### Nuprl Lemma : non_neg_sum

`∀[n:ℕ]. ∀[f:ℕn ⟶ ℤ].  0 ≤ Σ(f[x] | x < n) supposing ∀x:ℕn. (0 ≤ f[x])`

Proof

Definitions occuring in Statement :  sum: `Σ(f[x] | x < k)` int_seg: `{i..j-}` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` le: `A ≤ B` all: `∀x:A. B[x]` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` so_lambda: `λ2x.t[x]` so_apply: `x[s]` nat: `ℕ` all: `∀x:A. B[x]` implies: `P `` Q` false: `False` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` le: `A ≤ B` less_than': `less_than'(a;b)` decidable: `Dec(P)` or: `P ∨ Q` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` lelt: `i ≤ j < k` less_than: `a < b` guard: `{T}`
Lemmas referenced :  sum_wf add_nat_wf nat_wf zero-le-nat lelt_wf decidable__lt subtype_rel_self int_seg_subtype subtype_rel_dep_function int_term_value_subtract_lemma int_formula_prop_not_lemma itermSubtract_wf intformnot_wf subtract_wf decidable__le false_wf primrec0_lemma le_wf all_wf primrec_wf less_than'_wf less_than_wf ge_wf int_formula_prop_wf int_formula_prop_less_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_and_lemma intformless_wf itermVar_wf itermConstant_wf intformle_wf intformand_wf satisfiable-full-omega-tt nat_properties int_seg_wf sum-as-primrec
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule lemma_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality lambdaEquality applyEquality natural_numberEquality setElimination rename hypothesis lambdaFormation intWeakElimination independent_isectElimination dependent_pairFormation int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll independent_functionElimination productElimination independent_pairEquality addEquality because_Cache axiomEquality equalityTransitivity equalitySymmetry functionEquality dependent_set_memberEquality unionElimination

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[f:\mBbbN{}n  {}\mrightarrow{}  \mBbbZ{}].    0  \mleq{}  \mSigma{}(f[x]  |  x  <  n)  supposing  \mforall{}x:\mBbbN{}n.  (0  \mleq{}  f[x])

Date html generated: 2016_05_14-AM-07_31_51
Last ObjectModification: 2016_01_14-PM-09_56_55

Theory : int_2

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