### Nuprl Lemma : sum-l_sum

`∀[n:ℕ]. ∀[a:ℕn ⟶ ℤ].  (Σ(a[i] | i < n) = l_sum(map(λi.a[i];upto(n))) ∈ ℤ)`

Proof

Definitions occuring in Statement :  l_sum: `l_sum(L)` upto: `upto(n)` sum: `Σ(f[x] | x < k)` map: `map(f;as)` int_seg: `{i..j-}` nat: `ℕ` uall: `∀[x:A]. B[x]` so_apply: `x[s]` lambda: `λx.A[x]` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat: `ℕ` so_apply: `x[s]` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` prop: `ℙ` uimplies: `b supposing a` sq_type: `SQType(T)` all: `∀x:A. B[x]` implies: `P `` Q` guard: `{T}` squash: `↓T` ge: `i ≥ j ` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` so_lambda: `λ2x.t[x]` label: `...\$L... t` true: `True` subtype_rel: `A ⊆r B` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  l_sum-sum int_seg_wf upto_wf lelt_wf l_member_wf subtype_base_sq int_subtype_base nat_wf sum_wf squash_wf true_wf nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf length_upto le_wf equal_wf select_upto int_seg_properties decidable__equal_int intformeq_wf int_formula_prop_eq_lemma decidable__lt intformless_wf int_formula_prop_less_lemma iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality setElimination rename because_Cache hypothesis lambdaEquality applyEquality functionExtensionality hypothesisEquality dependent_set_memberEquality setEquality instantiate cumulativity intEquality independent_isectElimination dependent_functionElimination equalityTransitivity equalitySymmetry independent_functionElimination sqequalRule functionEquality isect_memberEquality axiomEquality imageElimination unionElimination dependent_pairFormation int_eqEquality voidElimination voidEquality independent_pairFormation computeAll universeEquality productElimination imageMemberEquality baseClosed

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[a:\mBbbN{}n  {}\mrightarrow{}  \mBbbZ{}].    (\mSigma{}(a[i]  |  i  <  n)  =  l\_sum(map(\mlambda{}i.a[i];upto(n))))

Date html generated: 2017_04_17-AM-08_38_49
Last ObjectModification: 2017_02_27-PM-04_57_16

Theory : list_1

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