Nuprl Lemma : regular-upto-regularize
∀f:ℕ+ ⟶ ℤ. ∀k,m:ℕ. ((↑regular-upto(k;m + 1;f))
⇒ {∀n:ℕ. ((n ≤ m)
⇒ (regularize(k;f) n ~ f n))})
Proof
Definitions occuring in Statement :
regularize: regularize(k;f)
,
regular-upto: regular-upto(k;n;f)
,
nat_plus: ℕ+
,
nat: ℕ
,
assert: ↑b
,
guard: {T}
,
le: A ≤ B
,
all: ∀x:A. B[x]
,
implies: P
⇒ Q
,
apply: f a
,
function: x:A ⟶ B[x]
,
add: n + m
,
natural_number: $n
,
int: ℤ
,
sqequal: s ~ t
Definitions unfolded in proof :
all: ∀x:A. B[x]
,
implies: P
⇒ Q
,
guard: {T}
,
regularize: regularize(k;f)
,
ifthenelse: if b then t else f fi
,
btrue: tt
,
member: t ∈ T
,
prop: ℙ
,
uall: ∀[x:A]. B[x]
,
nat: ℕ
,
ge: i ≥ j
,
decidable: Dec(P)
,
or: P ∨ Q
,
uimplies: b supposing a
,
not: ¬A
,
satisfiable_int_formula: satisfiable_int_formula(fmla)
,
exists: ∃x:A. B[x]
,
false: False
,
top: Top
,
and: P ∧ Q
,
uiff: uiff(P;Q)
,
sq_type: SQType(T)
,
iff: P
⇐⇒ Q
,
rev_implies: P
⇐ Q
,
int_seg: {i..j-}
,
lelt: i ≤ j < k
Lemmas referenced :
le_wf,
assert_wf,
regular-upto_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
nat_plus_wf,
nat_wf,
subtype_base_sq,
bool_wf,
bool_subtype_base,
eqtt_to_assert,
assert-regular-upto,
int_seg_wf,
int_seg_properties,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
lelt_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
because_Cache,
dependent_set_memberEquality,
addEquality,
natural_numberEquality,
dependent_functionElimination,
unionElimination,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
functionExtensionality,
applyEquality,
functionEquality,
instantiate,
cumulativity,
productElimination,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}f:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}. \mforall{}k,m:\mBbbN{}. ((\muparrow{}regular-upto(k;m + 1;f)) {}\mRightarrow{} \{\mforall{}n:\mBbbN{}. ((n \mleq{} m) {}\mRightarrow{} (regularize(k;f) n \msim{} f n))\})
Date html generated:
2017_10_03-AM-09_07_41
Last ObjectModification:
2017_09_11-PM-01_45_52
Theory : reals
Home
Index