### Nuprl Lemma : fps-geometric-slice

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

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

Proof

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: `ℕ+` nat: `ℕ` valueall-type: `valueall-type(T)` ifthenelse: `if b then t else f fi ` eq_int: `(i =z j)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` remainder: `n rem m` divide: `n ÷ m` natural_number: `\$n` universe: `Type` equal: `s = t ∈ T` crng: `CRng` rng_one: `1`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` nat: `ℕ` implies: `P `` Q` false: `False` ge: `i ≥ j ` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` not: `¬A` top: `Top` and: `P ∧ Q` prop: `ℙ` guard: `{T}` nat_plus: `ℕ+` int_seg: `{i..j-}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` subtype_rel: `A ⊆r B` le: `A ≤ B` less_than': `less_than'(a;b)` less_than: `a < b` squash: `↓T` nequal: `a ≠ b ∈ T ` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` ifthenelse: `if b then t else f fi ` uiff: `uiff(P;Q)` bfalse: `ff` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` 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

Latex:
\mforall{}[X:Type]
\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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