Nuprl Lemma : fps-deriv-commutes
∀[X:Type]. ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f,g:PowerSeries(X;r)]. ∀[x,y:X]. (ddf/dx/dy = ddf/dy/dx ∈ PowerSeries(X;r))
Proof
Definitions occuring in Statement :
fps-deriv: df/dx,
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-deriv: df/dx,
fps-coeff: f[b],
crng: CRng,
rng: Rng,
subtype_rel: A ⊆r B,
nat: ℕ,
power-series: PowerSeries(X;r),
true: True,
infix_ap: x f y,
squash: ↓T,
prop: ℙ,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
eqof: eqof(d),
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A,
top: Top
Lemmas referenced :
fps-ext,
fps-deriv_wf,
bag_wf,
rng_car_wf,
int-to-ring_wf,
bag-count_wf,
nat_wf,
cons-bag_wf,
rng_times_wf,
equal_wf,
squash_wf,
true_wf,
rng_times_assoc,
subtype_rel_self,
iff_weakening_equal,
ifthenelse_wf,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
and_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
crng_times_comm,
rng_wf,
add_functionality_wrt_eq,
bag-count-cons,
bag-append-assoc2,
cons-bag-as-append,
bag-append-assoc,
bag-append-assoc-comm,
single-bag_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
productElimination,
independent_isectElimination,
lambdaFormation,
sqequalRule,
isect_memberEquality,
axiomEquality,
setElimination,
rename,
addEquality,
applyEquality,
lambdaEquality,
natural_numberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
instantiate,
independent_functionElimination,
intEquality,
unionElimination,
equalityElimination,
dependent_set_memberEquality,
independent_pairFormation,
applyLambdaEquality,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
cumulativity,
voidElimination,
universeEquality,
voidEquality
Latex:
\mforall{}[X:Type]. \mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[f,g:PowerSeries(X;r)]. \mforall{}[x,y:X].
(ddf/dx/dy = ddf/dy/dx)
Date html generated:
2018_05_21-PM-10_17_20
Last ObjectModification:
2018_05_19-PM-04_19_12
Theory : power!series
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