### Nuprl Lemma : fps-geometric-slice_lemma

`∀[X:Type]`
`  ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[m:ℕ]. ∀[n:ℕ+m + 1]. ∀[g:PowerSeries(X;r)].`
`    [(1÷(1-g))]_m = ([(1÷(1-g))]_m - n*g) ∈ 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-mul: `(f*g)` fps-sub: `(f-g)` fps-one: `1` power-series: `PowerSeries(X;r)` deq: `EqDecider(T)` int_seg: `{i..j-}` nat: `ℕ` valueall-type: `valueall-type(T)` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` subtract: `n - m` add: `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` 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` 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` prop: `ℙ` true: `True` int_seg: `{i..j-}` subtype_rel: `A ⊆r B` guard: `{T}` iff: `P `⇐⇒` Q` and: `P ∧ Q` rev_implies: `P `` Q` implies: `P `` Q` fps-slice: `[f]_n` all: `∀x:A. B[x]` bool: `𝔹` unit: `Unit` it: `⋅` uiff: `uiff(P;Q)` nat: `ℕ` ge: `i ≥ j ` lelt: `i ≤ j < k` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` 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]` ring_p: `IsRing(T;plus;zero;neg;times;one)` group_p: `IsGroup(T;op;id;inv)` cand: `A c∧ B` comm: `Comm(T;op)` fps-summation: `fps-summation(r;b;x.f[x])` bor: `p ∨bq` nequal: `a ≠ b ∈ T ` bag-member: `x ↓∈ bs` bag-no-repeats: `bag-no-repeats(T;bs)` decidable: `Dec(P)` single-bag: `{x}` bag-append: `as + bs` append: `as @ bs` so_lambda: `so_lambda(x,y,z.t[x; y; z])` so_apply: `x[s1;s2;s3]` sq_or: `a ↓∨ b` rev_uimplies: `rev_uimplies(P;Q)` sq_stable: `SqStable(P)` upto: `upto(n)` eq_int: `(i =z j)`
Lemmas referenced :  fps-rng_wf crng_properties rng_properties fps-mul-slice 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_properties full-omega-unsat intformand_wf intformeq_wf itermConstant_wf itermVar_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_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_wf deq_wf valueall-type_wf intformless_wf itermAdd_wf int_formula_prop_less_lemma int_term_value_add_lemma 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 fps-zero_wf fps-add-comm bag-summation-filter fps-add_wf bor_wf bag-summation-equal ifthenelse_wf bag-member_wf fps-sub-slice fps-ext bag-null_wf assert-bag-null bag-size_wf fps-slice-slice intformnot_wf int_formula_prop_not_lemma neg_id_fps mon_ident_fps fps-neg_wf mul_zero_fps bag-extensionality-no-repeats decidable__int_equal bag-filter_wf subtype_rel_bag bag-append_wf single-bag_wf bag-filter-no-repeats subtype_rel_list no_repeats_upto decidable__le le_wf equal-wf-base-T list_subtype_base int_subtype_base no_repeats_wf list_ind_cons_lemma list_ind_nil_lemma cons_wf nil_wf no_repeats_cons no_repeats_singleton equal-wf-base member_singleton l_member_wf bag-member-filter or_wf bag-member-append bag-member-single assert_of_bor sq_stable__bag-member bag-member-from-upto decidable__lt decidable__equal_int bag-summation-append bag-summation-single itermSubtract_wf int_term_value_subtract_lemma mul_over_plus_fps mul_over_minus_fps mul_one_fps mul_comm_fps mon_assoc_fps abmonoid_ac_1_fps abmonoid_comm_fps iabgrp_op_inv_assoc_fps
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 lambdaEquality imageElimination natural_numberEquality imageMemberEquality baseClosed instantiate productElimination independent_functionElimination lambdaFormation unionElimination equalityElimination addEquality approximateComputation dependent_pairFormation int_eqEquality intEquality dependent_functionElimination isect_memberEquality voidElimination voidEquality independent_pairFormation promote_hyp cumulativity hyp_replacement axiomEquality universeEquality impliesFunctionality setEquality dependent_set_memberEquality productEquality baseApply closedConclusion addLevel inlFormation inrFormation orFunctionality functionEquality equalityUniverse levelHypothesis

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

Date html generated: 2018_05_21-PM-09_58_06
Last ObjectModification: 2018_05_19-PM-04_14_52

Theory : power!series

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