Nuprl Lemma : absval-diff-product-bound
∀u,v,x,y:ℕ. ((|u - v| * |x - y|) ≤ |(imax(u;v) * imax(x;y)) - imin(u;v) * imin(x;y)|)
Proof
Definitions occuring in Statement :
imin: imin(a;b),
imax: imax(a;b),
absval: |i|,
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x],
multiply: n * m,
subtract: n - m
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
nat: ℕ,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
int_seg: {i..j-},
squash: ↓T,
prop: ℙ,
guard: {T},
lelt: i ≤ j < k,
and: P ∧ Q,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
implies: P ⇒ Q,
not: ¬A,
top: Top,
true: True,
subtype_rel: A ⊆r B,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
sq_type: SQType(T),
le: A ≤ B,
less_than': less_than'(a;b),
subtract: n - m,
absval: |i|,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
rev_uimplies: rev_uimplies(P;Q)
Lemmas referenced :
int_seg_wf,
nat_wf,
subtype_base_sq,
set_subtype_base,
le_wf,
int_subtype_base,
absval-non-neg,
subtract_wf,
equal_wf,
squash_wf,
true_wf,
absval_pos,
int_seg_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermSubtract_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_subtract_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
iff_weakening_equal,
mul_preserves_le,
int_seg_subtype_nat,
false_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
lelt_wf,
mul-distributes,
mul-distributes-right,
add-associates,
minus-one-mul,
mul-associates,
mul-swap,
one-mul,
add-swap,
itermAdd_wf,
int_term_value_add_lemma,
decidable__equal_int,
absval_wf,
imax_wf,
imin_wf,
add-mul-special,
zero-mul,
le_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
decidable__lt,
intformeq_wf,
int_formula_prop_eq_lemma,
le_weakening,
absval-diff-symmetry,
imax_unfold,
imin_unfold,
le_functionality
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
instantiate,
cumulativity,
independent_isectElimination,
sqequalRule,
intEquality,
lambdaEquality,
dependent_set_memberEquality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
because_Cache,
productElimination,
dependent_functionElimination,
unionElimination,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
imageMemberEquality,
baseClosed,
independent_functionElimination,
sqequalIntensionalEquality,
multiplyEquality,
baseApply,
closedConclusion,
minusEquality,
addEquality,
applyLambdaEquality,
equalityElimination,
promote_hyp
Latex:
\mforall{}u,v,x,y:\mBbbN{}. ((|u - v| * |x - y|) \mleq{} |(imax(u;v) * imax(x;y)) - imin(u;v) * imin(x;y)|)
Date html generated:
2017_04_14-AM-09_13_55
Last ObjectModification:
2017_02_27-PM-03_51_48
Theory : int_2
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