Nuprl Lemma : fps-geometric-slice1
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[m:ℕ]. ∀[g:PowerSeries(X;r)]. ([(1÷(1-[g]_1))]_m = ([g]_1)^(m) ∈ PowerSeries(X;r))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-exp: (f)^(n)
,
fps-slice: [f]_n
,
fps-div: (f÷g)
,
fps-sub: (f-g)
,
fps-one: 1
,
power-series: PowerSeries(X;r)
,
deq: EqDecider(T)
,
nat: ℕ
,
valueall-type: valueall-type(T)
,
uimplies: b supposing a
,
uall: ∀[x:A]. B[x]
,
natural_number: $n
,
universe: Type
,
equal: s = t ∈ T
,
crng: CRng
,
rng_one: 1
Definitions unfolded in proof :
uall: ∀[x:A]. B[x]
,
member: t ∈ T
,
uimplies: b supposing a
,
nat_plus: ℕ+
,
less_than: a < b
,
squash: ↓T
,
less_than': less_than'(a;b)
,
true: True
,
and: P ∧ Q
,
prop: ℙ
,
nat: ℕ
,
all: ∀x:A. B[x]
,
implies: P
⇒ Q
,
bool: 𝔹
,
unit: Unit
,
it: ⋅
,
btrue: tt
,
uiff: uiff(P;Q)
,
ifthenelse: if b then t else f fi
,
bfalse: ff
,
exists: ∃x:A. B[x]
,
or: P ∨ Q
,
sq_type: SQType(T)
,
guard: {T}
,
bnot: ¬bb
,
assert: ↑b
,
false: False
,
nequal: a ≠ b ∈ T
,
ge: i ≥ j
,
satisfiable_int_formula: satisfiable_int_formula(fmla)
,
top: Top
,
subtype_rel: A ⊆r B
,
iff: P
⇐⇒ Q
,
rev_implies: P
⇐ Q
Lemmas referenced :
fps-geometric-slice,
less_than_wf,
power-series_wf,
nat_wf,
crng_wf,
deq_wf,
valueall-type_wf,
fps-slice_wf,
fps-exp_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
nat_properties,
satisfiable-full-omega-tt,
intformnot_wf,
intformeq_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
squash_wf,
true_wf,
rem-one,
div-one,
iff_weakening_equal,
fps-slice-slice
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
cumulativity,
isect_memberEquality,
axiomEquality,
because_Cache,
universeEquality,
setElimination,
rename,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
lambdaEquality,
intEquality,
voidEquality,
computeAll,
applyEquality,
imageElimination
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[m:\mBbbN{}]. \mforall{}[g:PowerSeries(X;r)]. ([(1\mdiv{}(1-[g]\_1))]\_m = ([g]\_1)\^{}(m))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_59_03
Last ObjectModification:
2017_07_26-PM-06_33_40
Theory : power!series
Home
Index