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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