Nuprl Lemma : fps-compose-single
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[x:X]. ∀[b:bag(X)]. ∀[f:PowerSeries(X;r)].
<b>(x:=f) = (<(b|¬x)>*(f)^(#((b|x)))) ∈ PowerSeries(X;r) supposing f[{}] = 0 ∈ |r|
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-compose: g(x:=f),
fps-exp: (f)^(n),
fps-mul: (f*g),
fps-single: <c>,
fps-coeff: f[b],
power-series: PowerSeries(X;r),
bag-co-restrict: (b|¬x),
bag-restrict: (b|x),
bag-size: #(bs),
empty-bag: {},
bag: bag(T),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_zero: 0,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
crng: CRng,
rng: Rng,
fps-sub: (f-g)
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
fps-compose-single-general,
fps-mul_wf,
fps-single_wf,
bag-co-restrict_wf,
fps-exp_wf,
bag-size_wf,
bag-restrict_wf,
subtype_rel_self,
iff_weakening_equal,
valueall-type_wf,
power-series_wf,
crng_wf,
deq_wf,
rng_car_wf,
fps-coeff_wf,
empty-bag_wf,
rng_zero_wf,
bag_wf,
fps-one_wf,
fps-scalar-mul-zero,
fps-sub_wf,
fps-scalar-mul_wf,
neg_id_fps,
mon_ident_fps,
fps-add_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
thin,
lambdaEquality,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
because_Cache,
independent_isectElimination,
natural_numberEquality,
sqequalRule,
imageMemberEquality,
baseClosed,
instantiate,
productElimination,
independent_functionElimination,
setElimination,
rename,
isect_memberEquality,
axiomEquality,
universeEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[x:X]. \mforall{}[b:bag(X)]. \mforall{}[f:PowerSeries(X;r)].
<b>(x:=f) = (<(b|\mneg{}x)>*(f)\^{}(\#((b|x)))) supposing f[\{\}] = 0
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-10_10_24
Last ObjectModification:
2018_05_19-PM-04_15_20
Theory : power!series
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