Nuprl Lemma : rleq-iff
∀x,y:ℝ. (x ≤ y ⇐⇒ ∀n:ℕ+. ∃N:ℕ+. ∀m:{N...}. (((-2) * m) ≤ (n * ((y m) - x m))))
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
real: ℝ,
int_upper: {i...},
nat_plus: ℕ+,
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
multiply: n * m,
subtract: n - m,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
real: ℝ,
rleq: x ≤ y,
iff: P ⇐⇒ Q,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
int_upper: {i...},
le: A ≤ B,
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
exists: ∃x:A. B[x],
rsub: x - y,
rminus: -(x),
rnonneg2: rnonneg2(x),
decidable: Dec(P),
or: P ∨ Q,
false: False,
uiff: uiff(P;Q),
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
top: Top
Lemmas referenced :
subtract-is-int-iff,
false_wf,
int_formula_prop_wf,
int_term_value_minus_lemma,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermMinus_wf,
itermAdd_wf,
itermSubtract_wf,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
minus-is-int-iff,
add-is-int-iff,
multiply-is-int-iff,
decidable__le,
nat_plus_properties,
int_upper_properties,
radd-bdd-diff,
rminus_wf,
radd_wf,
rnonneg2_functionality,
real_wf,
less_than_wf,
less_than_transitivity1,
subtract_wf,
le_wf,
int_upper_wf,
exists_wf,
nat_plus_wf,
all_wf,
rnonneg2_wf,
iff_wf,
rnonneg_wf,
regular-int-seq_wf,
rsub_wf,
rnonneg-iff
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
addLevel,
productElimination,
independent_pairFormation,
impliesFunctionality,
lemma_by_obid,
isectElimination,
hypothesis,
dependent_set_memberEquality,
hypothesisEquality,
natural_numberEquality,
independent_functionElimination,
applyEquality,
because_Cache,
sqequalRule,
lambdaEquality,
multiplyEquality,
minusEquality,
independent_isectElimination,
dependent_functionElimination,
addEquality,
dependent_pairFormation,
unionElimination,
pointwiseFunctionality,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
baseApply,
closedConclusion,
baseClosed,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}x,y:\mBbbR{}. (x \mleq{} y \mLeftarrow{}{}\mRightarrow{} \mforall{}n:\mBbbN{}\msupplus{}. \mexists{}N:\mBbbN{}\msupplus{}. \mforall{}m:\{N...\}. (((-2) * m) \mleq{} (n * ((y m) - x m))))
Date html generated:
2016_05_18-AM-07_14_59
Last ObjectModification:
2016_01_17-AM-01_55_34
Theory : reals
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