Nuprl Lemma : sum_of_geometric_prog
∀[r:CRng]. ∀[a:|r|]. ∀[n:ℕ]. (((1 +r (-r a)) * (Σ(r) 0 ≤ i < n. a ↑r i)) = (1 +r (-r (a ↑r n))) ∈ |r|)
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
rng_nexp: e ↑r n,
rng_sum: rng_sum,
crng: CRng,
rng_one: 1,
rng_times: *,
rng_minus: -r,
rng_plus: +r,
rng_car: |r|,
nat: ℕ,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
apply: f a,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
decidable: Dec(P),
or: P ∨ Q,
crng: CRng,
rng: Rng,
squash: ↓T,
infix_ap: x f y,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
so_apply: x[s],
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat_plus: ℕ+
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
nat_wf,
rng_car_wf,
crng_wf,
equal_wf,
squash_wf,
true_wf,
rng_times_wf,
infix_ap_wf,
rng_plus_wf,
rng_one_wf,
rng_minus_wf,
rng_sum_unroll_base,
rng_nexp_wf,
int_seg_subtype_nat,
false_wf,
int_seg_wf,
rng_nexp_zero,
iff_weakening_equal,
rng_times_over_plus,
rng_zero_wf,
rng_times_over_minus,
rng_times_zero,
rng_minus_zero,
rng_plus_inv,
rng_plus_zero,
rng_sum_unroll_hi,
le_wf,
rng_sum_wf,
rng_nexp_unroll,
rng_times_one,
crng_times_comm,
rng_plus_assoc,
rng_plus_ac_1,
rng_plus_comm,
rng_plus_inv_assoc
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
unionElimination,
because_Cache,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
productElimination,
dependent_set_memberEquality
Latex:
\mforall{}[r:CRng]. \mforall{}[a:|r|]. \mforall{}[n:\mBbbN{}]. (((1 +r (-r a)) * (\mSigma{}(r) 0 \mleq{} i < n. a \muparrow{}r i)) = (1 +r (-r (a \muparrow{}r n))))
Date html generated:
2017_10_01-AM-08_19_40
Last ObjectModification:
2017_02_28-PM-02_04_21
Theory : rings_1
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