Nuprl Lemma : bdd-diff-regular
∀[x,y:ℕ+ ⟶ ℤ]. ∀[k,l:ℕ+].
(∀n:ℕ+. (|(x n) - y n| ≤ ((2 * k) + (2 * l)))) supposing (bdd-diff(x;y) and k-regular-seq(x) and l-regular-seq(y))
Proof
Definitions occuring in Statement :
bdd-diff: bdd-diff(f;g),
regular-int-seq: k-regular-seq(f),
absval: |i|,
nat_plus: ℕ+,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
bdd-diff: bdd-diff(f;g),
exists: ∃x:A. B[x],
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
prop: ℙ,
nat: ℕ,
true: True,
top: Top,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rev_uimplies: rev_uimplies(P;Q),
subtract: n - m,
regular-int-seq: k-regular-seq(f),
so_lambda: λ2x.t[x],
so_apply: x[s],
gt: i > j,
uiff: uiff(P;Q),
less_than': less_than'(a;b)
Lemmas referenced :
le-add-cancel,
minus-one-mul-top,
mul-associates,
mul-commutes,
mul-swap,
mul-distributes,
minus-zero,
condition-implies-le,
less-iff-le,
not-lt-2,
false_wf,
and_wf,
decidable__lt,
pos_mul_arg_bounds,
all_functionality_wrt_uimplies,
int_formula_prop_less_lemma,
intformless_wf,
int_formula_prop_and_lemma,
intformand_wf,
multiply_functionality_wrt_le,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
itermConstant_wf,
itermMultiply_wf,
itermAdd_wf,
set_subtype_base,
int_subtype_base,
nat_plus_subtype_nat,
absval_pos,
absval_mul,
left_mul_subtract_distrib,
add-zero,
zero-add,
zero-mul,
add-swap,
add-mul-special,
add-commutes,
minus-one-mul,
add-associates,
minus-minus,
minus-add,
int-triangle-inequality,
add_functionality_wrt_le,
le_weakening,
le_functionality,
iff_weakening_equal,
absval_sym,
add_functionality_wrt_eq,
true_wf,
squash_wf,
le_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
itermVar_wf,
intformle_wf,
intformnot_wf,
satisfiable-full-omega-tt,
less_than_wf,
decidable__le,
nat_properties,
nat_plus_properties,
nat_wf,
regular-int-seq_wf,
bdd-diff_wf,
subtract_wf,
absval_wf,
less_than'_wf,
nat_plus_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
because_Cache,
lemma_by_obid,
hypothesis,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
independent_pairEquality,
isectElimination,
addEquality,
multiplyEquality,
natural_numberEquality,
setElimination,
rename,
applyEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
functionEquality,
intEquality,
voidElimination,
minusEquality,
voidEquality,
dependent_set_memberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
imageElimination,
imageMemberEquality,
baseClosed,
universeEquality,
independent_functionElimination,
sqequalIntensionalEquality,
independent_pairFormation,
inlFormation
Latex:
\mforall{}[x,y:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}]. \mforall{}[k,l:\mBbbN{}\msupplus{}].
(\mforall{}n:\mBbbN{}\msupplus{}. (|(x n) - y n| \mleq{} ((2 * k) + (2 * l)))) supposing
(bdd-diff(x;y) and
k-regular-seq(x) and
l-regular-seq(y))
Date html generated:
2016_05_18-AM-06_47_36
Last ObjectModification:
2016_01_17-AM-01_47_24
Theory : reals
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