### Nuprl Lemma : int-prod-split

`∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ]. ∀[m:ℕn + 1].  (Π(f[x] | x < n) = (Π(f[x] | x < m) * Π(f[x + m] | x < n - m)) ∈ ℤ)`

Proof

Definitions occuring in Statement :  int-prod: `Π(f[x] | x < k)` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` so_apply: `x[s]` function: `x:A ⟶ B[x]` multiply: `n * m` subtract: `n - m` add: `n + m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` uimplies: `b supposing a` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` all: `∀x:A. B[x]` top: `Top` and: `P ∧ Q` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` int_seg: `{i..j-}` decidable: `Dec(P)` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` subtract: `n - m` lelt: `i ≤ j < k` int-prod: `Π(f[x] | x < k)` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` bfalse: `ff` subtype_rel: `A ⊆r B` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` rev_implies: `P `` Q` iff: `P `⇐⇒` Q` lt_int: `i <z j` le: `A ≤ B` less_than': `less_than'(a;b)` true: `True`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename intWeakElimination Error :lambdaFormation_alt,  natural_numberEquality independent_isectElimination approximateComputation independent_functionElimination Error :dependent_pairFormation_alt,  Error :lambdaEquality_alt,  int_eqEquality dependent_functionElimination Error :isect_memberEquality_alt,  voidElimination sqequalRule independent_pairFormation Error :universeIsType,  axiomEquality Error :isectIsTypeImplies,  Error :inhabitedIsType,  Error :functionIsTypeImplies,  addEquality Error :functionIsType,  unionElimination instantiate cumulativity intEquality because_Cache equalityTransitivity equalitySymmetry hypothesis_subsumption productElimination equalityElimination Error :equalityIsType4,  baseApply closedConclusion baseClosed applyEquality promote_hyp Error :equalityIsType1,  multiplyEquality Error :dependent_set_memberEquality_alt,  Error :productIsType,  minusEquality

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[f:\mBbbN{}n  {}\mrightarrow{}  \mBbbZ{}].  \mforall{}[m:\mBbbN{}n  +  1].    (\mPi{}(f[x]  |  x  <  n)  =  (\mPi{}(f[x]  |  x  <  m)  *  \mPi{}(f[x  +  m]  |  x  <  n  -  m)))

Date html generated: 2019_06_20-PM-01_18_37
Last ObjectModification: 2018_10_15-PM-02_14_19

Theory : int_2

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