Nuprl Lemma : omral_scale_non_zero_vals
∀g:OCMon. ∀r:CDRng. ∀k:|g|. ∀v:|r|. ∀ps:(|g| × |r|) List.
((¬↑(0 ∈b map(λx.(snd(x));ps))) ⇒ (¬↑(0 ∈b map(λx.(snd(x));<k,v>* ps))))
Proof
Definitions occuring in Statement :
omral_scale: <k,v>* ps,
mem: a ∈b as,
map: map(f;as),
list: T List,
assert: ↑b,
pi2: snd(t),
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
lambda: λx.A[x],
product: x:A × B[x],
add_grp_of_rng: r↓+gp,
cdrng: CDRng,
rng_zero: 0,
rng_car: |r|,
ocmon: OCMon,
dset_of_mon: g↓set,
grp_car: |g|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
cdrng: CDRng,
crng: CRng,
rng: Rng,
so_lambda: λ2x.t[x],
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
abdmonoid: AbDMon,
dset_of_mon: g↓set,
set_car: |p|,
pi1: fst(t),
add_grp_of_rng: r↓+gp,
grp_car: |g|,
dmon: DMon,
pi2: snd(t),
so_apply: x[s],
top: Top,
omral_scale: <k,v>* ps,
ycomb: Y,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3],
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
set_eq: =b,
grp_eq: =b,
not: ¬A,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
infix_ap: x f y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
exists: ∃x:A. B[x],
sq_type: SQType(T),
bnot: ¬bb
Lemmas referenced :
cdrng_is_abdmonoid,
list_induction,
grp_car_wf,
rng_car_wf,
not_wf,
assert_wf,
mem_wf,
dset_of_mon_wf,
abdmonoid_wf,
rng_zero_wf,
map_wf,
set_car_wf,
dset_of_mon_wf0,
omral_scale_wf,
list_wf,
map_nil_lemma,
list_ind_nil_lemma,
mem_nil_lemma,
map_cons_lemma,
mem_cons_lemma,
cdrng_wf,
ocmon_wf,
false_wf,
list_ind_cons_lemma,
not_over_or,
equal_wf,
or_wf,
iff_transitivity,
bor_wf,
infix_ap_wf,
bool_wf,
rng_eq_wf,
iff_weakening_uiff,
assert_of_bor,
assert_of_rng_eq,
cdrng_subtype_drng,
rng_times_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
cons_wf,
grp_op_wf,
uiff_transitivity,
equal-wf-T-base,
bnot_wf,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productEquality,
setElimination,
rename,
because_Cache,
hypothesis,
sqequalRule,
productElimination,
lambdaEquality,
functionEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
addLevel,
impliesFunctionality,
independent_pairFormation,
orFunctionality,
impliesLevelFunctionality,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
independent_pairEquality,
baseClosed
Latex:
\mforall{}g:OCMon. \mforall{}r:CDRng. \mforall{}k:|g|. \mforall{}v:|r|. \mforall{}ps:(|g| \mtimes{} |r|) List.
((\mneg{}\muparrow{}(0 \mmember{}\msubb{} map(\mlambda{}x.(snd(x));ps))) {}\mRightarrow{} (\mneg{}\muparrow{}(0 \mmember{}\msubb{} map(\mlambda{}x.(snd(x));<k,v>* ps))))
Date html generated:
2017_10_01-AM-10_05_41
Last ObjectModification:
2017_03_03-PM-01_12_02
Theory : polynom_3
Home
Index