Nuprl Lemma : lookup_omral_times
∀g:OCMon. ∀r:CDRng. ∀ps,qs:|omral(g;r)|. ∀z:|g|.
(((ps ** qs)[z]) = (msFor{r↓+gp} x ∈ dom(ps). msFor{r↓+gp} y ∈ dom(qs). when (x * y) =b z. ((ps[x]) * (qs[y]))) ∈ |r|)
Proof
Definitions occuring in Statement :
omral_times: ps ** qs,
omral_dom: dom(ps),
omralist: omral(g;r),
lookup: as[k],
mset_for: mset_for,
infix_ap: x f y,
all: ∀x:A. B[x],
equal: s = t ∈ T,
rng_when: rng_when,
add_grp_of_rng: r↓+gp,
cdrng: CDRng,
rng_times: *,
rng_zero: 0,
rng_car: |r|,
oset_of_ocmon: g↓oset,
ocmon: OCMon,
grp_op: *,
grp_eq: =b,
grp_car: |g|,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
subtype_rel: A ⊆r B,
dset: DSet,
cdrng: CDRng,
crng: CRng,
abgrp: AbGrp,
grp: Group{i},
iabmonoid: IAbMonoid,
imon: IMonoid,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
implies: P ⇒ Q,
rng: Rng,
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
set_car: |p|,
pi1: fst(t),
omralist: omral(g;r),
oalist: oal(a;b),
dset_set: dset_set,
mk_dset: mk_dset(T, eq),
dset_list: s List,
set_prod: s × t,
add_grp_of_rng: r↓+gp,
grp_id: e,
pi2: snd(t),
grp_car: |g|,
omon: OMon,
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
infix_ap: x f y,
so_apply: x[s1;s2],
guard: {T},
rng_car: |r|,
omral_times: ps ** qs,
ycomb: Y,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
omral_dom: dom(ps),
null_mset: 0{s},
oal_dom: dom(ps),
mk_mset: mk_mset(as),
mset_inj: mset_inj{s}(x),
mset_sum: a + b,
append: as @ bs,
set_eq: =b,
grp_op: *,
squash: ↓T,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
mset_mem: mset_mem
Lemmas referenced :
grp_car_wf,
set_car_wf,
omralist_wf,
dset_wf,
cdrng_wf,
ocmon_wf,
cdrng_is_abdmonoid,
add_grp_of_rng_wf_b,
subtype_rel_sets,
grp_sig_wf,
monoid_p_wf,
grp_op_wf,
grp_id_wf,
inverse_wf,
grp_inv_wf,
comm_wf,
set_wf,
omralist_ind_a,
equal_wf,
rng_car_wf,
lookup_wf,
oset_of_ocmon_wf0,
rng_zero_wf,
omral_times_wf,
subtype_rel_self,
list_wf,
mset_for_wf,
oset_of_ocmon_wf,
ulinorder_wf,
assert_wf,
grp_le_wf,
bool_wf,
grp_eq_wf,
band_wf,
qoset_subtype_dset,
poset_subtype_qoset,
loset_subtype_poset,
subtype_rel_transitivity,
loset_wf,
poset_wf,
qoset_wf,
rng_when_wf,
rng_times_wf,
add_grp_of_rng_wf,
omral_dom_wf,
list_ind_nil_lemma,
map_nil_lemma,
lookup_nil_lemma,
mset_for_null_lemma,
not_wf,
before_wf,
ocmon_subtype_omon,
map_wf,
set_prod_wf,
dset_of_mon_wf,
abdmonoid_dmon,
list_ind_cons_lemma,
map_cons_lemma,
lookup_cons_pr_lemma,
mset_for_inj_lemma,
omon_inc,
squash_wf,
true_wf,
omral_plus_wf,
omral_scale_wf,
rng_plus_wf,
mset_for_functionality,
ifthenelse_wf,
rng_wf,
ite_rw_true,
assert_of_mon_eq,
mset_mem_wf,
abmonoid_subtype_iabmonoid,
abdmonoid_abmonoid,
abdmonoid_wf,
abmonoid_wf,
iabmonoid_wf,
infix_ap_wf,
mon_subtype_grp_sig,
dmon_subtype_mon,
ocmon_subtype_abdmonoid,
dmon_wf,
mon_wf,
dset_of_mon_wf0,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
iff_weakening_equal,
ite_rw_false,
lookup_omral_plus,
omral_scale_wf2,
omral_times_wf2,
rng_before_imp_before_all,
ball_char,
grp_blt_wf,
assert_of_grp_blt,
mem_wf,
grp_lt_transitivity_2,
grp_leq_weakening_eq,
grp_lt_irreflexivity,
lookup_omral_scale_c
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
sqequalHypSubstitution,
dependent_functionElimination,
hypothesisEquality,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
because_Cache,
applyEquality,
lambdaEquality,
sqequalRule,
instantiate,
setEquality,
cumulativity,
independent_isectElimination,
productEquality,
dependent_set_memberEquality,
productElimination,
functionEquality,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
equalityTransitivity,
equalitySymmetry,
imageElimination,
universeEquality,
equalityUniverse,
levelHypothesis,
natural_numberEquality,
imageMemberEquality,
baseClosed,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
allFunctionality,
addLevel,
impliesFunctionality
Latex:
\mforall{}g:OCMon. \mforall{}r:CDRng. \mforall{}ps,qs:|omral(g;r)|. \mforall{}z:|g|.
(((ps ** qs)[z])
= (msFor\{r\mdownarrow{}+gp\} x \mmember{} dom(ps)
msFor\{r\mdownarrow{}+gp\} y \mmember{} dom(qs)
when (x * y) =\msubb{} z.
((ps[x]) * (qs[y]))))
Date html generated:
2018_05_22-AM-07_46_59
Last ObjectModification:
2018_05_19-AM-08_28_26
Theory : polynom_3
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