Nuprl Lemma : satisfies-shadow-inequalities
∀[n:ℕ]. ∀[ineqs:{L:ℤ List| ||L|| = n ∈ ℤ} List]. ∀[i:ℕ+n]. ∀[xs:ℤ List].
(∀as∈shadow-inequalities(i;ineqs).xs\i ⋅ as ≥0) supposing (∀as∈ineqs.xs ⋅ as ≥0)
Proof
Definitions occuring in Statement :
shadow-inequalities: shadow-inequalities(i;ineqs),
list-delete: as\i,
satisfies-integer-inequality: xs ⋅ as ≥0,
l_all: (∀x∈L.P[x]),
length: ||as||,
list: T List,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
l_all: (∀x∈L.P[x]),
all: ∀x:A. B[x],
satisfies-integer-inequality: xs ⋅ as ≥0,
and: P ∧ Q,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
ge: i ≥ j ,
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
implies: P ⇒ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
sq_type: SQType(T),
guard: {T},
sq_stable: SqStable(P),
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
subtract: n - m,
less_than': less_than'(a;b),
true: True,
shadow-inequalities: shadow-inequalities(i;ineqs),
has-value: (a)↓,
top: Top,
less_than: a < b,
cons: [a / b],
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
bfalse: ff,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
list-delete: as\i
Lemmas referenced :
member-less_than,
le_witness_for_triv,
l_all_wf,
list_wf,
equal-wf-base,
list_subtype_base,
int_subtype_base,
set_subtype_base,
le_wf,
istype-int,
satisfies-integer-inequality_wf,
l_member_wf,
int_seg_wf,
istype-nat,
subtype_rel_list,
equal-wf-base-T,
subtype_base_sq,
select_wf,
sq_stable__le,
set_wf,
equal_wf,
less_than_transitivity1,
length_wf,
le_weakening,
list-set-type,
map_wf,
list-delete_wf,
filter_wf5,
eq_int_wf,
decidable__le,
istype-false,
not-le-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
le-add-cancel,
list-valueall-type,
int-valueall-type,
evalall-reduce,
value-type-has-value,
list-value-type,
l_all_append,
subtract_wf,
set-value-type,
squash_wf,
true_wf,
length-shadow-vec,
false_wf,
iff_weakening_equal,
shadow-vec_wf,
lelt_wf,
lt_int_wf,
equal_functionality_wrt_subtype_rel2,
less_than_wf,
eager-product-map_wf,
istype-universe,
length-list-delete,
int_seg_subtype_nat,
subtype_rel_self,
istype-void,
filter_type,
l_all_implies_l_all_filter,
assert_wf,
l_all_eager_product-map,
le_weakening2,
assert_of_lt_int,
add-swap,
minus-minus,
le_antisymmetry_iff,
less-iff-le,
not-lt-2,
decidable__lt,
le-add-cancel2,
not-ge-2,
ge_wf,
hd_wf,
product_subtype_list,
list-cases,
length_of_nil_lemma,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
bool_cases_sqequal,
eqff_to_assert,
less_than_irreflexivity,
istype-top,
eqtt_to_assert,
length_of_cons_lemma,
reduce_hd_cons_lemma,
spread_cons_lemma,
shadow-vec-property,
l_all_iff,
member_filter_2,
member_map,
assert_of_eq_int,
add-zero,
top_wf,
int-dot-select,
zero-mul,
integer-dot-product_wf,
eager-append-is-append
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality_alt,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
axiomEquality,
hypothesis,
extract_by_obid,
isectElimination,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
functionIsTypeImplies,
inhabitedIsType,
universeIsType,
setEquality,
intEquality,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
because_Cache,
natural_numberEquality,
setIsType,
equalityIstype,
sqequalBase,
isect_memberEquality_alt,
isectIsTypeImplies,
lambdaFormation,
lambdaEquality,
instantiate,
cumulativity,
independent_functionElimination,
imageMemberEquality,
imageElimination,
lambdaFormation_alt,
dependent_set_memberEquality_alt,
unionElimination,
independent_pairFormation,
voidElimination,
addEquality,
Error :memTop,
minusEquality,
callbyvalueReduce,
universeEquality,
equalityUniverse,
levelHypothesis,
dependent_set_memberEquality,
isect_memberEquality,
voidEquality,
dependent_pairFormation,
productEquality,
equalityIsType4,
productIsType,
hyp_replacement,
promote_hyp,
hypothesis_subsumption,
equalityIsType1,
equalityIsType2,
dependent_pairFormation_alt,
axiomSqEquality,
lessCases,
equalityElimination,
isect_memberFormation
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[ineqs:\{L:\mBbbZ{} List| ||L|| = n\} List]. \mforall{}[i:\mBbbN{}\msupplus{}n]. \mforall{}[xs:\mBbbZ{} List].
(\mforall{}as\mmember{}shadow-inequalities(i;ineqs).xs\mbackslash{}i \mcdot{} as \mgeq{}0) supposing (\mforall{}as\mmember{}ineqs.xs \mcdot{} as \mgeq{}0)
Date html generated:
2020_05_19-PM-09_39_28
Last ObjectModification:
2020_01_04-PM-07_58_43
Theory : omega
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