Nuprl Lemma : fps-rng_wf
∀[X:Type]. ∀[eq:EqDecider(X)]. ∀[r:CRng]. (fps-rng(r) ∈ CRng) supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-rng: fps-rng(r),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
member: t ∈ T,
universe: Type,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
crng: CRng,
rng: Rng,
prop: ℙ,
fps-rng: fps-rng(r),
rng_sig: RngSig,
ring_p: IsRing(T;plus;zero;neg;times;one),
and: P ∧ Q,
group_p: IsGroup(T;op;id;inv),
cand: A c∧ B,
rng_car: |r|,
pi1: fst(t),
rng_plus: +r,
pi2: snd(t),
rng_zero: 0,
rng_minus: -r,
infix_ap: x f y,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
rng_times: *,
rng_one: 1,
monoid_p: IsMonoid(T;op;id),
inverse: Inverse(T;op;id;inv),
bilinear: BiLinear(T;pl;tm),
assoc: Assoc(T;op),
fps-coeff: f[b],
power-series: PowerSeries(X;r),
fps-add: (f+g),
fps-zero: 0,
ident: Ident(T;op;id),
fps-neg: -(f),
comm: Comm(T;op),
fps-one: 1,
fps-mul: (f*g),
so_lambda: λ2x.t[x],
all: ∀x:A. B[x],
so_apply: x[s],
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),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
top: Top,
bag-null: bag-null(bs),
empty-bag: {}
Lemmas referenced :
comm_wf,
rng_car_wf,
rng_times_wf,
crng_wf,
deq_wf,
valueall-type_wf,
ring_p_wf,
rng_plus_wf,
rng_zero_wf,
rng_minus_wf,
rng_one_wf,
bool_wf,
unit_wf2,
fps-one_wf,
fps-mul_wf,
fps-neg_wf,
fps-zero_wf,
fps-add_wf,
btrue_wf,
power-series_wf,
crng_properties,
rng_properties,
equal_wf,
squash_wf,
true_wf,
istype-universe,
fps-add-comm,
subtype_rel_self,
iff_weakening_equal,
fps-add-assoc,
bag_wf,
fps-mul-assoc,
rng_plus_comm,
fps-mul-comm,
bag-split,
bag-null_wf,
bag-partitions_wf,
bag-summation-append,
eqtt_to_assert,
assert-bag-null,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
equal-wf-T-base,
bag-filter_wf,
bnot_wf,
bag-summation_wf,
crng_all_properties,
infix_ap_wf,
bag-append-ident,
empty-bag_wf,
bag-partitions-with-one-given,
subtype_rel_bag,
iff_wf,
assert-bag-eq,
bag-eq_wf,
pi2_wf,
iff_imp_equal_bool,
bag-summation-single,
pi1_wf_top,
subtype_rel_product,
top_wf,
istype-void,
null_nil_lemma,
bag-summation-is-zero,
null_wf3,
bag-subtype-list,
assert_of_null,
assert_elim,
bfalse_wf,
btrue_neq_bfalse,
subtype_rel_list,
bag-member_wf,
rng_plus_zero,
fps-coeff_wf,
rng_times_zero,
assoc_wf,
bag-summation-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
dependent_set_memberEquality_alt,
universeIsType,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality_alt,
universeEquality,
productEquality,
unionEquality,
functionEquality,
inlEquality,
independent_isectElimination,
lambdaEquality,
dependent_pairEquality,
productElimination,
independent_pairFormation,
applyEquality,
lambdaEquality_alt,
imageElimination,
inhabitedIsType,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
independent_functionElimination,
independent_pairEquality,
isect_memberEquality,
isect_memberFormation,
functionExtensionality,
functionExtensionality_alt,
dependent_functionElimination,
productIsType,
lambdaFormation_alt,
unionElimination,
equalityElimination,
dependent_pairFormation_alt,
equalityIsType1,
promote_hyp,
cumulativity,
voidElimination,
equalityIsType3,
hyp_replacement,
applyLambdaEquality,
setEquality,
setIsType,
impliesFunctionality,
addLevel,
lambdaFormation,
dependent_pairFormation,
voidEquality
Latex:
\mforall{}[X:Type]. \mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. (fps-rng(r) \mmember{} CRng) supposing valueall-type(X)
Date html generated:
2019_10_16-AM-11_34_24
Last ObjectModification:
2018_10_11-PM-02_48_30
Theory : power!series
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