Nuprl Lemma : fps-mul-ucont
∀[X:Type]. ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[g:PowerSeries(X;r)]. fps-ucont(X;eq;r;f.(f*g)) supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-ucont: fps-ucont(X;eq;r;f.G[f]),
fps-mul: (f*g),
power-series: PowerSeries(X;r),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
universe: Type,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
member: t ∈ T,
valueall-type: valueall-type(T),
has-value: (a)↓,
prop: ℙ,
fps-ucont: fps-ucont(X;eq;r;f.G[f]),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
fps-restrict: fps-restrict(eq;r;f;d),
fps-mul: (f*g),
fps-coeff: f[b],
crng: CRng,
rng: Rng,
so_lambda: λ2x.t[x],
power-series: PowerSeries(X;r),
top: Top,
so_apply: x[s],
pi1: fst(t),
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
iff: P ⇐⇒ Q,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
rev_implies: P ⇐ Q,
pi2: snd(t),
cand: A c∧ B,
sub-bag: sub-bag(T;as;bs),
ring_p: IsRing(T;plus;zero;neg;times;one),
group_p: IsGroup(T;op;id;inv)
Lemmas referenced :
equal-wf-base,
base_wf,
bag-summation-equal,
bag_wf,
rng_car_wf,
rng_plus_wf,
rng_zero_wf,
bag-partitions_wf,
rng_times_wf,
pi1_wf_top,
pi2_wf,
deq-sub-bag_wf,
bool_wf,
eqtt_to_assert,
assert-deq-sub-bag,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
sub-bag_wf,
bag-member_wf,
rng_plus_comm2,
all_wf,
power-series_wf,
fps-coeff_wf,
fps-mul_wf,
fps-restrict_wf,
crng_wf,
deq_wf,
valueall-type_wf,
bag-member-partitions,
bag-append_wf,
crng_properties,
rng_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomSqleEquality,
hypothesis,
extract_by_obid,
because_Cache,
equalityTransitivity,
equalitySymmetry,
rename,
lambdaFormation,
dependent_pairFormation,
productEquality,
cumulativity,
setElimination,
independent_isectElimination,
lambdaEquality,
applyEquality,
productElimination,
independent_pairEquality,
voidElimination,
voidEquality,
unionElimination,
equalityElimination,
dependent_functionElimination,
independent_functionElimination,
promote_hyp,
instantiate,
independent_pairFormation,
universeEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[g:PowerSeries(X;r)]. fps-ucont(X;eq;r;f.(f*g))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-10_11_15
Last ObjectModification:
2017_07_26-PM-06_34_39
Theory : power!series
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