### 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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