Nuprl Lemma : rpoly-nth-deriv-linear
∀[d,n:ℕ]. ∀[a,b:ℕd + 1 ⟶ ℝ]. ∀[x:ℝ].
(rpoly-nth-deriv(n;d;λi.((a i) + (b i));x) = (rpoly-nth-deriv(n;d;a;x) + rpoly-nth-deriv(n;d;b;x)))
Proof
Definitions occuring in Statement :
rpoly-nth-deriv: rpoly-nth-deriv(n;d;a;x),
req: x = y,
radd: a + b,
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rpoly-nth-deriv: rpoly-nth-deriv(n;d;a;x),
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
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,
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,
rpolynomial: (Σi≤n. a_i * x^i),
rev_uimplies: rev_uimplies(P;Q),
pointwise-req: x[k] = y[k] for k ∈ [n,m],
so_apply: x[s],
less_than': less_than'(a;b),
le: A ≤ B,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
lelt: i ≤ j < k,
int_seg: {i..j-},
subtract: n - m,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
ge: i ≥ j ,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
nat_plus: ℕ+,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
rneq: x ≠ y
Lemmas referenced :
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
real_term_polynomial,
itermSubtract_wf,
itermConstant_wf,
itermAdd_wf,
int-to-real_wf,
req-iff-rsub-is-0,
radd_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
req_witness,
rpoly-nth-deriv_wf,
int_seg_wf,
real_wf,
nat_wf,
rsum_linearity1,
req_inversion,
req_weakening,
req_functionality,
rsum_functionality,
false_wf,
int_seg_subtype_nat,
rnexp_wf,
rmul_wf,
rsum_wf,
lelt_wf,
decidable__lt,
add-zero,
zero-mul,
add-mul-special,
add-swap,
minus-one-mul,
add-associates,
add-commutes,
int_subtype_base,
le_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
subtract_wf,
poly-nth-deriv_wf,
radd_functionality,
rmul_functionality,
nat_plus_wf,
equal-wf-base,
int_formula_prop_eq_lemma,
intformeq_wf,
nat_plus_properties,
nequal_wf,
subtype_rel_sets,
fact_wf,
int-rdiv_wf,
poly-nth-deriv-req,
int-rdiv-req,
rless_wf,
rless-int,
rdiv_wf,
rmul-rdiv-cancel,
rmul-ac,
rmul_comm,
rmul-assoc,
rmul-distrib,
rdiv_functionality,
uiff_transitivity,
req_wf,
rmul_preserves_req
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
sqequalRule,
dependent_functionElimination,
natural_numberEquality,
computeAll,
lambdaEquality,
intEquality,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
applyEquality,
functionExtensionality,
addEquality,
isect_memberEquality,
functionEquality,
multiplyEquality,
independent_pairFormation,
voidEquality,
int_eqEquality,
dependent_set_memberEquality,
baseClosed,
applyLambdaEquality,
setEquality,
inrFormation
Latex:
\mforall{}[d,n:\mBbbN{}]. \mforall{}[a,b:\mBbbN{}d + 1 {}\mrightarrow{} \mBbbR{}]. \mforall{}[x:\mBbbR{}].
(rpoly-nth-deriv(n;d;\mlambda{}i.((a i) + (b i));x)
= (rpoly-nth-deriv(n;d;a;x) + rpoly-nth-deriv(n;d;b;x)))
Date html generated:
2017_10_03-PM-00_16_02
Last ObjectModification:
2017_07_28-AM-08_38_16
Theory : reals
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