Nuprl Lemma : fps-single-slice
∀[X:Type]. ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[c:bag(X)]. ∀[n:ℤ].
([<c>]_n = if (#(c) =z n) then <c> else 0 fi ∈ PowerSeries(X;r))
Proof
Definitions occuring in Statement :
fps-slice: [f]_n,
fps-single: <c>,
fps-zero: 0,
power-series: PowerSeries(X;r),
bag-size: #(bs),
bag: bag(T),
deq: EqDecider(T),
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
int: ℤ,
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
fps-zero: 0,
fps-single: <c>,
fps-coeff: f[b],
fps-slice: [f]_n,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
crng: CRng,
rng: Rng,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
nequal: a ≠ b ∈ T ,
squash: ↓T,
true: True,
nat: ℕ
Lemmas referenced :
fps-ext,
fps-slice_wf,
fps-single_wf,
ifthenelse_wf,
eq_int_wf,
bag-size_wf,
power-series_wf,
fps-zero_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
bag-eq_wf,
assert-bag-eq,
rng_one_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bag_wf,
neg_assert_of_eq_int,
squash_wf,
true_wf,
nat_wf,
rng_zero_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,
applyEquality,
because_Cache,
sqequalRule,
productElimination,
independent_isectElimination,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
lambdaEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
intEquality,
isect_memberEquality,
axiomEquality,
universeEquality
Latex:
\mforall{}[X:Type]. \mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[c:bag(X)]. \mforall{}[n:\mBbbZ{}].
([<c>]\_n = if (\#(c) =\msubz{} n) then <c> else 0 fi )
Date html generated:
2018_05_21-PM-09_56_03
Last ObjectModification:
2017_07_26-PM-06_32_52
Theory : power!series
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