Nuprl Lemma : sum_of_geometric_prog

[r:CRng]. ∀[a:|r|]. ∀[n:ℕ].  (((1 +r (-r a)) (r) 0 ≤ i < n. a ↑i)) (1 +r (-r (a ↑n))) ∈ |r|)

This theorem is one of freek's list of 100 theorems


Definitions occuring in Statement :  rng_nexp: e ↑n rng_sum: rng_sum crng: CRng rng_one: 1 rng_times: * rng_minus: -r rng_plus: +r rng_car: |r| nat: uall: [x:A]. B[x] infix_ap: y apply: a natural_number: $n equal: t ∈ T
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T nat: implies:  Q false: False ge: i ≥  uimplies: 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: decidable: Dec(P) or: P ∨ Q crng: CRng rng: Rng squash: T infix_ap: y so_lambda: λ2x.t[x] subtype_rel: A ⊆B le: A ≤ B less_than': less_than'(a;b) so_apply: x[s] true: True guard: {T} iff: ⇐⇒ Q rev_implies:  Q nat_plus: +
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

\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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