Nuprl Lemma : reg-seq-nexp_wf
∀[x:ℝ]. ∀[k:ℕ+]. (reg-seq-nexp(x;k) ∈ {f:ℕ+ ⟶ ℤ| (k * ((canon-bnd(x)^k - 1 ÷ 2^k - 1) + 1)) + 1-regular-seq(f)} )
Proof
Definitions occuring in Statement :
reg-seq-nexp: reg-seq-nexp(x;k),
canon-bnd: canon-bnd(x),
real: ℝ,
regular-int-seq: k-regular-seq(f),
fastexp: i^n,
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
divide: n ÷ m,
multiply: n * m,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
nat_plus: ℕ+,
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
and: P ∧ Q,
prop: ℙ,
guard: {T},
subtype_rel: A ⊆r B,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
reg-seq-nexp: reg-seq-nexp(x;k),
real: ℝ,
top: Top,
int_upper: {i...},
regular-int-seq: k-regular-seq(f),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_stable: SqStable(P),
le: A ≤ B,
squash: ↓T,
true: True,
sq_type: SQType(T),
ge: i ≥ j ,
cand: A c∧ B,
less_than: a < b,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
rev_uimplies: rev_uimplies(P;Q),
less_than': less_than'(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
bfalse: ff,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b
Latex:
\mforall{}[x:\mBbbR{}]. \mforall{}[k:\mBbbN{}\msupplus{}].
(reg-seq-nexp(x;k) \mmember{} \{f:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}| (k * ((canon-bnd(x)\^{}k - 1 \mdiv{} 2\^{}k - 1) + 1)) + 1-regular-seq(f)\} )
Date html generated:
2020_05_20-AM-10_55_39
Last ObjectModification:
2020_01_03-AM-00_54_03
Theory : reals
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