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
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