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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