Nuprl Lemma : IVT-rpolynomial1
∀n:ℕ. ∀a:ℕn + 1 ⟶ ℝ.
(((Σi≤n. a_i * r0^i) < r0) ⇒ (r0 < (Σi≤n. a_i * r1^i)) ⇒ (∃x:{x:ℝ| x ∈ [r0, r1]} . ((Σi≤n. a_i * x^i) = r0)))
Proof
Definitions occuring in Statement :
rccint: [l, u],
i-member: r ∈ I,
rpolynomial: (Σi≤n. a_i * x^i),
rless: x < y,
req: x = y,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
prop: ℙ,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
uimplies: b supposing a,
r-ap: f(x),
top: Top,
guard: {T},
sq_exists: ∃x:A [B[x]],
exists: ∃x:A. B[x],
sq_stable: SqStable(P),
nat: ℕ,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
int_seg: {i..j-},
rless: x < y,
nat_plus: ℕ+,
ge: i ≥ j ,
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False
Lemmas referenced :
IVT-locally-non-constant,
int-to-real_wf,
rless-int,
rless_wf,
rpolynomial_wf,
real_wf,
i-member_wf,
rccint_wf,
req_wf,
member_rccint_lemma,
istype-void,
rleq_weakening_rless,
rpolynomial-locally-non-zero,
rleq_wf,
sq_stable__req,
int_seg_wf,
istype-nat,
req_weakening,
nat_plus_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__lt,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
istype-le,
istype-less_than,
rpolynomial_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
natural_numberEquality,
hypothesis,
productElimination,
independent_functionElimination,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
hypothesisEquality,
baseClosed,
dependent_set_memberEquality_alt,
universeIsType,
lambdaEquality_alt,
setElimination,
rename,
setIsType,
independent_isectElimination,
because_Cache,
isect_memberEquality_alt,
voidElimination,
dependent_pairFormation_alt,
productIsType,
imageElimination,
functionIsType,
addEquality,
applyEquality,
unionElimination,
approximateComputation,
int_eqEquality
Latex:
\mforall{}n:\mBbbN{}. \mforall{}a:\mBbbN{}n + 1 {}\mrightarrow{} \mBbbR{}.
(((\mSigma{}i\mleq{}n. a\_i * r0\^{}i) < r0)
{}\mRightarrow{} (r0 < (\mSigma{}i\mleq{}n. a\_i * r1\^{}i))
{}\mRightarrow{} (\mexists{}x:\{x:\mBbbR{}| x \mmember{} [r0, r1]\} . ((\mSigma{}i\mleq{}n. a\_i * x\^{}i) = r0)))
Date html generated:
2019_10_30-AM-09_13_13
Last ObjectModification:
2019_02_13-AM-11_20_28
Theory : reals
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