Nuprl Lemma : regular-less
∀[x,y:ℝ].  ∀n:ℕ+. ((x n) + 4 < y n 
⇒ (∀m:ℕ+. ((x m) ≤ ((y m) + 4))))
Proof
Definitions occuring in Statement : 
real: ℝ
, 
nat_plus: ℕ+
, 
less_than: a < b
, 
uall: ∀[x:A]. B[x]
, 
le: A ≤ B
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
apply: f a
, 
add: n + m
, 
natural_number: $n
Definitions unfolded in proof : 
uall: ∀[x:A]. B[x]
, 
member: t ∈ T
, 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
real: ℝ
, 
uimplies: b supposing a
, 
and: P ∧ Q
, 
cand: A c∧ B
, 
nat_plus: ℕ+
, 
sq_stable: SqStable(P)
, 
regular-int-seq: k-regular-seq(f)
, 
squash: ↓T
, 
le: A ≤ B
, 
prop: ℙ
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
less_than: a < b
, 
false: False
, 
uiff: uiff(P;Q)
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
not: ¬A
, 
top: Top
, 
subtype_rel: A ⊆r B
, 
sq_type: SQType(T)
, 
guard: {T}
, 
ifthenelse: if b then t else f fi 
, 
btrue: tt
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
bfalse: ff
Lemmas referenced : 
assert_of_bnot, 
iff_weakening_uiff, 
iff_transitivity, 
eqff_to_assert, 
assert_of_lt_int, 
eqtt_to_assert, 
bool_subtype_base, 
bool_wf, 
subtype_base_sq, 
bool_cases, 
int_term_value_minus_lemma, 
itermMinus_wf, 
minus-is-int-iff, 
not_wf, 
bnot_wf, 
assert_wf, 
subtract-is-int-iff, 
int_subtype_base, 
lt_int_wf, 
real_wf, 
less_than'_wf, 
nat_plus_wf, 
false_wf, 
int_formula_prop_wf, 
int_formula_prop_less_lemma, 
int_term_value_subtract_lemma, 
int_term_value_constant_lemma, 
int_term_value_add_lemma, 
int_term_value_var_lemma, 
int_term_value_mul_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_not_lemma, 
int_formula_prop_and_lemma, 
intformless_wf, 
itermSubtract_wf, 
itermConstant_wf, 
itermAdd_wf, 
itermVar_wf, 
itermMultiply_wf, 
intformle_wf, 
intformnot_wf, 
intformand_wf, 
satisfiable-full-omega-tt, 
multiply-is-int-iff, 
add-is-int-iff, 
less_than_wf, 
decidable__le, 
nat_plus_properties, 
absval_ifthenelse, 
subtract_wf, 
sq_stable__le, 
mul_cancel_in_le, 
mul_preserves_lt
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
isect_memberFormation, 
introduction, 
cut, 
lambdaFormation, 
lemma_by_obid, 
sqequalHypSubstitution, 
isectElimination, 
thin, 
addEquality, 
applyEquality, 
setElimination, 
rename, 
hypothesisEquality, 
natural_numberEquality, 
independent_isectElimination, 
hypothesis, 
multiplyEquality, 
independent_functionElimination, 
dependent_functionElimination, 
sqequalRule, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
independent_pairFormation, 
because_Cache, 
productElimination, 
dependent_set_memberEquality, 
unionElimination, 
pointwiseFunctionality, 
equalityTransitivity, 
equalitySymmetry, 
promote_hyp, 
baseApply, 
closedConclusion, 
dependent_pairFormation, 
lambdaEquality, 
int_eqEquality, 
intEquality, 
isect_memberEquality, 
voidElimination, 
voidEquality, 
computeAll, 
independent_pairEquality, 
axiomEquality, 
instantiate, 
cumulativity, 
impliesFunctionality
Latex:
\mforall{}[x,y:\mBbbR{}].    \mforall{}n:\mBbbN{}\msupplus{}.  ((x  n)  +  4  <  y  n  {}\mRightarrow{}  (\mforall{}m:\mBbbN{}\msupplus{}.  ((x  m)  \mleq{}  ((y  m)  +  4))))
Date html generated:
2016_05_18-AM-06_47_40
Last ObjectModification:
2016_01_17-AM-01_46_47
Theory : reals
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