Nuprl Lemma : gen-divisors-sum-int-ring
∀[n:ℕ+]. ∀[f:ℕ+n + 1 ⟶ ℤ]. (Σ i|n. f[i] = Σ i|n. f[i] ∈ ℤ)
Proof
Definitions occuring in Statement :
gen-divisors-sum: Σ i|n. f[i],
divisors-sum: Σ i|n. f[i] ,
int_seg: {i..j-},
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T,
int_ring: ℤ-rng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
divisors-sum: Σ i|n. f[i] ,
gen-divisors-sum: Σ i|n. f[i],
int_ring: ℤ-rng,
rng_zero: 0,
pi2: snd(t),
pi1: fst(t),
rng_plus: +r,
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,
squash: ↓T,
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,
decidable: Dec(P),
subtract: n - m,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ
Lemmas referenced :
from-upto_wf,
list-subtype-bag,
le_wf,
less_than_wf,
int_seg_wf,
nat_plus_wf,
equal_wf,
squash_wf,
true_wf,
bag-summation-from-upto,
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,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
sum_wf,
nat_plus_subtype_nat,
itermAdd_wf,
int_term_value_add_lemma,
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,
iff_weakening_equal,
nat_wf,
decidable__equal_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
addEquality,
setElimination,
rename,
because_Cache,
hypothesis,
applyEquality,
setEquality,
intEquality,
productEquality,
hypothesisEquality,
independent_isectElimination,
lambdaEquality,
functionEquality,
isect_memberEquality,
axiomEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
remainderEquality,
productElimination,
lambdaFormation,
dependent_pairFormation,
int_eqEquality,
dependent_functionElimination,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
baseClosed,
unionElimination,
equalityElimination,
functionExtensionality,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
baseApply,
closedConclusion,
dependent_set_memberEquality,
imageMemberEquality
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[f:\mBbbN{}\msupplus{}n + 1 {}\mrightarrow{} \mBbbZ{}]. (\mSigma{} i|n. f[i] = \mSigma{} i|n. f[i] )
Date html generated:
2018_05_21-PM-07_31_40
Last ObjectModification:
2017_07_26-PM-05_06_52
Theory : general
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