Nuprl Lemma : fps-div-coeff_wf
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f,g:PowerSeries(X;r)]. ∀[x:|r|]. ∀[b:bag(X)]. (fps-div-coeff(eq;r;f;g;x;b) ∈ |r|)
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-div-coeff: fps-div-coeff(eq;r;f;g;x;b),
power-series: PowerSeries(X;r),
bag: bag(T),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type,
crng: CRng,
rng_car: |r|
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},
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
less_than': less_than'(a;b),
fps-div-coeff: fps-div-coeff(eq;r;f;g;x;b),
infix_ap: x f y,
crng: CRng,
rng: Rng,
so_lambda: λ2x.t[x],
so_apply: x[s],
pi2: snd(t),
pi1: fst(t),
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
uiff: uiff(P;Q),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
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,
le_wf,
bag-size_wf,
nat_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,
rng_times_wf,
rng_plus_wf,
fps-coeff_wf,
rng_minus_wf,
crng_properties,
rng_all_properties,
bag_wf,
pi2_wf,
bag-summation_wf,
infix_ap_wf,
decidable__lt,
lelt_wf,
rng_plus_comm2,
equal_wf,
itermAdd_wf,
int_term_value_add_lemma,
rng_car_wf,
power-series_wf,
crng_wf,
deq_wf,
valueall-type_wf,
bag-append_wf,
pi1_wf_top,
bag-subtype,
bag-partitions_wf,
bag-member_wf,
subtype_rel_bag,
subtype_rel_sets,
bag-member-partitions,
bag-filter_wf,
bnot_wf,
bag-null_wf,
assert_wf,
set_wf,
bag-size-append,
null-bag-size,
eq_int_wf,
not_wf,
equal-wf-T-base,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
assert_of_eq_int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
thin,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
cumulativity,
applyEquality,
because_Cache,
productElimination,
unionElimination,
applyLambdaEquality,
hypothesis_subsumption,
dependent_set_memberEquality,
setEquality,
productEquality,
independent_pairEquality,
imageElimination,
addEquality,
universeEquality,
hyp_replacement,
baseClosed,
impliesFunctionality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[f,g:PowerSeries(X;r)]. \mforall{}[x:|r|]. \mforall{}[b:bag(X)].
(fps-div-coeff(eq;r;f;g;x;b) \mmember{} |r|)
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_55_30
Last ObjectModification:
2017_07_26-PM-06_32_43
Theory : power!series
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