Nuprl Lemma : regularize-2-regular
∀f:ℤ ⟶ ℤ. 2-regular-seq(regularize(f))
Proof
Definitions occuring in Statement :
regularize: regularize(f),
regular-int-seq: k-regular-seq(f),
all: ∀x:A. B[x],
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
absval: |i|,
subtract: n - m,
true: True,
has-value: (a)↓,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
lelt: i ≤ j < k,
ge: i ≥ j ,
nat_plus: ℕ+,
int_seg: {i..j-},
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
not: ¬A,
so_apply: x[s],
so_lambda: λ2x.t[x],
false: False,
assert: ↑b,
bnot: ¬bb,
guard: {T},
sq_type: SQType(T),
or: P ∨ Q,
prop: ℙ,
exists: ∃x:A. B[x],
bfalse: ff,
ifthenelse: if b then t else f fi ,
uimplies: b supposing a,
and: P ∧ Q,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
implies: P ⇒ Q,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
member: t ∈ T,
regularize: regularize(f),
regular-int-seq: k-regular-seq(f),
all: ∀x:A. B[x],
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
less_than: a < b,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
cand: A c∧ B
Lemmas referenced :
absval-minus,
le_transitivity,
itermMinus_wf,
int_term_value_minus_lemma,
absval-diff-symmetry,
mul_preserves_le,
mul_bounds_1a,
mul-swap,
absval_pos,
decidable__assert,
all_wf,
rem_bounds_absval_le,
multiply_functionality_wrt_le,
int-triangle-inequality,
add_functionality_wrt_eq,
div_rem_sum2,
subtype_rel_sets,
nequal_wf,
mul_cancel_in_le,
itermSubtract_wf,
int_term_value_subtract_lemma,
absval_unfold,
lt_int_wf,
assert_of_lt_int,
top_wf,
squash_wf,
true_wf,
absval_mul,
iff_weakening_equal,
le_weakening,
le_functionality,
itermMultiply_wf,
int_term_value_mul_lemma,
zero-mul,
add-mul-special,
one-mul,
mul-commutes,
mul-associates,
mul-distributes,
regular-upto_wf,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
mu-property,
bnot_wf,
nat_wf,
not_wf,
assert_wf,
assert_of_bnot,
exists_wf,
mu_wf,
less_than_wf,
int_subtype_base,
assert-regular-upto,
false_wf,
le_wf,
set_subtype_base,
int_seg_properties,
nat_properties,
nat_plus_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
itermAdd_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
decidable__le,
decidable__lt,
lelt_wf,
int_seg_wf,
value-type-has-value,
int-value-type,
subtract_wf,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
add-associates,
add-zero,
zero-add,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
add-swap,
minus-zero,
le-add-cancel2,
minus-minus,
minus-one-mul,
minus-one-mul-top,
and_wf,
uall_wf,
isect_wf,
nat_plus_subtype_nat,
nat_plus_wf,
absval_wf
Rules used in proof :
multiplyEquality,
functionEquality,
equalityEquality,
minusEquality,
callbyvalueReduce,
computeAll,
voidEquality,
isect_memberEquality,
int_eqEquality,
addEquality,
natural_numberEquality,
dependent_set_memberEquality,
intEquality,
levelHypothesis,
rename,
setElimination,
cumulativity,
isect_memberFormation,
uallFunctionality,
independent_pairFormation,
introduction,
existsFunctionality,
addLevel,
lambdaEquality,
voidElimination,
independent_functionElimination,
instantiate,
dependent_functionElimination,
promote_hyp,
dependent_pairFormation,
independent_isectElimination,
productElimination,
equalitySymmetry,
equalityTransitivity,
equalityElimination,
unionElimination,
hypothesis,
because_Cache,
applyEquality,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
lemma_by_obid,
cut,
sqequalRule,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
universeEquality,
imageElimination,
baseClosed,
imageMemberEquality,
sqequalAxiom,
lessCases,
setEquality,
divideEquality,
remainderEquality,
impliesFunctionality
Latex:
\mforall{}f:\mBbbZ{} {}\mrightarrow{} \mBbbZ{}. 2-regular-seq(regularize(f))
Date html generated:
2016_05_18-AM-10_51_02
Last ObjectModification:
2016_01_17-AM-00_19_19
Theory : reals
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