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
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