Nuprl Lemma : bdd-diff-iff-eventual
∀f,g:ℕ+ ⟶ ℤ. (∃m:ℕ+. ∃B:ℕ. ∀n:{m...}. (|(f n) - g n| ≤ B) ⇐⇒ bdd-diff(f;g))
Proof
Definitions occuring in Statement :
bdd-diff: bdd-diff(f;g),
absval: |i|,
int_upper: {i...},
nat_plus: ℕ+,
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
subtract: n - m,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
int_upper: {i...},
nat: ℕ,
le: A ≤ B,
guard: {T},
uimplies: b supposing a,
subtype_rel: A ⊆r B,
so_apply: x[s],
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
bdd-diff: bdd-diff(f;g),
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
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,
cand: A c∧ B,
less_than': less_than'(a;b),
true: True,
less_than: a < b,
squash: ↓T
Lemmas referenced :
exists_wf,
nat_plus_wf,
nat_wf,
all_wf,
int_upper_wf,
le_wf,
absval_wf,
subtract_wf,
less_than_transitivity1,
less_than_wf,
bdd-diff_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
subtype_rel_sets,
nat_plus_properties,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermVar_wf,
intformless_wf,
itermConstant_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
imax_wf,
imax-list_wf,
map-length,
length-from-upto,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
decidable__lt,
itermSubtract_wf,
int_term_value_subtract_lemma,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
bool_subtype_base,
assert-bnot,
imax_ub,
imax-list-ub,
map_wf,
false_wf,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
from-upto_wf,
l_exists_iff,
l_member_wf,
member-map,
member-from-upto
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
setElimination,
rename,
because_Cache,
applyEquality,
functionExtensionality,
hypothesisEquality,
dependent_set_memberEquality,
productElimination,
natural_numberEquality,
independent_isectElimination,
functionEquality,
intEquality,
dependent_functionElimination,
unionElimination,
instantiate,
cumulativity,
independent_functionElimination,
dependent_pairFormation,
setEquality,
applyLambdaEquality,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
inlFormation,
inrFormation,
productEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}f,g:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}. (\mexists{}m:\mBbbN{}\msupplus{}. \mexists{}B:\mBbbN{}. \mforall{}n:\{m...\}. (|(f n) - g n| \mleq{} B) \mLeftarrow{}{}\mRightarrow{} bdd-diff(f;g))
Date html generated:
2017_10_02-PM-07_13_04
Last ObjectModification:
2017_07_28-AM-07_20_00
Theory : reals
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