Nuprl Lemma : partition-lemma
∀e:ℝ
((r0 < e)
⇒ (∀n:ℕ+. ∀f:ℕn ⟶ ℝ.
∀x:ℝ. ∃i:ℕn. (|x - f i| ≤ e) supposing f 0≤x≤f (n - 1) supposing ∀i:ℕn - 1. r0≤(f (i + 1)) - f i≤e))
Proof
Definitions occuring in Statement :
rbetween: x≤y≤z,
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
int_seg: {i..j-},
nat_plus: ℕ+,
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtract: n - m,
uimplies: b supposing a,
member: t ∈ T,
rbetween: x≤y≤z,
and: P ∧ Q,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
not: ¬A,
false: False,
uall: ∀[x:A]. B[x],
int_seg: {i..j-},
uiff: uiff(P;Q),
lelt: i ≤ j < k,
nat_plus: ℕ+,
rless: x < y,
sq_exists: ∃x:A [B[x]],
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
less_than: a < b,
squash: ↓T,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
real: ℝ,
sq_stable: SqStable(P),
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
absval: |i|,
req_int_terms: t1 ≡ t2,
rsub: x - y,
rge: x ≥ y,
rgt: x > y,
cand: A c∧ B,
itermConstant: "const"
Lemmas referenced :
less_than'_wf,
rsub_wf,
int_seg_wf,
add-member-int_seg2,
nat_plus_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
lelt_wf,
subtract_wf,
int-to-real_wf,
false_wf,
rbetween_wf,
real_wf,
all_wf,
add-subtract-cancel,
sq_stable__less_than,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
isect_wf,
exists_wf,
rleq_wf,
rabs_wf,
primrec-wf-nat-plus,
nat_plus_wf,
rless_wf,
rleq_antisymmetry,
req-iff-rsub-is-0,
rleq-int,
rless_transitivity2,
rleq_weakening_rless,
rleq_functionality,
rabs_functionality,
rsub_functionality,
req_weakening,
req_transitivity,
rabs-int,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
trivial-rsub-rless,
rless-cases,
subtype_rel_self,
satisfiable-full-omega-tt,
int_seg_subtype,
subtype_rel_dep_function,
trivial-rless-radd,
radd_wf,
rabs-difference-symmetry,
rabs-of-nonneg,
radd-preserves-rleq,
rminus_wf,
uiff_transitivity,
radd_comm,
radd-ac,
radd_functionality,
radd-rminus-both,
radd-zero-both,
rleq_functionality_wrt_implies,
rsub_functionality_wrt_rleq,
rleq_weakening_equal,
subtract-add-cancel,
add-associates,
add-swap,
add-commutes,
zero-add,
squash_wf,
true_wf,
and_wf,
equal_wf,
rmax_lb,
rabs-as-rmax,
radd-rminus-assoc,
rmul_wf,
rless-int,
rmul_reverses_rleq_iff,
real_term_polynomial,
itermMultiply_wf,
itermMinus_wf,
real_term_value_mul_lemma,
real_term_value_minus_lemma,
real_term_value_add_lemma,
rleq_weakening,
radd_functionality_wrt_rless1,
regular-int-seq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
sqequalRule,
isect_memberFormation,
introduction,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
voidElimination,
extract_by_obid,
isectElimination,
applyEquality,
functionExtensionality,
natural_numberEquality,
hypothesis,
because_Cache,
setElimination,
rename,
independent_isectElimination,
dependent_set_memberEquality,
independent_pairFormation,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
minusEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
baseClosed,
functionEquality,
addEquality,
imageElimination,
unionElimination,
computeAll,
hyp_replacement,
setEquality
Latex:
\mforall{}e:\mBbbR{}
((r0 < e)
{}\mRightarrow{} (\mforall{}n:\mBbbN{}\msupplus{}. \mforall{}f:\mBbbN{}n {}\mrightarrow{} \mBbbR{}.
\mforall{}x:\mBbbR{}. \mexists{}i:\mBbbN{}n. (|x - f i| \mleq{} e) supposing f 0\mleq{}x\mleq{}f (n - 1)
supposing \mforall{}i:\mBbbN{}n - 1. r0\mleq{}(f (i + 1)) - f i\mleq{}e))
Date html generated:
2019_10_29-AM-10_49_02
Last ObjectModification:
2019_01_27-PM-07_16_01
Theory : reals
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