Nuprl Lemma : shadow-vec-property
∀[as,bs:ℤ List]. ∀[i:ℕ||as||].
(∀[xs:ℤ List]
0 ≤ shadow-vec(i;as;bs) ⋅ xs\i supposing (||xs|| = ||as|| ∈ ℤ) ∧ (0 ≤ as ⋅ xs) ∧ (0 ≤ bs ⋅ xs)) supposing
((bs[i] ≤ 0) and
(0 ≤ as[i]) and
(||as|| = ||bs|| ∈ ℤ))
Proof
Definitions occuring in Statement :
shadow-vec: shadow-vec(i;as;bs),
list-delete: as\i,
integer-dot-product: as ⋅ bs,
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
and: P ∧ Q,
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,
and: P ∧ Q,
all: ∀x:A. B[x],
le: A ≤ B,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
cand: A c∧ B,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
guard: {T},
nat: ℕ,
int-vec-mul: a * as,
shadow-vec: shadow-vec(i;as;bs),
has-value: (a)↓,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
uiff: uiff(P;Q),
top: Top
Lemmas referenced :
add_functionality_wrt_le,
select_wf,
le_reflexive,
le_witness_for_triv,
list_subtype_base,
int_subtype_base,
istype-le,
integer-dot-product_wf,
sq_stable__le,
less_than_transitivity1,
length_wf,
le_weakening,
istype-int,
int_seg_wf,
list_wf,
minus-one-mul,
zero-add,
add-commutes,
add-mul-special,
zero-mul,
mul_preserves_le,
minus-one-mul-top,
int-dot-mul-left,
mul-associates,
mul-commutes,
add_nat_wf,
int-vec-mul_wf,
int-dot-add-left,
map-length,
list-valueall-type,
int-valueall-type,
evalall-reduce,
value-type-has-value,
list-value-type,
int-vec-add_wf,
int-dot-select,
int_seg_subtype_nat,
istype-false,
less_than_wf,
squash_wf,
true_wf,
length-int-vec-add,
length-int-vec-mul,
equal_wf,
istype-universe,
iff_weakening_equal,
subtype_rel_self,
subtype_base_sq,
select-int-vec-mul,
lelt_wf,
set_subtype_base,
add-is-int-iff,
multiply-is-int-iff,
false_wf,
select-int-vec-add,
list-delete_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
extract_by_obid,
isectElimination,
because_Cache,
minusEquality,
independent_isectElimination,
hypothesis,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
productIsType,
equalityIstype,
inhabitedIsType,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
applyEquality,
intEquality,
sqequalBase,
setElimination,
rename,
isect_memberEquality_alt,
isectIsTypeImplies,
independent_pairFormation,
natural_numberEquality,
independent_functionElimination,
imageMemberEquality,
imageElimination,
universeIsType,
multiplyEquality,
Error :memTop,
dependent_set_memberEquality_alt,
lambdaFormation_alt,
callbyvalueReduce,
lambdaEquality_alt,
instantiate,
universeEquality,
cumulativity,
dependent_set_memberEquality,
lambdaEquality,
voidEquality,
voidElimination,
isect_memberEquality,
lambdaFormation
Latex:
\mforall{}[as,bs:\mBbbZ{} List]. \mforall{}[i:\mBbbN{}||as||].
(\mforall{}[xs:\mBbbZ{} List]
0 \mleq{} shadow-vec(i;as;bs) \mcdot{} xs\mbackslash{}i
supposing (||xs|| = ||as||) \mwedge{} (0 \mleq{} as \mcdot{} xs) \mwedge{} (0 \mleq{} bs \mcdot{} xs)) supposing
((bs[i] \mleq{} 0) and
(0 \mleq{} as[i]) and
(||as|| = ||bs||))
Date html generated:
2020_05_19-PM-09_38_13
Last ObjectModification:
2019_12_31-PM-00_59_38
Theory : omega
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