Nuprl Lemma : fps-div-unique
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f,g:PowerSeries(X;r)]. ∀[x:|r|].
∀q:PowerSeries(X;r)
q = (f÷g) ∈ PowerSeries(X;r) supposing ((g*q) = f ∈ PowerSeries(X;r)) ∧ ((g[{}] * x) = 1 ∈ |r|)
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-div: (f÷g),
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,
all: ∀x:A. B[x],
and: P ∧ Q,
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,
all: ∀x:A. B[x],
and: P ∧ Q,
prop: ℙ,
crng: CRng,
rng: Rng,
infix_ap: x f y,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
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,
ring_p: IsRing(T;plus;zero;neg;times;one),
monoid_p: IsMonoid(T;op;id),
ident: Ident(T;op;id)
Lemmas referenced :
fps-div-property,
equal_wf,
power-series_wf,
fps-mul_wf,
rng_car_wf,
rng_times_wf,
fps-coeff_wf,
empty-bag_wf,
rng_one_wf,
crng_wf,
deq_wf,
valueall-type_wf,
squash_wf,
true_wf,
fps-div_wf,
fps-one_wf,
iff_weakening_equal,
fps-mul-assoc,
fps-mul-comm,
fps-rng_wf,
crng_properties,
rng_properties
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
lambdaFormation,
productElimination,
equalityTransitivity,
equalitySymmetry,
productEquality,
cumulativity,
setElimination,
rename,
applyEquality,
because_Cache,
sqequalRule,
isect_memberEquality,
lambdaEquality,
dependent_functionElimination,
axiomEquality,
universeEquality,
imageElimination,
imageMemberEquality,
baseClosed,
natural_numberEquality,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[f,g:PowerSeries(X;r)]. \mforall{}[x:|r|].
\mforall{}q:PowerSeries(X;r). q = (f\mdiv{}g) supposing ((g*q) = f) \mwedge{} ((g[\{\}] * x) = 1)
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_55_41
Last ObjectModification:
2017_07_26-PM-06_32_46
Theory : power!series
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