Nuprl Lemma : mul-div-bounds
∀[a,b:ℤ]. ∀[m:ℤ-o]. (|(a * (b ÷ m)) - b * (a ÷ m)| ≤ (|a| + |b|))
Proof
Definitions occuring in Statement :
absval: |i|,
int_nzero: ℤ-o,
uall: ∀[x:A]. B[x],
le: A ≤ B,
divide: n ÷ m,
multiply: n * m,
subtract: n - m,
add: n + m,
int: ℤ
Definitions unfolded in proof :
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
less_than: a < b,
le: A ≤ B,
sq_type: SQType(T),
uiff: uiff(P;Q),
or: P ∨ Q,
decidable: Dec(P),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
guard: {T},
true: True,
nat: ℕ,
squash: ↓T,
subtype_rel: A ⊆r B,
prop: ℙ,
and: P ∧ Q,
top: Top,
all: ∀x:A. B[x],
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
le_weakening,
add_functionality_wrt_le,
int-triangle-inequality2,
le_transitivity,
le_functionality,
nat_properties,
equal-wf-base,
mul_preserves_le,
int_formula_prop_less_lemma,
int_formula_prop_le_lemma,
intformless_wf,
intformle_wf,
decidable__le,
rem_bounds_absval,
int_nzero_wf,
less_than'_wf,
false_wf,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
int_term_value_mul_lemma,
itermAdd_wf,
itermSubtract_wf,
itermMultiply_wf,
multiply-is-int-iff,
add-is-int-iff,
decidable__equal_int,
int_subtype_base,
subtype_base_sq,
iff_weakening_equal,
nat_wf,
absval_mul,
true_wf,
squash_wf,
le_wf,
equal-wf-T-base,
absval_nat_plus,
int_formula_prop_wf,
int_formula_prop_not_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_and_lemma,
intformnot_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformand_wf,
full-omega-unsat,
int_nzero_properties,
subtract_wf,
absval_wf,
mul_cancel_in_le,
mul_preserves_eq,
div_rem_sum
Rules used in proof :
applyLambdaEquality,
remainderEquality,
axiomEquality,
independent_pairEquality,
closedConclusion,
baseApply,
promote_hyp,
pointwiseFunctionality,
unionElimination,
cumulativity,
instantiate,
productElimination,
universeEquality,
imageMemberEquality,
imageElimination,
baseClosed,
addEquality,
applyEquality,
independent_pairFormation,
sqequalRule,
voidEquality,
voidElimination,
isect_memberEquality,
dependent_functionElimination,
intEquality,
int_eqEquality,
lambdaEquality,
dependent_pairFormation,
independent_functionElimination,
approximateComputation,
natural_numberEquality,
lambdaFormation,
rename,
setElimination,
divideEquality,
multiplyEquality,
independent_isectElimination,
hypothesis,
equalitySymmetry,
equalityTransitivity,
because_Cache,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[a,b:\mBbbZ{}]. \mforall{}[m:\mBbbZ{}\msupminus{}\msupzero{}]. (|(a * (b \mdiv{} m)) - b * (a \mdiv{} m)| \mleq{} (|a| + |b|))
Date html generated:
2017_09_29-PM-05_57_31
Last ObjectModification:
2017_09_06-PM-01_47_38
Theory : int_2
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