Nuprl Lemma : divisors-sum_wf

`∀[n:ℕ+]. ∀[f:ℕ+n + 1 ⟶ ℤ].  (Σ i|n. f[i]  ∈ ℤ)`

Proof

Definitions occuring in Statement :  divisors-sum: `Σ i|n. f[i] ` int_seg: `{i..j-}` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` so_apply: `x[s]` member: `t ∈ T` function: `x:A ⟶ B[x]` add: `n + m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` divisors-sum: `Σ i|n. f[i] ` subtype_rel: `A ⊆r B` so_lambda: `λ2x.t[x]` nat_plus: `ℕ+` int_seg: `{i..j-}` nequal: `a ≠ b ∈ T ` guard: `{T}` lelt: `i ≤ j < k` and: `P ∧ Q` not: `¬A` implies: `P `` Q` uimplies: `b supposing a` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` all: `∀x:A. B[x]` top: `Top` prop: `ℙ` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` so_apply: `x[s]` decidable: `Dec(P)` or: `P ∨ Q` subtract: `n - m` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b`
Lemmas referenced :  sum_wf nat_plus_subtype_nat eq_int_wf int_seg_properties nat_plus_properties satisfiable-full-omega-tt intformand_wf intformeq_wf itermAdd_wf itermVar_wf itermConstant_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_add_lemma int_term_value_var_lemma int_term_value_constant_lemma int_formula_prop_le_lemma int_formula_prop_wf equal-wf-base int_subtype_base bool_wf eqtt_to_assert assert_of_eq_int int_seg_wf add-member-int_seg2 decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma add-subtract-cancel decidable__lt intformless_wf int_formula_prop_less_lemma lelt_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int nat_plus_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut sqequalRule extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality applyEquality hypothesis lambdaEquality remainderEquality setElimination rename because_Cache addEquality natural_numberEquality productElimination lambdaFormation independent_isectElimination dependent_pairFormation int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll baseApply closedConclusion baseClosed unionElimination equalityElimination functionExtensionality dependent_set_memberEquality equalityTransitivity equalitySymmetry promote_hyp instantiate cumulativity independent_functionElimination axiomEquality functionEquality

Latex:
\mforall{}[n:\mBbbN{}\msupplus{}].  \mforall{}[f:\mBbbN{}\msupplus{}n  +  1  {}\mrightarrow{}  \mBbbZ{}].    (\mSigma{}  i|n.  f[i]    \mmember{}  \mBbbZ{})

Date html generated: 2018_05_21-PM-07_31_24
Last ObjectModification: 2017_07_26-PM-05_06_38

Theory : general

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