Nuprl Lemma : bdd-diff-regular-int-seq
∀[k,b:ℕ]. ∀[f:{f:ℕ+ ⟶ ℤ| k-regular-seq(f)} ]. ∀[g:ℕ+ ⟶ ℤ].
k + b-regular-seq(g) supposing ∀n:ℕ+. (|(f n) - g n| ≤ (2 * b))
Proof
Definitions occuring in Statement :
regular-int-seq: k-regular-seq(f),
absval: |i|,
nat_plus: ℕ+,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
set: {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,
regular-int-seq: k-regular-seq(f),
all: ∀x:A. B[x],
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
nat: ℕ,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
false: False,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
top: Top,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
rev_uimplies: rev_uimplies(P;Q),
guard: {T},
subtract: n - m,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
absval_pos,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_term_value_add_lemma,
itermConstant_wf,
itermMultiply_wf,
itermAdd_wf,
nat_plus_subtype_nat,
absval-diff-symmetry,
multiply_functionality_wrt_le,
absval_mul,
iff_weakening_equal,
left_mul_subtract_distrib,
add_functionality_wrt_eq,
true_wf,
squash_wf,
zero-add,
zero-mul,
add-mul-special,
add-commutes,
add-swap,
minus-one-mul,
add-associates,
int-triangle-inequality,
add_functionality_wrt_le,
le_weakening,
le_functionality,
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,
regular-int-seq_wf,
set_wf,
le_wf,
all_wf,
nat_wf,
less_than'_wf,
nat_plus_wf,
subtract_wf,
absval_wf,
sq_stable__le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
setElimination,
thin,
rename,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
multiplyEquality,
hypothesisEquality,
applyEquality,
hypothesis,
because_Cache,
sqequalRule,
natural_numberEquality,
addEquality,
independent_functionElimination,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality,
dependent_functionElimination,
productElimination,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
functionEquality,
intEquality,
voidElimination,
voidEquality,
minusEquality,
dependent_set_memberEquality,
unionElimination,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
computeAll,
universeEquality
Latex:
\mforall{}[k,b:\mBbbN{}]. \mforall{}[f:\{f:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}| k-regular-seq(f)\} ]. \mforall{}[g:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}].
k + b-regular-seq(g) supposing \mforall{}n:\mBbbN{}\msupplus{}. (|(f n) - g n| \mleq{} (2 * b))
Date html generated:
2016_05_18-AM-06_46_41
Last ObjectModification:
2016_01_17-AM-01_46_16
Theory : reals
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