Nuprl Lemma : fps-elim-x-atom
∀[X:Type]. ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[x,y:X].
(atom(y)(x:=0) = if eq x y then 0 else atom(y) fi ∈ PowerSeries(X;r))
Proof
Definitions occuring in Statement :
fps-elim-x: f(x:=0),
fps-atom: atom(x),
fps-zero: 0,
power-series: PowerSeries(X;r),
deq: EqDecider(T),
ifthenelse: if b then t else f fi ,
uall: ∀[x:A]. B[x],
apply: f a,
universe: Type,
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
deq: EqDecider(T),
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
fps-atom: atom(x),
fps-zero: 0,
fps-coeff: f[b],
fps-elim-x: f(x:=0),
fps-single: <c>,
fps-elim: fps-elim(x),
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
eqof: eqof(d),
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,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
fps-ext,
fps-elim-x_wf,
fps-atom_wf,
ifthenelse_wf,
power-series_wf,
fps-zero_wf,
bag-deq-member_wf,
bool_wf,
eqtt_to_assert,
assert-bag-deq-member,
safe-assert-deq,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bag-eq_wf,
single-bag_wf,
assert-bag-eq,
bag_wf,
rng_zero_wf,
bag-member_wf,
rng_one_wf,
crng_wf,
deq_wf,
bag-member-single,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
cumulativity,
hypothesis,
applyEquality,
setElimination,
rename,
productElimination,
independent_isectElimination,
lambdaFormation,
sqequalRule,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
isect_memberEquality,
axiomEquality,
universeEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[X:Type]. \mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[x,y:X].
(atom(y)(x:=0) = if eq x y then 0 else atom(y) fi )
Date html generated:
2018_05_21-PM-09_59_27
Last ObjectModification:
2017_07_26-PM-06_33_52
Theory : power!series
Home
Index