Nuprl Lemma : rless-iff-rleq
∀x,y:ℝ. (x < y ⇐⇒ ∃m:ℕ+. (x ≤ (y - (r1/r(m)))))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rsub: x - y,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
so_apply: x[s],
itermConstant: "const",
req_int_terms: t1 ≡ t2,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rless: x < y,
sq_exists: ∃x:{A| B[x]},
rge: x ≥ y,
rsub: x - y
Lemmas referenced :
rless_wf,
exists_wf,
nat_plus_wf,
rleq_wf,
rsub_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_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_var_lemma,
int_formula_prop_wf,
real_wf,
radd-preserves-rless,
rminus_wf,
rless_functionality,
radd_wf,
real_term_polynomial,
itermSubtract_wf,
itermAdd_wf,
itermMinus_wf,
real_term_value_const_lemma,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_minus_lemma,
real_term_value_var_lemma,
req-iff-rsub-is-0,
req_weakening,
small-reciprocal-real,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rsub_functionality_wrt_rleq,
rleq_weakening_rless,
rleq_weakening,
radd-preserves-rleq,
rless_transitivity1,
uiff_transitivity,
rleq_functionality,
radd_functionality,
radd-rminus-assoc,
radd_comm,
radd-zero-both,
rmul_preserves_rless,
itermMultiply_wf,
int_term_value_mul_lemma,
rmul_wf,
rmul-rdiv-cancel2,
rmul-int
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
sqequalRule,
lambdaEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
independent_isectElimination,
inrFormation,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
addLevel,
levelHypothesis,
promote_hyp,
dependent_set_memberEquality,
equalityTransitivity,
equalitySymmetry,
multiplyEquality,
lemma_by_obid
Latex:
\mforall{}x,y:\mBbbR{}. (x < y \mLeftarrow{}{}\mRightarrow{} \mexists{}m:\mBbbN{}\msupplus{}. (x \mleq{} (y - (r1/r(m)))))
Date html generated:
2017_10_03-AM-09_05_59
Last ObjectModification:
2017_07_28-AM-07_41_51
Theory : reals
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