Nuprl Lemma : fps-elim-hom
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[x:X].
(FunThru2op(PowerSeries(X;r);PowerSeries(X;r);λf,g. (f+g);λf,g. (f+g);fps-elim(x))
∧ FunThru2op(PowerSeries(X;r);PowerSeries(X;r);λf,g. (f*g);λf,g. (f*g);fps-elim(x))
∧ ((fps-elim(x) 1) = 1 ∈ PowerSeries(X;r)))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-elim: fps-elim(x),
fps-mul: (f*g),
fps-add: (f+g),
fps-one: 1,
power-series: PowerSeries(X;r),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
apply: f a,
lambda: λx.A[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng,
fun_thru_2op: FunThru2op(A;B;opa;opb;f)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
cand: A c∧ B,
fun_thru_2op: FunThru2op(A;B;opa;opb;f),
infix_ap: x f y,
uiff: uiff(P;Q),
all: ∀x:A. B[x],
fps-elim: fps-elim(x),
fps-add: (f+g),
fps-coeff: f[b],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
crng: CRng,
rng: Rng,
power-series: PowerSeries(X;r),
fps-mul: (f*g),
fps-one: 1,
true: True,
squash: ↓T,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
pi1: fst(t),
pi2: snd(t),
so_apply: x[s],
sq_or: a ↓∨ b,
ring_p: IsRing(T;plus;zero;neg;times;one),
group_p: IsGroup(T;op;id;inv),
top: Top
Lemmas referenced :
fps-ext,
fps-elim_wf,
fps-add_wf,
bag-deq-member_wf,
bool_wf,
eqtt_to_assert,
assert-bag-deq-member,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bag-member_wf,
rng_plus_wf,
bag_wf,
power-series_wf,
fps-mul_wf,
fps-one_wf,
bag-null_wf,
assert-bag-null,
bag-member-empty-iff,
equal-wf-T-base,
rng_zero_wf,
rng_one_wf,
crng_wf,
deq_wf,
valueall-type_wf,
rng_car_wf,
squash_wf,
true_wf,
rng_plus_zero,
iff_weakening_equal,
bag-summation-is-zero,
bag-partitions_wf,
rng_times_wf,
rng_plus_comm2,
bag-member-partitions,
rng_times_zero,
bag-member-append,
crng_properties,
rng_properties,
bag-summation-equal,
pi1_wf_top,
pi2_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
applyEquality,
cumulativity,
hypothesis,
productElimination,
independent_isectElimination,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
setElimination,
rename,
isect_memberEquality,
axiomEquality,
independent_pairFormation,
hyp_replacement,
applyLambdaEquality,
baseClosed,
independent_pairEquality,
universeEquality,
natural_numberEquality,
lambdaEquality,
imageElimination,
imageMemberEquality,
productEquality,
voidEquality,
inlFormation,
inrFormation
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[x:X].
(FunThru2op(PowerSeries(X;r);PowerSeries(X;r);\mlambda{}f,g. (f+g);\mlambda{}f,g. (f+g);fps-elim(x))
\mwedge{} FunThru2op(PowerSeries(X;r);PowerSeries(X;r);\mlambda{}f,g. (f*g);\mlambda{}f,g. (f*g);fps-elim(x))
\mwedge{} ((fps-elim(x) 1) = 1))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-09_59_09
Last ObjectModification:
2017_07_26-PM-06_33_41
Theory : power!series
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