Nuprl Lemma : rless-int-fractions
∀a,b:ℤ. ∀c,d:ℕ+. ((r(a)/r(c)) < (r(b)/r(d)) ⇐⇒ a * d < b * c)
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rless: x < y,
int-to-real: r(n),
nat_plus: ℕ+,
less_than: a < b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
multiply: n * m,
int: ℤ
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],
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
not: ¬A,
top: Top,
rless: x < y,
sq_exists: ∃x:{A| B[x]},
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
rmul-rdiv-cancel,
rmul-ac,
rmul_functionality,
rmul-assoc,
req_inversion,
req_functionality,
uiff_transitivity,
rmul-int,
rmul_comm,
rmul-rdiv-cancel2,
rless_functionality,
req_weakening,
req_wf,
rmul_wf,
rmul_preserves_rless,
nat_plus_wf,
less_than_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
satisfiable-full-omega-tt,
decidable__lt,
nat_plus_properties,
rless-int,
int-to-real_wf,
rdiv_wf,
rless_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
independent_isectElimination,
sqequalRule,
inrFormation,
dependent_functionElimination,
because_Cache,
productElimination,
independent_functionElimination,
natural_numberEquality,
unionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
multiplyEquality,
promote_hyp,
addLevel
Latex:
\mforall{}a,b:\mBbbZ{}. \mforall{}c,d:\mBbbN{}\msupplus{}. ((r(a)/r(c)) < (r(b)/r(d)) \mLeftarrow{}{}\mRightarrow{} a * d < b * c)
Date html generated:
2016_05_18-AM-07_27_34
Last ObjectModification:
2016_01_17-AM-02_00_26
Theory : reals
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