Nuprl Lemma : proportional-round-property
∀[k,l:ℕ+]. ∀[r:ℚ]. |(k * r) - l * proportional-round(r;k;l)| < l
Proof
Definitions occuring in Statement :
qabs: |r|,
qless: r < s,
qsub: r - s,
qmul: r * s,
proportional-round: proportional-round(r;k;l),
rationals: ℚ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
multiply: n * m
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
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],
cand: A c∧ B,
not: ¬A,
qdiv: (r/s),
proportional-round: proportional-round(r;k;l),
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),
has-value: (a)↓,
has-valueall: has-valueall(a),
bfalse: ff,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
bool: 𝔹,
unit: Unit,
it: ⋅,
uiff: uiff(P;Q),
band: p ∧b q,
or: P ∨ Q,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
less_than: a < b,
squash: ↓T,
bor: p ∨bq,
rev_uimplies: rev_uimplies(P;Q),
true: True,
decidable: Dec(P),
subtract: n - m,
le: A ≤ B,
nat: ℕ,
ge: i ≥ j ,
gt: i > j,
int_lower: {...i}
Lemmas referenced :
assert-q_less-eq,
qabs_wf,
qsub_wf,
qmul_wf,
proportional-round_wf,
subtype_rel_set,
int-subtype-rationals,
iff_weakening_equal,
q-elim,
nat_plus_properties,
iff_weakening_uiff,
assert_wf,
qeq_wf2,
equal-wf-base,
rationals_wf,
int_subtype_base,
assert-qeq,
valueall-type-has-valueall,
int-valueall-type,
evalall-reduce,
product-valueall-type,
nat_plus_wf,
set-valueall-type,
less_than_wf,
mul_nzero,
full-omega-unsat,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
nequal_wf,
nat_plus_inc_int_nzero,
q_less_wf,
qless_witness,
isint-int,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
int_formula_prop_not_lemma,
intformnot_wf,
add-is-int-iff,
multiply-is-int-iff,
itermAdd_wf,
itermMultiply_wf,
int_term_value_add_lemma,
int_term_value_mul_lemma,
false_wf,
set_subtype_base,
equal_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
decidable__equal_int,
div_rem_sum,
subtype_rel_sets,
mul-associates,
mul-commutes,
mul-swap,
one-mul,
mul-distributes,
minus-one-mul,
add-associates,
add-mul-special,
zero-mul,
zero-add,
decidable__le,
not_wf,
mul_nat_plus,
le_wf,
rem_bounds_1,
int_term_value_minus_lemma,
itermMinus_wf,
decidable__lt,
int_formula_prop_le_lemma,
intformle_wf,
rem_bounds_2,
add-swap
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
because_Cache,
hypothesis,
sqequalRule,
multiplyEquality,
setElimination,
rename,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
productElimination,
independent_functionElimination,
dependent_functionElimination,
lambdaFormation_alt,
natural_numberEquality,
baseApply,
closedConclusion,
baseClosed,
universeIsType,
isintReduceTrue,
callbyvalueReduce,
sqleReflexivity,
minusEquality,
intEquality,
productEquality,
lambdaEquality_alt,
inhabitedIsType,
independent_pairEquality,
divideEquality,
dependent_set_memberEquality_alt,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
equalityIsType4,
hyp_replacement,
applyLambdaEquality,
addEquality,
unionElimination,
equalityElimination,
equalityIsType2,
promote_hyp,
instantiate,
cumulativity,
equalityIsType1,
pointwiseFunctionality,
imageElimination,
remainderEquality,
setEquality,
lambdaEquality,
dependent_pairFormation,
lambdaFormation,
dependent_set_memberEquality,
voidEquality,
isect_memberEquality,
functionIsType
Latex:
\mforall{}[k,l:\mBbbN{}\msupplus{}]. \mforall{}[r:\mBbbQ{}]. |(k * r) - l * proportional-round(r;k;l)| < l
Date html generated:
2019_10_16-PM-00_31_46
Last ObjectModification:
2018_10_10-PM-01_07_04
Theory : rationals
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