Nuprl Lemma : rleq-iff-bdd
∀[x,y:ℝ]. (x ≤ y ⇐⇒ ∃B:ℕ. ∀n:ℕ+. ((x n) ≤ ((y n) + B)))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
real: ℝ,
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
add: n + m
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
exists: ∃x:A. B[x],
nat: ℕ,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
prop: ℙ,
false: False,
real: ℝ,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
sq_exists: ∃x:A [B[x]],
int_upper: {i...},
ge: i ≥ j ,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T
Lemmas referenced :
rleq-iff4,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermConstant_wf,
istype-int,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
istype-le,
nat_plus_wf,
rleq_wf,
rleq-iff-not-rless,
rless_wf,
le_witness_for_triv,
istype-nat,
real_wf,
rless-iff2,
regular-less-iff,
istype-less_than,
nat_properties,
intformand_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
nat_plus_properties,
add-is-int-iff,
intformless_wf,
int_formula_prop_less_lemma,
false_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
independent_pairFormation,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_functionElimination,
dependent_pairFormation_alt,
dependent_set_memberEquality_alt,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
lambdaEquality_alt,
isect_memberEquality_alt,
voidElimination,
sqequalRule,
universeIsType,
functionIsType,
applyEquality,
setElimination,
rename,
addEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
productIsType,
int_eqEquality,
because_Cache,
imageElimination,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
baseClosed
Latex:
\mforall{}[x,y:\mBbbR{}]. (x \mleq{} y \mLeftarrow{}{}\mRightarrow{} \mexists{}B:\mBbbN{}. \mforall{}n:\mBbbN{}\msupplus{}. ((x n) \mleq{} ((y n) + B)))
Date html generated:
2019_10_29-AM-09_38_21
Last ObjectModification:
2019_02_13-PM-04_09_00
Theory : reals
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