Nuprl Lemma : absval-diff-product-bound2
∀u,v,x,y:ℤ. (|(u * v) - x * y| ≤ ((|u| * |v - y|) + (|y| * |u - x|)))
Proof
Definitions occuring in Statement :
absval: |i|,
le: A ≤ B,
all: ∀x:A. B[x],
multiply: n * m,
subtract: n - m,
add: n + m,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
squash: ↓T,
uall: ∀[x:A]. B[x],
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
uimplies: b supposing a,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
subtract: n - m
Lemmas referenced :
le_wf,
squash_wf,
true_wf,
absval_wf,
subtract_wf,
add_functionality_wrt_eq,
absval_mul,
subtype_rel_self,
iff_weakening_equal,
istype-int,
istype-void,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_functionality,
le_weakening,
int-triangle-inequality,
mul-distributes,
add-associates,
minus-one-mul,
mul-swap,
mul-commutes,
add-commutes,
add-mul-special,
zero-mul,
zero-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :lambdaFormation_alt,
cut,
applyEquality,
thin,
Error :lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
Error :universeIsType,
because_Cache,
multiplyEquality,
setElimination,
rename,
Error :inhabitedIsType,
sqequalRule,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
instantiate,
universeEquality,
productElimination,
independent_functionElimination,
addEquality,
Error :isect_memberEquality_alt,
voidElimination,
minusEquality,
dependent_functionElimination,
unionElimination,
approximateComputation,
Error :dependent_pairFormation_alt,
int_eqEquality
Latex:
\mforall{}u,v,x,y:\mBbbZ{}. (|(u * v) - x * y| \mleq{} ((|u| * |v - y|) + (|y| * |u - x|)))
Date html generated:
2019_06_20-PM-01_13_48
Last ObjectModification:
2019_02_14-PM-00_03_12
Theory : int_2
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