Nuprl Lemma : fps-exp-unroll
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[n:ℕ+]. ∀[f:PowerSeries(X;r)]. ((f)^(n) = ((f)^(n - 1)*f) ∈ PowerSeries(X;r))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-exp: (f)^(n),
fps-mul: (f*g),
power-series: PowerSeries(X;r),
deq: EqDecider(T),
nat_plus: ℕ+,
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
subtract: n - m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
fps-rng: fps-rng(r),
rng_car: |r|,
pi1: fst(t),
rng_times: *,
pi2: snd(t),
infix_ap: x f y,
fps-exp: (f)^(n)
Lemmas referenced :
rng_nexp_unroll,
fps-rng_wf,
crng_subtype_rng,
crng_wf,
deq_wf,
valueall-type_wf,
istype-universe
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
hypothesis,
applyEquality,
sqequalRule,
isect_memberEquality_alt,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
universeIsType,
instantiate,
universeEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[f:PowerSeries(X;r)]. ((f)\^{}(n) = ((f)\^{}(n - 1)*f))
supposing valueall-type(X)
Date html generated:
2020_05_20-AM-09_05_45
Last ObjectModification:
2020_02_04-PM-01_54_43
Theory : power!series
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