Nuprl Lemma : geom-sum-int
∀[a:ℤ]. ∀[n:ℕ]. (((1 - a) * Σ(a^i | i < n)) = (1 - a^n) ∈ ℤ)
Proof
Definitions occuring in Statement :
exp: i^n,
sum: Σ(f[x] | x < k),
nat: ℕ,
uall: ∀[x:A]. B[x],
multiply: n * m,
subtract: n - m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
integ_dom: IntegDom{i},
int_ring: ℤ-rng,
rng_car: |r|,
pi1: fst(t),
squash: ↓T,
prop: ℙ,
infix_ap: x f y,
nat: ℕ,
so_lambda: λ2x.t[x],
crng: CRng,
int_seg: {i..j-},
lelt: i ≤ j < k,
and: P ∧ Q,
le: A ≤ B,
less_than: a < b,
ge: i ≥ j ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
so_apply: x[s],
rng: Rng,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rng_times: *,
pi2: snd(t),
rng_plus: +r,
rng_one: 1,
rng_minus: -r,
subtract: n - m,
sq_type: SQType(T)
Lemmas referenced :
sum_of_geometric_prog,
int_ring_wf,
equal_wf,
squash_wf,
true_wf,
istype-universe,
rng_car_wf,
rng_times_wf,
rng_plus_wf,
rng_one_wf,
rng_minus_wf,
rng_sum-int,
rng_nexp_wf,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
subtype_rel_self,
int_seg_wf,
iff_weakening_equal,
sum_functionality,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
itermAdd_wf,
int_term_value_add_lemma,
exp_wf2,
rng_nexp-int,
istype-nat,
subtype_base_sq,
int_subtype_base,
decidable__equal_int,
intformeq_wf,
itermMinus_wf,
int_formula_prop_eq_lemma,
int_term_value_minus_lemma,
sum_wf,
minus-zero,
add-zero,
zero-add,
subtract-elim
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
hypothesisEquality,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
sqequalRule,
imageElimination,
universeIsType,
instantiate,
universeEquality,
because_Cache,
natural_numberEquality,
dependent_set_memberEquality_alt,
productElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
intEquality,
imageMemberEquality,
baseClosed,
addEquality,
lambdaFormation_alt,
axiomEquality,
isectIsTypeImplies,
cumulativity,
multiplyEquality,
functionIsType
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[n:\mBbbN{}]. (((1 - a) * \mSigma{}(a\^{}i | i < n)) = (1 - a\^{}n))
Date html generated:
2019_10_15-AM-11_23_21
Last ObjectModification:
2019_08_14-AM-11_01_55
Theory : general
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