Nuprl Lemma : sparse-signed-rep_wf
∀[m:ℤ]
(sparse-signed-rep(m) ∈ {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 :
sparse-signed-rep: sparse-signed-rep(m),
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,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
subtract: n - m,
add: n + m,
minus: -n,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
sq_exists: ∃x:A [B[x]],
sparse-signed-rep: sparse-signed-rep(m),
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
int_seg: {i..j-},
uimplies: b supposing a,
guard: {T},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
less_than: a < b,
squash: ↓T,
so_apply: x[s],
assert: ↑b,
ifthenelse: if b then t else f fi ,
btrue: tt,
select: L[n],
nil: [],
it: ⋅,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
less_than': less_than'(a;b),
cons: [a / b],
bfalse: ff,
uiff: uiff(P;Q)
Lemmas referenced :
sparse-signed-rep-exists-ext,
subtype_rel_self,
sq_exists_wf,
list_wf,
int_seg_wf,
equal-wf-base-T,
int_subtype_base,
power-sum_wf,
length_wf_nat,
select_wf,
int_seg_properties,
length_wf,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
less_than_wf,
not_wf,
equal-wf-T-base,
last_wf,
subtype_rel_list,
list-cases,
null_nil_lemma,
length_of_nil_lemma,
stuck-spread,
base_wf,
product_subtype_list,
null_cons_lemma,
length_of_cons_lemma,
false_wf,
all_wf,
subtract_wf,
or_wf,
subtract-is-int-iff,
itermSubtract_wf,
int_term_value_subtract_lemma,
itermAdd_wf,
int_term_value_add_lemma,
equal_wf,
evalall-reduce,
list-valueall-type,
set-valueall-type,
lelt_wf,
int-valueall-type
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
sqequalHypSubstitution,
isectElimination,
functionEquality,
intEquality,
minusEquality,
natural_numberEquality,
lambdaEquality,
productEquality,
hypothesisEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
productElimination,
dependent_functionElimination,
unionElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
imageElimination,
baseClosed,
lambdaFormation,
promote_hyp,
hypothesis_subsumption,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
baseApply,
closedConclusion,
addEquality,
axiomEquality
Latex:
\mforall{}[m:\mBbbZ{}]
(sparse-signed-rep(m) \mmember{} \{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_36_01
Last ObjectModification:
2018_05_19-PM-05_05_53
Theory : general
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