Nuprl Lemma : rpoly-nth-deriv_wf
∀[d,n:ℕ]. ∀[a:ℕd + 1 ⟶ ℝ]. ∀[x:ℝ]. (rpoly-nth-deriv(n;d;a;x) ∈ ℝ)
Proof
Definitions occuring in Statement :
rpoly-nth-deriv: rpoly-nth-deriv(n;d;a;x),
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
member: t ∈ T,
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 ,
bfalse: ff,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
prop: ℙ,
subtract: n - m,
sq_type: SQType(T)
Lemmas referenced :
lt_int_wf,
bool_wf,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
less_than_wf,
eqtt_to_assert,
assert_of_lt_int,
int-to-real_wf,
le_int_wf,
le_wf,
bnot_wf,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
rpolynomial_wf,
subtract_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_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_wf,
poly-nth-deriv_wf,
itermAdd_wf,
int_term_value_add_lemma,
subtype_base_sq,
int_subtype_base,
add-commutes,
add-associates,
minus-one-mul,
add-swap,
add-mul-special,
zero-mul,
add-zero,
equal_wf,
real_wf,
int_seg_wf,
nat_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
because_Cache,
hypothesis,
hypothesisEquality,
lambdaFormation,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
baseClosed,
independent_functionElimination,
productElimination,
independent_isectElimination,
natural_numberEquality,
dependent_set_memberEquality,
dependent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
addEquality,
instantiate,
cumulativity,
multiplyEquality,
axiomEquality,
functionEquality
Latex:
\mforall{}[d,n:\mBbbN{}]. \mforall{}[a:\mBbbN{}d + 1 {}\mrightarrow{} \mBbbR{}]. \mforall{}[x:\mBbbR{}]. (rpoly-nth-deriv(n;d;a;x) \mmember{} \mBbbR{})
Date html generated:
2017_10_03-PM-00_15_14
Last ObjectModification:
2017_07_28-AM-08_37_41
Theory : reals
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