Nuprl Lemma : reg-seq-adjust_wf
∀[n:ℕ+]. ∀[x:ℝ]. reg-seq-adjust(n;x) ∈ {f:ℕ+ ⟶ ℤ| if (n =z 1) then 1 else 4 fi -regular-seq(f)} supposing ∀i:ℕ+. (i <\000C n ⇒ (|x i| ≤ 4))
Proof
Definitions occuring in Statement :
reg-seq-adjust: reg-seq-adjust(n;x),
real: ℝ,
regular-int-seq: k-regular-seq(f),
absval: |i|,
nat_plus: ℕ+,
ifthenelse: if b then t else f fi ,
eq_int: (i =z j),
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
real: ℝ,
reg-seq-adjust: reg-seq-adjust(n;x),
nat_plus: ℕ+,
less_than: a < b,
and: P ∧ Q,
less_than': less_than'(a;b),
true: True,
squash: ↓T,
top: Top,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
all: ∀x:A. B[x],
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
le: A ≤ B,
decidable: Dec(P),
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
absval: |i|,
subtract: n - m
Lemmas referenced :
top_wf,
less_than_wf,
nat_plus_wf,
eq_int_wf,
bool_wf,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal_wf,
regular-int-seq_wf,
ifthenelse_wf,
all_wf,
le_wf,
absval_wf,
nat_wf,
real_wf,
lt_int_wf,
assert_of_lt_int,
nat_plus_properties,
full-omega-unsat,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
bdd-diff-regular-int-seq,
false_wf,
subtract_wf,
decidable__le,
intformnot_wf,
intformle_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
decidable__equal_int,
itermAdd_wf,
int_term_value_add_lemma,
add-is-int-iff,
itermSubtract_wf,
int_term_value_subtract_lemma,
and_wf,
le_functionality,
le_weakening,
int-triangle-inequality,
add_functionality_wrt_le,
squash_wf,
true_wf,
minus-one-mul,
add-mul-special,
zero-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
setElimination,
thin,
rename,
dependent_set_memberEquality,
sqequalRule,
lambdaEquality,
hypothesisEquality,
hypothesis,
lessCases,
independent_pairFormation,
isectElimination,
baseClosed,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
axiomSqEquality,
extract_by_obid,
isect_memberEquality,
because_Cache,
voidElimination,
voidEquality,
lambdaFormation,
imageElimination,
productElimination,
independent_functionElimination,
applyEquality,
unionElimination,
equalityElimination,
intEquality,
independent_isectElimination,
impliesFunctionality,
dependent_functionElimination,
axiomEquality,
functionEquality,
functionExtensionality,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
promote_hyp,
instantiate,
cumulativity,
hyp_replacement,
applyLambdaEquality,
addEquality,
minusEquality,
pointwiseFunctionality,
baseApply,
closedConclusion
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. \mforall{}[x:\mBbbR{}].
reg-seq-adjust(n;x) \mmember{} \{f:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}| if (n =\msubz{} 1) then 1 else 4 fi -regular-seq(f)\} supposing \mforall{}i:\mBbbN{}\msupplus{}. \000C(i < n {}\mRightarrow{} (|x i| \mleq{} 4))
Date html generated:
2019_10_16-PM-03_07_18
Last ObjectModification:
2018_08_20-PM-09_45_00
Theory : reals
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