Nuprl Lemma : rleq-int-fractions
∀[a,b:ℤ]. ∀[c,d:ℕ+]. uiff((r(a)/r(c)) ≤ (r(b)/r(d));(a * d) ≤ (b * c))
Proof
Definitions occuring in Statement :
rdiv: (x/y),
rleq: x ≤ y,
int-to-real: r(n),
nat_plus: ℕ+,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
multiply: n * m,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
le: A ≤ B,
nat_plus: ℕ+,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
le_witness_for_triv,
rleq_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
rless_wf,
istype-le,
nat_plus_wf,
rmul_preserves_rleq2,
rleq-int,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
rmul_wf,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
rleq_functionality,
req_transitivity,
rmul_functionality,
req_weakening,
rmul-rinv,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rmul-int,
rmul_preserves_rleq
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
productElimination,
equalityTransitivity,
hypothesis,
equalitySymmetry,
independent_isectElimination,
universeIsType,
hypothesisEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
inrFormation_alt,
dependent_functionElimination,
independent_functionElimination,
natural_numberEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
functionIsTypeImplies,
inhabitedIsType,
multiplyEquality,
independent_pairEquality,
isectIsTypeImplies
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[c,d:\mBbbN{}\msupplus{}]. uiff((r(a)/r(c)) \mleq{} (r(b)/r(d));(a * d) \mleq{} (b * c))
Date html generated:
2019_10_29-AM-09_58_15
Last ObjectModification:
2019_01_27-PM-07_29_01
Theory : reals
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