Nuprl Lemma : omral_times_non_zero_vals
∀g:OCMon. ∀r:CDRng. ∀ps,qs:(|g| × |r|) List. ((¬↑(0 ∈b map(λx.(snd(x));qs))) ⇒ (¬↑(0 ∈b map(λx.(snd(x));ps ** qs))))
Proof
Definitions occuring in Statement :
omral_times: ps ** qs,
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,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
ocmon: OCMon,
abmonoid: AbMon,
mon: Mon,
cdrng: CDRng,
crng: CRng,
rng: Rng,
or: P ∨ Q,
omral_times: ps ** qs,
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,
subtype_rel: A ⊆r B,
guard: {T},
rng_car: |r|,
pi1: fst(t),
set_car: |p|,
dset_of_mon: g↓set,
grp_car: |g|,
add_grp_of_rng: r↓+gp,
pi2: snd(t),
cons: [a / b],
le: A ≤ B,
less_than': less_than'(a;b),
colength: colength(L),
nil: [],
it: ⋅,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
squash: ↓T,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
decidable: Dec(P)
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
grp_car_wf,
rng_car_wf,
list-cases,
cdrng_is_abdmonoid,
list_ind_nil_lemma,
map_nil_lemma,
mem_nil_lemma,
not_wf,
assert_wf,
mem_wf,
dset_of_mon_wf,
abdmonoid_dmon,
rng_zero_wf,
subtype_rel_self,
set_car_wf,
dset_of_mon_wf0,
add_grp_of_rng_wf,
map_wf,
mon_subtype_grp_sig,
dmon_subtype_mon,
subtype_rel_transitivity,
abdmonoid_wf,
dmon_wf,
mon_wf,
grp_sig_wf,
product_subtype_list,
colength-cons-not-zero,
colength_wf_list,
istype-false,
le_wf,
subtract-1-ge-0,
subtype_base_sq,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
int_subtype_base,
spread_cons_lemma,
decidable__equal_int,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
decidable__le,
list_ind_cons_lemma,
nat_wf,
list_wf,
cdrng_wf,
ocmon_wf,
omral_scale_wf,
omral_times_wf,
omral_plus_non_zero_vals,
omral_scale_non_zero_vals
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
universeIsType,
functionIsTypeImplies,
inhabitedIsType,
productEquality,
because_Cache,
unionElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
applyEquality,
instantiate,
productIsType,
promote_hyp,
hypothesis_subsumption,
equalityIsType1,
dependent_set_memberEquality_alt,
applyLambdaEquality,
imageElimination,
equalityIsType4,
baseApply,
closedConclusion,
baseClosed,
intEquality
Latex:
\mforall{}g:OCMon. \mforall{}r:CDRng. \mforall{}ps,qs:(|g| \mtimes{} |r|) List.
((\mneg{}\muparrow{}(0 \mmember{}\msubb{} map(\mlambda{}x.(snd(x));qs))) {}\mRightarrow{} (\mneg{}\muparrow{}(0 \mmember{}\msubb{} map(\mlambda{}x.(snd(x));ps ** qs))))
Date html generated:
2019_10_16-PM-01_09_05
Last ObjectModification:
2018_10_08-PM-01_30_25
Theory : polynom_3
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