### Nuprl Lemma : int-prod-factor

`∀[n:ℕ]. ∀[f,g:ℕn ⟶ ℤ].  (Π(f[x] * g[x] | x < n) = (Π(f[x] | x < n) * Π(g[x] | x < n)) ∈ 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` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  member: `t ∈ T` top: `Top` uall: `∀[x:A]. B[x]` so_apply: `x[s]` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` uimplies: `b supposing a` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` prop: `ℙ` nat: `ℕ` so_lambda: `λ2x.t[x]` true: `True` ge: `i ≥ j ` int-prod: `Π(f[x] | x < k)` lt_int: `i <z j` subtract: `n - m` ifthenelse: `if b then t else f fi ` btrue: `tt` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` bfalse: `ff` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` squash: `↓T` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` primrec: `primrec(n;b;c)` less_than: `a < b` less_than': `less_than'(a;b)` has-value: `(a)↓`
Lemmas referenced :  int_seg_wf decidable__lt full-omega-unsat intformand_wf intformnot_wf intformless_wf itermVar_wf itermSubtract_wf itermConstant_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_subtract_lemma int_term_value_constant_lemma int_formula_prop_wf lelt_wf subtract_wf decidable__equal_int intformeq_wf int_formula_prop_eq_lemma decidable__le intformle_wf int_formula_prop_le_lemma int-prod_wf le_wf nat_wf nat_properties ge_wf less_than_wf primrec-unroll lt_int_wf bool_wf eqtt_to_assert assert_of_lt_int eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot squash_wf true_wf subtype_rel_self iff_weakening_equal top_wf itermMultiply_wf int_term_value_mul_lemma value-type-has-value int-value-type primrec_wf itermAdd_wf int_term_value_add_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity natural_numberEquality isect_memberEquality voidElimination voidEquality functionEquality cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin hypothesis intEquality because_Cache hypothesisEquality lambdaEquality applyEquality functionExtensionality setElimination rename dependent_set_memberEquality productElimination independent_pairFormation dependent_functionElimination unionElimination independent_isectElimination approximateComputation independent_functionElimination dependent_pairFormation int_eqEquality sqequalRule multiplyEquality equalityTransitivity equalitySymmetry isect_memberFormation intWeakElimination lambdaFormation axiomEquality equalityElimination promote_hyp instantiate cumulativity imageElimination universeEquality imageMemberEquality baseClosed addEquality minusEquality lessCases sqequalAxiom callbyvalueReduce

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[f,g:\mBbbN{}n  {}\mrightarrow{}  \mBbbZ{}].    (\mPi{}(f[x]  *  g[x]  |  x  <  n)  =  (\mPi{}(f[x]  |  x  <  n)  *  \mPi{}(g[x]  |  x  <  n)))

Date html generated: 2018_05_21-PM-00_29_45
Last ObjectModification: 2018_05_19-AM-06_54_57

Theory : int_2

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