Nuprl Lemma : fps-geometric-slice
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[m:ℕ]. ∀[n:ℕ+]. ∀[g:PowerSeries(X;r)].
[(1÷(1-g))]_m = if (m rem n =z 0) then (g)^(m ÷ n) else 0 fi ∈ PowerSeries(X;r)
supposing g = [g]_n ∈ PowerSeries(X;r)
supposing valueall-type(X)
This theorem is one of freek's list of 100 theorems
Proof
Definitions occuring in Statement :
fps-exp: (f)^(n),
fps-slice: [f]_n,
fps-div: (f÷g),
fps-sub: (f-g),
fps-one: 1,
fps-zero: 0,
power-series: PowerSeries(X;r),
deq: EqDecider(T),
nat_plus: ℕ+,
nat: ℕ,
valueall-type: valueall-type(T),
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
remainder: n rem m,
divide: n ÷ m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_one: 1
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
nat_plus: ℕ+,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
nequal: a ≠ b ∈ T ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
int_upper: {i...},
label: ...$L... t
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
equal_wf,
power-series_wf,
fps-slice_wf,
nat_plus_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
subtract_wf,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
le_wf,
decidable__lt,
lelt_wf,
itermAdd_wf,
int_term_value_add_lemma,
nat_wf,
crng_wf,
deq_wf,
valueall-type_wf,
fps-geometric-slice_lemma2,
squash_wf,
true_wf,
eq_int_wf,
nat_plus_properties,
equal-wf-base,
int_subtype_base,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
fps-exp_wf,
divide_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
fps-zero_wf,
iff_weakening_equal,
rem_base_case,
div_base_case,
fps-one_wf,
fps-exp-zero,
int_upper_subtype_nat,
nequal-le-implies,
zero-add,
fps-geometric-slice_lemma,
fps-mul_wf,
add-is-int-iff,
div_bounds_1,
ifthenelse_wf,
div_rec_case,
fps-exp-unroll,
add-subtract-cancel,
rem_rec_case,
mul_zero_fps
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
thin,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
cumulativity,
equalityTransitivity,
equalitySymmetry,
because_Cache,
productElimination,
unionElimination,
applyEquality,
applyLambdaEquality,
hypothesis_subsumption,
dependent_set_memberEquality,
addEquality,
universeEquality,
imageElimination,
remainderEquality,
baseClosed,
equalityElimination,
promote_hyp,
instantiate,
imageMemberEquality,
hyp_replacement,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[m:\mBbbN{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[g:PowerSeries(X;r)].
[(1\mdiv{}(1-g))]\_m = if (m rem n =\msubz{} 0) then (g)\^{}(m \mdiv{} n) else 0 fi supposing g = [g]\_n
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_58_59
Last ObjectModification:
2017_07_26-PM-06_33_39
Theory : power!series
Home
Index