Nuprl Lemma : rleq-int
∀n,m:ℤ. (r(n) ≤ r(m) ⇐⇒ n ≤ m)
Proof
Definitions occuring in Statement :
rleq: x ≤ y,
int-to-real: r(n),
le: A ≤ B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
rleq: x ≤ y,
iff: P ⇐⇒ Q,
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
real: ℝ,
prop: ℙ,
rev_implies: P ⇐ Q,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
uiff: uiff(P;Q)
Lemmas referenced :
rnonneg-int,
int_formula_prop_wf,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermSubtract_wf,
itermConstant_wf,
itermVar_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
le_wf,
iff_wf,
real_wf,
rnonneg_wf,
rsub-int,
subtract_wf,
int-to-real_wf,
rsub_wf,
rnonneg_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
addLevel,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairFormation,
impliesFunctionality,
lemma_by_obid,
dependent_functionElimination,
isectElimination,
hypothesisEquality,
hypothesis,
independent_isectElimination,
independent_functionElimination,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
intEquality,
unionElimination,
natural_numberEquality,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll
Latex:
\mforall{}n,m:\mBbbZ{}. (r(n) \mleq{} r(m) \mLeftarrow{}{}\mRightarrow{} n \mleq{} m)
Date html generated:
2016_05_18-AM-07_05_06
Last ObjectModification:
2016_01_17-AM-01_50_51
Theory : reals
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