Nuprl Lemma : derivative-rpolynomial
∀n:ℕ. ∀a:ℕn + 1 ⟶ ℝ. ∀I:Interval. d((Σi≤n. a_i * x^i))/dx = λx.rpoly-deriv(n;a;x) on I
Proof
Definitions occuring in Statement :
rpoly-deriv: rpoly-deriv(n;a;x),
derivative: d(f[x])/dx = λz.g[z] on I,
interval: Interval,
rpolynomial: (Σi≤n. a_i * x^i),
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
rpoly-deriv: rpoly-deriv(n;a;x),
poly-deriv: poly-deriv(a),
all: ∀x:A. B[x],
eq_int: (i =z j),
subtract: n - m,
ifthenelse: if b then t else f fi ,
btrue: tt,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
label: ...$L... t,
rfun: I ⟶ℝ,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
and: P ∧ Q,
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
ge: i ≥ j ,
r-ap: f(x),
rfun-eq: rfun-eq(I;f;g),
subtype_rel: A ⊆r B,
true: True,
squash: ↓T,
less_than': less_than'(a;b),
rev_uimplies: rev_uimplies(P;Q),
nat_plus: ℕ+,
itermConstant: "const",
req_int_terms: t1 ≡ t2,
real_term_value: real_term_value(f;t),
int_term_ind: int_term_ind,
itermSubtract: left (-) right,
itermAdd: left (+) right,
itermMultiply: left (*) right,
itermVar: vvar
Lemmas referenced :
interval_wf,
int_seg_wf,
real_wf,
all_wf,
subtract_wf,
derivative_wf,
rpolynomial_wf,
subtract-add-cancel,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
i-member_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
int-to-real_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
rmul_wf,
add-member-int_seg2,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
lelt_wf,
set_wf,
less_than_wf,
primrec-wf2,
nat_properties,
add-subtract-cancel,
nat_wf,
derivative_functionality,
req_weakening,
subtype_rel_self,
subtype_rel_dep_function,
top_wf,
false_wf,
derivative-const,
rpolynomial_unroll,
req_functionality,
rnexp_zero_lemma,
rnexp_wf,
radd_wf,
int_seg_subtype,
derivative-add,
derivative-rnexp,
derivative-const-mul,
rmul-ac,
int_subtype_base,
decidable__equal_int,
real_term_polynomial,
itermMultiply_wf,
req-iff-rsub-is-0
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
hypothesis,
functionEquality,
sqequalHypSubstitution,
isectElimination,
natural_numberEquality,
addEquality,
rename,
setElimination,
hypothesisEquality,
because_Cache,
lambdaEquality,
dependent_set_memberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
functionExtensionality,
applyEquality,
setEquality,
equalityElimination,
productElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
baseClosed,
imageMemberEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}. \mforall{}I:Interval. d((\mSigma{}i\mleq{}n. a\_i * x\^{}i))/dx = \mlambda{}x.rpoly-deriv(n;a;x) on I
Date html generated:
2017_10_03-PM-00_16_26
Last ObjectModification:
2017_07_28-AM-08_38_33
Theory : reals
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