Nuprl Lemma : fps-elim-x-elim-y
∀[X:Type]. ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[x,y:X]. ∀[g:PowerSeries(X;r)].
(g(x:=0)(y:=0) = g(y:=0)(x:=0) ∈ PowerSeries(X;r))
Proof
Definitions occuring in Statement :
fps-elim-x: f(x:=0),
power-series: PowerSeries(X;r),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
fps-elim-x: f(x:=0),
fps-coeff: f[b],
fps-elim: fps-elim(x),
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
crng: CRng,
rng: Rng,
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,
power-series: PowerSeries(X;r)
Lemmas referenced :
fps-ext,
fps-elim-x_wf,
bag-deq-member_wf,
bool_wf,
eqtt_to_assert,
assert-bag-deq-member,
rng_zero_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bag-member_wf,
bag_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
cumulativity,
hypothesis,
productElimination,
independent_isectElimination,
lambdaFormation,
sqequalRule,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
setElimination,
rename,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
applyEquality,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[X:Type]. \mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[x,y:X]. \mforall{}[g:PowerSeries(X;r)].
(g(x:=0)(y:=0) = g(y:=0)(x:=0))
Date html generated:
2018_05_21-PM-09_59_20
Last ObjectModification:
2017_07_26-PM-06_33_49
Theory : power!series
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