Nuprl Lemma : integer-approx_wf
∀[x:ℝ]. ∀[k:ℕ+]. (integer-approx(x;k) ∈ {n:ℤ| |x - r(n)| ≤ (r1 + (r1/r(k)))} )
Proof
Definitions occuring in Statement :
integer-approx: integer-approx(x;k),
rdiv: (x/y),
rleq: x ≤ y,
rabs: |x|,
rsub: x - y,
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,
integer-approx: integer-approx(x;k),
nequal: a ≠ b ∈ T ,
nat_plus: ℕ+,
not: ¬A,
implies: P ⇒ Q,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
real: ℝ,
guard: {T},
rneq: x ≠ y,
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
decidable: Dec(P),
subtype_rel: A ⊆r B,
rational-approx: (x within 1/n),
uiff: uiff(P;Q),
rdiv: (x/y),
req_int_terms: t1 ≡ t2,
sq_type: SQType(T),
true: True,
squash: ↓T,
nat: ℕ,
int_nzero: ℤ-o
Lemmas referenced :
nat_plus_properties,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
rational-approx-property,
rleq_functionality_wrt_implies,
rabs_wf,
rsub_wf,
int-to-real_wf,
radd_wf,
rational-approx_wf,
rleq_weakening_equal,
r-triangle-inequality2,
rdiv_wf,
rless-int,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
rless_wf,
radd_functionality_wrt_rleq,
rleq_wf,
nat_plus_wf,
real_wf,
rabs-rmul,
int_subtype_base,
istype-less_than,
rmul_wf,
int-rdiv_wf,
nat_plus_inc_int_nzero,
rinv_wf2,
rminus_wf,
itermSubtract_wf,
itermAdd_wf,
itermMinus_wf,
req_functionality,
rmul_functionality,
req_weakening,
rsub_functionality,
int-rdiv-req,
req_transitivity,
radd_functionality,
rmul-rinv,
rminus_functionality,
rmul-int,
req_inversion,
rminus-int,
radd-int,
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,
subtype_base_sq,
div_rem_sum,
decidable__equal_int,
add-is-int-iff,
multiply-is-int-iff,
int_term_value_add_lemma,
int_term_value_minus_lemma,
false_wf,
rabs-of-nonneg,
rleq-int,
decidable__le,
intformle_wf,
int_formula_prop_le_lemma,
rmul_preserves_rleq,
req_wf,
squash_wf,
true_wf,
iff_weakening_equal,
rless_functionality,
rleq_functionality,
rabs_functionality,
rabs-int,
absval_wf,
rem_bounds_absval_le,
nequal_wf,
subtype_rel_self,
rleq_weakening
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
dependent_set_memberEquality_alt,
divideEquality,
because_Cache,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
closedConclusion,
inrFormation_alt,
productElimination,
unionElimination,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
isectIsTypeImplies,
inhabitedIsType,
multiplyEquality,
applyEquality,
equalityIstype,
baseApply,
baseClosed,
sqequalBase,
applyLambdaEquality,
remainderEquality,
addEquality,
minusEquality,
instantiate,
cumulativity,
intEquality,
pointwiseFunctionality,
promote_hyp,
imageElimination,
imageMemberEquality,
universeEquality
Latex:
\mforall{}[x:\mBbbR{}]. \mforall{}[k:\mBbbN{}\msupplus{}]. (integer-approx(x;k) \mmember{} \{n:\mBbbZ{}| |x - r(n)| \mleq{} (r1 + (r1/r(k)))\} )
Date html generated:
2019_10_29-AM-10_08_58
Last ObjectModification:
2019_02_03-PM-02_38_42
Theory : reals
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