Nuprl Lemma : fps-deriv-mul
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f,g:PowerSeries(X;r)]. ∀[x:X].
(d(f*g)/dx = ((f*dg/dx)+(df/dx*g)) ∈ PowerSeries(X;r))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-deriv: df/dx,
fps-mul: (f*g),
fps-add: (f+g),
power-series: PowerSeries(X;r),
deq: EqDecider(T),
valueall-type: valueall-type(T),
uimplies: b supposing a,
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,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
crng: CRng,
rng: Rng,
implies: P ⇒ Q,
compose: f o g,
squash: ↓T,
prop: ℙ,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
nat: ℕ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
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,
top: Top,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
nequal: a ≠ b ∈ T ,
infix_ap: x f y,
ge: i ≥ j ,
le: A ≤ B,
fps-deriv: df/dx,
fps-add: (f+g)
Lemmas referenced :
fps-linear-ucont-equal,
fps-deriv_wf,
fps-mul_wf,
power-series_wf,
fps-add_wf,
rng_car_wf,
bag_wf,
crng_wf,
deq_wf,
valueall-type_wf,
fps-ucont-composition,
fps-deriv-ucont,
fps-mul-ucont,
fps-add-ucont-general,
equal_wf,
squash_wf,
true_wf,
fps-deriv-add,
subtype_rel_self,
iff_weakening_equal,
mul_over_plus_fps,
mul_comm_fps,
mon_assoc_fps,
abmonoid_ac_1_fps,
fps-scalar-mul_wf,
fps-deriv-scalar-mul,
fps-scalar-mul-mul,
fps-scalar-mul-add,
fps-single_wf,
fps-ucont_wf,
fps-mul-comm,
abmonoid_comm_fps,
bag-append_wf,
int-to-ring_wf,
bag-count_wf,
nat_wf,
bag-drop_wf,
fps-mul-single,
fps-deriv-single,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
bag-drop-append,
int_subtype_base,
int-to-ring-zero,
mon_ident_fps,
rng_wf,
fps-scalar-mul-zero,
bag-count-append,
decidable__equal_int,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermAdd_wf,
itermVar_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
decidable__lt,
assert_wf,
bnot_wf,
not_wf,
equal-wf-T-base,
intformless_wf,
int_formula_prop_less_lemma,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot,
bag-append-comm,
rng_plus_wf,
fps-scalar-mul-rng-add,
int-to-ring-add,
set_subtype_base,
le_wf,
decidable__le,
nat_properties,
intformle_wf,
int_formula_prop_le_lemma,
fps-zero_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
independent_isectElimination,
hypothesis,
hypothesisEquality,
sqequalRule,
lambdaEquality,
independent_pairFormation,
lambdaFormation,
setElimination,
rename,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
dependent_functionElimination,
independent_functionElimination,
applyEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
productElimination,
hyp_replacement,
functionEquality,
unionElimination,
equalityElimination,
dependent_pairFormation,
promote_hyp,
cumulativity,
voidElimination,
intEquality,
voidEquality,
approximateComputation,
int_eqEquality,
applyLambdaEquality,
impliesFunctionality,
dependent_set_memberEquality,
addEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[f,g:PowerSeries(X;r)]. \mforall{}[x:X].
(d(f*g)/dx = ((f*dg/dx)+(df/dx*g)))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-10_16_51
Last ObjectModification:
2018_05_19-PM-04_18_58
Theory : power!series
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