Nuprl Lemma : fps-mul-single
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[b,c:bag(X)]. ((<b>*<c>) = <b + c> ∈ PowerSeries(X;r)) supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-mul: (f*g),
fps-single: <c>,
power-series: PowerSeries(X;r),
bag-append: as + bs,
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
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,
power-series: PowerSeries(X;r),
all: ∀x:A. B[x],
crng: CRng,
rng: Rng,
uiff: uiff(P;Q),
fps-single: <c>,
fps-coeff: f[b],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
power-series_wf,
fps-mul-single-general,
fps-single_wf,
bag-append_wf,
subtype_rel_self,
iff_weakening_equal,
fps-ext,
bag-diff_wf,
bag_wf,
unit_wf2,
fps-coeff_wf,
rng_zero_wf,
rng_car_wf,
crng_wf,
deq_wf,
valueall-type_wf,
bag-diff-property,
bag-eq_wf,
bool_wf,
eqtt_to_assert,
assert-bag-eq,
rng_one_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
and_wf,
all_wf,
not_wf,
bag-append-cancel
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,
unionEquality,
lambdaFormation,
unionElimination,
setElimination,
rename,
dependent_functionElimination,
functionEquality,
isect_memberEquality,
axiomEquality,
universeEquality,
equalityElimination,
dependent_pairFormation,
promote_hyp,
voidElimination,
dependent_set_memberEquality,
independent_pairFormation,
applyLambdaEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[b,c:bag(X)]. ((<b>*<c>) = <b + c>) supposing valueall-type(X)
Date html generated:
2019_10_16-AM-11_34_33
Last ObjectModification:
2018_08_21-PM-01_59_48
Theory : power!series
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