### 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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