Nuprl Lemma : sparse-signed-rep-exists-ext
∀m:ℤ
(∃L:{-1..2-} List [((m = Σi<||L||.L[i]*2^i ∈ ℤ)
∧ (0 < ||L|| ⇒ (¬(last(L) = 0 ∈ ℤ)))
∧ (∀i:ℕ||L|| - 1. ((L[i] = 0 ∈ ℤ) ∨ (L[i + 1] = 0 ∈ ℤ))))])
Proof
Definitions occuring in Statement :
power-sum: Σi<n.a[i]*x^i,
last: last(L),
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
less_than: a < b,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
subtract: n - m,
add: n + m,
minus: -n,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
sparse-signed-rep-exists,
uniform-comp-nat-induction,
decidable__equal_int,
decidable__int_equal,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
uimplies: b supposing a,
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
guard: {T},
or: P ∨ Q,
squash: ↓T,
decidable__assert,
genrec-ap: genrec-ap,
ifthenelse: if b then t else f fi
Lemmas referenced :
sparse-signed-rep-exists,
lifting-strict-int_eq,
top_wf,
equal_wf,
has-value_wf_base,
base_wf,
is-exception_wf,
lifting-strict-decide,
uniform-comp-nat-induction,
decidable__equal_int,
decidable__int_equal,
decidable__assert
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
isectElimination,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
callbyvalueDecide,
hypothesisEquality,
equalityTransitivity,
equalitySymmetry,
unionEquality,
unionElimination,
sqleReflexivity,
dependent_functionElimination,
independent_functionElimination,
baseApply,
closedConclusion,
decideExceptionCases,
inrFormation,
because_Cache,
imageMemberEquality,
imageElimination,
exceptionSqequal,
inlFormation
Latex:
\mforall{}m:\mBbbZ{}
(\mexists{}L:\{-1..2\msupminus{}\} List [((m = \mSigma{}i<||L||.L[i]*2\^{}i)
\mwedge{} (0 < ||L|| {}\mRightarrow{} (\mneg{}(last(L) = 0)))
\mwedge{} (\mforall{}i:\mBbbN{}||L|| - 1. ((L[i] = 0) \mvee{} (L[i + 1] = 0))))])
Date html generated:
2018_05_21-PM-08_35_47
Last ObjectModification:
2017_07_26-PM-06_00_22
Theory : general
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