Nuprl Lemma : regular-iff-all-regular-upto
∀k:ℕ+. ∀x:ℕ+ ⟶ ℤ. (k-regular-seq(x) ⇐⇒ ∀b:ℕ+. (↑regular-upto(k;b;x)))
Proof
Definitions occuring in Statement :
regular-upto: regular-upto(k;n;f),
regular-int-seq: k-regular-seq(f),
nat_plus: ℕ+,
assert: ↑b,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
regular-int-seq: k-regular-seq(f),
nat_plus: ℕ+,
prop: ℙ,
nat: ℕ,
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,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
less_than: a < b,
squash: ↓T,
subtract: n - m
Latex:
\mforall{}k:\mBbbN{}\msupplus{}. \mforall{}x:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}. (k-regular-seq(x) \mLeftarrow{}{}\mRightarrow{} \mforall{}b:\mBbbN{}\msupplus{}. (\muparrow{}regular-upto(k;b;x)))
Date html generated:
2020_05_20-AM-11_05_12
Last ObjectModification:
2020_03_14-AM-09_31_45
Theory : reals
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