Nuprl Lemma : r-archimedean2
∀x:ℝ. ∃N:ℕ. ∀n:{N...}. (|(x/r(n + 1))| ≤ (r1/r(2)))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
int-to-real: r(n),
real: ℝ,
int_upper: {i...},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
add: n + m,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
rge: x ≥ y,
top: Top,
not: ¬A,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
prop: ℙ,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
uimplies: b supposing a,
and: P ∧ Q,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
int_upper: {i...},
so_apply: x[s],
subtype_rel: A ⊆r B,
rdiv: (x/y),
itermConstant: "const",
req_int_terms: t1 ≡ t2,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b
Lemmas referenced :
real_wf,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rmul_comm,
rmul-rdiv-cancel2,
req_weakening,
rleq_functionality,
uiff_transitivity,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
intformnot_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_properties,
rleq-int,
rless_wf,
rleq_wf,
rless-int,
rdiv_wf,
rmul_preserves_rleq,
rabs_wf,
int-to-real_wf,
rmul_wf,
r-archimedean,
int_upper_wf,
all_wf,
int_upper_properties,
decidable__lt,
intformand_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
rneq_wf,
squash_wf,
true_wf,
rabs-int,
iff_weakening_equal,
absval_pos,
le_wf,
rinv_wf2,
absval_wf,
rneq_functionality,
rabs-of-nonneg,
rabs-rdiv,
rless_functionality,
req_transitivity,
real_term_polynomial,
itermSubtract_wf,
itermMultiply_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
rmul-rinv3,
uiff_transitivity2,
rinv-mul-as-rdiv,
absval_unfold,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
less_than_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
hypothesis,
equalitySymmetry,
equalityTransitivity,
computeAll,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
lambdaEquality,
unionElimination,
rename,
setElimination,
addEquality,
baseClosed,
imageMemberEquality,
independent_pairFormation,
independent_functionElimination,
inrFormation,
sqequalRule,
independent_isectElimination,
because_Cache,
dependent_pairFormation,
productElimination,
hypothesisEquality,
natural_numberEquality,
isectElimination,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
applyEquality,
imageElimination,
universeEquality,
dependent_set_memberEquality,
minusEquality,
equalityElimination,
lessCases,
isect_memberFormation,
sqequalAxiom,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}x:\mBbbR{}. \mexists{}N:\mBbbN{}. \mforall{}n:\{N...\}. (|(x/r(n + 1))| \mleq{} (r1/r(2)))
Date html generated:
2017_10_03-AM-09_23_03
Last ObjectModification:
2017_07_28-AM-07_46_12
Theory : reals
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