Nuprl Lemma : rpoly-nth-deriv_functionality
∀[d,n:ℕ]. ∀[a,b:ℕd + 1 ⟶ ℝ]. ∀[x1,x2:ℝ].
(rpoly-nth-deriv(n;d;a;x1) = rpoly-nth-deriv(n;d;b;x2)) supposing ((x1 = x2) and (∀i:ℕd + 1. ((a i) = (b i))))
Proof
Definitions occuring in Statement :
rpoly-nth-deriv: rpoly-nth-deriv(n;d;a;x),
req: x = y,
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
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,
ifthenelse: if b then t else f fi ,
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),
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
subtract: n - m,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
so_apply: x[s],
pointwise-req: x[k] = y[k] for k ∈ [n,m],
int_seg: {i..j-},
lelt: i ≤ j < k,
nat_plus: ℕ+,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
rev_uimplies: rev_uimplies(P;Q),
rneq: x ≠ y,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
req_weakening,
int-to-real_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
less_than_wf,
poly-nth-deriv_wf,
subtract_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_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_add_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
le_wf,
int_subtype_base,
add-commutes,
add-associates,
minus-one-mul,
add-swap,
add-mul-special,
zero-mul,
add-zero,
rsum_functionality,
rmul_wf,
rnexp_wf,
int_seg_subtype_nat,
false_wf,
int_seg_wf,
rmul_functionality,
decidable__lt,
lelt_wf,
req_witness,
rpoly-nth-deriv_wf,
req_wf,
all_wf,
real_wf,
nat_wf,
poly-nth-deriv-req,
int-rdiv_wf,
fact_wf,
subtype_rel_sets,
nequal_wf,
nat_plus_properties,
intformeq_wf,
int_formula_prop_eq_lemma,
equal-wf-base,
nat_plus_wf,
req_functionality,
rdiv_wf,
rless-int,
rless_wf,
int-rdiv-req,
rdiv_functionality,
rnexp_functionality
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,
natural_numberEquality,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
independent_functionElimination,
voidElimination,
dependent_set_memberEquality,
addEquality,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
independent_pairFormation,
computeAll,
multiplyEquality,
applyEquality,
functionExtensionality,
functionEquality,
setEquality,
applyLambdaEquality,
baseClosed,
inrFormation
Latex:
\mforall{}[d,n:\mBbbN{}]. \mforall{}[a,b:\mBbbN{}d + 1 {}\mrightarrow{} \mBbbR{}]. \mforall{}[x1,x2:\mBbbR{}].
(rpoly-nth-deriv(n;d;a;x1) = rpoly-nth-deriv(n;d;b;x2)) supposing
((x1 = x2) and
(\mforall{}i:\mBbbN{}d + 1. ((a i) = (b i))))
Date html generated:
2017_10_03-PM-00_15_38
Last ObjectModification:
2017_07_28-AM-08_37_59
Theory : reals
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