### Nuprl Lemma : finite-partition

`∀n,k:ℕ. ∀c:ℕn ⟶ ℕk.`
`  ∃p:ℕk ⟶ (ℕ List)`
`   ((Σ(||p j|| | j < k) = n ∈ ℤ)`
`   ∧ (∀j:ℕk. ∀x,y:ℕ||p j||.  p j[x] > p j[y] supposing x < y)`
`   ∧ (∀j:ℕk. ∀x:ℕ||p j||.  (p j[x] < n c∧ ((c p j[x]) = j ∈ ℤ))))`

Proof

Definitions occuring in Statement :  sum: `Σ(f[x] | x < k)` select: `L[n]` length: `||as||` list: `T List` int_seg: `{i..j-}` nat: `ℕ` less_than: `a < b` uimplies: `b supposing a` cand: `A c∧ B` gt: `i > j` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` and: `P ∧ Q` apply: `f a` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` member: `t ∈ T` uall: `∀[x:A]. B[x]` nat: `ℕ` exists: `∃x:A. B[x]` and: `P ∧ Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` subtype_rel: `A ⊆r B` uimplies: `b supposing a` int_seg: `{i..j-}` prop: `ℙ` guard: `{T}` lelt: `i ≤ j < k` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` top: `Top` less_than: `a < b` squash: `↓T` gt: `i > j` cand: `A c∧ B` le: `A ≤ B` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` true: `True` so_apply: `x[s1;s2]` so_lambda: `λ2x y.t[x; y]` it: `⋅` nil: `[]` select: `L[n]` less_than': `less_than'(a;b)` uiff: `uiff(P;Q)` subtract: `n - m` bool: `𝔹` unit: `Unit` btrue: `tt` ifthenelse: `if b then t else f fi ` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` nequal: `a ≠ b ∈ T ` cons: `[a / b]`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :lambdaFormation_alt,  cut thin rename setElimination sqequalRule Error :functionIsType,  Error :universeIsType,  introduction extract_by_obid hypothesis sqequalHypSubstitution isectElimination natural_numberEquality hypothesisEquality Error :productIsType,  Error :equalityIsType4,  Error :inhabitedIsType,  Error :lambdaEquality_alt,  applyEquality functionExtensionality because_Cache Error :isectIsType,  independent_isectElimination productElimination dependent_functionElimination unionElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  int_eqEquality Error :isect_memberEquality_alt,  voidElimination independent_pairFormation Error :dependent_set_memberEquality_alt,  imageElimination Error :equalityIsType3,  equalityTransitivity equalitySymmetry applyLambdaEquality intEquality Error :setIsType,  functionEquality productEquality dependent_set_memberEquality computeAll isect_memberFormation imageMemberEquality universeEquality voidEquality isect_memberEquality baseClosed lambdaEquality dependent_pairFormation lambdaFormation addEquality minusEquality multiplyEquality equalityElimination Error :equalityIsType1,  promote_hyp instantiate cumulativity hyp_replacement Error :isect_memberFormation_alt,  pointwiseFunctionality baseApply closedConclusion

Latex:
\mforall{}n,k:\mBbbN{}.  \mforall{}c:\mBbbN{}n  {}\mrightarrow{}  \mBbbN{}k.
\mexists{}p:\mBbbN{}k  {}\mrightarrow{}  (\mBbbN{}  List)
((\mSigma{}(||p  j||  |  j  <  k)  =  n)
\mwedge{}  (\mforall{}j:\mBbbN{}k.  \mforall{}x,y:\mBbbN{}||p  j||.    p  j[x]  >  p  j[y]  supposing  x  <  y)
\mwedge{}  (\mforall{}j:\mBbbN{}k.  \mforall{}x:\mBbbN{}||p  j||.    (p  j[x]  <  n  c\mwedge{}  ((c  p  j[x])  =  j))))

Date html generated: 2019_06_20-PM-01_32_14
Last ObjectModification: 2018_10_05-AM-09_40_06

Theory : list_1

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