Nuprl Lemma : locally-non-zero-finite-deriv-seq
∀a,b:ℝ. ∀f:[a, b] ⟶ℝ.
((∀u,v:{v:ℝ| v ∈ [a, b]} .
((u < v)
⇒ (∃k:ℕ
∃F:ℕk + 1 ⟶ [a, b] ⟶ℝ
(finite-deriv-seq([a, b];k;i,x.F[i;x])
∧ (∀x:{x:ℝ| x ∈ [a, b]} . (F[0;x] = f(x)))
∧ (∃z:{z:ℝ| z ∈ [u, v]} . (r0 < Σ{|F[i;z]| | 0≤i≤k}))))))
⇒ locally-non-constant(f;a;b;r0))
Proof
Definitions occuring in Statement :
finite-deriv-seq: finite-deriv-seq(I;k;i,x.F[i; x]),
locally-non-constant: locally-non-constant(f;a;b;c),
r-ap: f(x),
rfun: I ⟶ℝ,
rccint: [l, u],
i-member: r ∈ I,
rsum: Σ{x[k] | n≤k≤m},
rless: x < y,
rabs: |x|,
req: x = y,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
nat: ℕ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
locally-non-constant: locally-non-constant(f;a;b;c),
member: t ∈ T,
top: Top,
and: P ∧ Q,
cand: A c∧ B,
guard: {T},
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
exists: ∃x:A. B[x],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
so_lambda: λ2x y.t[x; y],
label: ...$L... t,
rfun: I ⟶ℝ,
so_apply: x[s1;s2],
subtype_rel: A ⊆r B,
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
true: True,
i-member: r ∈ I,
rccint: [l, u],
sq_stable: SqStable(P),
so_lambda: λ2x.t[x],
so_apply: x[s],
rless: x < y,
sq_exists: ∃x:A [B[x]],
real: ℝ,
nat_plus: ℕ+,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
finite-deriv-seq: finite-deriv-seq(I;k;i,x.F[i; x]),
r-ap: f(x),
rfun-eq: rfun-eq(I;f;g),
subinterval: I ⊆ J
Lemmas referenced :
member_rccint_lemma,
istype-void,
rless_transitivity1,
rleq_weakening_rless,
rleq_wf,
rless_transitivity2,
int_seg_wf,
finite-deriv-seq_wf,
rccint_wf,
istype-false,
istype-le,
subtype_rel_self,
real_wf,
i-member_wf,
req_wf,
istype-less_than,
r-ap_wf,
sq_stable__rleq,
rless_wf,
int-to-real_wf,
rsum_wf,
rabs_wf,
rleq_transitivity,
sq_stable__less_than,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_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_less_lemma,
int_formula_prop_wf,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
rneq_wf,
primrec-wf2,
all_wf,
exists_wf,
istype-nat,
nat_properties,
rfun_wf,
sq_stable__i-member,
rless_functionality,
req_weakening,
rsum_single,
rabs_functionality,
rabs-positive-iff,
radd_wf,
radd-positive-implies,
rsum-split-first,
add-member-int_seg2,
subtract-add-cancel,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
add-subtract-cancel,
req_witness,
rsum-shift,
zero-add,
derivative_functionality,
small-reciprocal-rneq-zero,
non-zero-deriv-non-constant,
rfun_subtype_3,
derivative_functionality_wrt_subinterval,
subtype_rel_sets_simple
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
sqequalRule,
introduction,
extract_by_obid,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
hypothesisEquality,
independent_functionElimination,
independent_isectElimination,
isectElimination,
dependent_set_memberEquality_alt,
because_Cache,
productIsType,
universeIsType,
productElimination,
functionIsType,
natural_numberEquality,
addEquality,
lambdaEquality_alt,
applyEquality,
functionEquality,
setEquality,
setIsType,
inhabitedIsType,
imageMemberEquality,
baseClosed,
setElimination,
rename,
imageElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
productEquality,
closedConclusion,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}a,b:\mBbbR{}. \mforall{}f:[a, b] {}\mrightarrow{}\mBbbR{}.
((\mforall{}u,v:\{v:\mBbbR{}| v \mmember{} [a, b]\} .
((u < v)
{}\mRightarrow{} (\mexists{}k:\mBbbN{}
\mexists{}F:\mBbbN{}k + 1 {}\mrightarrow{} [a, b] {}\mrightarrow{}\mBbbR{}
(finite-deriv-seq([a, b];k;i,x.F[i;x])
\mwedge{} (\mforall{}x:\{x:\mBbbR{}| x \mmember{} [a, b]\} . (F[0;x] = f(x)))
\mwedge{} (\mexists{}z:\{z:\mBbbR{}| z \mmember{} [u, v]\} . (r0 < \mSigma{}\{|F[i;z]| | 0\mleq{}i\mleq{}k\}))))))
{}\mRightarrow{} locally-non-constant(f;a;b;r0))
Date html generated:
2019_10_30-AM-09_10_28
Last ObjectModification:
2018_11_12-PM-04_18_57
Theory : reals
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