Nuprl Lemma : fps-linear-ucont-equal
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[F,G:PowerSeries(X;r) ⟶ PowerSeries(X;r)].
F = G ∈ (PowerSeries(X;r) ⟶ PowerSeries(X;r))
supposing fps-ucont(X;eq;r;f.F[f])
∧ fps-ucont(X;eq;r;f.G[f])
∧ (∀f,g:PowerSeries(X;r). (F[(f+g)] = (F[f]+F[g]) ∈ PowerSeries(X;r)))
∧ (∀f,g:PowerSeries(X;r). (G[(f+g)] = (G[f]+G[g]) ∈ PowerSeries(X;r)))
∧ (∀c:|r|. ∀f:PowerSeries(X;r). (F[(c)*f] = (c)*F[f] ∈ PowerSeries(X;r)))
∧ (∀c:|r|. ∀f:PowerSeries(X;r). (G[(c)*f] = (c)*G[f] ∈ PowerSeries(X;r)))
∧ (∀b:bag(X). (F[<b>] = G[<b>] ∈ PowerSeries(X;r)))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-ucont: fps-ucont(X;eq;r;f.G[f]),
fps-scalar-mul: (c)*f,
fps-add: (f+g),
fps-single: <c>,
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],
all: ∀x:A. B[x],
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
all: ∀x:A. B[x],
fps-ucont: fps-ucont(X;eq;r;f.G[f]),
exists: ∃x:A. B[x],
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
crng: CRng,
rng: Rng,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
cand: A c∧ B,
comm: Comm(T;op),
infix_ap: x f y,
assoc: Assoc(T;op),
bag-summation: Σ(x∈b). f[x],
bag-accum: bag-accum(v,x.f[v; x];init;bs),
top: Top,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
cons-bag: x.b,
monoid_p: IsMonoid(T;op;id),
ident: Ident(T;op;id),
fps-restrict: fps-restrict(eq;r;f;d),
fps-coeff: f[b],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
sub-bag: sub-bag(T;as;bs)
Lemmas referenced :
fps-ext,
power-series_wf,
bag_wf,
fps-ucont_wf,
all_wf,
equal_wf,
fps-add_wf,
rng_car_wf,
fps-scalar-mul_wf,
fps-single_wf,
crng_wf,
deq_wf,
valueall-type_wf,
squash_wf,
true_wf,
iff_weakening_equal,
bag-append_wf,
fps-coeff_wf,
sub-bags_wf,
fps-restrict-summation,
fps-add-comm,
mon_assoc_fps,
bag_to_squash_list,
list_induction,
bag-summation_wf,
fps-zero_wf,
list-subtype-bag,
subtype_rel_self,
list_wf,
list_accum_nil_lemma,
empty-bag_wf,
rng_zero_wf,
fps-scalar-mul-zero,
single-bag_wf,
cons-bag-as-append,
bag-summation-append,
abmonoid_comm_fps,
mon_ident_fps,
and_wf,
bag-summation-single,
fps-restrict_wf,
deq-sub-bag_wf,
bool_wf,
eqtt_to_assert,
assert-deq-sub-bag,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
sub-bag_wf,
sub-bag_transitivity,
bag-append-comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
productElimination,
thin,
functionExtensionality,
rename,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
cumulativity,
hypothesis,
independent_isectElimination,
lambdaFormation,
dependent_functionElimination,
because_Cache,
productEquality,
sqequalRule,
lambdaEquality,
setElimination,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed,
independent_functionElimination,
independent_pairFormation,
promote_hyp,
hyp_replacement,
applyLambdaEquality,
voidElimination,
voidEquality,
equalityUniverse,
levelHypothesis,
independent_pairEquality,
dependent_set_memberEquality,
unionElimination,
equalityElimination,
dependent_pairFormation,
instantiate
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[F,G:PowerSeries(X;r) {}\mrightarrow{} PowerSeries(X;r)].
F = G
supposing fps-ucont(X;eq;r;f.F[f])
\mwedge{} fps-ucont(X;eq;r;f.G[f])
\mwedge{} (\mforall{}f,g:PowerSeries(X;r). (F[(f+g)] = (F[f]+F[g])))
\mwedge{} (\mforall{}f,g:PowerSeries(X;r). (G[(f+g)] = (G[f]+G[g])))
\mwedge{} (\mforall{}c:|r|. \mforall{}f:PowerSeries(X;r). (F[(c)*f] = (c)*F[f]))
\mwedge{} (\mforall{}c:|r|. \mforall{}f:PowerSeries(X;r). (G[(c)*f] = (c)*G[f]))
\mwedge{} (\mforall{}b:bag(X). (F[<b>] = G[<b>]))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-10_11_01
Last ObjectModification:
2017_07_26-PM-06_34_32
Theory : power!series
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