Nuprl Lemma : omral_dom_scale
∀g:OCMon. ∀r:CDRng. ∀k:|g|. ∀v:|r|. ∀ps:|omral(g;r)|. (↑(dom(<k,v>* ps) ⊆b fs-map(λk'.(k' * k), dom(ps))))
Proof
Definitions occuring in Statement :
omral_scale: <k,v>* ps,
omral_dom: dom(ps),
omralist: omral(g;r),
bsubmset: a ⊆b b,
fset_map: fs-map(f, a),
assert: ↑b,
infix_ap: x f y,
all: ∀x:A. B[x],
lambda: λx.A[x],
cdrng: CDRng,
rng_car: |r|,
oset_of_ocmon: g↓oset,
ocmon: OCMon,
grp_op: *,
grp_car: |g|,
set_car: |p|
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
ocmon: OCMon,
omon: OMon,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
abmonoid: AbMon,
mon: Mon,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
band: p ∧b q,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
uimplies: b supposing a,
bfalse: ff,
infix_ap: x f y,
so_apply: x[s],
cand: A c∧ B,
oset_of_ocmon: g↓oset,
dset_of_mon: g↓set,
set_car: |p|,
pi1: fst(t),
cdrng: CDRng,
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|,
dset: DSet,
finite_set: FiniteSet{s},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
crng: CRng,
rng: Rng,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
false: False,
fset_map: fs-map(f, a),
guard: {T},
dmon: DMon,
grp: Group{i},
rng_car: |r|,
grp_eq: =b,
rng_eq: =b,
omral_dom: dom(ps),
oal_dom: dom(ps),
true: True,
top: Top,
squash: ↓T,
compose: f o g,
mk_mset: mk_mset(as),
mset_mem: mset_mem,
mem: a ∈b as,
bexists: bexists,
rev_uimplies: rev_uimplies(P;Q),
set_eq: =b,
ball: ball,
loset: LOSet,
poset: POSet{i},
qoset: QOSet,
band_mon: <𝔹,∧b>
Lemmas referenced :
mem_bsubmset,
oset_of_ocmon_wf,
subtype_rel_sets,
abmonoid_wf,
ulinorder_wf,
grp_car_wf,
assert_wf,
infix_ap_wf,
bool_wf,
grp_le_wf,
equal_wf,
grp_eq_wf,
eqtt_to_assert,
cancel_wf,
grp_op_wf,
uall_wf,
monot_wf,
omral_dom_wf2,
omral_scale_wf2,
fset_map_wf,
set_car_wf,
oset_of_ocmon_wf0,
omral_dom_wf,
omralist_wf,
dset_wf,
finite_set_wf,
mset_mem_wf,
omral_scale_wf,
rng_car_wf,
cdrng_wf,
ocmon_wf,
decidable__assert,
assert_functionality_wrt_uiff,
fset_of_mset_wf,
mset_map_wf,
fset_of_mset_mem,
cdrng_properties,
add_grp_of_rng_wf_a,
grp_wf,
eqfun_p_wf,
mk_mset_wf,
map_wf,
set_prod_wf,
dset_of_mon_wf,
map_map,
squash_wf,
true_wf,
mset_wf,
mset_map_char,
uiff_transitivity,
not_wf,
bexists_wf,
set_eq_wf,
bnot_wf,
ball_wf,
assert_of_bnot,
bnot_thru_exists,
omral_scale_dom_pred,
mon_for_wf,
band_mon_wf,
loset_wf,
mon_for_map,
abmonoid_subtype_iabmonoid,
list_wf,
abmonoid_comm,
abdmonoid_abmonoid,
ocmon_subtype_abdmonoid,
subtype_rel_transitivity,
abdmonoid_wf,
iabmonoid_wf,
pi1_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
applyEquality,
sqequalRule,
instantiate,
hypothesis,
because_Cache,
lambdaEquality,
productEquality,
setElimination,
rename,
cumulativity,
universeEquality,
functionEquality,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
setEquality,
independent_pairFormation,
voidElimination,
promote_hyp,
dependent_set_memberEquality,
natural_numberEquality,
isect_memberEquality,
voidEquality,
imageElimination,
imageMemberEquality,
baseClosed,
independent_pairEquality
Latex:
\mforall{}g:OCMon. \mforall{}r:CDRng. \mforall{}k:|g|. \mforall{}v:|r|. \mforall{}ps:|omral(g;r)|.
(\muparrow{}(dom(<k,v>* ps) \msubseteq{}\msubb{} fs-map(\mlambda{}k'.(k' * k), dom(ps))))
Date html generated:
2017_10_01-AM-10_05_47
Last ObjectModification:
2017_03_03-PM-01_16_50
Theory : polynom_3
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