Nuprl Lemma : reduce-real_wf
∀[k:ℕ+]. ∀[x:ℝ]. ∀[b:{b:ℝ| r0 < b} ]. (reduce-real(x;b;k) ∈ {n:ℤ| |x - r(n) * b| ≤ (b + (b/r(k)))} )
Proof
Definitions occuring in Statement :
reduce-real: reduce-real(x;b;k),
rdiv: (x/y),
rleq: x ≤ y,
rless: x < y,
rabs: |x|,
rsub: x - y,
rmul: a * b,
radd: a + b,
int-to-real: r(n),
real: ℝ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
reduce-real: reduce-real(x;b;k),
uimplies: b supposing a,
rneq: x ≠ y,
or: P ∨ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
rless: x < y,
sq_exists: ∃x:A [B[x]],
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
rge: x ≥ y
Lemmas referenced :
sq_stable__rless,
int-to-real_wf,
integer-approx_wf,
rdiv_wf,
rless_wf,
rmul_preserves_rleq2,
rabs_wf,
rsub_wf,
radd_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,
zero-rleq-rabs,
rleq_wf,
rmul_wf,
real_wf,
nat_plus_wf,
rleq_weakening_rless,
req_weakening,
rminus_wf,
rinv_wf2,
itermSubtract_wf,
itermMultiply_wf,
itermAdd_wf,
itermMinus_wf,
rleq_weakening_equal,
req_functionality,
rabs-of-nonneg,
rleq_functionality,
req_transitivity,
rmul_functionality,
rabs_functionality,
radd_functionality,
rinv-mul-as-rdiv,
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_add_lemma,
real_term_value_minus_lemma,
real_term_value_const_lemma,
req_inversion,
rabs-rmul,
rmul-rinv3,
rleq_functionality_wrt_implies,
rdiv_functionality
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
setElimination,
thin,
rename,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
isectElimination,
natural_numberEquality,
hypothesis,
hypothesisEquality,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
because_Cache,
independent_isectElimination,
inrFormation_alt,
universeIsType,
applyEquality,
lambdaEquality_alt,
dependent_set_memberEquality_alt,
closedConclusion,
productElimination,
applyLambdaEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
promote_hyp,
inhabitedIsType,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
setIsType,
isectIsTypeImplies
Latex:
\mforall{}[k:\mBbbN{}\msupplus{}]. \mforall{}[x:\mBbbR{}]. \mforall{}[b:\{b:\mBbbR{}| r0 < b\} ]. (reduce-real(x;b;k) \mmember{} \{n:\mBbbZ{}| |x - r(n) * b| \mleq{} (b + (b/r(k)))\} \000C)
Date html generated:
2019_10_29-AM-10_09_22
Last ObjectModification:
2019_02_03-PM-03_15_46
Theory : reals
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