Nuprl Lemma : fps-deriv-single
∀[X:Type]. ∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[b:bag(X)]. ∀[x:X].
(d<b>/dx = (int-to-ring(r;(#x in b)))*<bag-drop(eq;b;x)> ∈ PowerSeries(X;r))
Proof
Definitions occuring in Statement :
fps-deriv: df/dx,
fps-scalar-mul: (c)*f,
fps-single: <c>,
power-series: PowerSeries(X;r),
bag-drop: bag-drop(eq;bs;a),
bag-count: (#x in bs),
bag: bag(T),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T,
int-to-ring: int-to-ring(r;n),
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
crng: CRng,
subtype_rel: A ⊆r B,
nat: ℕ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
all: ∀x:A. B[x],
fps-single: <c>,
fps-scalar-mul: (c)*f,
fps-coeff: f[b],
fps-deriv: df/dx,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
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,
infix_ap: x f y,
rng: Rng,
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
top: Top,
rev_uimplies: rev_uimplies(P;Q),
sq_or: a ↓∨ b,
ringeq_int_terms: t1 ≡ t2,
so_lambda: λ2x.t[x],
so_apply: x[s],
decidable: Dec(P),
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla)
Lemmas referenced :
fps-ext,
fps-deriv_wf,
fps-single_wf,
fps-scalar-mul_wf,
int-to-ring_wf,
bag-count_wf,
nat_wf,
bag-drop_wf,
bag-drop-property,
bag-eq_wf,
cons-bag_wf,
bool_wf,
eqtt_to_assert,
assert-bag-eq,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bag_wf,
crng_wf,
deq_wf,
rng_times_wf,
squash_wf,
true_wf,
rng_wf,
rng_one_wf,
subtype_rel_self,
iff_weakening_equal,
cons-bag-as-append,
bag-count-append,
single-bag_wf,
add-commutes,
add_functionality_wrt_eq,
bag-count-member-no-repeats,
bag-member-single,
bag-single-no-repeats,
bag-append-cancel,
bag-member_wf,
bag-member-cons,
rng_zero_wf,
itermAdd_wf,
itermMultiply_wf,
itermVar_wf,
itermConstant_wf,
itermMinus_wf,
ringeq-iff-rsub-is-0,
ring_polynomial_null,
ring_term_value_add_lemma,
ring_term_value_mul_lemma,
ring_term_value_var_lemma,
ring_term_value_const_lemma,
int-to-ring-zero,
ring_term_value_minus_lemma,
bag-append_wf,
not_wf,
or_functionality_wrt_iff,
set_subtype_base,
le_wf,
int_subtype_base,
decidable__le,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
int_formula_prop_wf,
bag-count-is-zero,
rng_car_wf,
rng_times_zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesisEquality,
hypothesis,
setElimination,
rename,
applyEquality,
lambdaEquality,
sqequalRule,
productElimination,
independent_isectElimination,
lambdaFormation,
dependent_functionElimination,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
isect_memberEquality,
axiomEquality,
universeEquality,
imageElimination,
intEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
addEquality,
voidEquality,
inlFormation,
approximateComputation,
int_eqEquality,
productEquality,
independent_pairFormation,
applyLambdaEquality,
dependent_set_memberEquality
Latex:
\mforall{}[X:Type]. \mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[b:bag(X)]. \mforall{}[x:X].
(d<b>/dx = (int-to-ring(r;(\#x in b)))*<bag-drop(eq;b;x)>)
Date html generated:
2018_05_21-PM-10_16_26
Last ObjectModification:
2018_05_19-PM-04_18_06
Theory : power!series
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