Nuprl Lemma : fps-deriv-compose
∀[X:Type]
∀[eq:EqDecider(X)]. ∀[r:CRng]. ∀[f,g:PowerSeries(X;r)]. ∀[x:X].
(df(x:=g)/dx = (df/dx(x:=g)*dg/dx) ∈ PowerSeries(X;r))
supposing valueall-type(X)
Proof
Definitions occuring in Statement :
fps-deriv: df/dx,
fps-compose: g(x:=f),
fps-mul: (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],
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
crng: CRng,
rng: Rng,
compose: f o g,
prop: ℙ,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
false: False,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
sq_type: SQType(T),
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
fps-sub: (f-g),
power-series: PowerSeries(X;r),
fps-zero: 0,
fps-coeff: f[b],
empty-bag: {},
fps-scalar-mul: (c)*f,
fps-neg: -(f),
infix_ap: x f y,
nat_plus: ℕ+,
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
fps-deriv: df/dx,
fps-mul: (f*g)
Lemmas referenced :
fps-linear-ucont-equal,
fps-deriv_wf,
fps-compose_wf,
power-series_wf,
fps-mul_wf,
equal_wf,
fps-compose-add,
fps-add_wf,
iff_weakening_equal,
fps-deriv-add,
fps-compose-scalar-mul,
fps-scalar-mul_wf,
fps-deriv-scalar-mul,
rng_car_wf,
bag_wf,
crng_wf,
deq_wf,
valueall-type_wf,
fps-ucont-composition,
fps-deriv-ucont,
fps-compose-ucont,
fps-mul-ucont,
squash_wf,
true_wf,
subtype_rel_self,
mul_over_plus_fps,
fps-scalar-mul-mul,
fps-single_wf,
int-to-ring_wf,
bag-count_wf,
nat_wf,
bag-drop_wf,
fps-deriv-single,
fps-sub_wf,
fps-coeff_wf,
empty-bag_wf,
fps-one_wf,
fps-compose-single-general,
fps-deriv-mul,
bag-co-restrict_wf,
fps-exp_wf,
bag-size_wf,
bag-restrict_wf,
fps-zero_wf,
subtype_base_sq,
set_subtype_base,
le_wf,
int_subtype_base,
decidable__le,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_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,
bag-co-restrict-property,
int-to-ring-zero,
fps-scalar-mul-zero,
mul_zero_fps,
mul_comm_fps,
mon_ident_fps,
fps-mul-assoc,
bag-drop-co-restrict,
bag-size-restrict,
decidable__equal_int,
rng_zero_wf,
fps-deriv-one,
fps-exp-zero,
bag-count-drop,
intformeq_wf,
itermAdd_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
bag-member-count,
fps-neg_wf,
fps-deriv-neg,
fps-ext,
rng_minus_wf,
nil_wf,
list-subtype-bag,
rng_times_zero,
rng_minus_zero,
intformless_wf,
int_formula_prop_less_lemma,
ge_wf,
less_than_wf,
subtract_wf,
fps-scalar-mul-one,
fps-exp-one,
int-to-ring-one,
mul_one_fps,
decidable__lt,
false_wf,
not-lt-2,
less-iff-le,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
subtract-add-cancel,
add-subtract-cancel,
fps-exp-unroll,
fps-mul-comm,
rng_plus_wf,
fps-scalar-mul-rng-add,
rng_one_wf,
fps-add-comm,
int-to-ring-add
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,
applyEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_functionElimination,
setElimination,
rename,
isect_memberEquality,
axiomEquality,
universeEquality,
dependent_functionElimination,
instantiate,
equalityUniverse,
levelHypothesis,
hyp_replacement,
applyLambdaEquality,
cumulativity,
intEquality,
unionElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
addEquality,
intWeakElimination,
minusEquality
Latex:
\mforall{}[X:Type]
\mforall{}[eq:EqDecider(X)]. \mforall{}[r:CRng]. \mforall{}[f,g:PowerSeries(X;r)]. \mforall{}[x:X].
(df(x:=g)/dx = (df/dx(x:=g)*dg/dx))
supposing valueall-type(X)
Date html generated:
2018_05_21-PM-10_17_15
Last ObjectModification:
2018_05_19-PM-04_19_35
Theory : power!series
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