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
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