Nuprl Lemma : polynomial-deriv-seq
∀I:Interval. ∀n:ℕ. ∀a:ℕn + 1 ⟶ ℝ. finite-deriv-seq(I;n;i,x.rpoly-nth-deriv(i;n;a;x))
Proof
Definitions occuring in Statement :
finite-deriv-seq: finite-deriv-seq(I;k;i,x.F[i; x]),
rpoly-nth-deriv: rpoly-nth-deriv(n;d;a;x),
interval: Interval,
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 :
all: ∀x:A. B[x],
finite-deriv-seq: finite-deriv-seq(I;k;i,x.F[i; x]),
rpoly-nth-deriv: rpoly-nth-deriv(n;d;a;x),
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
int_seg: {i..j-},
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 ,
guard: {T},
ge: i ≥ j ,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
decidable: Dec(P),
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
so_lambda: λ2x.t[x],
so_apply: x[s],
rpoly-deriv: rpoly-deriv(n;a;x),
nequal: a ≠ b ∈ T ,
poly-nth-deriv: poly-nth-deriv(n;a)
Lemmas referenced :
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
int_seg_properties,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
itermAdd_wf,
itermConstant_wf,
int_term_value_add_lemma,
int_term_value_constant_lemma,
derivative-rpolynomial,
subtract_wf,
decidable__le,
intformnot_wf,
intformle_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
le_wf,
poly-nth-deriv_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_dep_function,
int_seg_wf,
real_wf,
int_seg_subtype,
subtype_rel_self,
nat_wf,
interval_wf,
eq_int_wf,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
neg_assert_of_eq_int,
int_subtype_base,
decidable__equal_int,
primrec-unroll,
add-subtract-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
natural_numberEquality,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
addEquality,
dependent_set_memberEquality,
applyEquality,
functionEquality
Latex:
\mforall{}I:Interval. \mforall{}n:\mBbbN{}. \mforall{}a:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}. finite-deriv-seq(I;n;i,x.rpoly-nth-deriv(i;n;a;x))
Date html generated:
2017_10_03-PM-00_34_09
Last ObjectModification:
2017_07_28-AM-08_43_17
Theory : reals
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