Nuprl Lemma : fps-geometric-slice

  ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[m:ℕ]. ∀[n:ℕ+]. ∀[g:PowerSeries(X;r)].
    [(1÷(1-g))]_m if (m rem =z 0) then (g)^(m ÷ n) else fi  ∈ PowerSeries(X;r) 
    supposing [g]_n ∈ PowerSeries(X;r) 
  supposing valueall-type(X)

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


Definitions occuring in Statement :  fps-exp: (f)^(n) fps-slice: [f]_n fps-div: (f÷g) fps-sub: (f-g) fps-one: 1 fps-zero: 0 power-series: PowerSeries(X;r) deq: EqDecider(T) nat_plus: + nat: valueall-type: valueall-type(T) ifthenelse: if then else fi  eq_int: (i =z j) uimplies: supposing a uall: [x:A]. B[x] remainder: rem m divide: n ÷ m natural_number: $n universe: Type equal: t ∈ T crng: CRng rng_one: 1
Definitions unfolded in proof :  uall: [x:A]. B[x] member: t ∈ T uimplies: supposing a all: x:A. B[x] nat: implies:  Q false: False ge: i ≥  satisfiable_int_formula: satisfiable_int_formula(fmla) exists: x:A. B[x] not: ¬A top: Top and: P ∧ Q prop: guard: {T} nat_plus: + int_seg: {i..j-} lelt: i ≤ j < k decidable: Dec(P) or: P ∨ Q subtype_rel: A ⊆B le: A ≤ B less_than': less_than'(a;b) less_than: a < b squash: T nequal: a ≠ b ∈  bool: 𝔹 unit: Unit it: btrue: tt ifthenelse: if then else fi  uiff: uiff(P;Q) bfalse: ff sq_type: SQType(T) bnot: ¬bb assert: b true: True iff: ⇐⇒ Q rev_implies:  Q int_upper: {i...} label: ...$L... t
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 equal_wf power-series_wf fps-slice_wf nat_plus_wf int_seg_wf int_seg_properties decidable__le subtract_wf intformnot_wf itermSubtract_wf int_formula_prop_not_lemma int_term_value_subtract_lemma decidable__equal_int int_seg_subtype false_wf intformeq_wf int_formula_prop_eq_lemma le_wf decidable__lt lelt_wf itermAdd_wf int_term_value_add_lemma nat_wf crng_wf deq_wf valueall-type_wf fps-geometric-slice_lemma2 squash_wf true_wf eq_int_wf nat_plus_properties equal-wf-base int_subtype_base bool_wf eqtt_to_assert assert_of_eq_int fps-exp_wf divide_wf eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int fps-zero_wf iff_weakening_equal rem_base_case div_base_case fps-one_wf fps-exp-zero int_upper_subtype_nat nequal-le-implies zero-add fps-geometric-slice_lemma fps-mul_wf add-is-int-iff div_bounds_1 ifthenelse_wf div_rec_case fps-exp-unroll add-subtract-cancel rem_rec_case mul_zero_fps
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut thin lambdaFormation extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality hypothesis setElimination rename intWeakElimination natural_numberEquality independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality sqequalRule independent_pairFormation computeAll independent_functionElimination axiomEquality cumulativity equalityTransitivity equalitySymmetry because_Cache productElimination unionElimination applyEquality applyLambdaEquality hypothesis_subsumption dependent_set_memberEquality addEquality universeEquality imageElimination remainderEquality baseClosed equalityElimination promote_hyp instantiate imageMemberEquality hyp_replacement pointwiseFunctionality baseApply closedConclusion

    \mforall{}[eq:EqDecider(X)].  \mforall{}[r:CRng].  \mforall{}[m:\mBbbN{}].  \mforall{}[n:\mBbbN{}\msupplus{}].  \mforall{}[g:PowerSeries(X;r)].
        [(1\mdiv{}(1-g))]\_m  =  if  (m  rem  n  =\msubz{}  0)  then  (g)\^{}(m  \mdiv{}  n)  else  0  fi    supposing  g  =  [g]\_n 
    supposing  valueall-type(X)

Date html generated: 2018_05_21-PM-09_58_59
Last ObjectModification: 2017_07_26-PM-06_33_39

Theory : power!series

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