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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