Nuprl Lemma : rounded-numerator-property
∀[k:ℕ+]. ∀[r:ℚ]. |(k * r) - rounded-numerator(r;k)| < 1
Proof
Definitions occuring in Statement :
qabs: |r|,
qless: r < s,
qsub: r - s,
qmul: r * s,
rounded-numerator: rounded-numerator(r;k),
rationals: ℚ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
nat_plus: ℕ+,
cand: A c∧ B,
not: ¬A,
uiff: uiff(P;Q),
so_lambda: λ2x.t[x],
so_apply: x[s],
qdiv: (r/s),
rounded-numerator: rounded-numerator(r;k),
qmul: r * s,
qsub: r - s,
qabs: |r|,
q_less: q_less(r;s),
ifthenelse: if b then t else f fi ,
btrue: tt,
qpositive: qpositive(r),
qinv: 1/r,
qadd: r + s,
callbyvalueall: callbyvalueall,
evalall: evalall(t),
bfalse: ff,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
bor: p ∨bq,
band: p ∧b q,
rev_uimplies: rev_uimplies(P;Q),
less_than: a < b,
squash: ↓T,
true: True,
has-value: (a)↓,
has-valueall: has-valueall(a),
int_nzero: ℤ-o,
subtract: n - m,
decidable: Dec(P),
nat: ℕ,
le: A ≤ B,
int_lower: {...i},
gt: i > j,
ge: i ≥ j
Lemmas referenced :
assert-q_less-eq,
qabs_wf,
qsub_wf,
qmul_wf,
rounded-numerator_wf,
iff_weakening_equal,
q-elim,
nat_plus_properties,
assert-qeq,
int-subtype-rationals,
assert_wf,
qeq_wf2,
not_wf,
equal-wf-base,
rationals_wf,
int_subtype_base,
q_less_wf,
qless_witness,
subtype_rel_set,
less_than_wf,
nat_plus_wf,
valueall-type-has-valueall,
set-valueall-type,
int-valueall-type,
product-valueall-type,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
isint-int,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
intformnot_wf,
int_formula_prop_not_lemma,
itermAdd_wf,
itermMultiply_wf,
int_term_value_add_lemma,
int_term_value_mul_lemma,
multiply-is-int-iff,
add-is-int-iff,
false_wf,
evalall-reduce,
mul-associates,
mul-commutes,
mul-swap,
one-mul,
mul-distributes,
div_rem_sum,
nequal_wf,
div_rem_sum2,
minus-one-mul,
add-swap,
add-associates,
add-commutes,
add-mul-special,
zero-mul,
zero-add,
decidable__le,
rem_bounds_1,
mul_bounds_1a,
nat_plus_subtype_nat,
le_wf,
rem_bounds_2,
le_weakening2,
neg_mul_arg_bounds,
decidable__lt,
intformle_wf,
int_formula_prop_le_lemma,
gt_wf,
itermMinus_wf,
int_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
natural_numberEquality,
equalitySymmetry,
equalityTransitivity,
independent_isectElimination,
productElimination,
independent_functionElimination,
dependent_functionElimination,
setElimination,
rename,
addLevel,
impliesFunctionality,
baseClosed,
hyp_replacement,
applyLambdaEquality,
intEquality,
lambdaEquality,
isect_memberEquality,
isintReduceTrue,
callbyvalueReduce,
sqleReflexivity,
minusEquality,
productEquality,
lambdaFormation,
independent_pairEquality,
multiplyEquality,
divideEquality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation,
addEquality,
unionElimination,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
pointwiseFunctionality,
imageElimination,
baseApply,
closedConclusion,
dependent_set_memberEquality,
remainderEquality,
inrFormation
Latex:
\mforall{}[k:\mBbbN{}\msupplus{}]. \mforall{}[r:\mBbbQ{}]. |(k * r) - rounded-numerator(r;k)| < 1
Date html generated:
2018_05_21-PM-11_58_06
Last ObjectModification:
2018_05_19-PM-04_04_39
Theory : rationals
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