Nuprl Lemma : Rolles-theorem
∀a,b:ℝ.
((a < b)
⇒ (∀f,f':[a, b] ⟶ℝ.
(f'[x] continuous for x ∈ [a, b]
⇒ d(f[x])/dx = λx.f'[x] on [a, b]
⇒ (f[a] = f[b])
⇒ (∀e:ℝ. ((r0 < e) ⇒ (∃x:ℝ. ((x ∈ [a, b]) ∧ (|f'[x]| ≤ e))))))))
Proof
Definitions occuring in Statement :
derivative: d(f[x])/dx = λz.g[z] on I,
continuous: f[x] continuous for x ∈ I,
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
req: x = y,
int-to-real: r(n),
real: ℝ,
so_apply: x[s],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
rfun: I ⟶ℝ,
so_apply: x[s],
prop: ℙ,
top: Top,
and: P ∧ Q,
cand: A c∧ B,
uimplies: b supposing a,
guard: {T},
label: ...$L... t,
iff: P ⇐⇒ Q,
not: ¬A,
exists: ∃x:A. B[x],
inf: inf(A) = b,
lower-bound: lower-bound(A;b),
rrange: f[x](x∈I),
rset-member: x ∈ A,
i-member: r ∈ I,
rccint: [l, u],
nat_plus: ℕ+,
rneq: x ≠ y,
or: P ∨ Q,
rev_implies: P ⇐ Q,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
derivative: d(f[x])/dx = λz.g[z] on I,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
i-approx: i-approx(I;n),
iproper: iproper(I),
int_upper: {i...},
subtype_rel: A ⊆r B,
real: ℝ,
sq_stable: SqStable(P),
int_seg: {i..j-},
uiff: uiff(P;Q),
lelt: i ≤ j < k,
subtract: n - m,
le: A ≤ B,
rev_uimplies: rev_uimplies(P;Q),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
real_term_value: real_term_value(f;t),
int_term_ind: int_term_ind,
itermSubtract: left (-) right,
itermVar: vvar,
pointwise-req: x[k] = y[k] for k ∈ [n,m],
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
rleq: x ≤ y,
rnonneg: rnonneg(x),
rge: x ≥ y,
rsub: x - y,
rdiv: (x/y)
Lemmas referenced :
continuous-abs,
rccint_wf,
i-member_wf,
real_wf,
rless_wf,
int-to-real_wf,
req_wf,
member_rccint_lemma,
rleq_weakening_equal,
rleq_weakening_rless,
rleq_wf,
derivative_wf,
continuous_wf,
rfun_wf,
rccint-icompact,
not-rless,
range-inf_wf,
rabs_wf,
small-reciprocal-real,
range-inf-property,
icompact_wf,
req_weakening,
rless_transitivity1,
rdiv_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
all_wf,
rabs-nonzero-on-compact,
mul_nat_plus,
less_than_wf,
left_endpoint_rccint_lemma,
right_endpoint_rccint_lemma,
i-finite_wf,
simple-partition-exists,
rsum-telescopes,
subtract_wf,
sq_stable__less_than,
decidable__le,
intformle_wf,
itermSubtract_wf,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
le_wf,
int_seg_wf,
add-member-int_seg2,
subtract-add-cancel,
add-subtract-cancel,
lelt_wf,
itermAdd_wf,
int_term_value_add_lemma,
rsum_wf,
rsub_wf,
false_wf,
req_functionality,
squash_wf,
true_wf,
equal_wf,
iff_weakening_equal,
real_term_polynomial,
req-iff-rsub-is-0,
rsub_functionality,
rsum_linearity1,
rmul_wf,
rsum_functionality,
radd_wf,
itermMultiply_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req_inversion,
req_transitivity,
rabs-rsum,
rsum_functionality_wrt_rleq,
int_seg_properties,
int_term_value_mul_lemma,
set_wf,
less_than'_wf,
nat_plus_wf,
radd-preserves-rleq,
sq_stable__and,
sq_stable__rleq,
rleq_functionality,
rabs-of-nonneg,
rleq_functionality_wrt_implies,
rsum_linearity2,
rmul_functionality,
rmul_functionality_wrt_rleq2,
radd-zero-both,
radd-rminus-both,
radd_functionality,
radd-ac,
radd_comm,
uiff_transitivity,
rminus_wf,
rleq-int-fractions2,
sq_stable__i-member,
rmul_preserves_rleq2,
or_wf,
rless-implies-rless,
radd-preserves-req,
itermMinus_wf,
real_term_value_minus_lemma,
radd_comm_eq,
uiff_transitivity2,
rabs_functionality,
rabs-rminus,
rless-int-fractions,
rabs-bounds,
rleq-rmax,
rabs-as-rmax,
rmul_preserves_rless,
rinv_wf2,
rless_functionality,
rinv-mul-as-rdiv,
rminus_functionality,
rless_irreflexivity,
rmul_reverses_rleq,
rleq-int,
radd_functionality_wrt_rleq,
rless_functionality_wrt_implies,
rleq_weakening,
rless_transitivity2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
lambdaEquality,
applyEquality,
setElimination,
rename,
dependent_set_memberEquality,
setEquality,
independent_functionElimination,
natural_numberEquality,
because_Cache,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_isectElimination,
independent_pairFormation,
productEquality,
productElimination,
dependent_pairFormation,
inrFormation,
unionElimination,
int_eqEquality,
intEquality,
computeAll,
functionEquality,
imageMemberEquality,
baseClosed,
addEquality,
imageElimination,
functionExtensionality,
addLevel,
hyp_replacement,
equalitySymmetry,
equalityTransitivity,
applyLambdaEquality,
levelHypothesis,
multiplyEquality,
minusEquality,
independent_pairEquality,
axiomEquality,
universeEquality,
inlFormation,
isect_memberFormation,
orFunctionality,
promote_hyp,
comment,
orLevelFunctionality
Latex:
\mforall{}a,b:\mBbbR{}.
((a < b)
{}\mRightarrow{} (\mforall{}f,f':[a, b] {}\mrightarrow{}\mBbbR{}.
(f'[x] continuous for x \mmember{} [a, b]
{}\mRightarrow{} d(f[x])/dx = \mlambda{}x.f'[x] on [a, b]
{}\mRightarrow{} (f[a] = f[b])
{}\mRightarrow{} (\mforall{}e:\mBbbR{}. ((r0 < e) {}\mRightarrow{} (\mexists{}x:\mBbbR{}. ((x \mmember{} [a, b]) \mwedge{} (|f'[x]| \mleq{} e))))))))
Date html generated:
2017_10_03-PM-00_20_46
Last ObjectModification:
2017_07_28-AM-08_39_54
Theory : reals
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