Nuprl Lemma : rless-iff
∀x,y:ℝ. (x < y ⇐⇒ ∃n:ℕ+. ∀m:ℕ+. ((n ≤ m) ⇒ (m ≤ (n * ((y m) - x m)))))
Proof
Definitions occuring in Statement :
rless: x < y,
real: ℝ,
nat_plus: ℕ+,
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
apply: f a,
multiply: n * m,
subtract: n - m
Definitions unfolded in proof :
rless: x < y,
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
real: ℝ,
so_apply: x[s],
int_upper: {i...},
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
exists: ∃x:A. B[x],
sq_exists: ∃x:{A| B[x]},
sq_stable: SqStable(P),
squash: ↓T,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
less_than: a < b,
true: True,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
subtype_rel: A ⊆r B,
nat: ℕ,
ge: i ≥ j ,
rev_uimplies: rev_uimplies(P;Q),
subtract: n - m
Lemmas referenced :
regular-less-iff,
rless_wf,
exists_wf,
nat_plus_wf,
all_wf,
le_wf,
subtract_wf,
real_wf,
false_wf,
nat_plus_properties,
sq_stable__less_than,
decidable__le,
satisfiable-full-omega-tt,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
mul_nat_plus,
less_than_wf,
regular-consistency,
absval_unfold,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
top_wf,
add-is-int-iff,
intformand_wf,
intformless_wf,
itermConstant_wf,
itermSubtract_wf,
itermAdd_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
itermMinus_wf,
int_term_value_minus_lemma,
absval_wf,
nat_wf,
nat_plus_subtype_nat,
le_functionality,
multiply_functionality_wrt_le,
le_weakening,
multiply-is-int-iff,
itermMultiply_wf,
int_term_value_mul_lemma,
subtract-is-int-iff,
mul_preserves_le,
int_upper_wf,
decidable__lt,
not-lt-2,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
le-add-cancel,
less_than_transitivity1,
int_upper_properties,
subtype_rel_sets,
mul_cancel_in_lt
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
lambdaFormation,
hypothesis,
isectElimination,
thin,
hypothesisEquality,
independent_pairFormation,
productElimination,
independent_functionElimination,
lambdaEquality,
functionEquality,
setElimination,
rename,
because_Cache,
multiplyEquality,
applyEquality,
dependent_functionElimination,
dependent_set_memberEquality,
natural_numberEquality,
addEquality,
imageMemberEquality,
baseClosed,
imageElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
minusEquality,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
lessCases,
isect_memberFormation,
sqequalAxiom,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
instantiate,
cumulativity,
setEquality,
applyLambdaEquality
Latex:
\mforall{}x,y:\mBbbR{}. (x < y \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}\msupplus{}. \mforall{}m:\mBbbN{}\msupplus{}. ((n \mleq{} m) {}\mRightarrow{} (m \mleq{} (n * ((y m) - x m)))))
Date html generated:
2017_10_03-AM-08_24_48
Last ObjectModification:
2017_07_28-AM-07_23_29
Theory : reals
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