### Nuprl Lemma : gen-divisors-sum_wf

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

Proof

Definitions occuring in Statement :  gen-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` crng: `CRng` rng_car: `|r|`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` gen-divisors-sum: `Σ i|n. f[i]` nat_plus: `ℕ+` subtype_rel: `A ⊆r B` and: `P ∧ Q` prop: `ℙ` uimplies: `b supposing a` int_seg: `{i..j-}` lelt: `i ≤ j < k` crng: `CRng` rng: `Rng` so_lambda: `λ2x.t[x]` nequal: `a ≠ b ∈ T ` guard: `{T}` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` all: `∀x:A. B[x]` top: `Top` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` so_apply: `x[s]` bfalse: `ff` or: `P ∨ Q` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` cand: `A c∧ B`
Lemmas referenced :  from-upto_wf list-subtype-bag le_wf less_than_wf int_seg_wf bag-summation_wf rng_car_wf rng_plus_wf rng_zero_wf eq_int_wf int_seg_properties nat_plus_properties satisfiable-full-omega-tt intformand_wf intformeq_wf itermVar_wf itermConstant_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_eq_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 eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int rng_all_properties rng_plus_comm2 nat_plus_wf crng_wf
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality addEquality setElimination rename because_Cache hypothesis applyEquality setEquality intEquality productEquality hypothesisEquality independent_isectElimination lambdaEquality sqequalRule remainderEquality productElimination lambdaFormation dependent_pairFormation int_eqEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll baseClosed unionElimination equalityElimination functionExtensionality equalityTransitivity equalitySymmetry promote_hyp instantiate cumulativity independent_functionElimination axiomEquality functionEquality

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

Date html generated: 2018_05_21-PM-07_31_32
Last ObjectModification: 2017_07_26-PM-05_06_46

Theory : general

Home Index