Nuprl Lemma : qround-property
∀[k:ℕ+]. ∀[r:ℚ]. |r - qround(r;k)| < (1/2 * k)
Proof
Definitions occuring in Statement :
qabs: |r|,
qless: r < s,
qsub: r - s,
qdiv: (r/s),
qround: qround(r;k),
qmul: r * s,
rationals: ℚ,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
natural_number: $n
Definitions unfolded in proof :
decidable: Dec(P),
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
uiff: uiff(P;Q),
cand: A c∧ B,
or: P ∨ Q,
true: True,
iff: P ⇐⇒ Q,
prop: ℙ,
and: P ∧ Q,
all: ∀x:A. B[x],
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
guard: {T},
rev_implies: P ⇐ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_uimplies: rev_uimplies(P;Q),
qeq: qeq(r;s),
callbyvalueall: callbyvalueall,
evalall: evalall(t),
eq_int: (i =z j),
qsub: r - s
Lemmas referenced :
istype-less_than,
rounded-numerator_wf,
qless_wf,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__lt,
qless-int,
qmul-positive,
qabs-of-positive,
nat_plus_wf,
int-equal-in-rationals,
equal-wf-base,
iff_weakening_uiff,
int_subtype_base,
set_subtype_base,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
intformeq_wf,
intformand_wf,
full-omega-unsat,
nat_plus_properties,
qmul-mul,
istype-int,
less_than_wf,
rationals_wf,
subtype_rel_set,
qmul_wf,
int-subtype-rationals,
qdiv_wf,
qround_wf,
qsub_wf,
qabs_wf,
qless_witness,
squash_wf,
true_wf,
qround-eq,
subtype_rel_self,
iff_weakening_equal,
qabs-abs,
absval_unfold,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
istype-top,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
assert_wf,
itermMinus_wf,
int_term_value_minus_lemma,
qmul_preserves_qless,
qadd_wf,
qless_transitivity_2_qorder,
qle_weakening_eq_qorder,
qless_irreflexivity,
qabs-qmul,
assert-qeq,
qmul_comm_qrng,
qadd_comm_q,
qmul-qdiv-cancel,
qmul_over_plus_qrng,
qmul_over_minus_qrng,
qmul_ac_1_qrng,
qmul-qdiv-cancel6,
rounded-numerator-property,
qmul_assoc_qrng
Rules used in proof :
dependent_set_memberEquality_alt,
applyLambdaEquality,
hyp_replacement,
minusEquality,
productIsType,
unionElimination,
imageMemberEquality,
inlFormation_alt,
isectIsTypeImplies,
isect_memberEquality_alt,
inhabitedIsType,
productElimination,
multiplyEquality,
equalityTransitivity,
because_Cache,
equalitySymmetry,
sqequalBase,
baseClosed,
baseApply,
equalityIstype,
voidElimination,
universeIsType,
independent_pairFormation,
Error :memTop,
dependent_functionElimination,
int_eqEquality,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
lambdaFormation_alt,
rename,
setElimination,
independent_isectElimination,
lambdaEquality_alt,
intEquality,
sqequalRule,
applyEquality,
natural_numberEquality,
closedConclusion,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
imageElimination,
instantiate,
universeEquality,
equalityElimination,
lessCases,
axiomSqEquality,
promote_hyp,
cumulativity
Latex:
\mforall{}[k:\mBbbN{}\msupplus{}]. \mforall{}[r:\mBbbQ{}]. |r - qround(r;k)| < (1/2 * k)
Date html generated:
2020_05_20-AM-09_16_52
Last ObjectModification:
2019_12_13-AM-09_33_28
Theory : rationals
Home
Index