Nuprl Lemma : rounding-div-property
∀[a:ℤ]. ∀[n:ℕ+]. ((2 * |(n * [a ÷ n]) - a|) ≤ n)
Proof
Definitions occuring in Statement :
rounding-div: [b ÷ m],
absval: |i|,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
le: A ≤ B,
multiply: n * m,
subtract: n - m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rounding-div: [b ÷ m],
has-value: (a)↓,
uimplies: b supposing a,
nat_plus: ℕ+,
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
sq_type: SQType(T),
guard: {T},
nat: ℕ,
uiff: uiff(P;Q),
and: P ∧ Q,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
less_than': less_than'(a;b),
top: Top,
true: True,
squash: ↓T,
not: ¬A,
false: False,
prop: ℙ,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtract: n - m
Lemmas referenced :
value-type-has-value,
nat_plus_wf,
set-value-type,
less_than_wf,
int-value-type,
divrem-sq,
nat_plus_inc_int_nzero,
div_rem_sum,
divide_wfa,
rem_bounds_absval,
remainder_wfa,
subtype_base_sq,
int_subtype_base,
nat_wf,
set_subtype_base,
le_wf,
absval-non-neg,
absval_pos,
nat_plus_subtype_nat,
istype-le,
absval_strict_ubound,
istype-less_than,
absval_wf,
le_witness_for_triv,
istype-int,
decidable__lt,
istype-top,
istype-void,
absval_unfold,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
nat_plus_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermMultiply_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
itermMinus_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_term_value_minus_lemma,
int_formula_prop_wf,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
itermAdd_wf,
int_term_value_add_lemma,
minus-add,
minus-one-mul,
mul-commutes,
add-swap,
add-associates,
add-commutes,
add-mul-special,
zero-mul,
add-zero,
mul-distributes,
one-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
callbyvalueReduce,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
independent_isectElimination,
intEquality,
Error :lambdaEquality_alt,
natural_numberEquality,
hypothesisEquality,
Error :inhabitedIsType,
because_Cache,
applyEquality,
Error :lambdaFormation_alt,
dependent_functionElimination,
multiplyEquality,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
Error :dependent_set_memberEquality_alt,
productElimination,
Error :equalityIstype,
baseApply,
closedConclusion,
baseClosed,
sqequalBase,
setElimination,
rename,
Error :universeIsType,
Error :isect_memberEquality_alt,
Error :isectIsTypeImplies,
unionElimination,
lessCases,
axiomSqEquality,
independent_pairFormation,
voidElimination,
imageMemberEquality,
imageElimination,
minusEquality,
equalityElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality,
promote_hyp,
addEquality
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. ((2 * |(n * [a \mdiv{} n]) - a|) \mleq{} n)
Date html generated:
2019_06_20-PM-01_13_31
Last ObjectModification:
2019_03_06-AM-11_05_59
Theory : int_2
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