Nuprl Lemma : fps-add-ucont-general
∀[X:Type]. ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[F,G:PowerSeries(X;r) ⟶ PowerSeries(X;r)].
(fps-ucont(X;eq;r;f.F[f]) ⇒ fps-ucont(X;eq;r;f.G[f]) ⇒ fps-ucont(X;eq;r;f.(F[f]+G[f])))
Proof
Definitions occuring in Statement :
fps-ucont: fps-ucont(X;eq;r;f.G[f]),
fps-add: (f+g),
power-series: PowerSeries(X;r),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
so_apply: x[s],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
universe: Type,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
fps-ucont: fps-ucont(X;eq;r;f.G[f]),
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
fps-coeff: f[b],
fps-add: (f+g),
rng_plus: +r,
pi1: fst(t),
pi2: snd(t),
infix_ap: x f y,
top: Top,
prop: ℙ,
so_lambda: λ2x.t[x],
crng: CRng,
rng: Rng,
so_apply: x[s],
true: True,
squash: ↓T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
fps-restrict: fps-restrict(eq;r;f;d),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
iff: P ⇐⇒ Q,
ifthenelse: if b then t else f fi ,
power-series: PowerSeries(X;r),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
subtype_rel: A ⊆r B,
sub-bag: sub-bag(T;as;bs)
Lemmas referenced :
bag-append_wf,
top_wf,
power-series_wf,
all_wf,
equal_wf,
rng_car_wf,
fps-coeff_wf,
fps-add_wf,
fps-restrict_wf,
bag_wf,
fps-ucont_wf,
crng_wf,
deq_wf,
rng_plus_wf,
fps-ext,
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,
rng_zero_wf,
squash_wf,
true_wf,
subtype_rel_self,
iff_weakening_equal,
bag-append-assoc2,
bag-append-comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
dependent_pairFormation,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
lambdaEquality,
setElimination,
rename,
applyEquality,
functionEquality,
universeEquality,
because_Cache,
natural_numberEquality,
imageElimination,
independent_isectElimination,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
promote_hyp,
instantiate,
cumulativity,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[X:Type]. \mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[F,G:PowerSeries(X;r) {}\mrightarrow{} PowerSeries(X;r)].
(fps-ucont(X;eq;r;f.F[f]) {}\mRightarrow{} fps-ucont(X;eq;r;f.G[f]) {}\mRightarrow{} fps-ucont(X;eq;r;f.(F[f]+G[f])))
Date html generated:
2018_05_21-PM-10_11_06
Last ObjectModification:
2018_05_19-PM-04_15_35
Theory : power!series
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