Nuprl Lemma : rmaximum-select
∀n,m:ℤ. ∀x:{n..m + 1-} ⟶ ℝ. ∀e:ℝ. ((r0 < e) ⇒ (∃i:{n..m + 1-}. ((rmaximum(n;m;i.x[i]) - e) < x[i]))) supposing n ≤ m
Proof
Definitions occuring in Statement :
rmaximum: rmaximum(n;m;k.x[k]),
rless: x < y,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
uimplies: b supposing a,
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
cand: A c∧ B,
rge: x ≥ y,
req_int_terms: t1 ≡ t2,
true: True,
less_than': less_than'(a;b),
subtract: n - m,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
bfalse: ff,
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
uiff: uiff(P;Q),
so_apply: x[s],
nat_plus: ℕ+,
squash: ↓T,
sq_stable: SqStable(P),
real: ℝ,
subtype_rel: A ⊆r B,
lelt: i ≤ j < k,
sq_exists: ∃x:A [B[x]],
rless: x < y,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
sq_type: SQType(T),
ge: i ≥ j ,
guard: {T},
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
nat: ℕ,
rmaximum: rmaximum(n;m;k.x[k]),
prop: ℙ,
uall: ∀[x:A]. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
and: P ∧ Q,
le: A ≤ B,
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x]
Lemmas referenced :
req_weakening,
trivial-rless-radd,
rmax_strict_lb,
rleq_weakening_rless,
rleq_weakening_equal,
rless_functionality_wrt_implies,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
radd-zero,
radd_comm,
radd_wf,
rless_functionality,
radd-preserves-rless,
le_weakening2,
subtract-add-cancel,
rless-cases,
subtype_rel_self,
le-add-cancel,
add-commutes,
add-zero,
zero-add,
zero-mul,
add-mul-special,
minus-one-mul-top,
add-swap,
minus-one-mul,
minus-add,
add-associates,
condition-implies-le,
not-le-2,
false_wf,
le_reflexive,
int_seg_subtype,
subtype_rel_function,
assert_of_le_int,
bnot_of_lt_int,
assert_functionality_wrt_uiff,
eqff_to_assert,
bnot_wf,
le_int_wf,
assert_of_lt_int,
eqtt_to_assert,
assert_wf,
equal-wf-base,
uiff_transitivity,
bool_wf,
lt_int_wf,
primrec-unroll,
trivial-rsub-rless,
equal_wf,
primrec-wf2,
less_than_wf,
set_wf,
rmax_wf,
lelt_wf,
decidable__lt,
int_formula_prop_less_lemma,
intformless_wf,
nat_plus_properties,
sq_stable__less_than,
int_seg_properties,
primrec_wf,
rsub_wf,
exists_wf,
int-to-real_wf,
rless_wf,
real_wf,
int_seg_wf,
all_wf,
primrec0_lemma,
int_term_value_add_lemma,
int_formula_prop_eq_lemma,
itermAdd_wf,
intformeq_wf,
decidable__equal_int,
nat_properties,
int_subtype_base,
subtype_base_sq,
nat_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
subtract_wf,
decidable__le,
le_wf,
less_than'_wf
Rules used in proof :
multiplyEquality,
minusEquality,
closedConclusion,
baseApply,
equalityElimination,
functionExtensionality,
imageElimination,
baseClosed,
imageMemberEquality,
applyEquality,
addEquality,
functionEquality,
setElimination,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
cumulativity,
instantiate,
independent_pairFormation,
voidEquality,
isect_memberEquality,
int_eqEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
natural_numberEquality,
because_Cache,
dependent_set_memberEquality,
intEquality,
rename,
axiomEquality,
hypothesis,
isectElimination,
extract_by_obid,
voidElimination,
hypothesisEquality,
dependent_functionElimination,
lambdaEquality,
independent_pairEquality,
thin,
productElimination,
sqequalHypSubstitution,
sqequalRule,
introduction,
cut,
isect_memberFormation,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}n,m:\mBbbZ{}.
\mforall{}x:\{n..m + 1\msupminus{}\} {}\mrightarrow{} \mBbbR{}. \mforall{}e:\mBbbR{}. ((r0 < e) {}\mRightarrow{} (\mexists{}i:\{n..m + 1\msupminus{}\}. ((rmaximum(n;m;i.x[i]) - e) < x[i])))
supposing n \mleq{} m
Date html generated:
2018_05_22-PM-01_57_29
Last ObjectModification:
2018_05_21-AM-00_16_42
Theory : reals
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