Nuprl Lemma : rless-ibs
∀x,y:ℝ.
∃s:IBS
((x < y ⇐⇒ ∃n:ℕ. ((s n) = 1 ∈ ℤ))
∧ (∀n:ℕ. (((s n) = 0 ∈ ℤ) ⇒ (((y < x) ∧ (∀m:ℕ. ((s m) = 0 ∈ ℤ))) ∨ (|x - y| ≤ (r(4)/r(n + 1)))))))
Proof
Definitions occuring in Statement :
incr-binary-seq: IBS,
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
apply: f a,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
prop: ℙ,
incr-binary-seq: IBS,
subtype_rel: A ⊆r B,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
rev_implies: P ⇐ Q,
or: P ∨ Q,
nat: ℕ,
rneq: x ≠ y,
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top
Lemmas referenced :
rless_ibs_property,
rless_ibs_wf,
rless_wf,
istype-nat,
istype-int,
set_subtype_base,
lelt_wf,
int_subtype_base,
rleq_wf,
rabs_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
real_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
dependent_pairFormation_alt,
isectElimination,
sqequalRule,
productIsType,
functionIsType,
universeIsType,
equalityIstype,
applyEquality,
setElimination,
rename,
intEquality,
lambdaEquality_alt,
natural_numberEquality,
independent_isectElimination,
baseClosed,
sqequalBase,
equalitySymmetry,
because_Cache,
unionIsType,
inhabitedIsType,
equalityTransitivity,
productElimination,
closedConclusion,
addEquality,
inrFormation_alt,
independent_functionElimination,
unionElimination,
approximateComputation,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation
Latex:
\mforall{}x,y:\mBbbR{}.
\mexists{}s:IBS
((x < y \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}. ((s n) = 1))
\mwedge{} (\mforall{}n:\mBbbN{}. (((s n) = 0) {}\mRightarrow{} (((y < x) \mwedge{} (\mforall{}m:\mBbbN{}. ((s m) = 0))) \mvee{} (|x - y| \mleq{} (r(4)/r(n + 1)))))))
Date html generated:
2019_10_30-AM-11_25_17
Last ObjectModification:
2019_06_28-PM-01_56_01
Theory : real!vectors
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