Nuprl Lemma : derivative-rsum
∀[I:Interval]
∀n:ℕ. ∀m:{n...}.
∀[f,f':{n..m + 1-} ⟶ I ⟶ℝ].
((∀k:{n..m + 1-}. d(f[k;x])/dx = λx.f'[k;x] on I) ⇒ d(Σ{f[k;x] | n≤k≤m})/dx = λx.Σ{f'[k;x] | n≤k≤m} on I)
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
rfun: I ⟶ℝ,
interval: Interval,
rsum: Σ{x[k] | n≤k≤m},
int_upper: {i...},
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
rev_uimplies: rev_uimplies(P;Q),
r-ap: f(x),
rfun-eq: rfun-eq(I;f;g),
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
guard: {T},
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
so_apply: x[s],
subtype_rel: A ⊆r B,
so_apply: x[s1;s2],
rfun: I ⟶ℝ,
label: ...$L... t,
so_lambda: λ2x.t[x],
int_upper: {i...},
nat: ℕ,
prop: ℙ,
member: t ∈ T,
implies: P ⇒ Q,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x]
Lemmas referenced :
trivial-int-eq1,
decidable__equal_int,
int_subtype_base,
derivative-add,
rsum_unroll,
req_functionality,
derivative_functionality,
req_weakening,
int_formula_prop_eq_lemma,
intformeq_wf,
int_seg_properties,
radd_wf,
neg_assert_of_eq_int,
assert_of_eq_int,
eq_int_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
equal_wf,
eqff_to_assert,
int-to-real_wf,
assert_of_lt_int,
eqtt_to_assert,
bool_wf,
lt_int_wf,
decidable__le,
primrec-wf2,
less_than_wf,
set_wf,
lelt_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermSubtract_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_properties,
int_upper_properties,
rsum_wf,
subtract_wf,
le_wf,
interval_wf,
nat_wf,
int_upper_wf,
rfun_wf,
i-member_wf,
real_wf,
subtype_rel_self,
derivative_wf,
int_seg_wf,
all_wf
Rules used in proof :
cumulativity,
promote_hyp,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
instantiate,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
dependent_functionElimination,
independent_pairFormation,
productElimination,
dependent_set_memberEquality,
functionExtensionality,
setEquality,
functionEquality,
applyEquality,
hypothesisEquality,
lambdaEquality,
sqequalRule,
natural_numberEquality,
addEquality,
hypothesis,
because_Cache,
rename,
setElimination,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
lambdaFormation,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[I:Interval]
\mforall{}n:\mBbbN{}. \mforall{}m:\{n...\}.
\mforall{}[f,f':\{n..m + 1\msupminus{}\} {}\mrightarrow{} I {}\mrightarrow{}\mBbbR{}].
((\mforall{}k:\{n..m + 1\msupminus{}\}. d(f[k;x])/dx = \mlambda{}x.f'[k;x] on I)
{}\mRightarrow{} d(\mSigma{}\{f[k;x] | n\mleq{}k\mleq{}m\})/dx = \mlambda{}x.\mSigma{}\{f'[k;x] | n\mleq{}k\mleq{}m\} on I)
Date html generated:
2018_05_22-PM-02_45_43
Last ObjectModification:
2018_05_21-AM-00_54_12
Theory : reals
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