Nuprl Lemma : rless-iff8
∀x,y:ℝ.  (x < y 
⇐⇒ ∃m:{ℕ+| (x m) + 8 < y m})
Proof
Definitions occuring in Statement : 
rless: x < y
, 
real: ℝ
, 
nat_plus: ℕ+
, 
less_than: a < b
, 
all: ∀x:A. B[x]
, 
sq_exists: ∃x:{A| B[x]}
, 
iff: P 
⇐⇒ Q
, 
apply: f a
, 
add: n + m
, 
natural_number: $n
Definitions unfolded in proof : 
rless: x < y
, 
all: ∀x:A. B[x]
, 
iff: P 
⇐⇒ Q
, 
and: P ∧ Q
, 
implies: P 
⇒ Q
, 
member: t ∈ T
, 
prop: ℙ
, 
uall: ∀[x:A]. B[x]
, 
so_lambda: λ2x.t[x]
, 
real: ℝ
, 
so_apply: x[s]
, 
rev_implies: P 
⇐ Q
, 
sq_exists: ∃x:{A| B[x]}
, 
nat_plus: ℕ+
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
uimplies: b supposing a
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
false: False
, 
not: ¬A
, 
top: Top
, 
regular-int-seq: k-regular-seq(f)
, 
le: A ≤ B
, 
sq_stable: SqStable(P)
, 
squash: ↓T
, 
guard: {T}
, 
subtype_rel: A ⊆r B
, 
uiff: uiff(P;Q)
, 
less_than: a < b
, 
nat: ℕ
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
sq_type: SQType(T)
, 
bfalse: ff
Lemmas referenced : 
sq_exists_wf, 
nat_plus_wf, 
less_than_wf, 
real_wf, 
nat_plus_properties, 
decidable__lt, 
satisfiable-full-omega-tt, 
intformand_wf, 
intformnot_wf, 
intformless_wf, 
itermConstant_wf, 
itermAdd_wf, 
itermMultiply_wf, 
itermVar_wf, 
int_formula_prop_and_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_less_lemma, 
int_term_value_constant_lemma, 
int_term_value_add_lemma, 
int_term_value_mul_lemma, 
int_term_value_var_lemma, 
int_formula_prop_wf, 
equal_wf, 
sq_stable__le, 
intformeq_wf, 
int_formula_prop_eq_lemma, 
decidable__le, 
multiply-is-int-iff, 
int_subtype_base, 
add-is-int-iff, 
intformle_wf, 
int_formula_prop_le_lemma, 
false_wf, 
mul_preserves_le, 
le_wf, 
mul_cancel_in_lt, 
absval_ifthenelse, 
subtract_wf, 
lt_int_wf, 
subtract-is-int-iff, 
itermSubtract_wf, 
int_term_value_subtract_lemma, 
assert_wf, 
bnot_wf, 
not_wf, 
minus-is-int-iff, 
itermMinus_wf, 
int_term_value_minus_lemma, 
bool_cases, 
subtype_base_sq, 
bool_wf, 
bool_subtype_base, 
eqtt_to_assert, 
assert_of_lt_int, 
eqff_to_assert, 
iff_transitivity, 
iff_weakening_uiff, 
assert_of_bnot
Rules used in proof : 
sqequalSubstitution, 
sqequalRule, 
sqequalReflexivity, 
sqequalTransitivity, 
computationStep, 
lambdaFormation, 
independent_pairFormation, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
hypothesis, 
lambdaEquality, 
addEquality, 
applyEquality, 
setElimination, 
rename, 
hypothesisEquality, 
natural_numberEquality, 
because_Cache, 
dependent_set_memberFormation, 
dependent_set_memberEquality, 
multiplyEquality, 
dependent_functionElimination, 
unionElimination, 
independent_isectElimination, 
dependent_pairFormation, 
int_eqEquality, 
intEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
computeAll, 
equalityTransitivity, 
equalitySymmetry, 
independent_functionElimination, 
productElimination, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
baseApply, 
closedConclusion, 
pointwiseFunctionality, 
promote_hyp, 
instantiate, 
cumulativity, 
impliesFunctionality
Latex:
\mforall{}x,y:\mBbbR{}.    (x  <  y  \mLeftarrow{}{}\mRightarrow{}  \mexists{}m:\{\mBbbN{}\msupplus{}|  (x  m)  +  8  <  y  m\})
Date html generated:
2017_10_03-AM-08_24_57
Last ObjectModification:
2017_07_28-AM-07_23_32
Theory : reals
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