Nuprl Lemma : simple-partition-exists
∀a,b:ℝ.
((a ≤ b)
⇒ (∀e:ℝ
((r0 < e)
⇒ (∃M:ℕ+
∃g:ℕM + 1 ⟶ {x:ℝ| x ∈ [a, b]}
(((g 0) = a ∈ ℝ) ∧ ((g M) = b ∈ ℝ) ∧ (∀i:ℕM. (((g i) ≤ (g (i + 1))) ∧ (((g (i + 1)) - g i) ≤ e))))))))
Proof
Definitions occuring in Statement :
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
rless: x < y,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
full-partition: full-partition(I;p),
top: Top,
nat_plus: ℕ+,
partition: partition(I),
ge: i ≥ j ,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
prop: ℙ,
so_lambda: λ2x.t[x],
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
real: ℝ,
sq_stable: SqStable(P),
squash: ↓T,
uiff: uiff(P;Q),
subtract: n - m,
so_apply: x[s],
l_all: (∀x∈L.P[x]),
less_than: a < b,
guard: {T},
cand: A c∧ B,
select: L[n],
cons: [a / b],
left-endpoint: left-endpoint(I),
pi1: fst(t),
endpoints: endpoints(I),
rccint: [l, u],
outl: outl(x),
true: True,
rev_implies: P ⇐ Q,
rbetween: x≤y≤z,
rsub: x - y
Lemmas referenced :
rccint-icompact,
partition-exists,
rccint_wf,
length_of_cons_lemma,
right_endpoint_rccint_lemma,
subtract_wf,
length_wf,
real_wf,
append_wf,
cons_wf,
nil_wf,
length-append,
length_of_nil_lemma,
non_neg_length,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermSubtract_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
less_than_wf,
exists_wf,
int_seg_wf,
i-member_wf,
equal_wf,
false_wf,
lelt_wf,
sq_stable__less_than,
int-to-real_wf,
decidable__le,
member_rccint_lemma,
all_wf,
rleq_wf,
add-member-int_seg2,
add-subtract-cancel,
rsub_wf,
rless_wf,
full-partition-point-member,
full-partition_wf,
add-is-int-iff,
subtract-is-int-iff,
select_wf,
int_seg_properties,
left_endpoint_rccint_lemma,
squash_wf,
true_wf,
select_cons_tl,
le_wf,
length_append,
subtype_rel_list,
top_wf,
iff_weakening_equal,
length-singleton,
select_append_back,
select-cons-hd,
adjacent-full-partition-points,
radd-preserves-rleq,
radd_wf,
rminus_wf,
uiff_transitivity,
rleq_functionality,
radd_comm,
radd-ac,
req_weakening,
radd_functionality,
radd-rminus-both,
radd-zero-both,
rleq_transitivity,
partition-mesh_wf,
add_functionality_wrt_eq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_functionElimination,
hypothesis,
isectElimination,
dependent_pairFormation,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
addEquality,
setElimination,
rename,
natural_numberEquality,
unionElimination,
independent_isectElimination,
lambdaEquality,
int_eqEquality,
intEquality,
independent_pairFormation,
computeAll,
functionEquality,
because_Cache,
setEquality,
productEquality,
applyEquality,
functionExtensionality,
imageMemberEquality,
baseClosed,
imageElimination,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
baseApply,
closedConclusion,
universeEquality
Latex:
\mforall{}a,b:\mBbbR{}.
((a \mleq{} b)
{}\mRightarrow{} (\mforall{}e:\mBbbR{}
((r0 < e)
{}\mRightarrow{} (\mexists{}M:\mBbbN{}\msupplus{}
\mexists{}g:\mBbbN{}M + 1 {}\mrightarrow{} \{x:\mBbbR{}| x \mmember{} [a, b]\}
(((g 0) = a)
\mwedge{} ((g M) = b)
\mwedge{} (\mforall{}i:\mBbbN{}M. (((g i) \mleq{} (g (i + 1))) \mwedge{} (((g (i + 1)) - g i) \mleq{} e))))))))
Date html generated:
2017_10_03-AM-09_44_40
Last ObjectModification:
2017_07_28-AM-07_58_31
Theory : reals
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