Nuprl Lemma : fps-compose-atom
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[x,y:X]. ∀[f:PowerSeries(X;r)].
atom(y)(x:=f) = if eq y x then f else atom(y) fi ∈ PowerSeries(X;r) supposing f[{}] = 0 ∈ |r|
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-compose: g(x:=f),
fps-atom: atom(x),
fps-coeff: f[b],
power-series: PowerSeries(X;r),
empty-bag: {},
deq: EqDecider(T),
valueall-type: valueall-type(T),
ifthenelse: if b then t else f fi ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
universe: Type,
equal: s = t ∈ T,
crng: CRng,
rng_zero: 0,
rng_car: |r|
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
deq: EqDecider(T),
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
eqof: eqof(d),
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,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
crng: CRng,
rng: Rng,
fps-coeff: f[b],
fps-one: 1,
fps-scalar-mul: (c)*f,
fps-sub: (f-g),
fps-neg: -(f),
fps-add: (f+g),
power-series: PowerSeries(X;r),
infix_ap: x f y
Lemmas referenced :
bool_wf,
eqtt_to_assert,
safe-assert-deq,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
squash_wf,
true_wf,
power-series_wf,
fps-compose-atom-neq,
fps-atom_wf,
iff_weakening_equal,
rng_car_wf,
fps-coeff_wf,
empty-bag_wf,
rng_zero_wf,
crng_wf,
deq_wf,
valueall-type_wf,
fps-compose-atom-eq,
fps-compose_wf,
fps-ext,
fps-sub_wf,
fps-scalar-mul_wf,
fps-one_wf,
bag-null_wf,
assert-bag-null,
equal-wf-T-base,
bag_wf,
rng_plus_wf,
rng_minus_wf,
rng_one_wf,
rng_times_zero,
rng_minus_zero,
rng_plus_zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
applyEquality,
sqequalHypSubstitution,
setElimination,
thin,
rename,
hypothesisEquality,
hypothesis,
extract_by_obid,
lambdaFormation,
unionElimination,
equalityElimination,
isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
because_Cache,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
independent_functionElimination,
voidElimination,
cumulativity,
lambdaEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
universeEquality,
isect_memberEquality,
axiomEquality,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[x,y:X]. \mforall{}[f:PowerSeries(X;r)].
atom(y)(x:=f) = if eq y x then f else atom(y) fi supposing f[\{\}] = 0
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-10_06_11
Last ObjectModification:
2017_07_26-PM-06_34_12
Theory : power!series
Home
Index