Nuprl Lemma : fps-div-coeff-property
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f,g:PowerSeries(X;r)]. ∀[x:|r|].
(g*λb.fps-div-coeff(eq;r;f;g;x;b)) = f ∈ PowerSeries(X;r) supposing (g[{}] * x) = 1 ∈ |r|
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-div-coeff: fps-div-coeff(eq;r;f;g;x;b),
fps-mul: (f*g),
fps-coeff: f[b],
power-series: PowerSeries(X;r),
empty-bag: {},
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
infix_ap: x f y,
lambda: λx.A[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_one: 1,
rng_times: *,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
crng: CRng,
comm: Comm(T;op),
rng: Rng,
prop: ℙ,
and: P ∧ Q,
fps-mul: (f*g),
power-series: PowerSeries(X;r),
fps-coeff: f[b],
infix_ap: x f y,
fps-div-coeff: fps-div-coeff(eq;r;f;g;x;b),
so_lambda: λ2x.t[x],
pi1: fst(t),
pi2: snd(t),
so_apply: x[s],
cand: A c∧ B,
all: ∀x:A. B[x],
top: Top,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
ring_p: IsRing(T;plus;zero;neg;times;one),
group_p: IsGroup(T;op;id;inv),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
assert: ↑b,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
false: False,
not: ¬A
Lemmas referenced :
rng_plus_comm,
crng_properties,
rng_properties,
rng_all_properties,
ring_p_wf,
rng_car_wf,
rng_plus_wf,
rng_zero_wf,
rng_minus_wf,
rng_times_wf,
rng_one_wf,
bag_wf,
equal_wf,
fps-coeff_wf,
empty-bag_wf,
power-series_wf,
crng_wf,
deq_wf,
valueall-type_wf,
fps-div-coeff_wf,
bag-summation_wf,
bag-partitions_wf,
infix_ap_wf,
assert_wf,
bnot_wf,
bag-null_wf,
pi1_wf_top,
bag-filter_wf,
bag-summation-single,
squash_wf,
true_wf,
pi2_wf,
iff_weakening_equal,
and_wf,
bag-summation-append,
subtype_rel_bag,
single-bag_wf,
bag-split,
bag-append_wf,
empty_bag_append_lemma,
bag-partitions-with-one-given,
bag-eq_wf,
bool_wf,
eqtt_to_assert,
assert-bag-null,
iff_imp_equal_bool,
btrue_wf,
equal-wf-T-base,
assert-bag-eq,
iff_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bfalse_wf,
false_wf,
rng_times_assoc,
rng_times_over_plus,
rng_times_over_minus,
rng_times_one,
rng_plus_ac_1,
rng_plus_inv,
rng_plus_zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality,
productElimination,
sqequalRule,
functionExtensionality,
applyEquality,
because_Cache,
cumulativity,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
independent_isectElimination,
hyp_replacement,
applyLambdaEquality,
productEquality,
lambdaEquality,
independent_pairFormation,
setEquality,
dependent_functionElimination,
lambdaFormation,
independent_pairEquality,
voidElimination,
voidEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
instantiate,
isectEquality,
unionElimination,
equalityElimination,
addLevel,
impliesFunctionality,
dependent_pairFormation,
promote_hyp
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[f,g:PowerSeries(X;r)]. \mforall{}[x:|r|].
(g*\mlambda{}b.fps-div-coeff(eq;r;f;g;x;b)) = f supposing (g[\{\}] * x) = 1
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_55_37
Last ObjectModification:
2017_07_26-PM-06_32_44
Theory : power!series
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