### Nuprl Lemma : fps-geometric-slice_lemma2

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

Proof

Definitions occuring in Statement :  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)` int_seg: `{i..j-}` nat_plus: `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]` 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` crng: `CRng` rng: `Rng` fps-rng: `fps-rng(r)` rng_car: `|r|` pi1: `fst(t)` rng_plus: `+r` pi2: `snd(t)` rng_zero: `0` rng_minus: `-r` rng_times: `*` rng_one: `1` subtype_rel: `A ⊆r B` nat_plus: `ℕ+` le: `A ≤ B` and: `P ∧ Q` less_than': `less_than'(a;b)` false: `False` not: `¬A` implies: `P `` Q` prop: `ℙ` empty-bag: `{}` fps-one: `1` fps-sub: `(f-g)` fps-coeff: `f[b]` fps-neg: `-(f)` bag-null: `bag-null(bs)` fps-add: `(f+g)` ifthenelse: `if b then t else f fi ` btrue: `tt` squash: `↓T` true: `True` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` fps-slice: `[f]_n` all: `∀x:A. B[x]` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` int_seg: `{i..j-}` lelt: `i ≤ j < k` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` top: `Top` bfalse: `ff` or: `P ∨ Q` sq_type: `SQType(T)` bnot: `¬bb` assert: `↑b` infix_ap: `x f y` so_lambda: `λ2x.t[x]` so_apply: `x[s]` eq_int: `(i =z j)` upto: `upto(n)` fps-summation: `fps-summation(r;b;x.f[x])` from-upto: `[n, m)` lt_int: `i <z j` single-bag: `{x}` cand: `A c∧ B` ring_p: `IsRing(T;plus;zero;neg;times;one)` group_p: `IsGroup(T;op;id;inv)` comm: `Comm(T;op)` subtract: `n - m` nequal: `a ≠ b ∈ T ` bag-filter: `[x∈b|p[x]]` rev_uimplies: `rev_uimplies(P;Q)` decidable: `Dec(P)`
Lemmas referenced :  fps-rng_wf crng_properties rng_properties fps-mul-slice int_seg_subtype_nat false_wf fps-sub_wf fps-one_wf fps-div_wf rng_one_wf fps-div-property null_nil_lemma equal_wf squash_wf true_wf rng_car_wf fps-coeff_wf bag_wf power-series_wf crng_wf empty-bag_wf fps-slice_wf subtype_rel_self iff_weakening_equal bag_size_empty_lemma eq_int_wf bool_wf eqtt_to_assert assert_of_eq_int int_seg_properties nat_plus_properties full-omega-unsat intformand_wf intformeq_wf itermConstant_wf itermVar_wf intformless_wf intformle_wf int_formula_prop_and_lemma int_formula_prop_eq_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_le_lemma int_formula_prop_wf eqff_to_assert bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot neg_assert_of_eq_int rng_zero_wf rng_times_wf rng_plus_wf rng_minus_wf fps-summation_wf fps-mul_wf subtract_wf upto_wf list-subtype-bag int_seg_wf fps-one-slice nat_plus_wf deq_wf valueall-type_wf assert_wf bnot_wf not_wf equal-wf-T-base rng_times_over_plus rng_times_over_minus rng_times_zero rng_times_one rng_minus_zero rng_plus_zero bool_cases iff_transitivity iff_weakening_uiff assert_of_bnot int_subtype_base bag-summation-single fps-add_wf fps-zero_wf fps-add-comm btrue_wf intformnot_wf int_formula_prop_not_lemma fps-sub-slice fps-slice-slice neg_id_fps mon_ident_fps mul_one_fps bag-summation-filter bag-summation-equal ifthenelse_wf bag-member_wf bag-member-from-upto itermAdd_wf int_term_value_add_lemma fps-neg_wf mul_zero_fps lt_int_wf assert_of_lt_int filter_cons_lemma less_than_wf filter_nil_lemma filter_is_nil le_wf from-upto_wf l_all_iff l_member_wf equal-wf-base set_wf decidable__equal_int itermSubtract_wf int_term_value_subtract_lemma
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality independent_isectElimination hypothesis equalityTransitivity equalitySymmetry applyLambdaEquality setElimination rename sqequalRule because_Cache applyEquality natural_numberEquality independent_pairFormation lambdaFormation lambdaEquality imageElimination imageMemberEquality baseClosed instantiate productElimination independent_functionElimination unionElimination equalityElimination approximateComputation dependent_pairFormation int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality promote_hyp cumulativity hyp_replacement addEquality axiomEquality universeEquality impliesFunctionality callbyvalueReduce sqleReflexivity functionEquality setEquality productEquality

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

Date html generated: 2018_05_21-PM-09_58_25
Last ObjectModification: 2018_05_19-PM-04_14_46

Theory : power!series

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