Nuprl Lemma : satisfies-shadow_inequalities
∀[n:{2...}]
∀ineqs:{L:ℤ List| ||L|| = n ∈ ℤ} List
((∃xs:ℤ List. (∀as∈ineqs.xs ⋅ as ≥0)) ⇒ (∃xs:ℤ List. (∀as∈shadow_inequalities(ineqs).xs ⋅ as ≥0)))
Proof
Definitions occuring in Statement :
shadow_inequalities: shadow_inequalities(ineqs),
satisfies-integer-inequality: xs ⋅ as ≥0,
l_all: (∀x∈L.P[x]),
length: ||as||,
list: T List,
int_upper: {i...},
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
int_upper: {i...},
prop: ℙ,
so_apply: x[s],
or: P ∨ Q,
shadow_inequalities: shadow_inequalities(ineqs),
nil: [],
it: ⋅,
cons: [a / b],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
top: Top,
nat_plus: ℕ+,
le: A ≤ B,
and: P ∧ Q,
decidable: Dec(P),
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
less_than': less_than'(a;b),
true: True,
nat: ℕ,
listp: A List+,
guard: {T},
subtract: n - m,
int_seg: {i..j-},
sq_stable: SqStable(P),
lelt: i ≤ j < k,
squash: ↓T,
sq_type: SQType(T),
ge: i ≥ j ,
less_than: a < b,
has-value: (a)↓
Lemmas referenced :
set_wf,
list_wf,
equal_wf,
length_wf,
list-cases,
product_subtype_list,
l_all_wf,
equal-wf-base-T,
list_subtype_base,
int_subtype_base,
satisfies-integer-inequality_wf,
istype-int,
set_subtype_base,
le_wf,
l_member_wf,
int_upper_wf,
nil_wf,
l_all_nil,
istype-void,
l_all_wf_nil,
max_tl_coeffs_wf,
decidable__lt,
istype-false,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
less_than_wf,
length_of_cons_lemma,
non_neg_length,
length_wf_nat,
cons_wf,
index-of-min_wf,
subtype_rel_sets,
subtract_wf,
le_antisymmetry_iff,
condition-implies-le,
minus-add,
minus-minus,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-associates,
subtype_rel_set,
int_seg_wf,
all_wf,
select_wf,
sq_stable__le,
subtype_base_sq,
nat_wf,
decidable__le,
not-le-2,
subtype_rel_self,
le_reflexive,
one-mul,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
minus-zero,
add-zero,
omega-shadow,
int_upper_properties,
nat_properties,
value-type-has-value,
int-value-type,
list-delete_wf,
shadow-inequalities_wf,
add-member-int_seg2,
le-add-cancel2,
lelt_wf,
satisfies-shadow-inequalities,
upper_subtype_nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
Error :lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
intEquality,
hypothesis,
sqequalRule,
Error :lambdaEquality_alt,
hypothesisEquality,
setElimination,
rename,
Error :universeIsType,
dependent_functionElimination,
unionElimination,
promote_hyp,
hypothesis_subsumption,
Error :productIsType,
setEquality,
baseApply,
closedConclusion,
baseClosed,
because_Cache,
applyEquality,
independent_isectElimination,
Error :setIsType,
Error :inhabitedIsType,
Error :equalityIsType4,
natural_numberEquality,
Error :dependent_pairFormation_alt,
Error :isect_memberEquality_alt,
voidElimination,
Error :equalityIsType1,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
Error :dependent_set_memberEquality_alt,
independent_pairFormation,
sqequalIntensionalEquality,
addEquality,
minusEquality,
functionEquality,
imageMemberEquality,
imageElimination,
instantiate,
cumulativity,
multiplyEquality,
callbyvalueReduce
Latex:
\mforall{}[n:\{2...\}]
\mforall{}ineqs:\{L:\mBbbZ{} List| ||L|| = n\} List
((\mexists{}xs:\mBbbZ{} List. (\mforall{}as\mmember{}ineqs.xs \mcdot{} as \mgeq{}0))
{}\mRightarrow{} (\mexists{}xs:\mBbbZ{} List. (\mforall{}as\mmember{}shadow\_inequalities(ineqs).xs \mcdot{} as \mgeq{}0)))
Date html generated:
2019_06_20-PM-00_50_40
Last ObjectModification:
2018_10_03-AM-00_13_30
Theory : omega
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