Nuprl Lemma : rleq-rat2real
∀[q1,q2:ℚ]. uiff(rat2real(q1) ≤ rat2real(q2);q1 ≤ q2)
Proof
Definitions occuring in Statement :
rat2real: rat2real(q),
rleq: x ≤ y,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
qle: r ≤ s,
rationals: ℚ
Definitions unfolded in proof :
guard: {T},
rneq: x ≠ y,
or: P ∨ Q,
decidable: Dec(P),
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
rev_implies: P ⇐ Q,
bfalse: ff,
ifthenelse: if b then t else f fi ,
rat2real: rat2real(q),
mk-rational: mk-rational(a;b),
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
uiff: uiff(P;Q),
uimplies: b supposing a,
prop: ℙ,
and: P ∧ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
not: ¬A,
cand: A c∧ B,
nat_plus: ℕ+,
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
int-rdiv-req,
rleq_functionality,
int-rdiv_wf,
int-to-real_wf,
rless_wf,
rless-int,
rdiv_wf,
rleq-int-fractions,
istype-le,
int_term_value_mul_lemma,
int_formula_prop_le_lemma,
itermMultiply_wf,
intformle_wf,
decidable__le,
mk-rational-qdiv,
istype-less_than,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__lt,
qle-mk-rational,
le_wf,
nequal_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
full-omega-unsat,
mk-rational_wf,
le_witness_for_triv,
qle_witness,
qle_wf,
qdiv_wf,
rat2real_wf,
rleq_wf,
uiff_wf,
istype-assert,
assert-qeq,
int_subtype_base,
rationals_wf,
equal-wf-base,
int-subtype-rationals,
qeq_wf2,
assert_wf,
iff_weakening_uiff,
nat_plus_properties,
q-elim
Rules used in proof :
inrFormation_alt,
promote_hyp,
unionElimination,
multiplyEquality,
intEquality,
sqequalBase,
equalityIstype,
independent_pairFormation,
voidElimination,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
dependent_set_memberEquality_alt,
universeIsType,
functionIsTypeImplies,
equalityTransitivity,
lambdaEquality_alt,
inhabitedIsType,
isectIsTypeImplies,
isect_memberEquality_alt,
independent_pairEquality,
independent_isectElimination,
applyLambdaEquality,
equalitySymmetry,
hyp_replacement,
baseClosed,
because_Cache,
natural_numberEquality,
sqequalRule,
applyEquality,
independent_functionElimination,
lambdaFormation_alt,
rename,
setElimination,
hypothesis,
isectElimination,
productElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[q1,q2:\mBbbQ{}]. uiff(rat2real(q1) \mleq{} rat2real(q2);q1 \mleq{} q2)
Date html generated:
2019_10_29-AM-09_59_21
Last ObjectModification:
2019_10_17-AM-11_35_41
Theory : reals
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