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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