Nuprl Lemma : rless_ibs_property
∀x,y:ℝ.
((x < y ⇐⇒ ∃n:ℕ. ((rless_ibs(x;y) n) = 1 ∈ ℤ))
∧ (∀n:ℕ
(((rless_ibs(x;y) n) = 0 ∈ ℤ)
⇒ (((y < x) ∧ (∀m:ℕ. ((rless_ibs(x;y) m) = 0 ∈ ℤ))) ∨ (|x - y| ≤ (r(4)/r(n + 1)))))))
Proof
Definitions occuring in Statement :
rless_ibs: rless_ibs(x;y),
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],
uall: ∀[x:A]. B[x],
member: t ∈ T,
rless_ibs: rless_ibs(x;y),
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
real: ℝ,
rev_implies: P ⇐ Q,
nat: ℕ,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
prop: ℙ,
so_apply: x[s],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
uiff: uiff(P;Q),
guard: {T},
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
cand: A c∧ B,
true: True,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rneq: x ≠ y,
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
rless: x < y,
sq_exists: ∃x:A [B[x]],
less_than': less_than'(a;b)
Lemmas referenced :
rless_ibs_wf,
real_wf,
nat_plus_wf,
istype-less_than,
istype-nat,
istype-int,
bl-exists_wf,
int_seg_wf,
upto_wf,
lt_int_wf,
int_seg_properties,
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,
l_member_wf,
rless-iff2,
rless_wf,
subtract_wf,
nat_plus_properties,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
istype-le,
eqtt_to_assert,
assert-bl-exists,
l_exists_functionality,
assert_wf,
less_than_wf,
iff_weakening_uiff,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
l_exists_wf,
l_exists_iff,
subtract-add-cancel,
add-is-int-iff,
false_wf,
member_upto2,
int_subtype_base,
bool_cases,
iff_transitivity,
assert_of_bnot,
rational-approx-property,
uimplies_transitivity,
rleq_wf,
rabs_wf,
rsub_wf,
radd_wf,
rational-approx_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
r-triangle-inequality2,
radd_functionality_wrt_rleq,
rdiv_wf,
int-to-real_wf,
rless-int,
rleq_functionality,
rabs-difference-symmetry,
req_weakening,
radd-preserves-rleq,
rminus_wf,
rmul_wf,
rinv_wf2,
itermMinus_wf,
itermMultiply_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
rless_transitivity2,
rleq_weakening_rless,
rless_irreflexivity,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
implies-close-reals,
absval_ubound,
subtract-is-int-iff
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
sqequalRule,
independent_pairFormation,
inhabitedIsType,
universeIsType,
productElimination,
productIsType,
addEquality,
applyEquality,
setElimination,
rename,
natural_numberEquality,
equalityIstype,
because_Cache,
lambdaEquality_alt,
dependent_set_memberEquality_alt,
imageElimination,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
setIsType,
equalityElimination,
baseClosed,
equalityTransitivity,
equalitySymmetry,
sqequalBase,
promote_hyp,
instantiate,
cumulativity,
pointwiseFunctionality,
baseApply,
closedConclusion,
intEquality,
inrFormation_alt,
inlFormation_alt,
functionIsType,
minusEquality
Latex:
\mforall{}x,y:\mBbbR{}.
((x < y \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}. ((rless\_ibs(x;y) n) = 1))
\mwedge{} (\mforall{}n:\mBbbN{}
(((rless\_ibs(x;y) n) = 0)
{}\mRightarrow{} (((y < x) \mwedge{} (\mforall{}m:\mBbbN{}. ((rless\_ibs(x;y) m) = 0))) \mvee{} (|x - y| \mleq{} (r(4)/r(n + 1)))))))
Date html generated:
2019_10_30-AM-10_15_59
Last ObjectModification:
2019_06_28-PM-01_55_45
Theory : real!vectors
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