Nuprl Lemma : fps-restrict-empty
∀[X:Type]. ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f:PowerSeries(X;r)].
(fps-restrict(eq;r;f;{}) = (f[{}])*1 ∈ PowerSeries(X;r))
Proof
Definitions occuring in Statement :
fps-restrict: fps-restrict(eq;r;f;d),
fps-scalar-mul: (c)*f,
fps-one: 1,
fps-coeff: f[b],
power-series: PowerSeries(X;r),
empty-bag: {},
deq: EqDecider(T),
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,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
fps-one: 1,
fps-coeff: f[b],
fps-scalar-mul: (c)*f,
fps-restrict: fps-restrict(eq;r;f;d),
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
iff: P ⇐⇒ Q,
ifthenelse: if b then t else f fi ,
squash: ↓T,
prop: ℙ,
crng: CRng,
rng: Rng,
power-series: PowerSeries(X;r),
true: True,
subtype_rel: A ⊆r B,
guard: {T},
rev_implies: P ⇐ Q,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A
Lemmas referenced :
fps-ext,
fps-restrict_wf,
empty-bag_wf,
fps-scalar-mul_wf,
fps-coeff_wf,
fps-one_wf,
deq-sub-bag_wf,
bool_wf,
eqtt_to_assert,
assert-deq-sub-bag,
bag-null_wf,
assert-bag-null,
sub-bag-empty,
equal_wf,
squash_wf,
true_wf,
rng_car_wf,
rng_times_one,
iff_weakening_equal,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
equal-wf-T-base,
bag_wf,
sub-bag_wf,
rng_times_zero,
power-series_wf,
crng_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
cumulativity,
hypothesis,
productElimination,
independent_isectElimination,
lambdaFormation,
sqequalRule,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
because_Cache,
dependent_functionElimination,
independent_functionElimination,
applyEquality,
lambdaEquality,
imageElimination,
setElimination,
rename,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
voidElimination,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[X:Type]. \mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[f:PowerSeries(X;r)].
(fps-restrict(eq;r;f;\{\}) = (f[\{\}])*1)
Date html generated:
2018_05_21-PM-10_10_40
Last ObjectModification:
2017_07_26-PM-06_34_26
Theory : power!series
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