Nuprl Lemma : rpolynomial_unroll
∀[n:ℕ]. ∀[a:ℕn + 1 ⟶ ℝ]. ∀[x:ℝ].
((Σi≤n. a_i * x^i) = if (n =z 0) then a 0 else ((a n) * x^n) + (Σi≤n - 1. a_i * x^i) fi )
Proof
Definitions occuring in Statement :
rpolynomial: (Σi≤n. a_i * x^i),
rnexp: x^k1,
req: x = y,
rmul: a * b,
radd: a + b,
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rpolynomial: (Σi≤n. a_i * x^i),
nat: ℕ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
int_upper: {i...},
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
req_witness,
rpolynomial_wf,
int_seg_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
false_wf,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
lelt_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_upper_subtype_nat,
le_wf,
nequal-le-implies,
zero-add,
radd_wf,
rmul_wf,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
rnexp_wf,
subtract_wf,
int_upper_properties,
itermSubtract_wf,
int_term_value_subtract_lemma,
subtype_rel_dep_function,
real_wf,
int_seg_subtype,
subtract-add-cancel,
subtype_rel_self,
nat_wf,
rsum_wf,
int_seg_subtype_nat,
lt_int_wf,
assert_of_lt_int,
int-to-real_wf,
less_than_wf,
rnexp_zero_lemma,
rmul-one,
req_weakening,
req_functionality,
rsum_unroll,
radd_comm
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
functionExtensionality,
applyEquality,
natural_numberEquality,
addEquality,
setElimination,
rename,
hypothesis,
because_Cache,
lambdaFormation,
unionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
independent_pairFormation,
dependent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
hypothesis_subsumption,
functionEquality
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[a:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}]. \mforall{}[x:\mBbbR{}].
((\mSigma{}i\mleq{}n. a\_i * x\^{}i) = if (n =\msubz{} 0) then a 0 else ((a n) * x\^{}n) + (\mSigma{}i\mleq{}n - 1. a\_i * x\^{}i) fi )
Date html generated:
2017_10_03-AM-08_58_11
Last ObjectModification:
2017_07_28-AM-07_37_59
Theory : reals
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