Nuprl Lemma : rless-int
∀n,m:ℤ. (r(n) < r(m) ⇐⇒ n < m)
Proof
Definitions occuring in Statement :
rless: x < y,
int-to-real: r(n),
less_than: a < b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
int-to-real: r(n),
rless: x < y,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
so_apply: x[s],
rev_implies: P ⇐ Q,
sq_exists: ∃x:{A| B[x]},
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,
nat: ℕ,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
int_term_value_add_lemma,
itermAdd_wf,
le_wf,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
itermMultiply_wf,
itermConstant_wf,
mul_preserves_le,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__le,
nat_plus_properties,
decidable__lt,
less_than_wf,
nat_plus_wf,
sq_exists_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
independent_pairFormation,
cut,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
lambdaEquality,
addEquality,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
intEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
because_Cache,
dependent_set_memberEquality,
dependent_set_memberFormation,
introduction,
imageMemberEquality,
baseClosed
Latex:
\mforall{}n,m:\mBbbZ{}. (r(n) < r(m) \mLeftarrow{}{}\mRightarrow{} n < m)
Date html generated:
2016_05_18-AM-07_05_03
Last ObjectModification:
2016_01_17-AM-01_50_26
Theory : reals
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