf,
iff_weakening_uiff,
assert_of_band,
assert_of_bnot,
mem_bsubmset,
mset_for_functionality,
infix_ap_wf,
dset_of_mon_wf0,
lookup_merge,
mset_for_of_op,
mset_for_wf,
map_nil_lemma,
istype-void,
lookup_nil_lemma,
mset_for_null_lemma,
mon_when_wf,
iabmonoid_subtype_imon,
iabmonoid_wf,
imon_wf,
set_eq_wf,
ifthenelse_wf,
grp_eq_wf,
mset_wf,
null_mset_wf,
mset_inj_wf,
oal_dom_inj,
lookup_oal_inj,
uiff_transitivity,
equal-wf-T-base,
assert_of_mon_eq,
mset_for_mset_inj,
member_wf,
assert_of_dset_eq,
dist_hom_over_mset_for,
oalist_fact
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
functionIsType,
universeIsType,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
applyEquality,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
independent_pairFormation,
productElimination,
productIsType,
dependent_functionElimination,
instantiate,
independent_isectElimination,
because_Cache,
equalityIsType1,
lambdaEquality_alt,
inhabitedIsType,
isect_memberFormation_alt,
isect_memberEquality_alt,
axiomEquality,
imageElimination,
universeEquality,
independent_functionElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
promote_hyp,
cumulativity,
voidElimination,
productEquality,
unionIsType,
inlFormation_alt,
inrFormation_alt,
functionEquality,
functionExtensionality_alt,
dependent_set_memberEquality_alt
Latex:
\mforall{}s:LOSet. \mforall{}g:AbDMon. \mforall{}h:AbMon. \mforall{}f:|s| {}\mrightarrow{} MonHom(g,h).
(\mlambda{}ps:|oal(s;g)|. msFor\{h\} k \mmember{} dom(ps)
(f k (ps[k]))) = !v:|oal(s;g)| {}\mrightarrow{} |h|
(IsMonHom\{oal\_mon(s;g),h\}(v)
\mwedge{} (\mforall{}j:|s|. ((f j) = (v o (\mlambda{}w.inj(j,w))))))
Date html generated:
2019_10_16-PM-01_08_57
Last ObjectModification:
2018_10_08-PM-00_19_56
Theory : polynom_2
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