Nuprl Lemma : small-sparse-rep
∀r:{-2..3-}
(∃L:{-1..2-} List [((r = Σi<||L||.L[i]*2^i ∈ ℤ)
∧ (||L|| = 2 ∈ 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,
select: L[n],
length: ||as||,
list: T List,
int_seg: {i..j-},
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
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 :
all: ∀x:A. B[x],
member: t ∈ T,
int_seg: {i..j-},
decidable: Dec(P),
or: P ∨ Q,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
sq_exists: ∃x:A [B[x]],
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
true: True,
subtract: n - m,
power-sum: Σi<n.a[i]*x^i,
cand: A c∧ B,
sum: Σ(f[x] | x < k),
sum_aux: sum_aux(k;v;i;x.f[x]),
select: L[n],
cons: [a / b],
exp: i^n,
primrec: primrec(n;b;c),
nat: ℕ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s]
Lemmas referenced :
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
int_seg_properties,
int_seg_subtype_special,
int_seg_cases,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
cons_wf,
istype-false,
le_wf,
less_than_wf,
nil_wf,
length_of_cons_lemma,
length_of_nil_lemma,
sum_wf,
select_wf,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
exp_wf2,
int_seg_subtype_nat,
select-cons-hd,
list_subtype_base,
set_subtype_base,
lelt_wf,
length_wf_nat,
subtract_wf,
length_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
intformeq_wf,
int_formula_prop_eq_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
minusEquality,
natural_numberEquality,
unionElimination,
instantiate,
isectElimination,
cumulativity,
intEquality,
independent_isectElimination,
because_Cache,
independent_functionElimination,
equalityTransitivity,
equalitySymmetry,
hypothesis_subsumption,
sqequalRule,
productElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
dependent_set_memberFormation_alt,
closedConclusion,
dependent_set_memberEquality_alt,
imageMemberEquality,
baseClosed,
productIsType,
multiplyEquality,
addEquality,
applyEquality,
inlFormation_alt,
equalityIsType4,
inhabitedIsType,
baseApply,
functionIsType,
unionIsType,
inrFormation_alt
Latex:
\mforall{}r:\{-2..3\msupminus{}\}
(\mexists{}L:\{-1..2\msupminus{}\} List [((r = \mSigma{}i<||L||.L[i]*2\^{}i)
\mwedge{} (||L|| = 2)
\mwedge{} (\mforall{}i:\mBbbN{}||L|| - 1. ((L[i] = 0) \mvee{} (L[i + 1] = 0))))])
Date html generated:
2019_10_15-AM-11_27_10
Last ObjectModification:
2018_10_11-PM-10_14_48
Theory : general
Home
Index