Nuprl Lemma : fps-compose-fps-product
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[x:X]. ∀[f:PowerSeries(X;r)]. ∀[T:Type]. ∀[b:bag(T)]. ∀[G:T ⟶ PowerSeries(X;r)].
(Π(i∈b).G[i](x:=f) = Π(i∈b).G[i](x:=f) ∈ PowerSeries(X;r))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-compose: g(x:=f),
fps-product: Π(x∈b).f[x],
power-series: PowerSeries(X;r),
bag: bag(T),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
fps-product: Π(x∈b).f[x],
bag-product: Πx ∈ b. f[x],
squash: ↓T,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
subtype_rel: A ⊆r B,
cand: A c∧ B,
implies: P ⇒ Q,
empty-bag: {},
all: ∀x:A. B[x],
cons-bag: x.b,
prop: ℙ,
infix_ap: x f y,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
top: Top,
assoc: Assoc(T;op),
comm: Comm(T;op),
monoid_p: IsMonoid(T;op;id),
ident: Ident(T;op;id)
Lemmas referenced :
bag_to_squash_list,
list_induction,
equal_wf,
power-series_wf,
fps-compose_wf,
bag-summation_wf,
fps-mul_wf,
fps-one_wf,
list-subtype-bag,
list_wf,
bag_wf,
crng_wf,
deq_wf,
valueall-type_wf,
squash_wf,
true_wf,
mul_assoc_fps,
iff_weakening_equal,
fps-mul-comm,
single-bag_wf,
bag-summation-empty,
fps-compose-one,
cons-bag-as-append,
bag-summation-single,
bag-summation-append,
mul_one_fps,
fps-compose-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
imageElimination,
productElimination,
promote_hyp,
hypothesis,
rename,
sqequalRule,
lambdaEquality,
cumulativity,
because_Cache,
independent_isectElimination,
applyEquality,
functionExtensionality,
independent_pairFormation,
independent_functionElimination,
lambdaFormation,
dependent_functionElimination,
hyp_replacement,
equalitySymmetry,
applyLambdaEquality,
functionEquality,
isect_memberEquality,
axiomEquality,
universeEquality,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
voidElimination,
voidEquality,
equalityUniverse,
levelHypothesis,
independent_pairEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[x:X]. \mforall{}[f:PowerSeries(X;r)]. \mforall{}[T:Type]. \mforall{}[b:bag(T)].
\mforall{}[G:T {}\mrightarrow{} PowerSeries(X;r)].
(\mPi{}(i\mmember{}b).G[i](x:=f) = \mPi{}(i\mmember{}b).G[i](x:=f))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-10_10_10
Last ObjectModification:
2017_07_26-PM-06_34_19
Theory : power!series
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